Thermodynamics-based stability criteria for constitutive equations of isotropic hyperelastic solids
Thermodynamics-based stability criteria for constitutive equations of isotropic hyperelastic solids
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基于热力学的各向同性超弹性固体本构方程的稳定性准则
DOI：
10.1016/j.jmps.2018.09.038
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发表时间：
2019
影响因子：
5.3
通讯作者：
Spearot, Douglas
中科院分区：
文献类型：
--
作者：
Upadhyay, Kshitiz;Subhash, Ghatu;Spearot, Douglas
利用连续介质力学的基本平衡定律和场方程，导出了各向同性有限弹性固体的广义热力学稳定性判据，并将其用于建立多项式形式的超弹性本构方程的本构不等式。个人的热力学本构不等式（称为T-C不等式）的neo-Hookean，Mooney Rivlin，和三参数广义Rivlin模型下的三个纯均匀变形模式，即，单轴压缩，单轴拉伸和剪切（简单和纯），并与两个常用的adscientific不等式，Baker-Ericksen（B-E）和E-不等式进行比较。由T-C不等式定义的稳定模型常数的范围由N维坐标空间中的区域表示（N是模型常数的总数），该区域被定义为稳定区域（ROS）。结果表明，ROs是材料变形的函数，随着极限应变的变化而变化，从一个代表热力学稳定性必要条件的初始大区域收缩到一个与E-不等式定义的ROs等价的无穷大极限应变下的收敛区域.通过研究不同变形模式下ROS的演化，讨论了T-C不等式在实验程序的选择、错误实验数据和模型常数的过滤等方面的意义。它还表明，虽然E-不等式是过度限制的超弹性材料与小到中等的极限应变，最近的实验证据支持的观察，B-E不等式是不准确的中等到大的极限应变条件下。所提出的数学框架的适用性，以其他超弹性应变能密度的形式，如指数/对数函数，证明通过调查的Fung-Demiray模型的热力学稳定性。结果表明，通常假设的限制，Fung-Demiray模型常数必须是积极的，可以放松，使一些典型的材料行为下的小到中等的极限应变也可以模拟。
A generalized thermodynamic stability criterion for isotropic finite elastic solids is derived using the fundamental balance laws and field equations of continuum mechanics, which is then used to formulate constitutive inequalities for the polynomial form of hyperelastic constitutive equations. Individual thermodynamic constitutive inequalities (called T-C inequalities) are derived for the neo-Hookean, Mooney Rivlin, and three-parameter generalized Rivlin models under three pure homogeneous deformation modes, namely, uniaxial compression, uniaxial tension and shear (simple and pure), and are compared against two commonly used adscititious inequalities, the Baker-Ericksen (B-E) and E-inequalities. The range of stable model constants as defined by the T-C inequalities is represented by a region in an N-dimensional coordinate space (N is the total number of model constants), which is defined as theRegion of Stability(ROS). It is shown that theROSis a function of material deformation and evolves with the limiting strain, shrinking from an initially large region representing the necessary condition of thermodynamic stability to a converged region under infinite limiting strain that is equivalent to theROSdefined by the E-inequalities. By investigating the evolution of theROSunder different deformation modes, the implication of T-C inequalities on the selection of experimental routines and filtering of erroneous test data and model constants is discussed. It is also demonstrated that while the E-inequalities are over-restrictive for hyperelastic materials with small to moderate limiting strains, an observation supported by recent experimental evidence, the B-E inequalities are inaccurate under moderate to large limiting strain conditions. The applicability of the proposed mathematical framework to other hyperelastic strain energy density forms, such as exponential/logarithmic functions, is demonstrated by investigating the thermodynamic stability of the Fung-Demiray model. It is shown that the commonly assumed restriction that the Fung-Demiray model constants must be positive can be relaxed so that some typical material behaviors under small to moderate limiting strains can also be modeled.
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