It is a long-standing challenge to predict the thermo-mechanically coupled behaviors of initially stressed soft elastomers since most of the existing theories ignore the influences of thermoelastic deformation histories. The constitutive equations may be completely different even for the same initial stresses, if the latter is originated from isothermal and adiabatic deformations, respectively. In this paper, we establish a general framework for deriving constitutive equations for soft elastomers with arbitrary initial states. Instead of using the virtual stress-free configuration, we define the natural state by imposing the stress-free condition and the natural temperature condition. The derivations are based on a new proposed intrinsic embedding method of initial states, in which an additive decomposition of material strains is employed and the material coordinates can be properly defined. Once the natural-state-based free energy density and internal constraint are specified, the required constitutive equations can be accordingly obtained. We then derive the explicit formulations of the Cauchy stress and the entropy by linearization. On this basis, the embedding of initial states in Saint Venant–Kirchhoff, Blatz–Ko, Mooney–Rivlin, Neo-Hookean, Gent, and exponential form elastomers are detailed discussed. The influences brought by the initial stresses, the initial temperature, and the internal constraint on the elastic coefficients are analyzed separately. The new proposed constitutive equations show quantitative agreement with the classical theories under isothermal circumstances and fill a theoretical blank in this field under non-isothermal circumstances. Our approaches significantly improve the current constitutive theory of soft materials and may shed some light on the theoretical modeling of multi-field coupling problems.

由于大多数现有理论都忽略了热弹性变形历史的影响，因此预测初始应力软弹性体的热机械耦合行为是一个长期存在的挑战。即使对于相同的初始应力，本构方程也可能完全不同，如果后者分别源自等温变形和绝热变形。在本文中，我们建立了一个通用框架，用于推导具有任意初始状态的软弹性体的本构方程。我们没有使用虚拟无应力配置，而是通过施加无应力条件和自然温度条件来定义自然状态。推导基于一种新提出的初始状态的内在嵌入方法，其中采用了材料应变的加性分解，并且可以正确定义材料坐标。一旦指定了基于自然状态的自由能密度和内部约束，就可以相应地获得所需的本构方程。然后，我们通过线性化推导出柯西应力和熵的显式公式。在此基础上，详细讨论了 Saint Venant-Kirchhoff、Blatz-Ko、Mooney-Rivlin、Neo-Hookean、Gent 和指数形式弹性体中初始态的嵌入。分别分析了初始应力、初始温度和内约束对弹性系数的影响。新提出的本构方程在等温条件下与经典理论在定量上吻合，填补了非等温条件下该领域的理论空白。我们的方法显着改进了当前软材料的本构理论，并可能为多场耦合问题的理论建模提供一些启示。