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Testing Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations

Sergey N. Korobeynikov, Alexey Yu. Larichkin, Patrizio Neff

Abstract

We test some Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations (cf., Thiel et al. Int. J. Non-linear Mech. 112: 57--72, 2019) and show that (1) the components of the Cauchy stress tensor for any Cauchy/Green isotropic elastic material under left finite simple shear (LFSS) deformation are equal to the components of the rotated Cauchy stress tensor for the same material under right finite simple shear (RFSS) deformation; (2) for any Hill's linear isotropic hyperelastic material model based on a symmetrically physical (SP) strain measure, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (3) for any two-power Ogden's isotropic hyperelastic material model based on a SP strain function, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (4) for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates under LFSS deformation, the behavior of the Cauchy stress tensor components as a function of the shear parameter is qualitatively similar to that for the same materials under simple shear deformation. In addition, we confirm the results of Lin (Lin R.C. ZAMP, 75: 191, 2024) showing that for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates without initial stresses under RFSS deformation, the Cauchy stress tensor components coincide with those for the Hencky isotropic hyperelastic material.

Testing Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations

Abstract

We test some Hooke-like isotropic hyper-/hypo-elastic material models under finite simple shear deformations (cf., Thiel et al. Int. J. Non-linear Mech. 112: 57--72, 2019) and show that (1) the components of the Cauchy stress tensor for any Cauchy/Green isotropic elastic material under left finite simple shear (LFSS) deformation are equal to the components of the rotated Cauchy stress tensor for the same material under right finite simple shear (RFSS) deformation; (2) for any Hill's linear isotropic hyperelastic material model based on a symmetrically physical (SP) strain measure, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (3) for any two-power Ogden's isotropic hyperelastic material model based on a SP strain function, LFSS and RFSS deformations lead to Eulerian and Lagrangian pure shear stresses, respectively; (4) for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates under LFSS deformation, the behavior of the Cauchy stress tensor components as a function of the shear parameter is qualitatively similar to that for the same materials under simple shear deformation. In addition, we confirm the results of Lin (Lin R.C. ZAMP, 75: 191, 2024) showing that for some Hooke-like isotropic hypoelastic materials with constitutive relations based on corotational stress rates without initial stresses under RFSS deformation, the Cauchy stress tensor components coincide with those for the Hencky isotropic hyperelastic material.
Paper Structure (12 sections, 7 theorems, 157 equations, 12 figures, 7 tables)

This paper contains 12 sections, 7 theorems, 157 equations, 12 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operations on tensors and spectral representation of a symmetric second-order tensor
  4. Local body deformations and basic kinematics
  5. Hooke-like isotropic hyper-/hypo-elastic material models
  6. Kinematics of finite simple shear deformations
  7. Testing some isotropic hyper-/hypo-elastic material models
  8. Testing some HLIH material models based on SP strain functions
  9. One- and two-power Ogden's models
  10. Hooke-like isotropic hypoelastic material models
  11. Conclusions
  12. Constitutive relations for the compressible isotropic hyperelastic Mooney-Rivlin material model

Key Result

Proposition 1

Let $\mathbf{h},\,\mathbf{H}\in \mathcal{T}^2$ be Eulerian and Lagrangian tensors, respectively, which are objective counterparts of each other. Then the Oldroyd tensor rates $\mathbf{h}^{\sharp}$, $\mathbf{h}^{\flat}$ and $\mathbf{H}^{\sharp}$, $\mathbf{H}^{\flat}$ are objective counterparts of eac

Figures (12)

  • Figure 1: Pure shear stress state in the homogeneous deformable sample in the current configuration.
  • Figure 2: Different types of shear deformation: (a) pure shear stretch for infinitesimal strains; (b) simple shear; (c) pure shear stretch for finite strains.
  • Figure 3: Plots of the scale functions $f(\lambda)$ generating the strain tensors given in Table \ref{['t1']} with (a) standard and (b) logarithmic scales for the stretch $\lambda$.
  • Figure 4: LFSS (a) and RFSS (b) deformations of an initially cubic sample with sides of length 1 under plane strain conditions for $\alpha = 0.5$.
  • Figure 5: LFSS (a) and RFSS (b) deformations of an initially cubic sample with sides of length 1 under plane strain conditions for $\alpha = 0.5,\,1.0,\,1.5$
  • ...and 7 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Remark 2
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 5 more