Soft tissue systems typically do not have a global stress-free configuration, and the configuration taken as the reference normally contains a pre-existing strain field which we refer to as the initial strain. This article presents a constitutive theory for hyperelastic solids with initial strains. The notion of a material metric is introduced. A constitutive description with the material metric describing the natural geometry is developed. Covariant conditions are introduced into the constitutive equation. It is found that the covariant conditions embody the notion of local natural configuration and imply a well-known invariance principle in elasticity. A representation theorem for covariant anisotropic functions is established. This theorem enables the derivation of covariant anisotropic functions using existing Euclidean representation theorems. Examples of covariant constitutive representations for isotropic, transversely isotropic, and general anisotropic materials are presented.
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