Responses of any arbitrary initially stressed reference and the stress-free reference

TL;DR

This work develops three strategies to obtain the constitutive response of any arbitrarily stressed or stress-free reference from the response of a known stressed reference. By reframing the problem as a change of reference configuration, it derives both implicit and Green-elastic formulations, including direct, generalized, and imaginary-reference inversions, and demonstrates their consistency through Treloar rubber data. A generalized energy model based on invariants of the initial stress and fractional powers of the right Cauchy stretch is formulated and validated across multiple references, with a two-parameter fit sufficing to capture the behavior. The study also uncovers universal relations in implicit elasticity and clarifies when reference choice affects the functional form of the constitutive law. Overall, the results provide a practically applicable framework for characterizing materials with initial stresses without destructive testing, linking implicit elasticity to traditional Green-elastic concepts and enabling cross-reference predictions.

Abstract

The constitutive relation for an initially stressed reference is often determined by using the response of a virtual stress-free reference. However, identifying the constitutive relation of the original stress-free body can be challenging without conducting destructive tests. This paper presents three approaches for determining the response of a stress-free reference -- or any arbitrary initially stressed reference -- when the response of a particular initially stressed reference is known. Unlike standard practice, these approaches of changing reference configurations do not begin with a known stress-free state. The first and third approaches directly derive the constitutive relations of one stressed reference from another. {The first approach is applicable to a specific constitutive relation of the known initially stressed state,} while the third approach extends the first and is applicable to any constitutive form. The second approach uses any general response of a given stressed reference to identify the stress-free material. The response of the stress-free material is further analyzed and processed to determine the response of any stressed reference. We observe that even when the known initially stressed state is Green elastic, the arbitrarily stressed or stress-free references may exhibit implicit elasticity. (complete abstract is available in the published version)

Figures (7)

• Figure 1: The configurations required in Section \ref{['given']} and Section \ref{['stress--free']}. Top: the deformation gradient ${\boldsymbol F}_1$ maps a stressed reference $\mathfrak{R}_1$ to the current configuration $\mathfrak{C}$, with Cauchy stress $\boldsymbol{\sigma}$. Bottom: the deformation gradient $\hat{{\boldsymbol F}}$ maps a vector from the stress-free reference$\mathfrak{R}_0$ to the configuration $\mathfrak{R}_1$. This deformation $\hat{{\boldsymbol F}}$ is associated with initial/ residual strain. Note: when Cauchy stress $\boldsymbol {\sigma}=0$ in the current configuration, $\mathfrak{C}$ is equivalent to $\mathfrak{R}_0$. This particular case presents us access with the stress-free reference without destructive testing.
• Figure 2: The constitutive relation for the initially stressed configurations $\mathfrak{R}_1$, with initial stress $\boldsymbol {\tau}_1$, is known. The goal is to find the response from the reference $\mathfrak{R}$ (where the initial stress $\boldsymbol {\tau}$ can be arbitrary).
• Figure 3: The strategy for addressing the solution in Section \ref{['checksec']}. Top: we select $\mathfrak{R}_1$ as the reference, and $\mathfrak{R}$ as the current, where the deformation gradient is ${\boldsymbol F}_1$. Bottom:$\mathfrak{R}$ is chosen as the reference and $\mathfrak{R}_1$ as the current, with deformation gradient ${\boldsymbol F}={\boldsymbol F}_1^{-1}$. The constitutive relation \ref{['consti']} is known for the case at the Top and determined for the case at the Bottom.
• Figure 4: Using the response of the stress-free reference $\mathfrak{R}_0$, derived in Sections \ref{['next']} and \ref{['next-1']}, the constitutive relations for an arbitrarily stressed reference $\mathfrak{R}$ are formulated in Sections \ref{['nextc']} and \ref{['nextc-1']}. The configuration $\mathfrak{R}_1$ is no longer needed in this process. The deformation gradient $\boldsymbol{F}_{tot}$ can be expressed as $\boldsymbol{{\boldsymbol F}\tilde{F}}$ through multiplicative decomposition.
• Figure 5: The general approach of Section \ref{['3rd']}. The initial stress $\boldsymbol {\tau}_1$ in $\mathfrak{R}_1$ is transformed to $\boldsymbol {\tau}_1'={\boldsymbol Q}\boldsymbol {\tau}_1{\boldsymbol Q}^T$ resulting in an imaginary stressed configuration $\mathfrak{R}_1'$, such that the stress $\boldsymbol {\tau}_1'$ becomes coaxial with $\boldsymbol {\tau}$ in $\mathfrak{R}$. Due to this co-axiality and given that the stress-free material is isotropic, the deformation from $\mathfrak{R}_1'$ to $\mathfrak{R}$ must be a pure stretch ${\boldsymbol V}_1$, coaxial with both $\boldsymbol {\tau}_1'$ and $\boldsymbol {\tau}$. This pure stretch has six independent components determined by inverting the associated constitutive relation. Since $\mathfrak{R}_1'$ represents an imaginary configuration that cannot and need not be physically accessed, the stress-field $\boldsymbol {\tau}_1'$ is not required to be self-balancing. Only the kinematic relations and the constitutive relation are required in the present context.