Modelling of initially stressed solids: structure of the energy density in the incompressible limit
arxiv(2024)
摘要
This study addresses the modelling of elastic bodies, particularly when the
relaxed configuration is unknown or non-existent. We adopt the theory of
initially stressed materials, incorporating the deformation gradient and stress
state of the reference configuration (initial stress tensor) into the response
function. We show that for the theory to be applicable, the response function
of the relaxed material is invertible up to an element of the material symmetry
group. Additionally, we establish that commonly imposed constitutive
restrictions, namely the initial stress compatibility condition and initial
stress reference independence, naturally arise when assuming an initial stress
generated solely from elastic distortion. The paper delves into modelling
aspects concerning incompressible materials, showcasing the expressibility of
strain energy density as a function of the deviatoric part of the initial
stress tensor and the isochoric part of the deformation gradient. This not only
reduces the number of independent invariants in the energy functional, but also
enhances numerical robustness in finite element simulations. The findings of
this research hold significant implications for modelling materials with
initial stress, extending potential applications to areas such as
mechanobiology, soft robotics, and 4D printing.
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