初始应变或应力材料的弹性储存能：限制条件与三阶展开
The elastic stored energy of initially strained, or stressed, materials: restrictions and third-order expansions
AI总结：
本研究探讨了初始应变或应力材料的弹性储存能函数需满足的限制条件，以避免能量不守恒等非物理行为，并通过三阶渐近展开证明其与经典弱非线性弹性的一致性。
摘要：
在工程应用和生物领域中广泛遇到的各种材料，其特点是即使在无外部变形或施加力的情况下，也存在非零的内部应力分布。这些初始应力源于初始应变或微观结构变化，如热膨胀、制造过程、体积生长、重塑等。对此类材料进行建模的一种常见本构选择是扩展连续介质力学场论中的经典方法，将初始应力或应变显式地纳入弹性储存能函数中。在此，我们讨论为什么这些能量函数需要满足某些限制条件以避免非物理行为，如能量不守恒，并且我们从经典弹性假设出发推导出所需的限制条件。为了说明其必要性，我们进行了严格的渐近展开，以证明这些依赖于应变和初始应力的储存能函数的限制条件，对于与经典三阶弱非线性弹性的应变能函数的一致性而言是必需的。
英文摘要：
A large variety of materials, widely encountered both in engineering applications and in the biological realm, are characterised by a non-vanishing internal stress distribution, even in the absence of external deformations or applied forces. These initial stresses are due to initial strains or microstructural changes, such as thermal expansion, manufacturing processes, volumetric growth, remodelling, etc. A common constitutive choice for modelling such materials extends the classical approach in the field theory of continuum mechanics to include, explicitly, the initial stress or strain in the elastic stored energy function. Here, we discuss why these energy functions need to satisfy some restrictions to avoid unphysical behaviours, such as non-conservation of energy, {and we derive the required restrictions from the classical assumptions of elasticity.} To illustrate their need, we perform a rigorous asymptotic expansion for proving that these restrictions on stored energy functions that depend on the strain and initial stress are required for consistency with strain energy functions of classical third-order weakly nonlinear elasticity.
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