Anti-plane Shear in Hyperelasticity

Abstract

We reconsider anti-plane shear deformations of the form φ(x) = (x ₁ , x ₂ , x ₃ + u(x ₁, x ₂)) based on prior work of Knowles and relate the existence of anti-plane shear deformations to fundamental constitutive concepts of elasticity theory like polyconvexity, rank-one convexity, and tension-compression symmetry. In addition, we provide finite element simulations to visualize our theoretical findings.

Appendix

Recall that in the isotropic case, the Cauchy-stress tensor can always be expressed in the form

(36)

with scalar-valued functions β _i depending on the invariants of B. In the hyperelastic isotropic case, β ₀, β ₁, and β ₋₁ are given by

$$\displaystyle \begin{aligned} \beta_0=\frac{2}{\sqrt{I_3}}\left(I_2\hspace{0.07em}\frac{\partial W}{\partial I_2}+I_3\hspace{0.07em}\frac{\partial W}{\partial I_3}\right),\quad \beta_1=\frac{2}{\sqrt{I_3}}\hspace{0.07em}\frac{\partial W}{\partial I_1},\quad \beta_{-1}=-2\sqrt{I_3}\hspace{0.07em}\frac{\partial W}{\partial I_2}.{} \end{aligned} $$

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Lemma 8

Let $\varphi \colon \Omega \to \mathbb {R}$ , φ(x) = (x ₁ + γ x ₂, x ₂, x ₃) be a simple shear deformation, with $\gamma \in \mathbb {R}$ denoting the amount of shear. Then the Cauchy shear stress σ ₁₂ of an arbitrary isotropic energy function W(I ₁, I ₂, I ₃) is monotone as a scalar-valued function depending on the amount of shear for positive γ if and only if W is APS-convex.

Proof

We consider the Cauchy-stress tensor for an arbitrary material which is stress-free in the reference configuration:

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In the case of simple shear we compute [6, p.41]

(39)

(40)

Therefore, the Cauchy shear stress component σ ₁₂ is a scalar-valued function depending on the amount of shear γ, given by

(41)

The positivity of the Cauchy shear stress is already implied by the (weak) empirical inequalities β ₁ > 0 , β ₋₁ ≤ 0. The condition for shear-monotonicity is given by

(42)

which is equivalent to APS-convexity condition (APS2) of the energy function W(I ₁, I ₂, I ₃) . □

Remark 11

The empirical inequalities (27) state that β ₀ ≤ 0 , β ₁ > 0 , β ₋₁ ≤ 0. In the case of APS-deformations (I ₁ = I ₂ = 3 + γ ² , I ₃ = 1), Pucci et al. [38, eq.(4.3)] obtain the inequality

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“where p, q are real numbers such that p > 0 and q ≠ 0”, by a “simple manipulation of the empirical inequalities (27) and [the stress-free reference configuration]”. In [38, Remark III], it is pointed out correctly that in the case of $p=1\,,q^2=\frac {1}{2}$ (they erroneously use q = 1) the resulting constitutive inequality

$$\displaystyle \begin{aligned} 0&<2\left((I_1-3)\hspace{0.07em}(h^*)'(3+\gamma^2)+\frac{1}{2}\hspace{0.07em}h^*(3+\gamma^2)\right)\\ &=(h^*)'(3+\gamma^2)\cdot 2\hspace{0.07em}\gamma^2+h^*(3+\gamma^2)=\frac{\mathrm{d}}{\mathrm{d}\gamma}\left[\gamma\hspace{0.07em}h^*(3+\gamma^2)\right] \end{aligned} $$

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is equivalent to APS-convexity by Eq. (APS3) with

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We are, however, not able to reproduce a proof of inequality (43), see also the counterexample in Remark 9.

Lemma 9

Let W be a sufficiently smooth isotropic energy function such that the induced Cauchy-stress response satisfies the (weak) empirical inequalities. Then for sufficiently small shear deformations (i.e., within a neighborhood of the identity ), the Cauchy shear stress is a monotone function of the amount of shear.

Proof

In Lemma 8, we already computed the Cauchy shear stress corresponding to a simple shear to be σ ₁₂(γ) = (β ₁ − β ₋₁) γ, with $\gamma \in \mathbb {R}$ denoting the amount of shear. The monotonicity of this map** is equivalent to

$$\displaystyle \begin{aligned} 0&<\frac{\mathrm{d}}{\mathrm{d}\gamma}\sigma_{12}(\gamma)=\frac{\mathrm{d}}{\mathrm{d}\gamma}\left[(\beta_1(3+\gamma^2)-\beta_{-1}(3+\gamma^2))\hspace{0.07em}\gamma\right]\\ &=\left(\beta_1^{\prime}(3+\gamma^2)-\beta_{-1}^{\prime}(3+\gamma^2)\right)2\hspace{0.07em}\gamma^2+\beta_1(3+\gamma^2)-\beta_{-1}(3+\gamma^2)\,. \end{aligned} $$

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According to the (weak) empirical inequalities, β ₁(3) − β ₋₁(3)=: μ > 0. Therefore, β ₁(3 + γ ²) − β ₋₁(3 + γ ²) ≥ ε > 0 for sufficiently small $\gamma \in \mathbb {R}$. If W and thus β ₁, β ₋₁ are sufficiently smooth, then $\beta _1^{\prime }-\beta _2^{\prime }$ is locally Lipschitz-continuous, and thus within a compact neighborhood of ,

for every sufficiently small shear deformation, i.e., sufficiently small γ. □