Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli
Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli
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DOI：
10.1103/physrevb.104.214105
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发表时间：
2021-05
影响因子：
3.7
通讯作者：
V. Levitas
中科院分区：
文献类型：
--
作者：
V. Levitas
提出了预应力单晶弹性的一般非线性理论。定义了各种类型的弹性模量，确定了它们的重要性，并给出了它们之间的关系。特别是，B 模量存在于柯西应力的 Jaumann 目标时间导数与变形率之间的关系中，并广泛用于各种有限元代码的计算算法中。概述并说明了超位错的复杂非线性弹性问题的简化线性解决方案的可能应用。充分考虑和分析了有限旋转的影响。定义了不同约束下不同类型的体积模量和剪切模量，并将其与多晶聚集体的有效性能联系起来。详细推导了相对于预应力结构的小变形的弹性能和应力-应变关系的表达式。在初始静水载荷下，根据从单晶或多晶获得的静水载荷下广义张量状态方程的存在性，推导出弹性模量和柔量的一般一致性条件。结果表明，B模量可以从吉布斯能量的表达式中找到。然而，由吉布斯能量定义的高阶弹性模量没有任何意义，因为它们不直接参与任何已知的方程，如应力应变关系和波传播方程。 B的偏投影也可以从等容小应变增量的弹性能表达式中找到，并且可以从一致性条件中找到B的缺失分量。对已知著作中的许多不一致和错误进行了分析。
A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined, and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect of finite rotations is fully taken into account and analyzed. Different types of the bulk and shear moduli under different constraints are defined and connected to the effective properties of polycrystalline aggregates. Expressions for elastic energy and stress-strain relationships for small distortions with respect to a pre-stressed configuration are derived in detail. Under initial hydrostatic load, general consistency conditions for elastic moduli and compliances are derived that follow from the existence of the generalized tensorial equation of state under hydrostatic loading obtained from single crystal or polycrystal. It is shown that B moduli can be found from the expression for the Gibbs energy. However, higher-order elastic moduli defined from the Gibbs energy do not have any meaning since they do not directly participate in any known equations, like stress-strain relationships and wave propagation equation. The deviatoric projection of B can also be found from the expression for the elastic energy for isochoric small strain increments, and the missing components of B can be found from the consistency conditions. Numerous inconsistencies and errors in the known works are analyzed.