连续介质力学中某些物理量的近似和大变形弹塑性定义的比较
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摘要
连续介质力学中,其物理量通常表示为张量函数,因此张量函数及其导数的研究在连续介质力学和计算力学等领域中是一个非常重要的问题。张量函数一般可以表示为两种形式,即主轴表示和抽象表示。由于抽象表示脱离了坐标系,使推导过程清晰、表达整齐统一,因而得到了众多力学家的重视。但是许多张量函数及其导数的表示不便在工程中直接应用,因此能够适用于工程计算的张量函数及其导数的近似表达得到了人们的关注。
随着工业技术的进步,人们对预估材料力学响应的精度要求越来越高,需要发展更有效、更符合实际的材料本构关系。因此出现了不同的弹塑性大变形本构定义,而面临众多的大变形弹塑性定义,定义之间的差别是人们比较关心的问题。
本文对张量函数及其导数的近似表达式以及三种大变形弹塑性定义的差别进行研究,主要工作和取得的进展如下：
1)对于连续介质力学中常见的三类张量函数,即开方、对数和指数张量函数进行泰勒展开,对其余项进行误差分析,得到误差最小的展开点。
2)利用上面的结论,推导出右伸长张量U、转动张量R和Hencky对数应变H以及指数函数的近似表示。该近似表示不但形式简洁,精确度高,而且计算速度远远快于精确表达式。
3)给出右伸长张量U、Hencky对数应变H以及指数函数关于右Cauchy-Green应变张量C的导数的近似表示,该导数的近似表示依然具有表达简洁,精确度高、计算速度快的优点,而且无需考虑自变量张量的特征值相等与否。
4)将Simo-Ortiz定义、Moran-Ortiz-Shih定义与小变形弹塑性推广得到的大变形定义进行比较,利用张量函数的相关知识,通过一个简单剪切问题,给出不同定义之间的数量级关系。
The physical quantity in continuum mechanics is usually expressed as a tensor function. So the research of the tensor functions and their derivatives is a very important issue in continuum mechanics and computational mechanics. The principal axis representations and abstract representations are two kinds of representations of tensor functions. An important character of the abstract representations is coordinate-free, which makes the derivations clear and formulations concise. Therefore, it has been the interest of many scientists working in the fields of theroretical and applied mechanics. But the representation of many tensor functions and their derivatives can't be applied in engineering directly, so the approximate expressions for tensor functions and their derivatives which can be applied in engineering calculations get more and more attentions.
With the development of industrial technology, more precise estimates of mechanical responses of materials are required and more effective and realistic constitutive relations are needed. Then, many different elasto-plastic large deformation constitutive definitions appeared. Facing so many elasto-plastic large deformation definitions, the differences of different definitions are widely concerned.
The approximate expressions of tensor functions and their derivatives and the difference of three elasto-plastic large deformation definitions are studied in this paper. The main work and achievements are as follows
1) The isotropic expression of square root tensors, logarithmic strain tensors and exponential tensor (which are three commonly met tensors in continuum mechanics) are got by Taylor series expanding, the reminders of expansions are analysed and the expanding points with the minimum errors are got.
2) Using the conclusions drawn above, the approximate expressions of right stretch tensor U, rotation tensor R, Hencky logarithmic strain tensor H and exponential tensor are deduced. These approximate expressions not only have simple representations and high precision, but also calculate much faster than exact expressions.
3) The approximate expressions of derivatives of right stretch tensor U, Hencky logarithmic strain tensor H and exponential tensor on right Cauchy-Green strain tensor are deduced. These approximate expressions also have simple representations, high precisions and faster calculating rates. And do not need to consider the eigenvalues of independent variable tensors weather equal or not.
4) Comparing the Simo-Ortiz definition, Moran-Ortiz-Shih definition and the large deformation definition generalized from small elasto-plastic deformation, the quantitative relationships of the three definitions are given by applying tensor function to a simple shear deformation.
随着工业技术的进步,人们对预估材料力学响应的精度要求越来越高,需要发展更有效、更符合实际的材料本构关系。因此出现了不同的弹塑性大变形本构定义,而面临众多的大变形弹塑性定义,定义之间的差别是人们比较关心的问题。
本文对张量函数及其导数的近似表达式以及三种大变形弹塑性定义的差别进行研究,主要工作和取得的进展如下：
1)对于连续介质力学中常见的三类张量函数,即开方、对数和指数张量函数进行泰勒展开,对其余项进行误差分析,得到误差最小的展开点。
2)利用上面的结论,推导出右伸长张量U、转动张量R和Hencky对数应变H以及指数函数的近似表示。该近似表示不但形式简洁,精确度高,而且计算速度远远快于精确表达式。
3)给出右伸长张量U、Hencky对数应变H以及指数函数关于右Cauchy-Green应变张量C的导数的近似表示,该导数的近似表示依然具有表达简洁,精确度高、计算速度快的优点,而且无需考虑自变量张量的特征值相等与否。
4)将Simo-Ortiz定义、Moran-Ortiz-Shih定义与小变形弹塑性推广得到的大变形定义进行比较,利用张量函数的相关知识,通过一个简单剪切问题,给出不同定义之间的数量级关系。
The physical quantity in continuum mechanics is usually expressed as a tensor function. So the research of the tensor functions and their derivatives is a very important issue in continuum mechanics and computational mechanics. The principal axis representations and abstract representations are two kinds of representations of tensor functions. An important character of the abstract representations is coordinate-free, which makes the derivations clear and formulations concise. Therefore, it has been the interest of many scientists working in the fields of theroretical and applied mechanics. But the representation of many tensor functions and their derivatives can't be applied in engineering directly, so the approximate expressions for tensor functions and their derivatives which can be applied in engineering calculations get more and more attentions.
With the development of industrial technology, more precise estimates of mechanical responses of materials are required and more effective and realistic constitutive relations are needed. Then, many different elasto-plastic large deformation constitutive definitions appeared. Facing so many elasto-plastic large deformation definitions, the differences of different definitions are widely concerned.
The approximate expressions of tensor functions and their derivatives and the difference of three elasto-plastic large deformation definitions are studied in this paper. The main work and achievements are as follows
1) The isotropic expression of square root tensors, logarithmic strain tensors and exponential tensor (which are three commonly met tensors in continuum mechanics) are got by Taylor series expanding, the reminders of expansions are analysed and the expanding points with the minimum errors are got.
2) Using the conclusions drawn above, the approximate expressions of right stretch tensor U, rotation tensor R, Hencky logarithmic strain tensor H and exponential tensor are deduced. These approximate expressions not only have simple representations and high precision, but also calculate much faster than exact expressions.
3) The approximate expressions of derivatives of right stretch tensor U, Hencky logarithmic strain tensor H and exponential tensor on right Cauchy-Green strain tensor are deduced. These approximate expressions also have simple representations, high precisions and faster calculating rates. And do not need to consider the eigenvalues of independent variable tensors weather equal or not.
4) Comparing the Simo-Ortiz definition, Moran-Ortiz-Shih definition and the large deformation definition generalized from small elasto-plastic deformation, the quantitative relationships of the three definitions are given by applying tensor function to a simple shear deformation.
引文
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