Merge remote-tracking branch 'gitea/master'
This commit is contained in:
commit
f633043fed
81
.obsidian/plugins/copilot/data.json
vendored
81
.obsidian/plugins/copilot/data.json
vendored
@ -243,5 +243,84 @@
|
||||
"promptUsageTimestamps": {},
|
||||
"embeddingRequestsPerMin": 90,
|
||||
"embeddingBatchSize": 16,
|
||||
"isPlusUser": false
|
||||
"isPlusUser": false,
|
||||
"inlineEditCommands": [
|
||||
{
|
||||
"name": "Fix grammar and spelling",
|
||||
"prompt": "<instruction>Fix the grammar and spelling of the text below. Preserve all formatting, line breaks, and special characters. Do not add or remove any content. Return only the corrected text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true
|
||||
},
|
||||
{
|
||||
"name": "Translate to Chinese",
|
||||
"prompt": "<instruction>Translate the text below into Chinese:\n 1. Preserve the meaning and tone\n 2. Maintain appropriate cultural context\n 3. Keep formatting and structure\n Return only the translated text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true,
|
||||
"modelKey": "gemma3:12b|ollama"
|
||||
},
|
||||
{
|
||||
"name": "Summarize",
|
||||
"prompt": "<instruction>Create a bullet-point summary of the text below. Each bullet point should capture a key point. Return only the bullet-point summary.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true
|
||||
},
|
||||
{
|
||||
"name": "Simplify",
|
||||
"prompt": "<instruction>Simplify the text below to a 6th-grade reading level (ages 11-12). Use simple sentences, common words, and clear explanations. Maintain the original key concepts. Return only the simplified text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true
|
||||
},
|
||||
{
|
||||
"name": "Emojify",
|
||||
"prompt": "<instruction>Add relevant emojis to enhance the text below. Follow these rules:\n 1. Insert emojis at natural breaks in the text\n 2. Never place two emojis next to each other\n 3. Keep all original text unchanged\n 4. Choose emojis that match the context and tone\n Return only the emojified text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true
|
||||
},
|
||||
{
|
||||
"name": "Make shorter",
|
||||
"prompt": "<instruction>Reduce the text below to half its length while preserving these elements:\n 1. Main ideas and key points\n 2. Essential details\n 3. Original tone and style\n Return only the shortened text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true
|
||||
},
|
||||
{
|
||||
"name": "Make longer",
|
||||
"prompt": "<instruction>Expand the text below to twice its length by:\n 1. Adding relevant details and examples\n 2. Elaborating on key points\n 3. Maintaining the original tone and style\n Return only the expanded text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true
|
||||
},
|
||||
{
|
||||
"name": "Generate table of contents",
|
||||
"prompt": "<instruction>Generate a hierarchical table of contents for the text below. Use appropriate heading levels (H1, H2, H3, etc.). Include page numbers if present. Return only the table of contents.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "Generate glossary",
|
||||
"prompt": "<instruction>Create a glossary of important terms, concepts, and phrases from the text below. Format each entry as \"Term: Definition\". Sort entries alphabetically. Return only the glossary.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "Remove URLs",
|
||||
"prompt": "<instruction>Remove all URLs from the text below. Preserve all other content and formatting. URLs may be in various formats (http, https, www). Return only the text with URLs removed.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "Rewrite as tweet",
|
||||
"prompt": "<instruction>Rewrite the text below as a single tweet with these requirements:\n 1. Maximum 280 characters\n 2. Use concise, impactful language\n 3. Maintain the core message\n Return only the tweet text.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "Rewrite as tweet thread",
|
||||
"prompt": "<instruction>Convert the text below into a Twitter thread following these rules:\n 1. Each tweet must be under 240 characters\n 2. Start with \"THREAD START\" on its own line\n 3. Separate tweets with \"\n\n---\n\n\"\n 4. End with \"THREAD END\" on its own line\n 5. Make content engaging and clear\n Return only the formatted thread.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "Explain like I am 5",
|
||||
"prompt": "<instruction>Explain the text below in simple terms that a 5-year-old would understand:\n 1. Use basic vocabulary\n 2. Include simple analogies\n 3. Break down complex concepts\n Return only the simplified explanation.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "Rewrite as press release",
|
||||
"prompt": "<instruction>Transform the text below into a professional press release:\n 1. Use formal, journalistic style\n 2. Include headline and dateline\n 3. Follow inverted pyramid structure\n Return only the press release format.</instruction>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": false
|
||||
},
|
||||
{
|
||||
"name": "check formula",
|
||||
"prompt": "<instruct>check formula in latex \nreturn only corrected formula in latex </instruct>\n\n<text>{copilot-selection}</text>",
|
||||
"showInContextMenu": true,
|
||||
"modelKey": "gemma3:12b|ollama"
|
||||
}
|
||||
]
|
||||
}
|
@ -7,22 +7,18 @@
|
||||
{"id":"ccedc7c8493efa72","type":"text","text":"根据海龟方法实时监测,提示","x":-80,"y":640,"width":250,"height":60},
|
||||
{"id":"d9e469f53ba4e8e8","type":"text","text":"steamlit","x":-420,"y":860,"width":250,"height":60},
|
||||
{"id":"9ac780ab8e6b0995","type":"text","text":"单个标的 两年 mysql {code}表\n每日最新追加","x":540,"y":380,"width":250,"height":100},
|
||||
{"id":"9eaa6f04dcf05226","type":"text","text":"class turtle 准备数据","x":170,"y":920,"width":250,"height":50},
|
||||
{"id":"4d88d50956995c5c","type":"text","text":"class Turtle_on_time","x":580,"y":915,"width":250,"height":60},
|
||||
{"id":"3dc3fc3a8007b525","type":"text","text":"获取实时数据,仅用来判断","x":260,"y":640,"width":250,"height":60},
|
||||
{"id":"46280e783fc464cc","x":830,"y":550,"width":250,"height":60,"type":"text","text":"买出/买入"},
|
||||
{"id":"f56929c1932b177f","x":1200,"y":457,"width":250,"height":60,"type":"text","text":"send_email()"},
|
||||
{"id":"71c674c78f469276","x":1200,"y":550,"width":250,"height":60,"type":"text","text":"等待回复"},
|
||||
{"id":"564cb83a55900bb9","x":1202,"y":645,"width":250,"height":60,"type":"text","text":"记录价格"}
|
||||
{"id":"992ad295131397e5","type":"text","text":"多Turtle的支持","x":260,"y":760,"width":250,"height":60},
|
||||
{"id":"9eaa6f04dcf05226","type":"text","text":"class turtle 准备数据","x":-40,"y":1020,"width":250,"height":50},
|
||||
{"id":"d862322c10fde2a5","type":"text","text":"如何提示, 与steamlit的结合\n- 邮件方案\n\t发-容易,回信 - 不容易\n- 公众号方案\n\t比较优雅\n\n发出信号后,是暂停等待操作完成,还是继续监测?","x":-110,"y":748,"width":320,"height":225},
|
||||
{"id":"3de77ead7112f4e8","x":1080,"y":820,"width":610,"height":430,"type":"text","text":"\t\t{\"id\":\"46280e783fc464cc\",\"x\":830,\"y\":550,\"width\":250,\"height\":60,\"type\":\"text\",\"text\":\"买出/买入\"},\n\t\t{\"id\":\"f56929c1932b177f\",\"x\":1200,\"y\":457,\"width\":250,\"height\":60,\"type\":\"text\",\"text\":\"send_email()\"},\n\t\t{\"id\":\"71c674c78f469276\",\"x\":1200,\"y\":550,\"width\":250,\"height\":60,\"type\":\"text\",\"text\":\"等待回复\"},\n\t\t{\"id\":\"564cb83a55900bb9\",\"x\":1202,\"y\":645,\"width\":250,\"height\":60,\"type\":\"text\",\"text\":\"记录价格\"}"}
|
||||
],
|
||||
"edges":[
|
||||
{"id":"c35374c532b0eeff","fromNode":"0b73f4540fbacbef","fromSide":"right","toNode":"6d7e97110fbb9fe3","toSide":"left"},
|
||||
{"id":"ea95de7df4aecf5a","fromNode":"6d7e97110fbb9fe3","fromSide":"right","toNode":"be8785f5c0ab6e70","toSide":"left"},
|
||||
{"id":"06c6defefd6e745a","fromNode":"6d7e97110fbb9fe3","fromSide":"bottom","toNode":"842b08dd71af6d2e","toSide":"top"},
|
||||
{"id":"9c80b8410a03208e","fromNode":"842b08dd71af6d2e","fromSide":"bottom","toNode":"ccedc7c8493efa72","toSide":"top"},
|
||||
{"id":"aedfd6cb63d28d1d","fromNode":"be8785f5c0ab6e70","fromSide":"right","toNode":"9ac780ab8e6b0995","toSide":"left"},
|
||||
{"id":"1a7d49aa65061e88","fromNode":"f56929c1932b177f","fromSide":"bottom","toNode":"71c674c78f469276","toSide":"top"},
|
||||
{"id":"5fe491f7948445cd","fromNode":"71c674c78f469276","fromSide":"bottom","toNode":"564cb83a55900bb9","toSide":"top"},
|
||||
{"id":"bac795506eca56c6","fromNode":"46280e783fc464cc","fromSide":"right","toNode":"f56929c1932b177f","toSide":"top"}
|
||||
{"id":"aedfd6cb63d28d1d","fromNode":"be8785f5c0ab6e70","fromSide":"right","toNode":"9ac780ab8e6b0995","toSide":"left"}
|
||||
]
|
||||
}
|
@ -5470,59 +5470,67 @@ $$
|
||||
$$
|
||||
|
||||
Subtracting the first equation of Eq. 13 from Eq. 16 yields $(\delta l)^{2}-(\delta l_{o})^{2}=$ $2(d\mathbf{x})^{\mathrm{T}}\boldsymbol{\varepsilon}_{m}d\mathbf{x}$ , or
|
||||
从方程13的第一个方程中减去方程16,得到 $(\delta l)^{2}-(\delta l_{o})^{2}=$ $2(d\mathbf{x})^{\mathrm{T}}\boldsymbol{\varepsilon}_{m}d\mathbf{x}$ ,或者
|
||||
|
||||
$$
|
||||
\frac{1}{2}[(\delta l)^{2}-(\delta l_{o})^{2}]=(d\mathbf{x})^{\mathrm{T}}\varepsilon_{m}d\mathbf{x}
|
||||
$$
|
||||
|
||||
where $\varepsilon_{m}$ is a $3\times3$ symmetric matrix called the Lagrangian strain tensor, and is defined as
|
||||
其中 $\varepsilon_{m}$ 是一个 $3\times3$ 的对称矩阵,称为**拉格朗日应变张量**,其定义如下:
|
||||
|
||||
$$
|
||||
{\bf\varepsilon}_{{m}}=\frac{1}{2}\{[{\bar{\bf J}}^{\mathrm{T}}+{\bar{\bf J}}]+{\bar{\bf J}}^{\mathrm{T}}{\bar{\bf J}}\}=\frac{1}{2}({\bf J}^{\mathrm{T}}{\bf J}-{\bf I})
|
||||
$$
|
||||
|
||||
Using matrix multiplications, it can be verified that the components $\varepsilon_{i j}$ of the matrix $\varepsilon_{m}$ are given by
|
||||
Using matrix multiplications, it can be verified that the components $\varepsilon_{i j}$ of the matrix $\varepsilon_{m}$ are given by
|
||||
使用矩阵乘法,可以验证矩阵 $\varepsilon_{m}$ 的分量 $\varepsilon_{i j}$ 由...给出。
|
||||
|
||||
$$
|
||||
\varepsilon_{i j}=\frac{1}{2}\left(u_{i,j}+u_{j,i}+\sum_{k=1}^{3}u_{k,i}u_{k,j}\right),\ \ i,j=1,2,3
|
||||
$$
|
||||
|
||||
The components $\varepsilon_{i j}$ that arise naturally in the analysis of deformation are called the strain components. Therefore, the strain components are, in general, nonlinear functions of the spatial derivatives of the displacement. Because of the symmetry of the strain tensor, it is sufficient to identify only the following six components: $\varepsilon_{11},\,\varepsilon_{22},\,\varepsilon_{33},\,\varepsilon_{12},\,\varepsilon_{13}$ , and $\varepsilon_{23}$ , which form the strain vector ε, that is,
|
||||
在变形分析中自然出现的成分 $\varepsilon_{i j}$ 被称为应变成分。因此,应变成分通常是非线性的,是位移空间导数的函数。由于应变张量的对称性,只需确定以下六个成分就足够了:$\varepsilon_{11},\,\varepsilon_{22},\,\varepsilon_{33},\,\varepsilon_{12},\,\varepsilon_{13}$ , 和 $\varepsilon_{23}$ ,它们构成应变矢量 ε,即,
|
||||
|
||||
$$
|
||||
\begin{array}{r}{\varepsilon=[\varepsilon_{11}\quad\varepsilon_{22}\quad\varepsilon_{33}\quad\varepsilon_{12}\quad\varepsilon_{13}\quad\varepsilon_{23}]^{\mathrm{T}}}\end{array}
|
||||
$$
|
||||
|
||||
Thus, the strain vector $\varepsilon$ can be written in a compact form as
|
||||
因此,应力矢量 $\varepsilon$ 可以写成紧凑形式如下:
|
||||
|
||||
$$
|
||||
\varepsilon=\mathbf{D}\mathbf{u}
|
||||
$$
|
||||
|
||||
where $\mathbf{D}$ is a differential operator defined according to Eq. 19.
|
||||
其中 $\mathbf{D}$ 是根据公式 19 定义的微分算子。
|
||||
|
||||
Small Strains It was previously shown that the gradient of the displacement vector ¯J can be written as $\bar{\mathbf{J}}=\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r}$ , where $\bar{\mathbf{J}}_{s}$ is symmetric and $\bar{\mathbf{J}}_{r}$ is antisymmetric. In the case of small strains and rotations, the squares and products of ${\bf\bar{J}}_{s}$ and $\bar{\mathbf{J}}_{r}$ can be
|
||||
Small Strains It was previously shown that the gradient of the displacement vector $\bar{\mathbf{J}}$ can be written as $\bar{\mathbf{J}}=\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r}$ , where $\bar{\mathbf{J}}_{s}$ is symmetric and $\bar{\mathbf{J}}_{r}$ is antisymmetric. In the case of small strains and rotations, the squares and products of ${\bf\bar{J}}_{s}$ and $\bar{\mathbf{J}}_{r}$ can be neglected, that is $\mathbf{\bar{J}}^{\mathrm{T}}\mathbf{\bar{J}}\approx\mathbf{0}$ , and to the same order of approximation the strain tensor reduces to
|
||||
小应变情况
|
||||
|
||||
此前已证明,位移矢量 $\bar{\mathbf{J}}$ 的梯度可以写成 $\bar{\mathbf{J}}=\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r}$ ,其中 $\bar{\mathbf{J}}_{s}$ 是对称张量,$\bar{\mathbf{J}}_{r}$ 是反对称张量。在小应变和小旋转的情况下,${\bf\bar{J}}_{s}$ 和 $\bar{\mathbf{J}}_{r}$ 的平方和乘积可以忽略,即 $\mathbf{\bar{J}}^{\mathrm{T}}\mathbf{\bar{J}}\approx\mathbf{0}$ ,并且在相同的近似阶数下,应变张量简化为
|
||||
|
||||
neglected, that is $\mathbf{\bar{J}}^{\mathrm{T}}\mathbf{\bar{J}}\approx\mathbf{0}$ , and to the same order of approximation the strain tensor reduces to
|
||||
|
||||
$$
|
||||
\boldsymbol{\varepsilon}_{m}\approx\frac{1}{2}[\bar{\mathbf{J}}^{\mathrm{T}}+\bar{\mathbf{J}}]
|
||||
$$
|
||||
|
||||
which is the form of the strain tensor often used in engineering applications. In this special case, the differential operator of Eq. 21 reduces to
|
||||
|
||||
这通常是应力张量的形式,在工程应用中经常使用。在这种特殊情况下,公式21中的微分算子简化为:
|
||||
$$
|
||||
\mathbf{D}=\frac{1}{2}\left[\begin{array}{l l l}{2\frac{\partial}{\partial x_{1}}}&{0}&{0}\\ {0}&{2\frac{\partial}{\partial x_{2}}}&{0}\\ {0}&{0}&{2\frac{\partial}{\partial x_{3}}}\\ {\frac{\partial}{\partial x_{2}}}&{\frac{\partial}{\partial x_{1}}}&{0}\\ {\frac{\partial}{\partial x_{3}}}&{0}&{\frac{\partial}{\partial x_{1}}}\\ {0}&{\frac{\partial}{\partial x_{3}}}&{\frac{\partial}{\partial x_{2}}}\end{array}\right]
|
||||
$$
|
||||
|
||||
and $\varepsilon_{11},\,\varepsilon_{22}$ , and $\varepsilon_{33}$ can be recognized as the normal strains while $\varepsilon_{12},\,\varepsilon_{13}$ , and $\varepsilon_{23}$ are recognized as the shear strains. Note that $|\partial u_{i}/\partial x_{j}|\ll1$ implies that the strains and rotations are small. There are, however, some applications in which the strains are small everywhere but the rotations are large. An example of these applications is the bending of a long thin flexible beam.
|
||||
|
||||
and $\varepsilon_{11}$、$\varepsilon_{22}$ 和 $\varepsilon_{33}$ 可以被认为是正应变,而 $\varepsilon_{12}$、$\varepsilon_{13}$ 和 $\varepsilon_{23}$ 则被认为是剪应变。需要注意的是, $|\partial u_{i}/\partial x_{j}|\ll1$ 表明应变和转动都很小。然而,在某些应用中,应变处处都很小,但转动却很大。弯曲细长柔性梁就是一个例子。
|
||||
Example 4.2 For the displacement of the body given in Example 1, find the strain components.
|
||||
|
||||
例 4.2 对于例 1 中给定的位移,求应变分量。
|
||||
Solution Using Eq. 19 and the spatial derivatives of the displacement given in Example 1, one has
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{e_{11}=\frac{1}{2}\big\{u_{1,1}+u_{1,1}+(u_{1,1})^{2}+(u_{2,1})^{2}+(u_{3,1})^{2}\big\}}\\ &{\phantom{m m m m m m m m}=\frac{1}{2}\big[2(k_{2}+(k_{2})^{2}+(k_{4}+2k_{3}x_{1}+3k_{6}(x_{1}))^{2})^{\top}\big]}\\ &{\phantom{m m m m m m m m m m}}\\ {e_{22}=\frac{1}{2}\big[u_{2,2}+u_{2,2}+(u_{4,2})^{2}+(u_{2,2})^{2}+(u_{3,2})^{2}\big]=0}\\ &{e_{33}=\frac{1}{2}\big[u_{3,3}+u_{3,3}+(u_{1,3})^{2}+(u_{2,3})^{2}+(u_{3,3})^{2}\big]=0}\\ &{e_{12}=\frac{1}{2}\big[u_{1,2}+u_{2,1}+u_{1,1}u_{1,2}+u_{2,1}u_{2,2}+u_{1,1}u_{3,2}\big]}\\ &{\phantom{m m m m m m m}=\frac{1}{2}\big[k_{4}+2k_{3}x_{1}+3k_{6}(x_{1})^{2}\big]}\\ &{e_{13}=\frac{1}{2}\big[u_{1,3}+u_{1,1}+u_{1,1}u_{1,3}+u_{2,1}u_{2,3}+u_{1,1}u_{3,3}\big]=0}\\ &{e_{23}=\frac{1}{2}\big[u_{2,3}+u_{3,2}+u_{1,3}u_{1,3}+u_{2,2}u_{3,2}+u_{3,3}^{2}(u_{3,3})\big]=0}\end{array}
|
||||
\begin{array}{r l}&{e_{11}=\frac{1}{2}\big\{u_{1,1}+u_{1,1}+(u_{1,1})^{2}+(u_{2,1})^{2}+(u_{3,1})^{2}\big\}}\\ &{\phantom{m m m m m m m m}=\frac{1}{2}\big[2(k_{2}+(k_{2})^{2}+(k_{4}+2k_{3}x_{1}+3k_{6}(x_{1}))^{2})^{\top}\big]}\\ &{\phantom{m m m m m m m m m m}}\\ &{e_{22}=\frac{1}{2}\big[u_{2,2}+u_{2,2}+(u_{4,2})^{2}+(u_{2,2})^{2}+(u_{3,2})^{2}\big]=0}\\ &{e_{33}=\frac{1}{2}\big[u_{3,3}+u_{3,3}+(u_{1,3})^{2}+(u_{2,3})^{2}+(u_{3,3})^{2}\big]=0}\\ &{e_{12}=\frac{1}{2}\big[u_{1,2}+u_{2,1}+u_{1,1}u_{1,2}+u_{2,1}u_{2,2}+u_{1,1}u_{3,2}\big]}\\ &{\phantom{m m m m m m m}=\frac{1}{2}\big[k_{4}+2k_{3}x_{1}+3k_{6}(x_{1})^{2}\big]}\\ &{e_{13}=\frac{1}{2}\big[u_{1,3}+u_{1,1}+u_{1,1}u_{1,3}+u_{2,1}u_{2,3}+u_{1,1}u_{3,3}\big]=0}\\ &{e_{23}=\frac{1}{2}\big[u_{2,3}+u_{3,2}+u_{1,3}u_{1,3}+u_{2,2}u_{3,2}+u_{3,3}^{2}(u_{3,3})\big]=0}\end{array}
|
||||
$$
|
||||
|
||||
Therefore, the vector of strains $\varepsilon$ of Eq. 20 is given by
|
||||
@ -5538,16 +5546,19 @@ $$
|
||||
$$
|
||||
|
||||
Another Form for the Strain Components Using the definition of the Lagrangian strain in terms of the Jacobian matrix J (Eq. 18), one can show that the components of the nonlinear Lagrangian strain tensor can be written as follows:
|
||||
利用雅可比矩阵 J (式 18) 对拉格朗日应变的概念进行定义,可以证明非线性拉格朗日应变张量的分量可以写成如下形式:
|
||||
|
||||
$$
|
||||
\varepsilon_{m}={\frac{1}{2}}\left({\bf J}^{\mathrm{T}}{\bf J}-{\bf I}\right)={\frac{1}{2}}\left[\begin{array}{c c c}{\left(\xi_{1}^{\mathrm{T}}\xi_{1}-1\right)}&{\left|\xi_{1}\right|\xi_{2}\left|\cos\alpha_{12}}&{\left|\xi_{1}\right|\left|\xi_{3}\right|\cos\alpha_{13}}\\ {\left|\xi_{1}\right|\left|\xi_{2}\right|\cos\alpha_{12}}&{\left(\xi_{2}^{\mathrm{T}}\xi_{2}-1\right)}&{\left|\xi_{2}\right|\left|\xi_{3}\right|\cos\alpha_{23}}\\ {\left|\xi_{1}\right|\left|\xi_{3}\right|\cos\alpha_{13}}&{\left|\xi_{2}\right|\left|\xi_{3}\right|\cos\alpha_{23}}&{\left(\xi_{3}^{\mathrm{T}}\xi_{3}-1\right)}\end{array}\right]
|
||||
\varepsilon_{m}={\frac{1}{2}}\left(\mathbf{J}^{\mathrm{T}}\mathbf{J}-\mathbf{I}\right)={\frac{1}{2}}\left[\begin{array}{c c c}{\left(\xi_{,1}^{\mathrm{T}}\xi_{,1}-1\right)}&{\left|\xi_{,1}\right|\xi_{,2}\cos\alpha_{12}}&{\left|\xi_{,1}\right|\left|\xi_{,3}\right|\cos\alpha_{13}}\\ {\left|\xi_{,1}\right|\left|\xi_{,2}\right|\cos\alpha_{12}}&{\left(\xi_{,2}^{\mathrm{T}}\xi_{,2}-1\right)}&{\left|\xi_{,2}\right|\left|\xi_{,3}\right|\cos\alpha_{23}}\\ {\left|\xi_{,1}\right|\left|\xi_{,3}\right|\cos\alpha_{13}}&{\left|\xi_{,2}\right|\left|\xi_{,3}\right|\cos\alpha_{23}}&{\left(\xi_{,3}^{\mathrm{T}}\xi_{,3}-1\right)}\end{array}\right]
|
||||
$$
|
||||
|
||||
where ξ,i = ∂∂xξ , and $\alpha_{i j}$ is the angle between the vectors $\xi_{i}$ and $\xi_{j}$ . The elements of the strain tensor of Eq. 24 give a clear physical interpretation of the normal and shear strain components.
|
||||
where $\xi_{,i} = \partial \xi/\partial x_{j}$, and $\alpha_{i j}$ is the angle between the vectors $\xi_{i}$ and $\xi_{j}$ . The elements of the strain tensor of Eq. 24 give a clear physical interpretation of the normal and shear strain components.
|
||||
其中,$\xi_{,i} = \partial \xi/\partial x_{j}$,且 $\alpha_{ij}$ 是向量 $\xi_{i}$ 和 $\xi_{j}$ 之间的夹角。公式 24 中的应变张量元素,可以清晰地解释法向应变和剪切应变的分量。
|
||||
|
||||
# 4.3 PHYSICAL INTERPRETATION OF STRAINS
|
||||
|
||||
The physical interpretation of the strains can also be provided in terms of the extension of the line element $P_{o}\,Q_{o}$ (Fig. 4.2), defined as
|
||||
The physical interpretation of the strains can also be provided in terms of the extension of the line element $P_{o}\,Q_{o}$ (Fig. 4.2), defined as
|
||||
对应于线元 $P_{o}\,Q_{o}$ (图 4.2) 的伸长,也可以用来解释应变,其定义为:
|
||||
|
||||
$$
|
||||
e=\delta l-\delta l_{o}
|
||||
@ -5559,7 +5570,8 @@ $$
|
||||
\varepsilon=\frac{e}{\delta l_{o}}=\frac{\delta l}{\delta l_{o}}-1
|
||||
$$
|
||||
|
||||
Let $\mathbf{n}$ be the vector of direction cosines along the line $P_{o}\,Q_{o}$ in the undeformed state, that is
|
||||
Let $\mathbf{n}$ be the vector of direction cosines along the line $P_{o}\,Q_{o}$ in the undeformed state, that is
|
||||
令 $\mathbf{n}$ 为线 $P_{o}\,Q_{o}$ 在未变形状态下的方向余弦向量,即
|
||||
|
||||
$$
|
||||
\mathbf{n}=\frac{d\mathbf{x}}{\delta l_{o}}=\frac{1}{\delta l_{o}}[d x_{1}\quad d x_{2}\quad d x_{3}]^{\mathrm{T}}
|
||||
@ -5595,26 +5607,31 @@ $$
|
||||
\varepsilon=-1\pm(1+\bar{\varepsilon}_{m})^{1/2}
|
||||
$$
|
||||
|
||||
The second solution is physically impossible because it does not represent the rigid body motion. Hence
|
||||
The second solution is physically impossible because it does not represent the rigid body motion. Hence
|
||||
第二个解决方案在物理上是不可行的,因为它不代表刚体的运动。因此,
|
||||
|
||||
$$
|
||||
\begin{array}{l}{{\displaystyle{\varepsilon=-1+(1+\bar{\varepsilon}_{m})^{1/2}=-1+1+\frac{1}{2}\bar{\varepsilon}_{m}-\frac{1}{8}(\bar{\varepsilon}_{m})^{2}+\cdot\cdot\cdot}}}\\ {{\displaystyle{\phantom{=}}}}\\ {{\displaystyle{\phantom{=}=\frac{1}{2}\bar{\varepsilon}_{m}-\frac{1}{8}(\bar{\varepsilon}_{m})^{2}+\cdot\cdot\cdot}}}\end{array}
|
||||
$$
|
||||
|
||||
where the binomial theorem has been used. Equation 33 represents the strain in the general case of large deformation. If, however, the strain components are assumed to be small, that is, $(\bar{\varepsilon}_{m})^{2}\approx0$ , Eq. 33 reduces to $\varepsilon\approx\bar{\varepsilon}_{m}/2$ , which by using Eq. 31 yields
|
||||
其中二项式定理已被应用。公式33表示大变形一般情况下的应变。然而,如果应变分量被假定为较小,即$(\bar{\varepsilon}_{m})^{2} \approx 0$,则公式33简化为$\varepsilon \approx \bar{\varepsilon}_{m}/2$,通过使用公式31可得。
|
||||
|
||||
$$
|
||||
\boldsymbol\varepsilon\approx{\mathbf{n}}^{{\mathrm{T}}}\boldsymbol\varepsilon_{m}{\mathbf{n}}
|
||||
$$
|
||||
|
||||
One can also show by directly using Eq. 29 that the definition of strain in the case of large deformation theory (Eq. 33) does not differ greatly from the definition of Eq. 34 unless the relative elongation $e$ of Eq. 25 is large. Equation 34 implies that the strain along a line element whose direction cosines in the undeformed state with respect to three orthogonal axes $\mathbf{X}_{1}$ , $\mathbf{X}_{2}$ , $\mathbf{X}_{3}$ are defined by the vector n can be determined if the strain components ${\boldsymbol{\varepsilon}}=[\varepsilon_{11}~\varepsilon_{22}~\varepsilon_{33}~\varepsilon_{12}~\varepsilon_{13}~\varepsilon_{23}]^{\mathrm{T}}$ are known.
|
||||
也可以通过直接使用公式 29 证明,在大型变形理论中应变定义(公式 33)与公式 34 的定义差异不大,除非公式 25 中的相对伸长量 $e$ 较大。公式 34 意味着,如果已知应变分量 ${\boldsymbol{\varepsilon}}=[\varepsilon_{11}~\varepsilon_{22}~\varepsilon_{33}~\varepsilon_{12}~\varepsilon_{13}~\varepsilon_{23}]^{\mathrm{T}}$,则可以确定沿一条在未变形状态下相对于三个正交轴 $\mathbf{X}_{1}$ , $\mathbf{X}_{2}$ , $\mathbf{X}_{3}$ 的方向余弦由向量 n 定义的线元上的应变。
|
||||
|
||||
Simple Example The preceding development can be exemplified by considering the case in which the element and the extension are along the $\mathbf{X}_{1}$ direction. In this case, the vector $d\mathbf{x}$ has the components $d\mathbf{x}=[d x_{1}\quad0\quad0]^{\mathrm{T}}.$ . The length of the line segment in the undeformed state can then be written as $\delta l_{o}=\sqrt{(d{\bf x})^{\mathrm{T}}(d{\bf x})}=$ $d x_{1}$ . If higher-order terms are neglected in Eq. 33, the strain can be written as $\varepsilon=\bar{\varepsilon}_{m}/2$ , or $\boldsymbol\varepsilon={\mathbf{n}}^{{\mathrm{T}}}\boldsymbol\varepsilon_{m}{\mathbf{n}}$ In this special case, one can verify that $\mathbf{n}=[1\,\,\,0\,\,\,0]^{\mathrm{T}}$ and $\varepsilon=[2u_{1,1}+(u_{1,1})^{2}+(u_{2,1})^{2}+(u_{3,1})^{2}]/2$ . If the assumption that the displacement gradients are small is used, one may neglect second-order terms and write $\varepsilon=u_{1,1}=\partial u_{1}/\partial x_{1}$ , which is the same expression used in textbooks on the strength of materials.
|
||||
Simple Example The preceding development can be exemplified by considering the case in which the element and the extension are along the $\mathbf{X}_{1}$ direction. In this case, the vector $d\mathbf{x}$ has the components $d\mathbf{x}=[d x_{1}\quad0\quad0]^{\mathrm{T}}.$ . The length of the line segment in the undeformed state can then be written as $\delta l_{o}=\sqrt{(d{\bf x})^{\mathrm{T}}(d{\bf x})}=$ $d x_{1}$ . If higher-order terms are neglected in Eq. 33, the strain can be written as $\varepsilon=\bar{\varepsilon}_{m}/2$ , or $\boldsymbol\varepsilon={\mathbf{n}}^{{\mathrm{T}}}\boldsymbol\varepsilon_{m}{\mathbf{n}}$ In this special case, one can verify that $\mathbf{n}=[1\,\,\,0\,\,\,0]^{\mathrm{T}}$ and $\varepsilon=[2u_{1,1}+(u_{1,1})^{2}+(u_{2,1})^{2}+(u_{3,1})^{2}]/2$ . If the assumption that the displacement gradients are small is used, one may neglect second-order terms and write $\varepsilon=u_{1,1}=\partial u_{1}/\partial x_{1}$ , which is the same expression used in textbooks on the strength of materials.
|
||||
|
||||
简单示例 先前的推导可以通过考虑元素和延长方向沿 $\mathbf{X}_{1}$ 方向的情况来加以说明。在这种情况下,矢量 $d\mathbf{x}$ 的分量为 $d\mathbf{x}=[d x_{1}\quad0\quad0]^{\mathrm{T}}.$。未变形状态下线段的长度可以写成 $\delta l_{o}=\sqrt{(d{\bf x})^{\mathrm{T}}(d{\bf x})}=$ $d x_{1}$。如果忽略公式 33 中的高阶项,应变可以写成 $\varepsilon=\bar{\varepsilon}_{m}/2$,或 $\boldsymbol\varepsilon={\mathbf{n}}^{{\mathrm{T}}}\boldsymbol\varepsilon_{m}{\mathbf{n}}$。在这种特殊情况下,可以验证 $\mathbf{n}=[1\,\,\,0\,\,\,0]^{\mathrm{T}}$ 且 $\varepsilon=[2u_{1,1}+(u_{1,1})^{2}+(u_{2,1})^{2}+(u_{3,1})^{2}]/2$。如果采用位移梯度较小的假设,可以忽略二阶项,写成 $\varepsilon=u_{1,1}=\partial u_{1}/\partial x_{1}$,这与材料力学教科书中的表达式相同。
|
||||
|
||||
# 4.4 RIGID BODY MOTION
|
||||
|
||||
In the case of a general rigid body displacement, the vector $\xi$ can be written as $\boldsymbol{\xi}=\mathbf{R}+\mathbf{Ax}$ , where $\mathbf{R}$ is the translation of the reference point and $\mathbf{A}$ is the orthogonal transformation matrix that defines the body orientation. It follows, in the case of rigid body displacement, that
|
||||
|
||||
就一般刚体位移而言,向量 $\xi$ 可以写成 $\boldsymbol{\xi}=\mathbf{R}+\mathbf{Ax}$ ,其中 $\mathbf{R}$ 是参考点的平移向量,$\mathbf{A}$ 是定义刚体姿态的正交变换矩阵。由此可见,在刚体位移的情况下,
|
||||
$$
|
||||
\mathbf{u}=\boldsymbol{\xi}-\mathbf{x}=\mathbf{R}+(\mathbf{A}-\mathbf{I})\mathbf{x}
|
||||
$$
|
||||
@ -5625,16 +5642,21 @@ $$
|
||||
\mathbf{J}=\mathbf{A},\ \ \ \ \ \bar{\mathbf{J}}=\mathbf{A}-\mathbf{I}
|
||||
$$
|
||||
|
||||
which demonstrate that $\mathbf{J}$ and $\bar{\mathbf{J}}$ do not remain constant in the case of a general rigid body motion, and therefore, they are not an appropriate measure of the deformation. Note that in this case, the Lagrangian strain tensor $\varepsilon_{m}$ is given by
|
||||
which demonstrate that $\mathbf{J}$ and $\bar{\mathbf{J}}$ do not remain constant in the case of a general rigid body motion, and therefore, they are not an appropriate measure of the deformation. Note that in this case, the Lagrangian strain tensor $\varepsilon_{m}$ is given by
|
||||
这表明 $\mathbf{J}$ 和 $\bar{\mathbf{J}}$ 在一般刚体运动情况下并非保持不变,因此它们不适合作为形变量来衡量。需要注意的是,在这种情况下,拉格朗日应变张量 $\varepsilon_{m}$ 由…给出。
|
||||
|
||||
$$
|
||||
\ \mathbf{\varepsilon}_{\!\varepsilon_{m}}={\frac{1}{2}}(\mathbf{J}^{\mathrm{T}}\mathbf{J}-\mathbf{I})=\mathbf{0}
|
||||
$$
|
||||
|
||||
and, therefore, $\varepsilon_{m}$ can be used as a deformation measure.
|
||||
|
||||
因此,$\varepsilon_{m}$ 可以作为一种变形量。
|
||||
Other Deformation Measures In continuum mechanics, several other deformation measures are often used. To briefly introduce these measures, we use Eq. 16 to write
|
||||
|
||||
|
||||
其他变形量
|
||||
在连续介质力学中,经常使用其他一些变形量。为了简要介绍这些量,我们使用公式 16 来表达
|
||||
|
||||
$$
|
||||
\left({\frac{\delta l}{\delta l_{o}}}\right)^{2}=\mathbf{n}^{\mathrm{T}}\mathbf{J}^{\mathrm{T}}\mathbf{J}\mathbf{n}=\mathbf{n}^{\mathrm{T}}\mathbf{C}_{r}\mathbf{n}
|
||||
$$
|
||||
@ -5646,36 +5668,42 @@ $$
|
||||
$$
|
||||
|
||||
is a symmetric tensor, called the right Cauchy–Green deformation tensor. The tensor $\mathbf{C}_{r}$ can be used as a measure of the deformation since in the case of a general rigid body displacement $\mathbf{C}_{r}=\mathbf{A}^{\mathrm{T}}\mathbf{A}=\mathbf{I},$ , and as a consequence, $\mathbf{C}_{r}$ remains constant throughout a rigid body motion. The Lagrangian strain tensor can be expressed in terms of $\mathbf{C}_{r}$ as
|
||||
是一个对称张量,称为右柯西-格林变形张量。张量 $\mathbf{C}_{r}$ 可以作为变形的度量,因为对于一般的刚体位移而言,$\mathbf{C}_{r}=\mathbf{A}^{\mathrm{T}}\mathbf{A}=\mathbf{I}$,因此,$\mathbf{C}_{r}$ 在刚体运动过程中保持不变。拉格朗日应变张量可以表示为 $\mathbf{C}_{r}$ 的函数,如下所示:
|
||||
|
||||
$$
|
||||
\boldsymbol{\varepsilon_{m}}=\frac{1}{2}(\mathbf{C}_{r}-\mathbf{I})
|
||||
$$
|
||||
|
||||
Another deformation measure is the left Cauchy–Green deformation tensor $\mathbf{C}_{l}$ defined as
|
||||
Another deformation measure is the left Cauchy–Green deformation tensor $\mathbf{C}_{l}$ defined as 另一种变形量是左柯西-格林变形张量 $\mathbf{C}_{l}$,其定义如下:
|
||||
|
||||
|
||||
$$
|
||||
\mathbf{C}_{l}=\mathbf{J}\mathbf{J}^{\mathrm{T}}
|
||||
$$
|
||||
|
||||
This tensor also remains constant and equal to the identity matrix in the case of rigid body motion. Another strain tensor $\varepsilon_{E}$ , called the Eulerian strain tensor, is defined in terms of $\mathbf{C}_{l}$ as
|
||||
This tensor also remains constant and equal to the identity matrix in the case of rigid body motion. Another strain tensor $\varepsilon_{E}$ , called the Eulerian strain tensor, is defined in terms of $\mathbf{C}_{l}$ as
|
||||
在刚体运动的情况下,此张量也保持恒定且等于单位矩阵。另一个应变张量 $\varepsilon_{E}$,称为欧拉应变张量,是根据 $\mathbf{C}_{l}$ 定义的。
|
||||
|
||||
$$
|
||||
\boldsymbol{\varepsilon_{E}}=\frac{1}{2}\big(\mathbf{I}-\mathbf{C}_{l}^{-1}\big)
|
||||
$$
|
||||
|
||||
In the case of rigid body motion, $\pmb{\varepsilon}_{E}=\pmb{\varepsilon}_{m}=\mathbf{0}$ . Furthermore, in the case of infinitesimal strains (small displacement gradients),
|
||||
在刚体运动的情况下,$\pmb{\varepsilon}_{E}=\pmb{\varepsilon}_{m}=\mathbf{0}$。 此外,对于微小应变(小位移梯度)的情况下,
|
||||
|
||||
$$
|
||||
\ \pmb{\varepsilon}_{E}=\pmb{\varepsilon}_{m}=\frac{1}{2}(\bar{\mathbf{J}}+\bar{\mathbf{J}}^{\mathrm{T}})
|
||||
$$
|
||||
|
||||
It is important, however, to point out that the infinitesimal strain tensor is not an exact measure of the deformation because it does not remain constant in the case of a rigid body motion. Recall that, in the case of rigid body motion, $\mathbf{J}=\mathbf{A}$ , and
|
||||
然而,需要指出的是,无穷小应变张量并非变形的精确度量,因为在刚体运动的情况下,它并不保持恒定。请回顾一下,在刚体运动的情况下,$\mathbf{J}=\mathbf{A}$,并且
|
||||
|
||||
$$
|
||||
\ \varepsilon_{m}={\frac{1}{2}}({\bar{\mathbf{J}}}^{\mathrm{T}}+{\bar{\mathbf{J}}})={\frac{1}{2}}(\mathbf{A}^{\mathrm{T}}+\mathbf{A}-2\mathbf{I})
|
||||
$$
|
||||
|
||||
It can be shown, however, that the elements of this tensor are of second order in the case of small rotations. For example, in the case of a simple rotation $\theta$ about the $\mathbf{X}_{3}$ axis, one has
|
||||
然而,可以证明,对于小旋转,此张量的元素是二阶的。例如,对于绕 $\mathbf{X}_{3}$ 轴进行简单旋转 $\theta$,则有
|
||||
|
||||
$$
|
||||
\begin{array}{r}{\mathbf{A}=\left[\begin{array}{c c c}{\cos\theta}&{-\sin\theta}&{0}\\ {\sin\theta}&{\cos\theta}&{0}\\ {0}&{0}&{1}\end{array}\right],}\end{array}
|
||||
@ -5688,34 +5716,42 @@ $$
|
||||
$$
|
||||
|
||||
which is of second order in the rotation $\theta$ since cos $\begin{array}{r}{\theta-1=-{\frac{\theta^{2}}{2!}}+{\frac{\theta^{4}}{4!}}+\cdot\cdot\cdot.}\end{array}$ .
|
||||
这是关于 $\theta$ 旋转的二阶项,因为 cos $\begin{array}{r}{\theta-1=-{\frac{\theta^{2}}{2!}}+{\frac{\theta^{4}}{4!}}+\cdot\cdot\cdot.}\end{array}$ 。
|
||||
|
||||
Decomposition of Displacement Using the polar decomposition theorem (Spencer 1980), it can be shown that the Jacobian matrix $\mathbf{J}$ can be written as
|
||||
位移分解:利用极分解定理(Spencer 1980),可以证明雅可比矩阵 $\mathbf{J}$ 可以表示为:
|
||||
|
||||
|
||||
$$
|
||||
\mathbf{J}=\mathbf{A}_{J}\mathbf{J}_{r}=\mathbf{J}_{l}\mathbf{A}_{J}
|
||||
$$
|
||||
|
||||
where $\mathbf{A}_{J}$ is an orthogonal rotation matrix, and $\mathbf{J}_{r}$ and $\mathbf{J}_{l}$ are symmetric positive definite matrices. The matrices $\mathbf{J}_{r}$ and $\mathbf{J}_{l}$ are called the right stretch and left stretch tensors, respectively. It follows from the preceding equation that
|
||||
其中 $\mathbf{A}_{J}$ 是一个正交旋转矩阵,而 $\mathbf{J}_{r}$ 和 $\mathbf{J}_{l}$ 是对称正定矩阵。矩阵 $\mathbf{J}_{r}$ 和 $\mathbf{J}_{l}$ 分别被称为右拉伸张量和左拉伸张量。根据上述方程可以推导出,
|
||||
|
||||
$$
|
||||
\mathbf{J}_{r}=\mathbf{A}_{J}^{\mathrm{T}}\mathbf{J}_{l}\mathbf{A}_{J},\ \ \ \ \mathbf{J}_{l}=\mathbf{A}_{J}\mathbf{J}_{r}\mathbf{A}_{J}^{\mathrm{T}}
|
||||
$$
|
||||
|
||||
In the special case of homogeneous motion, the Jacobian matrix $\mathbf{J}$ is assumed to be constant and independent of the spatial coordinates. In this special case, one has
|
||||
在匀速运动的特殊情况下,雅可比矩阵 $\mathbf{J}$ 被假定为常数,且与空间坐标无关。在这种特殊情况下,我们有
|
||||
|
||||
$$
|
||||
\boldsymbol{\xi}=\mathbf{J}\mathbf{x}
|
||||
$$
|
||||
|
||||
The motion of the body from the initial configuration $\mathbf{X}$ to the final configuration $\xi$ can be considered as two successive homogeneous motions. In the first motion, the coordinate vector $\mathbf{X}$ changes to $\mathbf{X}_{i}$ , and in the second motion, the coordinate vector $\mathbf{X}_{i}$ changes to $\xi$ , such that $\mathbf{x}_{i}=\mathbf{J}_{r}\mathbf{x}$ , $\xi=\mathbf{A}_{J}\mathbf{x}_{i}$ . It follows that
|
||||
躯体从初始构型 $\mathbf{X}$ 运动到最终构型 $\xi$ 可以被视为两个连续的齐次运动。在第一次运动中,坐标向量 $\mathbf{X}$ 变为 $\mathbf{X}_{i}$,在第二次运动中,坐标向量 $\mathbf{X}_{i}$ 变为 $\xi$, 使得 $\mathbf{x}_{i}=\mathbf{J}_{r}\mathbf{x}$, $\xi=\mathbf{A}_{J}\mathbf{x}_{i}$。由此可见,
|
||||
|
||||
$$
|
||||
\xi=\mathbf{A}_{J}\mathbf{x}_{i}=\mathbf{A}_{J}\mathbf{J}_{r}\mathbf{x}=\mathbf{J}\mathbf{x}
|
||||
$$
|
||||
|
||||
Therefore, any homogeneous displacement can be decomposed into a deformation described by the tensor $\mathbf{J}_{r}$ followed by a rotation described by the orthogonal tensor $\mathbf{A}_{J}$ . Similarly, if $\mathbf{J}_{l}$ is used instead of $\mathbf{J}_{r}$ , the displacement of the body can be considered as a rotation described by the orthogonal tensor $\mathbf{A}_{J}$ followed by a deformation defined by the tensor $\mathbf{J}_{l}$ .
|
||||
因此,任何齐次位移都可以分解为由张量 $\mathbf{J}_{r}$ 描述的变形,随后再由正交张量 $\mathbf{A}_{J}$ 描述的旋转。 类似地,如果使用 $\mathbf{J}_{l}$ 代替 $\mathbf{J}_{r}$,则可以认为该物体的位移是由正交张量 $\mathbf{A}_{J}$ 描述的旋转,随后再由张量 $\mathbf{J}_{l}$ 定义的变形。
|
||||
|
||||
In the case of nonhomogeneous deformation, one can write the relationship between the change in coordinates as $d\boldsymbol{\xi}=\mathbf{J}d\mathbf{x}$ (Spencer 1980). While J, in this case is a function of the spatial coordinates, the polar decomposition theorem can still be applied. In this case, the matrices $\mathbf{A}_{J},\mathbf{J}_{r}$ , and $\mathbf{J}_{l}$ are functions of the spatial coordinates, and the decomposition of the displacement can be regarded as decomposition of the displacements of infinitesimal volumes of the body.
|
||||
在非齐性变形的情况下,可以写出坐标变化与关系为 $d\boldsymbol{\xi}=\mathbf{J}d\mathbf{x}$ (Spencer 1980)。虽然 J 在此情况下是空间坐标的函数,但仍然可以应用极分解定理。在这种情况下,矩阵 $\mathbf{A}_{J},\mathbf{J}_{r}$ 和 $\mathbf{J}_{l}$ 都是空间坐标的函数,位移分解可以被视为对该物体的无限小体积的位移分解。
|
||||
|
||||
Note that the deformation measures $\mathbf{C}_{r}$ and $\mathbf{C}_{l}$ can be written as
|
||||
|
||||
@ -5724,8 +5760,10 @@ $$
|
||||
$$
|
||||
|
||||
Therefore, $\mathbf{C}_{r}$ is equivalent to $\mathbf{J}_{r}$ , while $\mathbf{C}_{l}$ is equivalent to $\mathbf{J}_{l}$ . It is, however, easier and more efficient to calculate $\mathbf{C}_{r}$ and $\mathbf{C}_{l}$ for a given J than to evaluate $\mathbf{J}_{r}$ and $\mathbf{J}_{l}$ from the polar decomposition theorem. For this reason $\mathbf{C}_{r}$ and $\mathbf{C}_{l}$ are often used, instead of $\mathbf{J}_{r}$ and $\mathbf{J}_{l}$ , as the deformation measures.
|
||||
因此,$\mathbf{C}_{r}$ 等同于 $\mathbf{J}_{r}$,而 $\mathbf{C}_{l}$ 等同于 $\mathbf{J}_{l}$。然而,对于给定的 J,计算 $\mathbf{C}_{r}$ 和 $\mathbf{C}_{l}$ 比从极分解定理推导出 $\mathbf{J}_{r}$ 和 $\mathbf{J}_{l}$ 更加简便高效。因此,$\mathbf{C}_{r}$ 和 $\mathbf{C}_{l}$ 通常被用作变形量,取代 $\mathbf{J}_{r}$ 和 $\mathbf{J}_{l}$。
|
||||
|
||||
Small Strains and Rotations Using Eq. 8, the matrix $\mathbf{J}$ can be written as $\mathbf{J}=\mathbf{I}+\bar{\mathbf{J}}=\mathbf{I}+\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r}$ , where $\bar{\mathbf{J}}_{s}$ and $\bar{\mathbf{J}}_{r}$ are defined by Eq. 9. In the case of small strains and rotations, higher order terms can be neglected, and the matrix $\mathbf{C}_{r}$ can be defined as
|
||||
利用公式 8,矩阵 $\mathbf{J}$ 可以写成 $\mathbf{J}=\mathbf{I}+\bar{\mathbf{J}}=\mathbf{I}+\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r}$ ,其中 $\bar{\mathbf{J}}_{s}$ 和 $\bar{\mathbf{J}}_{r}$ 由公式 9 定义。对于微小应变和微小转动的情况下,可以忽略高阶项,并且矩阵 $\mathbf{C}_{r}$ 可以被定义为
|
||||
|
||||
$$
|
||||
\begin{array}{r}{\boldsymbol{\mathbf{C}}_{r}=\mathbf{J}^{\mathrm{T}}\mathbf{J}=(\mathbf{I}+\bar{\mathbf{J}}_{s}-\bar{\mathbf{J}}_{r})(\mathbf{I}+\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r})\approx\mathbf{I}+2\bar{\mathbf{J}}_{s},}\end{array}
|
||||
@ -5738,6 +5776,7 @@ $$
|
||||
$$
|
||||
|
||||
The first of these two equations implies that $\mathbf{J}_{r}-\mathbf{I}$ reduces to the infinitesimal strain tensor in the case of small deformations. Using the same assumption, it can be shown that $\mathbf{J}_{l}-\mathbf{I}=\mathbf{J}_{r}-\mathbf{I}.$ . Note also that
|
||||
这两个方程中的第一个式子表明,在小变形情况下,$\mathbf{J}_{r}-\mathbf{I}$ 简化为无穷小应变张量。 采用同样的假设,可以证明 $\mathbf{J}_{l}-\mathbf{I}=\mathbf{J}_{r}-\mathbf{I}$。 另外需要注意的是,
|
||||
|
||||
$$
|
||||
\mathbf{A}_{J}=\mathbf{J}\mathbf{J}_{r}^{-1}\approx(\mathbf{I}+\bar{\mathbf{J}}_{s}+\bar{\mathbf{J}}_{r})(\mathbf{I}-\bar{\mathbf{J}}_{s})\approx\mathbf{I}+\bar{\mathbf{J}}_{r},
|
||||
@ -5753,18 +5792,21 @@ in the case of small rotations.
|
||||
|
||||
# 4.5 STRESS COMPONENTS
|
||||
|
||||
In this section, we consider the forces acting in the interior of a continuous body. Let $P$ be a point on the surface of the body, $\mathbf{n}$ be a unit vector directed along the outward normal to the surface at $P$ , and $\delta S$ be the area of an element of the surface
|
||||
In this section, we consider the forces acting in the interior of a continuous body. Let $P$ be a point on the surface of the body, $\mathbf{n}$ be a unit vector directed along the outward normal to the surface at $P$ , and $\delta S$ be the area of an element of the surface that contains $P$ . It is assumed that on the surface element with area $\delta S$ , the material outside the region under consideration exerts a force (Fig. 4.3)
|
||||
在本节中,我们将考虑作用于连续体内部的力。设 $P$ 为该物体的表面上的一点,$\mathbf{n}$ 为指向 $P$ 处表面外侧法线的单位向量,$\delta S$ 为包含 $P$ 的表面元素的面积。假设在面积为 $\delta S$ 的表面元素上,考察区域之外的材料施加了一个力(如图 4.3)。
|
||||
|
||||
|
||||

|
||||
Figure 4.3 Surface force.
|
||||
|
||||
that contains $P$ . It is assumed that on the surface element with area $\delta S$ , the material outside the region under consideration exerts a force (Fig. 4.3)
|
||||
|
||||
|
||||
$$
|
||||
\mathbf{f}=\sigma_{n}\mathbf{\Omega}\delta S
|
||||
$$
|
||||
|
||||
on the material in the region under consideration. The force vector $\mathbf{f}$ is called the surface force and the vector $\sigma_{n}$ is called the mean surface traction transmitted across the element of area $\delta S$ from the outside to the inside of the region under consideration. A surface traction equal in magnitude and opposite in direction to $\sigma_{n}$ is transmitted across the element with area δS from the inside to the outside of the part of the body under consideration. We make the assumption that as $\delta S$ tends to zero, $\sigma_{n}$ tends to a finite limit that is independent of the shape of the element with area $\delta S$ . The elastic force on an arbitrary surface through point $P$ can be written in terms of the elastic forces acting on three perpendicular surfaces of an infinitesimal volume containing point $P$ . To do this, we examine the forces acting on the elementary tetrahedron shown in Fig. 4.4. Let $\mathbf{f}_{1},\mathbf{f}_{2}$ , and $\mathbf{f}_{3}$ be, respectively, the force vectors acting on the surfaces whose outward normal is parallel to $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ . Let $\mathbf{n}$ be the vector of direction cosines of the outward normal to the arbitrary surface $\delta S$ . Then the areas of the other faces are
|
||||
关于所考虑区域内的材料而言。力矢量 $\mathbf{f}$ 被称为表面力,而矢量 $\sigma_{n}$ 被称为通过面积元素 $\delta S$ 从区域外部传递到内部的平均表面牵引力。一个大小相等、方向相反于 $\sigma_{n}$ 的表面牵引力,则通过面积为 δS 的元素从所考虑物体的内部传递到外部。我们假设当 $\delta S$ 趋近于零时,$\sigma_{n}$ 趋近于一个有限的极限,该极限与面积为 $\delta S$ 的元素的形状无关。通过点 $P$ 的任意表面的弹性力可以根据包含点 $P$ 的一个无穷小体积的三个相互垂直表面的弹性力来表示。为此,我们考察图 4.4 中所示的 elementary tetrahedron 所受的力。设 $\mathbf{f}_{1},\mathbf{f}_{2}$ , 和 $\mathbf{f}_{3}$ 分别是作用于外法线平行于 $\mathbf{X}_{1},\mathbf{X}_{2}$ , 和 $\mathbf{X}_{3}$ 的表面的力矢量。设 $\mathbf{n}$ 为任意表面 $\delta S$ 外法线的方向余弦矢量。那么其他面的面积是
|
||||
|
||||

|
||||
Figure 4.4 Surface forces on an elementary tetrahedron.
|
||||
@ -5774,12 +5816,14 @@ $$
|
||||
$$
|
||||
|
||||
where $n_{i},i=1,2,3$ are the components of $\mathbf{n}$ . The elastic force vectors exerted on the tetrahedron across its four faces are
|
||||
其中,$n_{i},i=1,2,3$ 是 $\mathbf{n}$ 的分量。作用于四面体四个面的弹性力向量为
|
||||
|
||||
$$
|
||||
\left.\begin{array}{l}{{\bf f}=\sigma_{n}\delta S,\ \ \ \ {\bf f}_{1}=-\sigma_{1}n_{1}\delta S}\\ {{\bf f}_{2}=-\sigma_{2}n_{2}\delta S,\ \ \ \ {\bf f}_{3}=-\sigma_{3}n_{3}\delta S}\end{array}\right\}
|
||||
$$
|
||||
|
||||
where $\sigma_{1},\sigma_{2}$ , and $\sigma_{3}$ are, respectively, the vectors of mean surface traction acting on the surfaces whose normals are in the directions $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ . The components of each surface traction $\sigma_{i}$ will be denoted as $\sigma_{i j}$ , $j=1,2,3$ , that is,
|
||||
where $\sigma_{1},\sigma_{2}$ , and $\sigma_{3}$ are, respectively, the vectors of mean surface traction acting on the surfaces whose normals are in the directions $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ . The components of each surface traction $\sigma_{i}$ will be denoted as $\sigma_{i j}$ , $j=1,2,3$ , that is,
|
||||
其中,$\sigma_{1},\sigma_{2}$ 和 $\sigma_{3}$ 分别是作用于法向量方向为 $\mathbf{X}_{1},\mathbf{X}_{2}$ 和 $\mathbf{X}_{3}$ 的表面的平均表面应力向量。每个表面应力 $\sigma_{i}$ 的分量将表示为 $\sigma_{i j}$,$j=1,2,3$,即,
|
||||
|
||||
$$
|
||||
\sigma_{i}=[\sigma_{i1}\quad\sigma_{i2}\quad\sigma_{i3}]^{\mathrm{T}},\quad i=1,2,3
|
||||
@ -5791,19 +5835,23 @@ $$
|
||||
\left.\begin{array}{c}{{\sigma_{1}}={\sigma_{11}}{\bf{i}}_{1}+{\sigma_{12}}{\bf{i}}_{2}+{\sigma_{13}}{\bf{i}}_{3}}\\ {{\sigma_{2}}={\sigma_{21}}{\bf{i}}_{1}+{\sigma_{22}}{\bf{i}}_{2}+{\sigma_{23}}{\bf{i}}_{3}}\\ {{\sigma_{3}}={\sigma_{31}}{\bf{i}}_{1}+{\sigma_{32}}{\bf{i}}_{2}+{\sigma_{33}}{\bf{i}}_{3}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
where $\mathbf{i}_{1},\mathbf{i}_{2}$ , and $\mathbf{i}_{3}$ are unit vectors in the $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ directions. It is also recognized that there is a body force whose mean value over the tetrahedron is $\mathbf{f}_{b}$ per unit volume. Examples of this kind of force are the gravitational and the magnetic forces. According to Newton’s second law, which states that the rate of change of momentum is proportional to the resultant force acting on the system, the equation of equilibrium of the tetrahedron can be written as
|
||||
where $\mathbf{i}_{1},\mathbf{i}_{2}$ , and $\mathbf{i}_{3}$ are unit vectors in the $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ directions. It is also recognized that there is a body force whose mean value over the tetrahedron is $\mathbf{f}_{b}$ per unit volume. Examples of this kind of force are the gravitational and the magnetic forces. According to Newton’s second law, which states that the rate of change of momentum is proportional to the resultant force acting on the system, the equation of equilibrium of the tetrahedron can be written as
|
||||
其中 $\mathbf{i}_{1},\mathbf{i}_{2}$ 和 $\mathbf{i}_{3}$ 分别是 $\mathbf{X}_{1},\mathbf{X}_{2}$ 和 $\mathbf{X}_{3}$ 方向上的单位向量。 此外,还应认识到,四面体上存在一个体力,其平均值是 $\mathbf{f}_{b}$ 每单位体积。 重力和磁力是此类力的例子。 根据牛顿第二定律,该定律指出动量变化率与作用于系统上的合力成正比,因此可以写出四面体的平衡方程为:
|
||||
|
||||
$$
|
||||
\mathbf{f}_{1}+\mathbf{f}_{2}+\mathbf{f}_{3}+\mathbf{f}+\mathbf{f}_{b}\delta\nu=\rho\mathbf{a}\;\delta\nu
|
||||
$$
|
||||
|
||||
where a, $\rho$ , and $\delta\nu$ are, respectively, the acceleration, mass density, and volume of the tetrahedron. Substituting Eq. 58 into Eq. 61 yields
|
||||
其中,a、ρ 和 δν 分别为四面体的加速度、质量密度和体积。将公式 58 代入公式 61 得到
|
||||
|
||||
$$
|
||||
\sigma_{n}=\sigma_{1}n_{1}+\sigma_{2}n_{2}+\sigma_{3}n_{3}+{\frac{\delta\nu}{\delta S}}(\rho\mathbf{a}-\mathbf{f}_{b})
|
||||
$$
|
||||
|
||||
We assume $\mathbf{n}$ and the point $P$ to be fixed and let $\delta S$ and $\delta V$ tend to zero. Since $\delta V$ is proportional to the cube and δS is proportional to the square of the linear dimension of the tetrahedron, we conclude that $\delta\nu/\delta S$ tends to zero as $\delta S$ approaches zero. Thus, in the limit one has
|
||||
我们假设 $\mathbf{n}$ 和点 $P$ 是固定的,并令 $\delta S$ 和 $\delta V$ 趋于零。由于 $\delta V$ 与四面体的线性尺寸的立方成正比,而 δS 与其平方成正比,因此我们得出结论:当 $\delta S$ 趋近于零时,$\delta\nu/\delta S$ 趋近于零。因此,在极限情况下,我们有:
|
||||
|
||||
|
||||
$$
|
||||
\sigma_{n}=\sigma_{1}n_{1}+\sigma_{2}n_{2}+\sigma_{3}n_{3}
|
||||
@ -5811,7 +5859,12 @@ $$
|
||||
|
||||
where $\sigma_{1},\sigma_{2},\sigma_{3}$ , and $\sigma_{n}$ are evaluated at $P$ . Equation 63 can be written in matrix form as
|
||||
|
||||
其中 $\sigma_{1},\sigma_{2},\sigma_{3}$ , 以及 $\sigma_{n}$ 在 $P$ 点被评估。 方程 63 可以写成矩阵形式如下:
|
||||
$$
|
||||
\sigma_{n}=\sigma_{m}\mathbf{n}
|
||||
$$
|
||||
where $\sigma_{m}$ is a $3\times3$ matrix defined as
|
||||
其中 $\sigma_{m}$ 是一个 $3\times3$ 矩阵,定义如下:
|
||||
|
||||
$$
|
||||
\sigma_{m}={\left[\begin{array}{l l l}{\sigma_{11}}&{\sigma_{21}}&{\sigma_{31}}\\ {\sigma_{12}}&{\sigma_{22}}&{\sigma_{32}}\\ {\sigma_{13}}&{\sigma_{23}}&{\sigma_{33}}\end{array}\right]}
|
||||
@ -5824,36 +5877,45 @@ $$
|
||||
$$
|
||||
|
||||
Therefore, the surface force f can be expressed in terms of the elements of the matrix $\sigma_{m}$ . These elements $\sigma_{i j}$ , $(i,j=1,2,3)$ ) are called the stress components. The components of the stress vectors $\sigma_{1},\sigma_{2}$ , and $\sigma_{3}$ represent the stress on the planes that are perpendicular, respectively, to the $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ axes (Eq. 60). Equation 63, which is called the Cauchy stress formula, gives the stress vector on an oblique plane with unit normal $\mathbf{n}$ .
|
||||
因此,表面力 f 可以用矩阵 $\sigma_{m}$ 的元素来表示。这些元素 $\sigma_{i j}$ ,($i,j=1,2,3$)被称为应力分量。应力向量 $\sigma_{1}$、$\sigma_{2}$ 和 $\sigma_{3}$ 的分量分别代表作用于垂直于 $\mathbf{X}_{1}$、$\mathbf{X}_{2}$ 和 $\mathbf{X}_{3}$ 轴的平面上的应力(式 60)。被称为 Cauchy 应力公式的式 63,给出了具有单位法向量 $\mathbf{n}$ 的倾斜平面上的应力向量。
|
||||
|
||||
By using Eq. 63 or 64, it can be shown that $\sigma_{m}$ is a tensor quantity. In the foregoing discussion, the stress components were defined with respect to the coordinate system $\mathbf{X}_{1},\mathbf{X}_{2}$ , and $\mathbf{X}_{3}$ . It is expected that the choice of the coordinate system will lead to a different set of stress components. Let $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ be another coordinate system. We now examine the relationship between the stress components $\sigma_{i j}$ associated with the coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ and the stress components $\bar{\sigma}_{i j}$ at the same point defined with respect to the coordinate system $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ . Let A be an orthogonal transformation matrix that defines the orientation of the coordinate system $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ with respect to the coordinate system $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ . One can then write the following equation:
|
||||
通过使用公式 63 或 64,可以证明 $\sigma_{m}$ 是一个张量。在前面的讨论中,应力分量是相对于坐标系 $\mathbf{X}_{1},\mathbf{X}_{2}$ , 和 $\mathbf{X}_{3}$ 定义的。预计坐标系的选取将导致一组不同的应力分量。设 $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ 为另一个坐标系。现在我们来考察与坐标系 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 相关的应力分量 $\sigma_{i j}$ 和在同一点,相对于坐标系 $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ 定义的应力分量 $\bar{\sigma}_{i j}$ 之间的关系。设 A 为一个正交变换矩阵,它定义了坐标系 $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ 相对于坐标系 $\mathbf{X}_{1}\mathbf{X}_{2}\mathbf{X}_{3}$ 的方向。 那么可以写出如下方程:
|
||||
|
||||
|
||||
$$
|
||||
\bar{\sigma}_{n}=\mathbf{A}^{\mathrm{T}}\sigma_{n}=\mathbf{A}^{\mathrm{T}}\sigma_{m}\mathbf{n}=\mathbf{A}^{\mathrm{T}}\sigma_{m}\mathbf{A}\bar{\mathbf{n}}
|
||||
$$
|
||||
|
||||
where $\bar{\bf n}=[\bar{n}_{1}\,\bar{n}_{2}\,\bar{n}_{3}]^{\mathrm{T}}$ is the normal to the surface whose components are defined with respect to the $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ coordinate system. Equation 67 can be written in a compact form as $\bar{\sigma}_{n}=\bar{\sigma}_{m}\bar{\mathbf{n}}$ , where $\bar{\sigma}_{m}$ is given by
|
||||
其中 $\bar{\bf n}=[\bar{n}_{1}\,\bar{n}_{2}\,\bar{n}_{3}]^{\mathrm{T}}$ 是法向量,其分量相对于 $\bar{\mathbf{X}}_{1}\bar{\mathbf{X}}_{2}\bar{\mathbf{X}}_{3}$ 坐标系定义。 方程 67 可以写成紧凑形式:$\bar{\sigma}_{n}=\bar{\sigma}_{m}\bar{\mathbf{n}}$ ,其中 $\bar{\sigma}_{m}$ 由...给出。
|
||||
|
||||
$$
|
||||
\bar{\sigma}_{m}=\mathbf{A}^{\mathrm{T}}\sigma_{m}\mathbf{A}
|
||||
$$
|
||||
|
||||
which demonstrates that $\sigma_{m}$ is indeed a second-order tensor.
|
||||
which demonstrates that $\sigma_{m}$ is indeed a second-order tensor.
|
||||
这表明 $\sigma_{m}$ 确实是一个二阶张量。
|
||||
|
||||
# 4.6 EQUATIONS OF EQUILIBRIUM
|
||||
# 4.6 EQUATIONS OF EQUILIBRIUM平衡方程式
|
||||
|
||||
In studying the mechanics of deformable bodies, a distinction is made between two kinds of forces: body forces acting on the element of volume (or mass) of the body such as gravitational, magnetic, and inertia forces, and surface forces acting on surface elements inside or on the boundary of the body such as contact forces and hydrostatic pressure. The resultant of the first kind of force follows from integration over the volume, whereas the second kind is the result of a surface integral. Thus, the condition for the dynamic equilibrium can be mathematically stated as
|
||||
In studying the mechanics of deformable bodies, a distinction is made between two kinds of forces: body forces acting on the element of volume (or mass) of the body such as gravitational, magnetic, and inertia forces, and surface forces acting on surface elements inside or on the boundary of the body such as contact forces and hydrostatic pressure. The resultant of the first kind of force follows from integration over the volume, whereas the second kind is the result of a surface integral. Thus, the condition for the dynamic equilibrium can be mathematically stated as 在研究变形体力学时,区分两种类型的力:作用于物体体积(或质量)元素的体力,例如重力、磁力和惯性力;以及作用于物体内部或边界表面的表面力,例如接触力和静水压力。第一种力的合力是通过体积积分得到的,而第二种力则是表面积分的结果。因此,动态平衡的条件可以数学上表述为:
|
||||
|
||||
|
||||
$$
|
||||
\int_{S}\sigma_{n}\,d S+\int_{\nu}\mathbf{f}_{b}\,d\nu=\int_{\nu}\rho\mathbf{a}\,d\nu
|
||||
$$
|
||||
|
||||
In the case of large deformation, one must distinguish between the density $\rho$ and volume $\nu$ in the current deformed configuration and the density $\rho_{o}$ and volume $V$ in the reference undeformed configuration. In the case of small deformation, on the other hand, such a distinction is not necessary because it is assumed that the small deformation does not have a significant effect on the material property and the volume of the continuum. Substituting Eq. 64 into Eq. 69 yields
|
||||
In the case of large deformation, one must distinguish between the density $\rho$ and volume $\nu$ in the current deformed configuration and the density $\rho_{o}$ and volume $V$ in the reference undeformed configuration. In the case of small deformation, on the other hand, such a distinction is not necessary because it is assumed that the small deformation does not have a significant effect on the material property and the volume of the continuum. Substituting Eq. 64 into Eq. 69 yields
|
||||
在发生大变形时,必须区分当前变形构型下的密度 $\rho$ 和体积 $\nu$,以及参考未变形构型下的密度 $\rho_{o}$ 和体积 $V$。而对于小变形的情况,则无需如此区分,因为假设小变形对材料属性和连续介质的体积影响不大。将公式 64 代入公式 69 得到:
|
||||
|
||||
|
||||
$$
|
||||
\int_{S}{\boldsymbol{\sigma}}_{m}\mathbf{n}\,d S+\int_{\nu}\mathbf{f}_{b}\,d\nu=\int_{\nu}{\boldsymbol{\rho}}\mathbf{a}\,d\nu
|
||||
$$
|
||||
|
||||
The surface integral can be transformed into a volume integral by use of the divergence theorem (Greenberg 1978), that is
|
||||
The surface integral can be transformed into a volume integral by use of the divergence theorem (Greenberg 1978), that is 散度定理可用于将面积积分转化为体积积分(Greenberg 1978),即:
|
||||
|
||||
|
||||
$$
|
||||
\int_{S}{\sigma_{m}\mathbf{n}\,d S}=\int_{\nu}{\sigma_{s}\,d\nu}
|
||||
@ -5872,20 +5934,25 @@ $$
|
||||
$$
|
||||
|
||||
This equation must hold in every region in the body, and hence the integrand must be zero throughout the body. This leads to
|
||||
这个方程在身体的每个区域都必须成立,因此积分项在整个身体范围内必须为零。由此得出
|
||||
|
||||
$$
|
||||
\sigma_{s}+\mathbf{f}_{b}=\rho\mathbf{a}
|
||||
$$
|
||||
|
||||
which is known as the equation of equilibrium. By using Eq. 72, we can write the components of Eq. 74 as
|
||||
which is known as the equation of equilibrium. By using Eq. 72, we can write the components of Eq. 74 as 这被称为平衡方程。利用公式72,我们可以将公式74的各分量写为:
|
||||
|
||||
$$
|
||||
\left.\begin{array}{l l}{\sigma_{11,1}+\sigma_{21,2}+\sigma_{31,3}+f_{b1}=\rho a_{1}}\\ {\sigma_{12,1}+\sigma_{22,2}+\sigma_{32,3}+f_{b2}=\rho a_{2}}\\ {\sigma_{13,1}+\sigma_{23,2}+\sigma_{33,3}+f_{b3}=\rho a_{3}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
where $(\mathbf{\varepsilon},i)$ denotes differentiation with respect to the spatial coordinate $x_{i};a_{1},a_{2}$ , and $a_{3}$ are the components of the acceleration vector; and $f_{b1},\,f_{b2}$ , and $f_{b3}$ are the components of the vector of the body force. It is important to note that the equations of equilibrium contain both time and spatial derivatives.
|
||||
where $(,i)$ denotes differentiation with respect to the spatial coordinate $x_{i};a_{1},a_{2}$ , and $a_{3}$ are the components of the acceleration vector; and $f_{b1},\,f_{b2}$ , and $f_{b3}$ are the components of the vector of the body force. It is important to note that the equations of equilibrium contain both time and spatial derivatives.
|
||||
其中 $(,i)$ 表示对空间坐标 $x_{i}$ 的求导;$a_{1}, a_{2}$ 和 $a_{3}$ 是加速度矢量的分量;而 $f_{b1}, f_{b2}$ 和 $f_{b3}$ 是体力矢量的分量。需要注意的是,平衡方程包含时域和空间域的导数。
|
||||
|
||||
Symmetry of the Stress Tensor In developing the differential equations of equilibrium we used the equilibrium of the forces. The condition that the resultant couple about the origin must be equal to zero can be used to prove the symmetry of the stress tensor. This condition can be expressed mathematically as
|
||||
Symmetry of the Stress Tensor In developing the differential equations of equilibrium we used the equilibrium of the forces. The condition that the resultant couple about the origin must be equal to zero can be used to prove the symmetry of the stress tensor. This condition can be expressed mathematically as
|
||||
应力张量对称性
|
||||
|
||||
在推导平衡微分方程时,我们使用了力的平衡。关于原点的合力矩必须为零这一条件,可用于证明应力张量的对称性。这一条件可以数学表达式为:
|
||||
|
||||
$$
|
||||
\int_{S}\mathbf{x}\times\sigma_{n}d S+\int_{\nu}\mathbf{x}\times\left(\mathbf{f}_{b}-\rho\mathbf{a}\right)d\nu=\mathbf{0}
|
||||
@ -5952,22 +6019,27 @@ $$
|
||||
$$
|
||||
|
||||
By using Eq. 74, Eq. 86 becomes $\int_{\nu}\mathbf{b}_{s}\ d\nu=\mathbf{0}$ . This equation must hold in every region in the body, and hence the integrand must be zero throughout the body. This leads to ${\bf b}_{s}={\bf0}$ . Using this equation and Eq. 84, one obtains $\sigma_{32}=\sigma_{23}$ , $\sigma_{13}=$ $\sigma_{31}$ , $\sigma_{21}=\sigma_{12}$ . This result can be written in a compact form as
|
||||
|
||||
利用公式 74,公式 86 变为 $\int_{\nu}\mathbf{b}_{s}\ d\nu=\mathbf{0}$ 。 该方程必须在身体的每个区域都成立,因此被积函数必须在整个身体范围内为零。 这导致 ${\bf b}_{s}={\bf0}$ 。 利用此方程和公式 84,可以得到 $\sigma_{32}=\sigma_{23}$ , $\sigma_{13}=$ $\sigma_{31}$ , $\sigma_{21}=\sigma_{12}$ 。 结果可以写成紧凑形式如下:
|
||||
$$
|
||||
\sigma_{i j}=\sigma_{j i}
|
||||
$$
|
||||
|
||||
which implies that the stress tensor is symmetric.
|
||||
|
||||
# 4.7 CONSTITUTIVE EQUATIONS
|
||||
# 4.7 CONSTITUTIVE EQUATIONS 本构方程
|
||||
|
||||
|
||||
The stress and strain tensors are insufficient for description of the mechanical behavior of deformable bodies. Body deformations depend on the applied forces, and the force-displacement relationship depends on the material of the body. To complete the specification of the mechanical properties of a material we require additional equations. These equations are called the constitutive equations and serve to distinguish one material from another. For convenience, we reproduce the stress and strain vectors, which are essential in the discussion that follows:
|
||||
The stress and strain tensors are insufficient for description of the mechanical behavior of deformable bodies. Body deformations depend on the applied forces, and the force-displacement relationship depends on the material of the body. To complete the specification of the mechanical properties of a material we require additional equations. These equations are called the constitutive equations and serve to distinguish one material from another. For convenience, we reproduce the stress and strain vectors, which are essential in the discussion that follows:
|
||||
应力与应变张量不足以描述可变形体的力学行为。 Body的变形取决于所施加的力,而力-位移关系则取决于材料本身。 为了完整地描述材料的力学性能,我们需要额外的方程。 这些方程被称为本构方程,用于区分不同的材料。 为了方便起见,我们在此重现应力与应变矢量,它们在以下讨论中至关重要:
|
||||
|
||||
|
||||
$$
|
||||
\left.\begin{array}{r l}&{\boldsymbol{\sigma}=[\sigma_{11}\quad\sigma_{22}\quad\sigma_{33}\quad\sigma_{12}\quad\sigma_{13}\quad\sigma_{23}]^{\mathrm{T}}}\\ &{\boldsymbol{\varepsilon}=[\varepsilon_{11}\quad\varepsilon_{22}\quad\varepsilon_{33}\quad\varepsilon_{12}\quad\varepsilon_{13}\quad\varepsilon_{23}]^{\mathrm{T}}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
It has been found experimentally that for most solid materials, the measured strains are proportional to the applied forces, provided the load does not exceed a given value, known as the elastic limit. This experimental observation can be stated as follows: The stress components at any point in the body are a linear function of the strain components. This statement is a generalization of Hooke’s law and does not apply to viscoelastic, plastic, or viscoplastic materials. The generalized form of Hooke’s law may thus be written as
|
||||
实验结果表明,对于大多数固体材料,测得的应变与施加的力成正比,前提是载荷不超过一个特定值,即弹性极限。这一实验观察可以表述为:物体的任一点处的应力分量是应变分量的线性函数。这是一个胡克定律的推广,不适用于粘弹性、塑性和粘塑性材料。因此,胡克定律的推广形式可以写成:
|
||||
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{\sigma_{11}=e_{11}\varepsilon_{11}+e_{12}\varepsilon_{22}+e_{13}\varepsilon_{33}+e_{14}\varepsilon_{12}+e_{15}\varepsilon_{13}+e_{16}\varepsilon_{23}}\\ &{\sigma_{22}=e_{21}\varepsilon_{11}+e_{22}\varepsilon_{22}+e_{23}\varepsilon_{33}+e_{24}\varepsilon_{12}+e_{25}\varepsilon_{13}+e_{26}\varepsilon_{23}}\\ &{\sigma_{33}=e_{31}\varepsilon_{11}+e_{32}\varepsilon_{22}+e_{33}\varepsilon_{33}+e_{34}\varepsilon_{12}+e_{35}\varepsilon_{13}+e_{36}\varepsilon_{23}}\\ &{\sigma_{12}=e_{41}\varepsilon_{11}+e_{42}\varepsilon_{22}+e_{43}\varepsilon_{33}+e_{44}\varepsilon_{12}+e_{45}\varepsilon_{13}+e_{46}\varepsilon_{23}}\\ &{\sigma_{13}=e_{51}\varepsilon_{11}+e_{52}\varepsilon_{22}+e_{53}\varepsilon_{33}+e_{54}\varepsilon_{12}+e_{55}\varepsilon_{13}+e_{56}\varepsilon_{23}}\\ &{\sigma_{23}=e_{61}\varepsilon_{11}+e_{62}\varepsilon_{22}+e_{63}\varepsilon_{33}+e_{64}\varepsilon_{12}+e_{65}\varepsilon_{13}+e_{66}\varepsilon_{23}}\end{array}
|
||||
@ -5985,9 +6057,11 @@ $$
|
||||
\mathbf{E}={\left[\begin{array}{l l l l l l}{e_{11}}&{e_{12}}&{e_{13}}&{e_{14}}&{e_{15}}&{e_{16}}\\ {e_{21}}&{e_{22}}&{e_{23}}&{e_{24}}&{e_{25}}&{e_{26}}\\ {e_{31}}&{e_{32}}&{e_{33}}&{e_{34}}&{e_{35}}&{e_{36}}\\ {e_{41}}&{e_{42}}&{e_{43}}&{e_{44}}&{e_{45}}&{e_{46}}\\ {e_{51}}&{e_{52}}&{e_{53}}&{e_{54}}&{e_{55}}&{e_{56}}\\ {e_{61}}&{e_{62}}&{e_{63}}&{e_{64}}&{e_{65}}&{e_{66}}\end{array}\right]}
|
||||
$$
|
||||
|
||||
Anisotropic Linearly Elastic Material Let $U$ be the strain energy per unit volume that represents the work done by internal stresses. On a unit cube, stresses represent forces, whereas strains represent displacements. Therefore, the work done
|
||||
Anisotropic Linearly Elastic Material Let $U$ be the strain energy per unit volume that represents the work done by internal stresses. On a unit cube, stresses represent forces, whereas strains represent displacements. Therefore, the work done by a force $\sigma$ during the motion $d\varepsilon$ can be written as $d U=\sigma^{\mathrm{T}}d\varepsilon$ , which implies that
|
||||
|
||||
各向异性线性弹性材料
|
||||
|
||||
by a force $\sigma$ during the motion $d\varepsilon$ can be written as $d U=\sigma^{\mathrm{T}}d\varepsilon$ , which implies that
|
||||
令 $U$ 表示内应力所做的功,单位体积的应变能。在一个单位立方体上,应力代表力,而应变代表位移。因此,力 $\sigma$ 在位移 $d\varepsilon$ 运动过程中所做的功可以写成 $d U=\sigma^{\mathrm{T}}d\varepsilon$,这意味着
|
||||
|
||||
$$
|
||||
\sigma=\left(\frac{\partial\boldsymbol{U}}{\partial\boldsymbol{\varepsilon}}\right)^{\mathrm{T}}
|
||||
@ -6024,18 +6098,23 @@ e_{i j}=e_{j i}
|
||||
$$
|
||||
|
||||
which shows that the matrix of the elastic coefficients is symmetric. Therefore, there are only 21 distinct elastic coefficients for a general anisotropic linearly elastic material. In terms of these coefficients, the matrix of elastic coefficients $\mathbf{E}$ can be written as
|
||||
这表明弹性系数矩阵是对称的。因此,对于一般的各向异性线性弹性材料,只有21个独立的弹性系数。用这些系数,弹性系数矩阵 $\mathbf{E}$ 可以表示为
|
||||
|
||||
$$
|
||||
\mathbf{E}={\left[\begin{array}{l l l l l}{e_{11}}&&&&\\ {e_{21}}&{e_{22}}&&{{\mathrm{symmetric}}}\\ {e_{31}}&{e_{32}}&{e_{33}}&&\\ {e_{41}}&{e_{42}}&{e_{43}}&{e_{44}}&\\ {e_{51}}&{e_{52}}&{e_{53}}&{e_{54}}&{e_{55}}&\\ {e_{61}}&{e_{62}}&{e_{63}}&{e_{64}}&{e_{65}}&{e_{66}}\end{array}\right]}
|
||||
$$
|
||||
|
||||
Material Symmetry In some structural materials, special kinds of symmetry may exist. The elastic coefficients, for example, may remain invariant under a coordinate transformation. For instance, consider the reflection with respect to the $\mathbf{X}_{1}\mathbf{X}_{2}$ plane given by the following transformation:
|
||||
Material Symmetry In some structural materials, special kinds of symmetry may exist. The elastic coefficients, for example, may remain invariant under a coordinate transformation. For instance, consider the reflection with respect to the $\mathbf{X}_{1}\mathbf{X}_{2}$ plane given by the following transformation:
|
||||
材料对称性
|
||||
|
||||
在某些结构材料中,可能存在特殊类型的对称性。例如,弹性系数在坐标变换下可能保持不变。例如,考虑关于 $\mathbf{X}_{1}\mathbf{X}_{2}$ 平面的反射,其变换如下:
|
||||
|
||||
$$
|
||||
\mathbf{A}={\left[\begin{array}{l l l}{1}&{0}&{0}\\ {0}&{1}&{0}\\ {0}&{0}&{-1}\end{array}\right]}
|
||||
$$
|
||||
|
||||
The transformed stresses and strains $\sigma_{m}^{\prime}$ and $\pmb{\varepsilon}_{m}^{\prime}$ are given, respectively, by
|
||||
转化后的应力 $\sigma_{m}^{\prime}$ 和应变 $\pmb{\varepsilon}_{m}^{\prime}$ 分别表示如下:
|
||||
|
||||
$$
|
||||
\sigma_{m}^{\prime}=\mathbf{A}^{\mathrm{T}}\sigma_{m}\mathbf{A},\;\;\pmb{\varepsilon}_{m}^{\prime}=\mathbf{A}^{\mathrm{T}}\pmb{\varepsilon}_{m}\mathbf{A}
|
||||
@ -6072,44 +6151,52 @@ $$
|
||||
$$
|
||||
|
||||
By comparing Eqs. 104 and 105 and using Eqs. 102 and 103, one gets $e_{15}=$ $-e_{15}$ , $e_{16}=-e_{16}$ , or $e_{15}=e_{16}=0$ . In a similar manner by considering other stress components, we find $e_{25}=e_{26}=e_{35}=e_{36}=e_{45}=e_{46}=0$ . Therefore, the elastic constants for a material that possesses a plane of elastic symmetry reduce to 13 elastic coefficients. If this plane of symmetry is the $\mathbf{X}_{1}\mathbf{X}_{2}$ plane, that is, the elastic properties are invariant under a reflection with respect to the $\mathbf{X}_{1}\mathbf{X}_{2}$ plane, the matrix $\mathbf{E}$ of elastic coefficients can be written as
|
||||
通过比较公式 104 和 105,并利用公式 102 和 103,可以得到 $e_{15}=$ $-e_{15}$ , $e_{16}=-e_{16}$ , 或者 $e_{15}=e_{16}=0$ 。 类似地,通过考虑其他应力分量,我们发现 $e_{25}=e_{26}=e_{35}=e_{36}=e_{45}=e_{46}=0$ 。 因此,对于具有弹性对称面的材料,其弹性常数简化为 13 个弹性系数。 如果这个弹性对称面是 $\mathbf{X}_{1}\mathbf{X}_{2}$ 平面,也就是说,弹性性质在关于 $\mathbf{X}_{1}\mathbf{X}_{2}$ 平面的反射下是不变的,那么弹性系数矩阵 $\mathbf{E}$ 可以写为
|
||||
|
||||
$$
|
||||
\mathbf{E}={\left[\begin{array}{l l l l l}{e_{11}}&{}&{}&{{\mathrm{symmetric}}}\\ {e_{21}}&{e_{22}}&{}&{}&{}\\ {e_{31}}&{e_{32}}&{e_{33}}&{}&{}\\ {e_{41}}&{e_{42}}&{e_{43}}&{e_{44}}&{}\\ {0}&{0}&{0}&{0}&{e_{55}}&{}\\ {0}&{0}&{0}&{0}&{e_{65}}&{e_{66}}\end{array}\right]}
|
||||
$$
|
||||
|
||||
If the material has two mutually orthogonal planes of elastic symmetry, one can show that $e_{41}=e_{42}=e_{43}=e_{65}=0$ and the matrix of elastic coefficients reduces to
|
||||
如果材料具有两个相互正交的弹性对称面,可以证明 $e_{41}=e_{42}=e_{43}=e_{65}=0$,并且弹性系数矩阵简化为
|
||||
|
||||
$$
|
||||
\mathbf{E}={\left[\begin{array}{l l l l l}{e_{11}}&{}&{}&{{\mathrm{symmetric}}}\\ {e_{21}}&{e_{22}}&{}&{}&{}\\ {e_{31}}&{e_{32}}&{e_{33}}&{}&{}\\ {0}&{0}&{0}&{e_{44}}&{}&{}\\ {0}&{0}&{0}&{0}&{e_{55}}&{}\\ {0}&{0}&{0}&{0}&{0}&{e_{66}}\end{array}\right]}
|
||||
$$
|
||||
|
||||
In some materials, the elastic coefficients $e_{i j}$ remain invariant under a rotation through an angle $\alpha$ about one of the axes, that is, the values of these coefficients are independent of the set of rectangular axes chosen. The transformation matrix $\mathbf{A}$ in this case is given by
|
||||
在某些材料中,弹性系数 $e_{i j}$ 在绕其中一个轴旋转角度 $\alpha$ 时保持不变,也就是说,这些系数的值与所选的直角坐标系无关。在这种情况下,变换矩阵 $\mathbf{A}$ 的表达式为:
|
||||
|
||||
$$
|
||||
\mathbf{A}={\left[\begin{array}{l l l}{\cos\alpha}&{-\sin\alpha}&{0}\\ {\sin\alpha}&{\cos\alpha}&{0}\\ {0}&{0}&{1}\end{array}\right]}
|
||||
$$
|
||||
|
||||
One may then write two equations similar to Eq. 100 and proceed as in the above case for different values of $\alpha$ to show that in the case of an isotropic solid there are only two independent constants, denoted as $\lambda$ and $\mu$ . We then have
|
||||
人们随后可以写出两个类似于公式100的方程,并像上述情况那样,对不同的$\alpha$值进行处理,以证明对于各向同性固体,只有两个独立的常数,分别用$\lambda$和$\mu$表示。 那么,我们就有
|
||||
|
||||
$$
|
||||
\left.\begin{array}{l}{{e_{12}=e_{13}=e_{21}=e_{23}=e_{31}=e_{32}=\lambda}}\\ {{e_{44}=e_{55}=e_{66}=2\mu}}\\ {{e_{11}=e_{22}=e_{33}=\lambda+2\mu}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
The two elastic constants, $\lambda$ and $\mu$ , are known as Lame’s constants.
|
||||
这两个弹性常数,λ 和 μ,被称为拉梅常数。
|
||||
|
||||
Homogeneous Isotropic Material If the material is homogeneous, $\lambda$ and $\mu$ are constants at all points. The matrix $\mathbf{E}$ of elastic coefficients can be written in the case of an isotropic material in terms of Lame’s constants as
|
||||
Homogeneous Isotropic Material If the material is homogeneous, $\lambda$ and $\mu$ are constants at all points. The matrix $\mathbf{E}$ of elastic coefficients can be written in the case of an isotropic material in terms of Lame’s constants as
|
||||
均质各向同性材料 如果材料是均质的,λ 和 μ 在所有点都是常数。对于各向同性材料,弹性系数矩阵 **E** 可以用拉梅常数表示为
|
||||
|
||||
$$
|
||||
\mathbf{E}={\left[\begin{array}{l l l l l l l}{\lambda+2\mu}&{\lambda}&{\lambda}&{0}&{0}&{0}\\ {\lambda}&{\lambda+2\mu}&{\lambda}&{0}&{0}&{0}\\ {\lambda}&{\lambda}&{\lambda+2\mu}&{0}&{0}&{0}\\ {0}&{0}&{0}&{2\mu}&{0}&{0}\\ {0}&{0}&{0}&{0}&{2\mu}&{0}\\ {0}&{0}&{0}&{0}&{0}&{2\mu}\end{array}\right]}
|
||||
$$
|
||||
|
||||
Using Eq. 90, one can then write the stress-strain relations in the following explicit form:
|
||||
Using Eq. 90, one can then write the stress-strain relations in the following explicit form:
|
||||
利用公式90,可以将其中的应力-应变关系写成如下显式形式:
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{\sigma_{11}=\lambda\varepsilon_{t}+2\mu\varepsilon_{11},\quad\;\sigma_{22}=\lambda\varepsilon_{t}+2\mu\varepsilon_{22},\quad\;\sigma_{33}=\lambda\varepsilon_{t}+2\mu\varepsilon_{33}}\\ &{\sigma_{12}=2\mu\varepsilon_{12},\quad\;\sigma_{13}=2\mu\varepsilon_{13},\quad\;\sigma_{23}=2\mu\varepsilon_{23}}\end{array}
|
||||
$$
|
||||
|
||||
where $\varepsilon_{t}=\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33}$ , which represents the change in volume of a unit cube, is called the dilation. The inverse of Eq. 111 gives
|
||||
where $\varepsilon_{t}=\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33}$ , which represents the change in volume of a unit cube, is called the dilation. The inverse of Eq. 111 gives
|
||||
其中 $\varepsilon_{t}=\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33}$,代表一个单位立方体体积的变化,被称为膨胀。 方程 111 的逆运算得到
|
||||
|
||||
$$
|
||||
\left.\begin{array}{l}{{\displaystyle\varepsilon_{11}=\frac{1}{E}[(1+\gamma)\sigma_{11}-\gamma\sigma_{t}],\quad\displaystyle\varepsilon_{22}=\frac{1}{E}[(1+\gamma)\sigma_{22}-\gamma\sigma_{t}]}}\\ {{\displaystyle\varepsilon_{33}=\frac{1}{E}[(1+\gamma)\sigma_{33}-\gamma\sigma_{t}],\quad\displaystyle\varepsilon_{12}=\frac{1}{2\mu}\sigma_{12}=\frac{1+\gamma}{E}\sigma_{12}}}\\ {{\displaystyle\varepsilon_{13}=\frac{1}{2\mu}\sigma_{13}=\frac{1+\gamma}{E}\sigma_{13},\quad\displaystyle\varepsilon_{23}=\frac{1}{2\mu}\sigma_{23}=\frac{1+\gamma}{E}\sigma_{23}}}\end{array}\right\}
|
||||
@ -6121,9 +6208,10 @@ $$
|
||||
\sigma_{t}=\sigma_{11}+\sigma_{22}+\sigma_{33},\quad E=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu},\quad\gamma=\frac{\lambda}{2(\lambda+\mu)}
|
||||
$$
|
||||
|
||||
The constants $\mu,E$ , and $\gamma$ are, respectively, called the modulus of rigidity, Young’s modulus, and Poisson’s ratio.
|
||||
The constants $\mu,E$ , and $\gamma$ are, respectively, called the modulus of rigidity, Young’s modulus, and Poisson’s ratio.
|
||||
常数 $\mu, E$ 和 $\gamma$ 分别被称为弹性模量、杨氏模量和泊松比。
|
||||
|
||||
# Example 4.3 In the case of two-dimensional analysis we have
|
||||
Example 4.3 In the case of two-dimensional analysis we have
|
||||
|
||||
$$
|
||||
\sigma_{23}=\sigma_{31}=0,\;\;\;\;\;\varepsilon_{23}=\varepsilon_{31}=0
|
||||
@ -6150,18 +6238,24 @@ $$
|
||||
$$
|
||||
|
||||
Nonlinear Material Models In this section, the linear Hookean material model is used as an example to demonstrate the development of the constitutive equations that relate the strains and stresses. The linear Hookean model, which defines linear relationships between the strains and the stresses, can be used in the case of small deformations. In the case of large deformations, however, other nonlinear material models such as the neo-Hookean and the Mooney-Rivlin material models can be used (Ogden, 1984; Shabana, 2012).
|
||||
非线性材料模型
|
||||
|
||||
在本节中,我们将线性胡克材料模型作为示例,展示如何推导描述应变和应力关系的本构方程。线性胡克模型,它定义了应变和应力之间的线性关系,适用于小变形情况。然而,对于大变形,可以使用其他非线性材料模型,例如新胡克模型和莫尼-里维林模型(Ogden, 1984; Shabana, 2012)。
|
||||
|
||||
# 4.8 VIRTUAL WORK AND ELASTIC FORCES
|
||||
|
||||
In the preceding section, the constitutive equations for a linear elastic material that relate the stresses and strains were obtained. It is, however, important to point out that in the case of large deformation, the Cauchy stress tensor is not associated with the Lagrangian strain tensor since Cauchy stress tensor is defined with respect to the current (deformed) configuration, while the Lagrangian strain tensor is defined with respect to the reference (undeformed) configuration. In this section, the partial differential equation of equilibrium (Eq. 74 or 75) will be used to derive the virtual work of the elastic forces. The analysis presented in this section shows that another symmetric stress tensor, the second Piola-Kirchhoff stress tensor is associated with the Lagrangian strain tensor $\varepsilon_{m}$ of Eq. 18. In the case of a linear elastic model, the second Piola-Kirchhoff strain tensor can be related to the Lagrangian strain tensor using the constitutive equations obtained in the preceding section. In order to simplify the derivation presented in this section, the tensor double product and the change of the volume of a material element are first discussed.
|
||||
In the preceding section, the constitutive equations for a linear elastic material that relate the stresses and strains were obtained. It is, however, important to point out that in the case of large deformation, the Cauchy stress tensor is not associated with the Lagrangian strain tensor since Cauchy stress tensor is defined with respect to the current (deformed) configuration, while the Lagrangian strain tensor is defined with respect to the reference (undeformed) configuration. In this section, the partial differential equation of equilibrium (Eq. 74 or 75) will be used to derive the virtual work of the elastic forces. The analysis presented in this section shows that another symmetric stress tensor, the second Piola-Kirchhoff stress tensor is associated with the Lagrangian strain tensor $\varepsilon_{m}$ of Eq. 18. In the case of a linear elastic model, the second Piola-Kirchhoff strain tensor can be related to the Lagrangian strain tensor using the constitutive equations obtained in the preceding section. In order to simplify the derivation presented in this section, the tensor double product and the change of the volume of a material element are first discussed.
|
||||
在上一节中,我们获得了描述线性弹性材料应力和应变关系的本构方程。然而,需要指出的是,对于大变形情况,柯西应力张量与拉格朗日应变张量之间不再相关,因为柯西应力张量是相对于当前(变形)构型定义的,而拉格朗日应变张量是相对于参考(未变形)构型定义的。在本节中,我们将使用平衡偏微分方程(方程 74 或 75)来推导弹性力的虚功。本节分析表明,另一个对称应力张量——第二皮奥拉-柯西夫应力张量,与拉格朗日应变张量 $\varepsilon_{m}$(如方程 18 所示)相关联。对于线性弹性模型,第二皮奥拉-柯西夫应力张量可以通过上一节获得的本构方程与拉格朗日应变张量建立联系。为了简化本节中的推导过程,我们首先讨论张量双积以及材料微元体积的变化。
|
||||
|
||||
Tensor Double Product (Contraction) If A and $\mathbf{B}$ are second order tensors, the double product or double contraction is defined as
|
||||
Tensor Double Product (Contraction) If A and $\mathbf{B}$ are second order tensors, the double product or double contraction is defined as
|
||||
张量双积 (收缩) 如果 A 和 $\mathbf{B}$ 是二阶张量,则双积或双收缩定义如下:
|
||||
|
||||
$$
|
||||
\mathbf{A}:\mathbf{B}=t r(\mathbf{A}^{\mathrm{T}}\mathbf{B})
|
||||
$$
|
||||
|
||||
where $t r$ denotes the trace of the matrix (sum of the diagonal elements). Using the properties of the trace, one can show that
|
||||
where $t r$ denotes the trace of the matrix (sum of the diagonal elements). Using the properties of the trace, one can show that
|
||||
其中,$t r$ 表示矩阵的迹(对角线元素的和)。利用迹的性质,可以证明
|
||||
|
||||
$$
|
||||
\mathbf{A}:\mathbf{B}=t r(\mathbf{A}^{\mathrm{T}}\mathbf{B})=t r(\mathbf{B}\mathbf{A}^{\mathrm{T}})=t r(\mathbf{B}^{\mathrm{T}}\mathbf{A})=t r(\mathbf{A}\mathbf{B}^{\mathrm{T}})=\sum_{i,j=1}^{3}A_{i j}B_{i j}
|
||||
@ -6170,20 +6264,24 @@ $$
|
||||
where $A_{i j}$ and $B_{i j}$ are, respectively, the elements of the tensors $\mathbf{A}$ and $\mathbf{B}$ .
|
||||
|
||||
Volume Change If $d V$ and $d\nu$ are, respectively, the volumes of a material element in the reference and current configurations, one can show that (Spencer 1980, Shabana 2012)
|
||||
体积变化 如果 $dV$ 和 $d\nu$ 分别是参考构型和当前构型中的材料微元的体积,则可以证明(Spencer 1980, Shabana 2012)。
|
||||
|
||||
$$
|
||||
d\nu=|{\bf J}|\,d V
|
||||
$$
|
||||
|
||||
where $\mathbf{J}$ is the matrix of position vector gradients (Eq. 15) and $|\mathbf{J}|$ is the determinant of $\mathbf{J}$ .
|
||||
其中 $\mathbf{J}$ 是位置向量梯度矩阵(式 15),而 $|\mathbf{J}|$ 是 $\mathbf{J}$ 的行列式。
|
||||
|
||||
Virtual Work In Eq. 74, $\mathbf{\sigma}_{\sigma_{s}}=d i\nu\,\mathbf{\sigma}_{m}$ , where ${\bf{\sigma}}_{m}$ is the Cauchy stress tensor. Therefore, Eq. 74 can be rewritten as
|
||||
Virtual Work In Eq. 74, $\mathbf{\sigma}_{\sigma_{s}}=d i\nu\,\mathbf{\sigma}_{m}$ , where ${\bf{\sigma}}_{m}$ is the Cauchy stress tensor. Therefore, Eq. 74 can be rewritten as
|
||||
虚功 在公式 74 中,$\mathbf{\sigma}_{\sigma_{s}}=d i\nu\,\mathbf{\sigma}_{m}$ ,其中 ${\bf{\sigma}}_{m}$ 是 Cauchy 应力张量。因此,公式 74 可以改写为
|
||||
|
||||
$$
|
||||
d i\nu\,\pmb{\upsigma}_{m}+\mathbf{f}_{b}=\rho\mathbf{a}
|
||||
$$
|
||||
|
||||
Multiplying this equation by $\delta\xi$ and integrating over the current volume, one obtains
|
||||
将此方程乘以 $\delta\xi$,并在体积上积分,可得
|
||||
|
||||
$$
|
||||
\int_{\nu}\left(d i\nu\,\pmb{\sigma}_{m}+\mathbf{f}_{b}-\rho\mathbf{a}\right)^{\mathrm{T}}\delta\xi d\nu=0
|
||||
@ -6207,27 +6305,32 @@ $$
|
||||
\int_{S}\mathbf{n}^{\mathrm{T}}\mathbf{\sigma}_{m}\delta{\boldsymbol{\xi}}\ d S-\int_{\nu}\mathbf{\sigma}_{m}:\delta\mathbf{J}\mathbf{J}^{-1}d\nu+\int_{\nu}\left(\mathbf{f}_{b}-\rho\mathbf{a}\right)^{\mathrm{T}}\delta{\boldsymbol{\xi}}\ d\nu=0
|
||||
$$
|
||||
|
||||
where $S$ is the current surface area and $\mathbf{n}$ is a unit normal to the surface. The first integral in the preceding equation represents the virtual work of the surface traction forces, the second integral is the virtual work of the internal elastic forces, and the third integral is the virtual work of the body and inertia forces. If the principle of conservation of mass $(\rho d\nu=\rho_{o}d V)$ or continuity condition is assumed, the virtual work of the inertia forces can be written as
|
||||
where $S$ is the current surface area and $\mathbf{n}$ is a unit normal to the surface. The first integral in the preceding equation represents the virtual work of the surface traction forces, the second integral is the virtual work of the internal elastic forces, and the third integral is the virtual work of the body and inertia forces. If the principle of conservation of mass $(\rho d\nu=\rho_{o}d V)$ or continuity condition is assumed, the virtual work of the inertia forces can be written as
|
||||
其中,$S$ 表示当前表面积,$\mathbf{n}$ 是表面的单位法向量。前一个积分代表表面力产生的虚功,第二个积分是内部弹性力产生的虚功,而第三个积分是质点和惯性力产生的虚功。如果假设质量守恒定律($\rho d\nu=\rho_{o}d V$)或连续性条件,则惯性力产生的虚功可以写为
|
||||
|
||||
$$
|
||||
\delta W_{i}=\int_{\nu}\rho\mathbf{a}^{\mathrm{T}}\delta\xi\ d\nu=\int_{V}\rho_{o}\mathbf{a}^{\mathrm{T}}\delta\xi\ d V
|
||||
$$
|
||||
|
||||
This equation is important in developing the inertia forces of the finite elements in the case of the large deformation analysis presented in Chapter 7 of this book.
|
||||
这个公式对于开发本书第七章中介绍的大变形分析中的有限元惯性力至关重要。
|
||||
|
||||
Virtual Work of the Elastic Forces The virtual work of the internal elastic forces defined by the second term of Eq. 121 is
|
||||
弹性力虚拟功 弹性力内部力虚拟功,由公式121的第二项定义的,是
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\int_{\nu}\mathbf{\sigma}_{\sigma_{m}}:\delta\mathbf{J}\mathbf{J}^{-1}~d\nu
|
||||
$$
|
||||
|
||||
Using Eq. 116, the integration can be performed using the volume at the reference configuration. The preceding equation can then be written as
|
||||
Using Eq. 116, the integration can be performed using the volume at the reference configuration. The preceding equation can then be written as 使
|
||||
用公式 116,可以利用参考构型下的体积进行积分。前述方程可以写成:
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\int_{V}|\mathbf{J}|\,\mathbf{\sigma}\sigma_{m}:\delta\mathbf{J}\mathbf{J}^{-1}\,d V
|
||||
$$
|
||||
|
||||
where $\mathbf{\sigma}_{\mathbf{\sigma}_{K m}}=|\mathbf{J}|\,\mathbf{\sigma}_{m}$ called the Kirchhoff stress tensor is a symmetric tensor and differs from Cauchy stress tensor by a scalar multiplier equal to the determinant of the matrix of the position vector gradients. In the case of small deformation, this determinant remains approximately equal to one, and Cauchy and Kirchhoff stress tensors do not differ significantly.
|
||||
其中,$\mathbf{\sigma}_{\mathbf{\sigma}_{K m}}=|\mathbf{J}|\,\mathbf{\sigma}_{m}$,被称为基尔霍夫应力张量,它是一个对称张量,与柯西应力张量之差是一个等于位置矢量梯度矩阵行列式的标量倍数。在小变形情况下,该行列式近似等于一,因此柯西应力张量和基尔霍夫应力张量差异不大。
|
||||
|
||||
Using Eq. 18, it is clear that
|
||||
|
||||
@ -6235,45 +6338,49 @@ $$
|
||||
\delta\,\pmb{\varepsilon}_{m}=\frac{1}{2}(\mathbf{J}^{\mathrm{T}}\delta\mathbf{J}+(\delta\mathbf{J}^{\mathrm{T}})\mathbf{J})
|
||||
$$
|
||||
|
||||
where $\varepsilon_{m}$ is the Lagrangian strain tensor. Using this equation, the virtual work of the elastic forces can be written in terms of the virtual changes of the components of the Lagrangian strain tensor as
|
||||
where $\varepsilon_{m}$ is the Lagrangian strain tensor. Using this equation, the virtual work of the elastic forces can be written in terms of the virtual changes of the components of the Lagrangian strain tensor as
|
||||
其中 $\varepsilon_{m}$ 是拉格朗日应变张量。利用此方程,弹性力的虚功可以表示为拉格朗日应变张量分量的虚拟变化,形式如下:
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\intop_{V}|\mathbf{J}|\,\mathbf{\sigma}\sigma_{m}:\mathbf{J}^{-1^{\mathrm{T}}}\delta\varepsilon_{m}\mathbf{J}^{-1}d V
|
||||
$$
|
||||
|
||||
This equation upon the use of the properties of the tensor double product (Eq. 115) can be written as
|
||||
This equation upon the use of the properties of the tensor double product (Eq. 115) can be written as 利用张量双积的性质(式115),该方程可以写成:
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\int_{V}\left(|\mathbf{J}|\,\mathbf{J}^{-1}\mathbf{\sigma}_{m}\mathbf{J}^{-1^{\mathrm{T}}}\right):\delta\varepsilon_{m}\,d V
|
||||
$$
|
||||
|
||||
which can be written as follows:
|
||||
which can be written as follows: 可以写成如下:
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\int_{V}\mathbf{\sigma}_{}\sigma_{P m}:\delta\boldsymbol{\varepsilon}_{m}\,d V
|
||||
$$
|
||||
|
||||
where $\mathbf{\nabla}\sigma_{P m}$ is the second Piola-Kirchhoff stress tensor defined as
|
||||
where $\mathbf{\nabla}\sigma_{P m}$ is the second Piola-Kirchhoff stress tensor defined as 其中 $\mathbf{\nabla}\sigma_{P m}$ 是第二类Piola-Kirchhoff应力张量,定义为
|
||||
|
||||
$$
|
||||
\pmb{\sigma}_{P m}=|\mathbf{J}|\,\mathbf{J}^{-1}\pmb{\sigma}_{m}\mathbf{J}^{-1^{\mathrm{T}}}
|
||||
$$
|
||||
|
||||
Clearly, the second Piola-Kirchhoff stress tensor is a symmetric tensor, and it is the stress tensor associated with the Lagrangian strain tensor.
|
||||
Clearly, the second Piola-Kirchhoff stress tensor is a symmetric tensor, and it is the stress tensor associated with the Lagrangian strain tensor. 显然,第二个Piola-Kirchhoff应力张量是一个对称张量,它是与拉格朗日应变张量相关的应力张量。
|
||||
|
||||
If the deformation is small, the matrix of the position vector gradients (Jacobian) J does not differ significantly from the identity matrix, and as a consequence it is acceptable not to distinguish between Cauchy stress tensor which is defined using the deformed configuration and the second Piola-Kirchhoff stress tensor associated with the reference undeformed configuration. In this book we will always use the Lagrangian strains with the understanding that the associated stress is the second Piola-Kirchhoff stress tensor. For the sake of simplicity of the notation, we will also use $\upsigma$ to denote the stress vector associated with $\mathbf{\nabla}\sigma_{P m}$ instead of Cauchy stress tensor since whenever there is a difference between the two tensors (case of large deformation), it is with the understanding that the second Piola-Kirchhoff stress tensor is the one to be used with the Lagrangian strain tensor. Using the tensor double product, and the fact that both $\mathbf{\delta}\mathbf{\sigma}_{\mathbf{}\mathbf{0}_{}P m}$ and $\varepsilon_{m}$ are symmetric tensors, one can use the definition of the tensor double product to show that Eq. 128 can be written as follows:
|
||||
如果变形很小,位置矢量梯度矩阵(雅可比矩阵)J与单位矩阵的差异并不显著,因此可以忽略考西应力张量(使用变形构型定义)和与参考未变形构型相关的第二皮奥拉-柯西福应力张量之间的区别。 在本书中,我们将始终使用拉格朗日应变,并理解其相关的应力是第二皮奥拉-柯西福应力张量。 为了简化记号,我们也将使用 $\upsigma$ 来表示与 $\mathbf{\nabla}\sigma_{P m}$ 相关的应力矢量,而不是考西应力张量,因为当两个张量之间存在差异(大变形情况)时,我们始终理解应使用第二皮奥拉-柯西福应力张量与拉格朗日应变张量。 利用张量双积,以及 $\mathbf{\delta}\mathbf{\sigma}_{\mathbf{}\mathbf{0}_{}P m}$ 和 $\varepsilon_{m}$ 都是对称张量的事实,可以利用张量双积的定义证明公式 128 可以写成如下形式:
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\int_{V}\left(\sigma_{11}\delta\varepsilon_{11}+\sigma_{22}\delta\varepsilon_{22}+\sigma_{33}\delta\varepsilon_{33}+2\sigma_{12}\delta\varepsilon_{12}+2\sigma_{13}\delta\varepsilon_{13}+2\sigma_{23}\delta\varepsilon_{23}\right)d V
|
||||
$$
|
||||
|
||||
where $\sigma_{i j}$ and $\epsilon_{i j}$ are, respectively, the elements of the second Piola-Kirchhof stress tensor and the Lagrangian strain tensor. Using the development presented in the preceding section, one can write $\mathbf{\sigma}_{\mathbf{\sigma}}\mathbf{\sigma}_{\mathbf{\sigma}}=\mathbf{E}\varepsilon$ , where $\upsigma$ and ε are the stress and strain vectors associated with the second Piola-Kirchhoff stress tensor and the Lagrangian strain tensor, and $\mathbf{E}$ is the matrix of elastic coefficients. One can then write the virtual work of the elastic forces given by Eq. 130 in the following form:
|
||||
where $\sigma_{i j}$ and $\epsilon_{i j}$ are, respectively, the elements of the second Piola-Kirchhof stress tensor and the Lagrangian strain tensor. Using the development presented in the preceding section, one can write $\mathbf{\sigma}_{\mathbf{\sigma}}\mathbf{\sigma}_{\mathbf{\sigma}}=\mathbf{E}\varepsilon$ , where $\upsigma$ and ε are the stress and strain vectors associated with the second Piola-Kirchhoff stress tensor and the Lagrangian strain tensor, and $\mathbf{E}$ is the matrix of elastic coefficients. One can then write the virtual work of the elastic forces given by Eq. 130 in the following form:
|
||||
其中,$\sigma_{i j}$ 和 $\epsilon_{i j}$ 分别是第二皮奥拉-基尔霍夫应力张量和拉格朗日应变张量的分量。利用前一节中提出的发展,可以写出 $\mathbf{\sigma}_{\mathbf{\sigma}}\mathbf{\sigma}_{\mathbf{\sigma}}=\mathbf{E}\varepsilon$,其中 $\upsigma$ 和 ε 分别是与第二皮奥拉-基尔霍夫应力张量和拉格朗日应变张量相关的应力矢量和应变矢量,而 $\mathbf{E}$ 是弹性系数矩阵。 那么,可以写出由公式 130 给出的弹性力的虚功,形式如下:
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\intop_{V}\mathbf{\boldsymbol{\varepsilon}}^{\mathrm{T}}\mathbf{E}\mathbf{I}_{t}\delta\mathbf{\boldsymbol{\varepsilon}}d V
|
||||
$$
|
||||
|
||||
where ${\bf{I}}_{t}$ is a diagonal matrix with dimension six. The first three diagonal elements are equal to one while the last three diagonal elements are equal to 2. This matrix is introduced in order to account for the “two” multipliers associated with the shear strains in the expression of the virtual work given by Eq. 128 or Eq. 130. Therefore, the virtual work of the elastic forces can be written as
|
||||
其中 ${\bf{I}}_{t}$ 是一个六维对角矩阵。前三个对角元素的值为一,而后三个对角元素的值为2。引入此矩阵是为了考虑在公式128或公式130给出的虚拟功表达式中,与剪切应变相关的“两个”乘数。因此,弹性力虚拟功可以写为
|
||||
|
||||
$$
|
||||
\delta W_{s}=-\int_{V}\varepsilon^{\mathrm{T}}\bar{\bf E}\;\delta\varepsilon d V
|
||||
@ -6281,7 +6388,7 @@ $$
|
||||
|
||||
where $\bar{\mathbf{E}}=\mathbf{E}\mathbf{I}_{t}$ is the modified matrix of elastic coefficients.
|
||||
|
||||
# Problems
|
||||
Problems
|
||||
|
||||
1. Determine whether $\mathbf{u}=[k(x_{2}-x_{1}),k(x_{1}-x_{2}),k x_{1}x_{2}]$ , where $k$ is a constant, represents continuously possible displacement components for a continuous medium. Consider $(x_{1},x_{2},x_{3})$ to be rectangular Cartesian coordinates of a point in the body.
|
||||
|
||||
@ -6337,31 +6444,43 @@ where $k_{1},k_{2}$ , and $k_{3}$ are constants. Determine whether or not these
|
||||
|
||||
# 5 FLOATING FRAME OF REFERENCE FORMULATION
|
||||
|
||||
In this chapter, approximation methods are used to formulate a finite set of dynamic equations of motion of multibody systems that contain interconnected deformable bodies. As shown in Chapter 3, the dynamic equations of motion of the rigid bodies in the multibody system can be defined in terms of the mass of the body, the inertia tensor, and the generalized forces acting on the body. On the other hand, the dynamic formulation of the system equations of motion of linear structural systems requires the definition of the system mass and stiffness matrices as well as the vector of generalized forces. In this chapter, the dynamic formulation of the equations of motion of deformable bodies that undergo large translational and rotational displacements are developed using the floating frame of reference formulation. It will be shown that the equations of motion of such systems can be written in terms of a set of inertia shape integrals in addition to the mass of the body, the inertia tensor, and the generalized forces that appear in the dynamic formulation of rigid body system equations of motion and the mass and stiffness matrices and the vector of generalized forces that appear in the dynamic equations of linear structural systems. These inertia shape integrals that depend on the assumed displacement field appear in the nonlinear terms that represent the inertia coupling between the reference motion and the elastic deformation of the body. It will be also shown that the deformable body inertia tensor depends on the elastic deformation of the body, and accordingly it is an implicit function of time.
|
||||
In this chapter, approximation methods are used to formulate a finite set of dynamic equations of motion of multibody systems that contain interconnected deformable bodies. As shown in Chapter 3, the dynamic equations of motion of the rigid bodies in the multibody system can be defined in terms of the mass of the body, the inertia tensor, and the generalized forces acting on the body. On the other hand, the dynamic formulation of the system equations of motion of linear structural systems requires the definition of the system mass and stiffness matrices as well as the vector of generalized forces. In this chapter, the dynamic formulation of the equations of motion of deformable bodies that undergo large translational and rotational displacements are developed using the floating frame of reference formulation. It will be shown that the equations of motion of such systems can be written in terms of **a set of inertia shape integrals** in addition to the mass of the body, the inertia tensor, and the generalized forces that appear in the dynamic formulation of rigid body system equations of motion and the mass and stiffness matrices and the vector of generalized forces that appear in the dynamic equations of linear structural systems. These inertia shape integrals that depend on the assumed displacement field appear in the nonlinear terms that represent the inertia coupling between the reference motion and the elastic deformation of the body. It will be also shown that the deformable body inertia tensor depends on the elastic deformation of the body, and accordingly it is an implicit function of time.
|
||||
本章将使用近似方法,推导包含互连变形体的多体系统的有限集动态运动方程。正如第三章所示,刚体多体系统的动态运动方程可以根据物体的质量、惯性张量以及作用在物体上的广义力来定义。另一方面,线性结构系统运动方程的动态公式需要定义系统质量矩阵和刚度矩阵,以及广义力向量。在本章中,我们将使用浮动参考系公式,推导经历大位移和旋转的变形体运动方程。将证明,此类系统的运动方程可以表示为一组惯性形状积分,除了物体的质量、惯性张量以及出现在刚体系统运动方程动态公式化中的广义力,以及出现在线性结构系统动态方程中的质量和刚度矩阵和广义力向量。这些取决于假设位移场的惯性形状积分出现在代表参考运动与物体弹性变形之间惯性耦合的非线性项中。还将证明,变形体的惯性张量取决于物体的弹性变形,因此它随时间呈隐式函数变化。
|
||||
|
||||
In the floating frame of reference formulation presented in this chapter, the configuration of each deformable body in the multibody system is identified by using two sets of coordinates: reference and elastic coordinates. Reference coordinates define the location and orientation of a selected body reference. Elastic coordinates, on the other hand, describe the body deformation with respect to the body reference. In order to avoid the computational difficulties associated with infinite-dimensional spaces, these coordinates are introduced by using classical approximation techniques such as Rayleigh–Ritz methods. The global position of an arbitrary point on the deformable body is thus defined by using a coupled set of reference and elastic coordinates. The kinetic energy of the deformable body is then developed and the inertia coupling between the reference motion and the elastic deformation is identified. The kinetic energy as well as the virtual work of the forces acting on the body are written in terms of the coupled sets of reference and elastic coordinates. Mechanical joints in the multibody system are formulated by using a set of nonlinear algebraic constraint equations that depend on the reference and elastic coordinates and possibly on time. These algebraic constraint equations can be used to identify a set of independent coordinates (system degrees of freedom) by using the generalized coordinate partitioning of the constraint Jacobian matrix, or can be adjoined to the system differential equations of motion by using the vector of Lagrange multipliers.
|
||||
在本书这一章所介绍的浮动参考系表述中,多体系统中每个可变形体的状态配置由两组坐标来确定:参考坐标和弹性坐标。参考坐标定义了选定体参考系的位姿;而弹性坐标则描述了相对于该体参考系的体变形。为了避免与无限维空间相关的计算困难,这些坐标是通过使用诸如瑞利–里兹方法等经典近似技术引入的。因此,可变形体上任意一点的全局位置由一组耦合的参考坐标和弹性坐标来定义。随后,推导了可变形体的动能,并识别了参考运动和弹性变形之间的惯性耦合。作用于该体的力的动能以及虚功,都以耦合的参考坐标和弹性坐标来表示。多体系统中的机械连接,则通过一组依赖于参考坐标和弹性坐标(以及可能依赖于时间)的非线性代数约束方程来构建。这些代数约束方程可以通过约束雅可比矩阵的广义坐标划分来识别一组独立的坐标(系统自由度),或者可以通过拉格朗日乘子向量附加到系统的运动微分方程中。
|
||||
|
||||
# 5.1 KINEMATIC DESCRIPTION
|
||||
|
||||
Multibody systems in general include two collections of bodies. One collection consists of bulky and compact solids that can be treated as rigid bodies, while the other collection includes typical structural components such as rods, beams, plates, and shells. As pointed out in previous chapters, rigid bodies have a finite number of degrees of freedom; for instance, a rigid body in space has six degrees of freedom that describe the location and orientation of the body with respect to the fixed frame of reference. On the other hand, structural components such as beams, plates, and shells have an infinite number of degrees of freedom that describe the displacement of each point on the component. As was shown in the preceding chapter, the behavior of such components is governed by a set of simultaneous partial differential equations. Using the separation of variables, the solution of these equations, if possible, leads to representation of the displacement field in terms of infinite series that can be written in the following form:
|
||||
多体系统通常包含两类构件。一类是由粗大、紧凑的实体构成,这些实体可以被视为刚体;另一类则包括典型的结构元件,如杆、梁、板和壳体。正如前几章所指出的,刚体具有有限的自由度;例如,在空间中的刚体有六个自由度,用于描述其相对于固定参考系的位姿。另一方面,梁、板和壳体等结构元件具有无限的自由度,用于描述元件上每个点的位移。正如前一章所示,这类构件的行为由一组联立偏微分方程支配。利用变量分离法,如果可行,这些方程的解可以导向位移场的表示,其形式为无穷级数,可以写成如下形式:
|
||||
|
||||
|
||||
$$
|
||||
{\begin{array}{l}{{\bar{u}}_{f1}=\displaystyle\sum_{k=1}^{\infty}a_{k}f_{k}\quad{\mathrm{where~}}f_{k}=f_{k}(x_{1},x_{2},x_{3})}\\ {{\bar{u}}_{f2}=\displaystyle\sum_{k=1}^{\infty}b_{k}g_{k}\quad{\mathrm{where~}}g_{k}=g_{k}(x_{1},x_{2},x_{3})\,\,;}\\ {{\bar{u}}_{f3}=\displaystyle\sum_{k=1}^{\infty}c_{k}h_{k}\quad{\mathrm{where~}}h_{k}=h_{k}(x_{1},x_{2},x_{3})}\end{array}}
|
||||
$$
|
||||
|
||||
where $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ are the components of the displacement of an arbitrary point that has coordinates $(x_{1},x_{2},x_{3})$ in the undeformed state. The vector of displacement $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}~\bar{u}_{f2}~\bar{u}_{f3}]^{\mathrm{T}}$ is space- and time-dependent. The coefficients $a_{k},\,b_{k}$ , and $c_{k}$ are assumed to depend only on time. These coefficients are called the coordinates, and the functions $f_{k},\,g_{k}$ , and $h_{k}$ are called the base functions. Each of the functions $f_{k},\,g_{k}$ , and $h_{k}$ must be admissible; that is, the function has to satisfy the kinematic constraints imposed on the boundary of the deformable body. It is also required that the infinite series of Eq. 1 converge to the limit functions $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ and that these limit functions give an accurate representation to the deformed shape.
|
||||
其中,$\bar{u}_{f1},\,\bar{u}_{f2}$ , 和 $\bar{u}_{f3}$ 分别是具有坐标 $(x_{1},x_{2},x_{3})$ 的任意点在未变形状态下的位移分量。位移矢量 $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}~\bar{u}_{f2}~\bar{u}_{f3}]^{\mathrm{T}}$ 是空间和时间相关的。系数 $a_{k},\,b_{k}$ , 和 $c_{k}$ 被假定仅随时间变化。这些系数被称为坐标,而函数 $f_{k},\,g_{k}$ , 和 $h_{k}$ 被称为基函数。每个函数 $f_{k},\,g_{k}$ , 和 $h_{k}$ 都必须是可容许的;也就是说,该函数必须满足施加在变形体边界上的运动约束。此外,还需要无限级数(如公式1所示)收敛于极限函数 $\bar{u}_{f1},\,\bar{u}_{f2}$ , 和 $\bar{u}_{f3}$,并且这些极限函数能够准确地表示变形后的形状。
|
||||
|
||||
Rayleigh–Ritz Approximation A simple example of Eq. 1 is the displacement representation that arises when one writes the partial differential equation of a vibrating beam and uses the separation of variables technique to solve this equation. In this particular case, the base functions are the eigenfunctions and the coordinates that are infinite in dimension are the time-dependent modal coordinates. Because of the computational difficulties encountered in dealing with infinite-dimensional spaces, classical approximation methods such as the Rayleigh–Ritz method and the Galerkin method are employed wherein the displacement of each point is expressed in terms of a finite number of coordinates. In this case the series of Eq. 1 are truncated, and this leads to
|
||||
Rayleigh–Ritz Approximation A simple example of Eq. 1 is the displacement representation that arises when one writes the partial differential equation of a vibrating beam and uses the separation of variables technique to solve this equation. In this particular case, the base functions are the eigenfunctions and the coordinates that are infinite in dimension are the time-dependent modal coordinates. Because of the computational difficulties encountered in dealing with infinite-dimensional spaces, classical approximation methods such as the Rayleigh–Ritz method and the Galerkin method are employed wherein the displacement of each point is expressed in terms of a finite number of coordinates. In this case the series of Eq. 1 are truncated, and this leads to
|
||||
瑞利-里兹逼近法
|
||||
|
||||
公式1的一个简单例子是振动梁的偏微分方程求解过程中出现的位移表示。通过变量分离法求解该方程时,基函数是特征函数,而无限维坐标则是随时间变化的模态坐标。由于处理无限维空间会遇到计算困难,因此采用诸如瑞利-里兹法和伽辽金法等经典逼近方法,其中每个点的位移用有限数量的坐标来表达。在这种情况下,公式1的级数被截断,这导致
|
||||
|
||||
$$
|
||||
\bar{u}_{f1}\approx\sum_{k=1}^{l}a_{k}\,f_{k},\quad\bar{u}_{f2}\approx\sum_{k=1}^{m}b_{k}\,g_{k},\quad\bar{u}_{f3}\approx\sum_{k=1}^{n}c_{k}h_{k}
|
||||
$$
|
||||
|
||||
The functions $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ represent, in this case, partial sums of the series of Eq. 1. For the approximation of Eq. 2 to be valid, the sequences of partial sums of Eq. 2 must converge to the limit functions of Eq. 1. In other words, we require the sequences of partial sums to be Cauchy sequences. A sequence of functions $(s_{1},s_{2},\ldots)$ is said to be a Cauchy sequence if, given a small number $\varepsilon>0$ , there exists a natural number $M(\varepsilon)$ such that if $n$ and $m$ are two arbitrary natural numbers that are greater than or equal to $M(\varepsilon)$ and $m>n$ , we have $|s_{m}-s_{n}|\,<\,\varepsilon$ . By assuming that the sequences of partial sums of the series in Eq. 1 are Cauchy sequences, and provided $l,m$ , and $n$ of Eq. 2 are relatively large, we are guaranteed that the approximation of Eq. 2 is acceptable.
|
||||
The functions $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ represent, in this case, partial sums of the series of Eq. 1. For the approximation of Eq. 2 to be valid, the sequences of partial sums of Eq. 2 must converge to the limit functions of Eq. 1. In other words, we require the sequences of partial sums to be Cauchy sequences. A sequence of functions $(s_{1},s_{2},\ldots)$ is said to be a Cauchy sequence if, given a small number $\varepsilon>0$ , there exists a natural number $M(\varepsilon)$ such that if $n$ and $m$ are two arbitrary natural numbers that are greater than or equal to $M(\varepsilon)$ and $m>n$ , we have $|s_{m}-s_{n}|\,<\,\varepsilon$ . By assuming that the sequences of partial sums of the series in Eq. 1 are Cauchy sequences, and provided $l,m$ , and $n$ of Eq. 2 are relatively large, we are guaranteed that the approximation of Eq. 2 is acceptable.
|
||||
函数 $\bar{u}_{f1},\,\bar{u}_{f2}$ , 和 $\bar{u}_{f3}$ 在本例中分别代表公式 1 中的级数的偏和。为了使公式 2 的近似有效,公式 2 的偏和序列必须收敛到公式 1 的极限函数。换句话说,我们要求偏和序列是柯西序列。一个函数序列 $(s_{1},s_{2},\ldots)$ 被称为柯西序列,如果给定一个任意小的数 $\varepsilon>0$ ,存在一个自然数 $M(\varepsilon)$,使得当 $n$ 和 $m$ 是两个任意自然数,且都大于或等于 $M(\varepsilon)$ 且 $m>n$ 时,有 $|s_{m}-s_{n}|\,<\,\varepsilon$ 。通过假设公式 1 中级数的偏和序列是柯西序列,并且在 $l,m$ 和 $n$ 在公式 2 中足够大的前提下,我们保证公式 2 的近似是可接受的。
|
||||
|
||||
Equation 2 implies also that approximations of the limit functions $\bar{u}_{f1},\ \bar{u}_{f2}$ , and $\bar{u}_{f3}$ can be obtained as linear combinations of the base functions $f_{k},\;g_{k}$ , and $h_{k}$ , respectively. This property, in addition to the fact that the sequences of partial sums of the series of Eq. 1 are Cauchy sequences, is called completeness; that is, completeness is achieved if the exact displacements, and their derivatives, can be matched arbitrarily closely if enough coordinates appear in the assumed displacement field. The assumed displacement field is, in general, either exact or stiff. This is mainly because the structure is permitted to deform only into the shapes described by the assumed displacement field.
|
||||
Equation 2 implies also that approximations of the limit functions $\bar{u}_{f1},\ \bar{u}_{f2}$ , and $\bar{u}_{f3}$ can be obtained as linear combinations of the base functions $f_{k},\;g_{k}$ , and $h_{k}$ , respectively. This property, in addition to the fact that the sequences of partial sums of the series of Eq. 1 are Cauchy sequences, is called completeness; that is, completeness is achieved if the exact displacements, and their derivatives, can be matched arbitrarily closely if enough coordinates appear in the assumed displacement field. The assumed displacement field is, in general, either exact or stiff. This is mainly because the structure is permitted to deform only into the shapes described by the assumed displacement field. 公式 2 也暗示,极限函数 $\bar{u}_{f1}$、$\bar{u}_{f2}$ 和 $\bar{u}_{f3}$ 的近似值可以分别作为基函数 $f_{k}$、$g_{k}$ 和 $h_{k}$ 的线性组合得到。 这一性质,以及公式 1 的级数偏和序列是柯西序列这一事实,合称为完备性;也就是说,如果足够多的坐标出现在假设的位移场中,能够任意接近地匹配精确位移及其导数,则实现了完备性。 假设的位移场通常要么是精确的,要么是刚性的。 这主要是因为结构仅被允许变形为假设的位移场所描述的形状。
|
||||
|
||||
Floating Frame of Reference In the development presented in the subsequent sections, we assume that the displacement field of Eq. 2 describes the deformation of the body with respect to a selected body reference as shown in Fig. 5.1. The motion of the body is then defined as the motion of its reference plus the motion of the material points on the body with respect to its reference. If the assumed displacement field contains rigid body modes, a set of reference conditions has to be imposed to define a unique displacement field with respect to the selected body reference. This subject is discussed in more detail in the following chapter where a finite element floating frame of reference formulation is presented.
|
||||
浮动参考系
|
||||
|
||||
在后续章节所述的发展过程中,我们假设公式2中的位移场描述了相对于所选的刚体参考系(如图5.1所示)的物体变形。然后,物体的运动被定义为其参考系的运动加上物体上各质点相对于其参考系的运动。如果假设的位移场包含刚体运动模式,则必须施加一组参考条件,以定义相对于所选刚体参考系的唯一位移场。本主题将在下一章中更详细地讨论,届时将介绍有限元浮动参考系公式。
|
||||
|
||||

|
||||
Figure 5.1 Deformable body coordinates.
|
||||
@ -6369,40 +6488,49 @@ Figure 5.1 Deformable body coordinates.
|
||||
One may write Eq. 2 in the following matrix form:
|
||||
|
||||
$$
|
||||
\bar{\ensuremath{\mathbf{u}}}_{f}=\mathbf{S}\ensuremath{\mathbf{q}}_{f}
|
||||
\bar{\mathbf{u}}_{f}=\mathbf{S}{\mathbf{q}}_{f}
|
||||
$$
|
||||
|
||||
where $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}\,\bar{u}_{f2}\,\bar{u}_{f3}]^{\mathrm{T}}$ is the deformation vector; $\mathbf{S}$ is the shape matrix whose elements are the base functions $f_{k},g_{k}$ , and $h_{k}$ ; and ${\bf q}_{f}$ is the vector of elastic coordinates that contains the time dependent coefficients $a_{k},b_{k}$ , and $c_{k}$ .
|
||||
where $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}\,\bar{u}_{f2}\,\bar{u}_{f3}]^{\mathrm{T}}$ is the **deformation vector**; $\mathbf{S}$ is the shape matrix whose elements are the base functions $f_{k},g_{k}$ , and $h_{k}$ ; and ${\bf q}_{f}$ is the vector of elastic coordinates that contains the time dependent coefficients $a_{k},b_{k}$ , and $c_{k}$ .
|
||||
其中 $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}\,\bar{u}_{f2}\,\bar{u}_{f3}]^{\mathrm{T}}$ 是 **变形矢量**;$\mathbf{S}$ 是形状矩阵,其元素为基函数 $f_{k},g_{k}$ 和 $h_{k}$;而 ${\bf q}_{f}$ 是弹性坐标矢量,包含随时间变化的系数 $a_{k},b_{k}$ 和 $c_{k}$。
|
||||
|
||||
To identify the configuration of deformable bodies, a set of generalized coordinates should be selected such that the location of an arbitrary point on the body can be described in terms of these generalized coordinates. To this end, we select a global coordinate system that is fixed in time and forms a single standard and as such serves to define the connectivity between different bodies in the multibody system. For an arbitrary body in the system, say, body $i,$ , we select a body reference $\mathbf{X}_{1}^{i}\mathbf{X}_{2}^{i}\mathbf{X}_{3}^{i}$ whose location and orientation with respect to the global coordinate system are defined by a set of coordinates called reference coordinates and denoted as $\mathbf{q}_{r}^{i}$ . The vector $\mathbf{q}_{r}^{i}$ can be written in a partitioned form as
|
||||
为了确定可变形体的构型,应选择一组广义坐标,使得该体上任意一点的位置能够用这些广义坐标来描述。为此,我们选择一个随时间固定的全局坐标系,作为单一的标准,并用于定义多体系统中不同体之间的连接关系。对于系统中的任意一个体,例如第 $i$ 个体,我们选择一个参考坐标系 $\mathbf{X}_{1}^{i}\mathbf{X}_{2}^{i}\mathbf{X}_{3}^{i}$,其相对于全局坐标系的位置和姿态由一组称为参考坐标并记为 $\mathbf{q}_{r}^{i}$ 的坐标来定义。向量 $\mathbf{q}_{r}^{i}$ 可以写成分块形式为:
|
||||
|
||||
$$
|
||||
{\bf q}_{r}^{i}=[{\bf R}^{i^{\textup T}}\quad{\boldsymbol{\Theta}}^{i^{\textup T}}]^{\textup T}
|
||||
$$
|
||||
|
||||
where $\mathbf{R}^{i}$ is a set of Cartesian coordinates that define the location of the origin of the body reference (Fig. 5.1) and ${\boldsymbol{\Theta}}^{i}$ is a set of rotational coordinates that describe the orientation of the selected body reference. The body coordinate system $\mathbf{X}_{1}^{i}\mathbf{X}_{2}^{i}\mathbf{X}_{3}^{i}$ is the floating frame of reference. The origin of this reference frame does not have to be rigidly attached to a material point on the deformable body. It is required, however, that there is no rigid body motion between the body and its coordinate system. It is also important to point out that the reference motion should not be interpreted as the rigid body motion, since different coordinate systems can be selected for the deformable body (Shabana 1996a). The floating frame of reference formulation, therefore, does not lead to a separation between the rigid body motion and the elastic deformation.
|
||||
其中,$\mathbf{R}^{i}$ 是定义物体参考系原点位置的笛卡尔坐标系集合 (图 5.1),${\boldsymbol{\Theta}}^{i}$ 是描述所选物体参考系姿态的旋转坐标系集合。物体坐标系 $\mathbf{X}_{1}^{i}\mathbf{X}_{2}^{i}\mathbf{X}_{3}^{i}$ 是运动坐标系。该参考系的坐标原点不必严格地固定在变形体的材料点上。但是,要求物体和其坐标系之间不存在刚体运动。需要注意的是,参考运动不应被解释为刚体运动,因为可以为变形体选择不同的坐标系 (Shabana 1996a)。因此,运动坐标系表述并不能将刚体运动和弹性变形分开。
|
||||
|
||||
Position Coordinates In this and the following chapter, the set of Cartesian reference coordinates is used to maintain the generality of the development. Other sets of coordinates such as joint variables can also be used with the formulation presented in this chapter by establishing the proper coordinate transformation. As pointed out in Chapter 2, three coordinates are required to define the location and orientation of the body reference in the two-dimensional analysis. These coordinates can be selected to be $R_{1}^{i},\,R_{2}^{i}$ , and $\theta^{i}$ , where $R_{1}^{i}$ and $R_{2}^{i}$ are the coordinates of the origin of the body reference and $\theta^{i}$ is the angular rotation of the body about the axis of rotation. In three-dimensional analysis, however, six independent coordinates are required. Three coordinates, $R_{1}^{i}$ , $R_{2}^{i}$ , and $R_{3}^{i}$ , define the location of the origin of the body reference, and three independent rotational coordinates define the orientation of this reference. This subject has been thoroughly investigated in Chapter 2, where it is pointed out that the orientation of the body reference can be identified using the three independent Euler angles, Rodriguez parameters, or the four dependent Euler parameters. If the body is rigid, the reference coordinates are sufficient for definition of the location of an arbitrary point on the body, and accordingly these coordinates completely describe the body kinematics. For rigid bodies, therefore, the configuration space of the body and the configuration space of its reference are the same and no conceptual difficulties arise in selecting the local reference frame of rigid bodies. For example, in the case of a rigid body, the global position of an arbitrary point $P$ on the rigid body can be written in the planar analysis as $\mathbf{r}_{P}^{i}=\mathbf{R}^{i}+\mathbf{A}^{i}\bar{\mathbf{u}}^{i}$ , where $\bar{\mathbf{u}}^{i}$ is the local position vector of point $P$ and $\mathbf{A}^{i}$ is the transformation matrix defined in the case of planar analysis as
|
||||
坐标位置
|
||||
|
||||
在本章及下一章中,使用笛卡尔参考坐标系以保持开发的普遍性。通过建立适当的坐标变换,可以使用诸如关节变量等其他坐标系,与本章提出的公式兼容。如第二章所述,需要三个坐标来定义二维分析中参考体的位姿。这些坐标可以选择为 $R_{1}^{i}$、 $R_{2}^{i}$ 和 $\theta^{i}$,其中 $R_{1}^{i}$ 和 $R_{2}^{i}$ 是参考体原点的坐标,而 $\theta^{i}$ 是参考体绕旋转轴的旋转角度。然而,在三维分析中,需要六个独立的坐标。三个坐标,$R_{1}^{i}$、 $R_{2}^{i}$ 和 $R_{3}^{i}$,定义了参考体原点的位置,而三个独立的旋转坐标定义了参考体的姿态。这一主题已在第二章中进行了彻底研究,指出参考体的姿态可以使用三个独立的欧拉角、罗德里格斯参数或四个相关的欧拉参数来识别。如果体是刚性的,则参考坐标足以定义体上任意一点的位置,因此这些坐标完全描述了体的运动学。因此,对于刚体,刚体的配置空间和其参考的配置空间是相同的,在选择刚体的局部参考系时不会产生概念上的困难。例如,在刚体的情况下,刚体上任意一点 $P$ 的全局位置可以在平面分析中表示为 $\mathbf{r}_{P}^{i}=\mathbf{R}^{i}+\mathbf{A}^{i}\bar{\mathbf{u}}^{i}$ ,其中 $\bar{\mathbf{u}}^{i}$ 是点 $P$ 的局部位置向量,而 $\mathbf{A}^{i}$ 是在平面分析中定义的变换矩阵。
|
||||
|
||||
$$
|
||||
\mathbf{A}^{i}={\left[\!\!\begin{array}{l l}{\cos\theta^{i}}&{-\sin\theta^{i}}\\ {\sin\theta^{i}}&{\cos\theta^{i}}\end{array}\!\!\right]}
|
||||
\mathbf{A}^{i}={\begin{bmatrix}{\cos\theta^{i}}&{-\sin\theta^{i}}\\ {\sin\theta^{i}}&{\cos\theta^{i}}\end{bmatrix}}
|
||||
$$
|
||||
Since the assumption of rigidity of the body $i$ implies that the distance between two arbitrary points on the body remains constant, one may conclude that the length of the vector $\bar{{\mathbf{u}}}^{i}$ remains constant and, as such, the components of this vector relative to the body coordinate system remain unchanged. Similar comments apply for the spatial analysis.
|
||||
假设物体 $i$ 具有刚体性质,这意味着该物体上任意两点之间的距离保持不变。因此,可以得出结论,向量 $\bar{{\mathbf{u}}}^{i}$ 的长度保持不变,并且相对于该物体的坐标系,该向量的分量也保持不变。 空间分析也适用类似的论述。
|
||||
|
||||
Since the assumption of rigidity of the body $i$ implies that the distance between two arbitrary points on the body remains constant, one may conclude that the length of the vector $\bar{\ensuremath{\mathbf{u}}}^{i}$ remains constant and, as such, the components of this vector relative to the body coordinate system remain unchanged. Similar comments apply for the spatial analysis.
|
||||
|
||||
When deformable bodies are considered, the distance between two arbitrary points on the deformable body does not, in general, remain constant because of the relative motion between the particles forming the body. In this case, the vector $\bar{\mathbf{u}}^{i}$ can be written as
|
||||
When deformable bodies are considered, the distance between two arbitrary points on the deformable body does not, in general, remain constant because of the relative motion between the particles forming the body. In this case, the vector $\bar{\mathbf{u}}^{i}$ can be written as 当考虑可变形体时,由于构成该体的粒子之间的相对运动,两个在可变形体上任意两点之间的距离通常不会保持恒定。 在这种情况下,向量 $\bar{\mathbf{u}}^{i}$ 可以表示为:
|
||||
|
||||
$$
|
||||
\bar{\mathbf{u}}^{i}=\bar{\mathbf{u}}_{o}^{i}+\bar{\mathbf{u}}_{f}^{i}=\bar{\mathbf{u}}_{o}^{i}+\mathbf{S}^{i}\mathbf{q}_{f}^{i}
|
||||
$$
|
||||
|
||||
where $\bar{\mathbf{u}}_{o}^{i}$ is the position of point $P$ in the undeformed state, $\mathbf{S}^{i}=\mathbf{S}^{i}(x_{1}^{i},x_{2}^{i},x_{3}^{i})$ is a space-dependent shape matrix, and ${\bf q}_{f}^{i}$ is the vector of time-dependent elastic generalized coordinates of the deformable body $i$ . One can then write the global position of an arbitrary point $P$ on body $i$ in the planar or the spatial case as
|
||||
其中,$\bar{\mathbf{u}}_{o}^{i}$ 为点 $P$ 在未变形状态下的位置,$\mathbf{S}^{i}=\mathbf{S}^{i}(x_{1}^{i},x_{2}^{i},x_{3}^{i})$ 是与空间相关的形变矩阵,${\bf q}_{f}^{i}$ 是可变形体 $i$ 的随时间变化的弹性广义坐标向量。 那么,在平面或空间情况下,可以写出任意点 $P$ 在体 $i$ 上的全局位置为:
|
||||
|
||||
|
||||
$$
|
||||
\mathbf{r}_{P}^{i}=\mathbf{R}^{i}+\mathbf{A}^{i}\bar{\mathbf{u}}^{i}=\mathbf{R}^{i}+\mathbf{A}^{i}\big(\bar{\mathbf{u}}_{o}^{i}+\mathbf{S}^{i}\mathbf{q}_{f}^{i}\big)
|
||||
$$
|
||||
|
||||
in which the global position of point $P$ is written in terms of the generalized reference and elastic coordinates of body $i$ . Therefore, we define the coordinates of body $i$ as
|
||||
in which the global position of point $P$ is written in terms of the generalized reference and elastic coordinates of body $i$ . Therefore, we define the coordinates of body $i$ as
|
||||
其中,点 $P$ 的全局位置用体 $i$ 的广义参考坐标和弹性坐标来表示。因此,我们定义体 $i$ 的坐标为:
|
||||
|
||||
$$
|
||||
\mathbf{q}^{i}=\left[\begin{array}{l}{\mathbf{q}_{r}^{i}}\\ {\mathbf{q}_{f}^{i}}\end{array}\right]
|
||||
@ -6415,29 +6543,38 @@ $$
|
||||
$$
|
||||
|
||||
where $\mathbf{R}^{i}$ and ${\boldsymbol{\Theta}}^{i}$ are the reference coordinates and $\mathbf{q}_{f}^{i}$ is the vector of elastic coordinates. Note that the vector $\bar{\mathbf{u}}_{o}^{i}$ of Eq. 6 can be written as
|
||||
其中,$\mathbf{R}^{i}$ 和 ${\boldsymbol{\Theta}}^{i}$ 分别为参考坐标,$\mathbf{q}_{f}^{i}$ 为弹性坐标矢量。需要注意的是,方程 6 中的矢量 $\bar{\mathbf{u}}_{o}^{i}$ 可以表示为
|
||||
|
||||
$$
|
||||
\bar{\mathbf{u}}_{o}^{i}=\left[x_{1}^{i}\quad x_{2}^{i}\quad x_{3}^{i}\right]^{\mathrm{T}}
|
||||
$$
|
||||
|
||||
where $x_{1}^{i},x_{2}^{i}$ , and $x_{3}^{i}$ are the coordinates of point $P$ , in the undeformed state, defined with respect to the body reference. Equation 6 can then be written as
|
||||
其中,$x_{1}^{i}, x_{2}^{i}$ 和 $x_{3}^{i}$ 分别是点 $P$ 在未变形状态下的坐标,它们是相对于物体的参考系定义的。 那么,方程 6 可以写成:
|
||||
|
||||
|
||||
$$
|
||||
\begin{array}{r}{\bar{\mathbf{u}}^{i}=\left[\begin{array}{l}{x_{1}^{i}}\\ {x_{2}^{i}}\\ {x_{3}^{i}}\end{array}\right]+\left[\begin{array}{l}{\mathbf{S}_{1}^{i}}\\ {\mathbf{S}_{2}^{i}}\\ {\mathbf{S}_{3}^{i}}\end{array}\right]\mathbf{q}_{f}^{i}}\end{array}
|
||||
$$
|
||||
|
||||
where $\mathbf{S}_{k}^{i}$ is the kth row of the body shape function.
|
||||
where $\mathbf{S}_{k}^{i}$ is the kth row of the body shape function.
|
||||
其中,$\mathbf{S}_{k}^{i}$ 为body shape function第 k 行。
|
||||
|
||||
|
||||

|
||||
|
||||
Figure 5.2 Two-dimensional beam.
|
||||
|
||||
assumed to be
|
||||
Example 5.1 The beam shown in Fig. 5.2 has length $l = 0.5 m$. The beam is initially straight, and its axis is parallel to the global $X_1$ axis. (Since we are considering only one beam in this example the superscript $i$ is omitted for simplicity). The origin of the beam reference is assumed to be rigidly attached to point $O^i$ , while the displacement field defined in the body coordinate system is assumed to be
|
||||
例 5.1 图 5.2 中所示的梁长度为 $l = 0.5 m$ 。 梁最初为笔直状态,其轴与全局 $X_1$ 轴平行。(由于本例中仅考虑一根梁,为了简化,省略了上标 $i$)。 梁的参考系原点被假定与点 $O^i$ 刚性连接,而体坐标系中定义的位移场为
|
||||
|
||||
$$
|
||||
\bar{\mathbf{u}}_{f}=\mathbf{S}\mathbf{q}_{f}=\left[\begin{array}{c c}{\bar{u}_{f1}}\\ {\bar{u}_{f2}}\end{array}\right]=\left[\begin{array}{c c}{\xi}&{0}\\ {0}&{3(\xi)^{2}-2(\xi)^{3}}\end{array}\right]\left[\begin{array}{c c}{q_{f1}}\\ {q_{f2}}\end{array}\right]
|
||||
$$
|
||||
|
||||
where $\bar{u}_{f1}$ and $\bar{u}_{f2}$ are the components of the displacement vector at any arbitrary point $x_{1}=x$ , $\bar{\mathbf{q}}_{f}=[q_{f1}\;q_{f2}]^{\mathrm{T}}$ is the vector of elastic coordinates, $\xi$ is a dimensionless quantity defined as $\xi=(x/l)$ , and the body shape function S is defined as
|
||||
其中,$\bar{u}_{f1}$ 和 $\bar{u}_{f2}$ 分别为任意一点 $x_{1}=x$ 处的位移矢量分量,$\bar{\mathbf{q}}_{f}=[q_{f1}\;q_{f2}]^{\mathrm{T}}$ 为弹性坐标矢量,$\xi$ 是一个无量纲量,定义为 $\xi=(x/l)$ ,且体形函数 S 定义为
|
||||
|
||||
|
||||
$$
|
||||
\mathbf{S}=\left[\begin{array}{l l}{\xi}&{0}\\ {0}&{3(\xi)^{2}-2(\xi)^{3}}\end{array}\right]
|
||||
@ -6573,7 +6710,7 @@ $$
|
||||
where $\omega^{i}$ is the angular velocity vector defined in the global, fixed frame of reference. It is clear from Eq. 25 that the central term on the right-hand side of Eq. 19 is a vector that is perpendicular to both $\omega^{i}$ and the vector $\mathbf{u}^{i}$ , which represents the position of $P$ relative to the origin of the body reference. Knowing that
|
||||
|
||||
$$
|
||||
-\,\mathbf{u}^{i}\times\,\mathbf{\omega}\mathbf{\omega}^{i}=-\tilde{\mathbf{u}}^{i}\,\mathbf{\omega}\mathbf{\omega}^{i}=\tilde{\mathbf{u}}^{i}^{\mathrm{T}}\mathbf{\omega}\mathbf{\omega}^{i}
|
||||
-\mathbf{u}^i \times \mathbf{\omega}^i = -\tilde{\mathbf{u}}^i \mathbf{\omega}^i = {\tilde{\mathbf{u}}^i}^T \mathbf{\omega}^i
|
||||
$$
|
||||
|
||||
where $\tilde{\mathbf{u}}^{i^{\mathrm{T}}}$ is the skew symmetric matrix defined as
|
||||
|
@ -1,10 +1,10 @@
|
||||
{
|
||||
"nodes":[
|
||||
{"id":"b698e88ca5fb9c51","type":"text","text":"状态指标:\n推进OKR的时候也要关注这些事情,它们是完成OKR的保障。\n\n\n效率状态 green","x":-96,"y":80,"width":456,"height":347},
|
||||
{"id":"58be7961ae7275a7","type":"text","text":"# 计划\n这周要做的3~5件重要的事情,这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\nP1 多体原理学习 YouTube课程 018\nP1 气动模块联合调试,跑通 no\nP1 稳态工况多体动力学求解方法\nP1 使用python搭建风电机组多体模型 刚性部件主动力、惯性力计算\nP1 柔性部件 叶片、塔架主动力惯性力算法\n\n","x":-620,"y":-307,"width":450,"height":347},
|
||||
{"id":"2b068bfe5df15a72","type":"text","text":"# 目标:多体动力学模块完善\n### 每周盘点一下它们\n\n\n关键结果:建模原理、建模方法掌握 (6.8/10)\n\n关键结果:对标Bladed模块完成 (7.2/10)\n\n关键结果:风机多体动力学文献调研情况完成 (5/10)","x":-96,"y":-307,"width":456,"height":347},
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{"id":"2b068bfe5df15a72","type":"text","text":"# 目标:多体动力学模块完善\n### 每周盘点一下它们\n\n\n关键结果:建模原理、建模方法掌握 (7.5/10)\n\n关键结果:对标Bladed模块完成 (7.2/10)\n\n关键结果:风机多体动力学文献调研情况完成 (5/10)","x":-96,"y":-307,"width":456,"height":347},
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{"id":"01ee5c157d0deeae","type":"text","text":"# 推进计划\n未来四周计划推进的重要事情\n\n文献调研启动\n\n建模重新推导\n\n\n","x":-620,"y":80,"width":456,"height":347},
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{"id":"1ebeabaf5c73ddbb","type":"text","text":"# 本月已完成\n\n多体原理学习 YouTube课程 018","x":-440,"y":520,"width":440,"height":340}
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{"id":"1ebeabaf5c73ddbb","type":"text","text":"# 本月已完成\n\n多体原理学习 YouTube课程 018\n气动模块联合调试,跑通\n使用python搭建风电机组多体模型 刚性部件主动力、惯性力计算 ","x":-440,"y":520,"width":440,"height":340},
|
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{"id":"58be7961ae7275a7","type":"text","text":"# 计划\n这周要做的3~5件重要的事情,这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\nP1 稳态工况多体动力学求解方法 --龙格库塔+ed_caloutput\nP1 柔性部件 叶片、塔架主动力惯性力算法\n- 变形体动力学 简略看看ing\n- 浮动坐标系方法 如何用与梁模型\n\nP1 编写Steady Operational Loads求解器\nP1 yaw 自由度再bug确认 已知原理了\n\n","x":-614,"y":-307,"width":450,"height":347}
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],
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"edges":[]
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}
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多体进展:
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### 多体理论学习
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- 凯恩方法学习 基于刚体。包含广义坐标、广义速率选取、绝对速度对广义速率求偏速度角速度、惯性张量计算,广义主动力与广义惯性力计算。本方法兼有矢量力学与分析力学的特点,可以明显减少计算步骤。缺点是非传统的推导方法不易被初学者习惯,建模过程中必须计算系统内各分体的加速度和惯性力,工作量也很繁重。
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- 对于风电机组模型的刚体:基础、机舱、轮毂,可以直接使用此方法建模。
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- 对于机组模型中的柔性体:塔架、叶片、低速轴等
|
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- 对于塔架和叶片,路线是基于浮动坐标系法,用模态坐标与形函数表示变形,动力学方程建立正在继续研究,推导出广义主动力和广义惯性力,开展后续扩展自由度的相关工作。
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|
||||
### 稳态计算转速、变桨控制算法编写
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### 近期工作
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Reference in New Issue
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