Thermo-mechanically coupled constitutive equations for soft elastomers with arbitrary initial states

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.