More on Anti-plane Shear

Abstract

We reconsider anti-plane shear deformations 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 A

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

$$\begin{aligned} \sigma =\beta _0\,{\mathbb {1}}+\beta _1\,B+\beta _{-1}\,B^{-1} \end{aligned}$$

(33)

with scalar-valued functions $\beta _i$ depending on the invariants of B. In the hyperelastic isotropic case, $\beta _0$, $\beta _1$ and $\beta _{-1}$ are given by

$$\begin{aligned} \beta _0=\frac{2}{\sqrt{I_3}}\left( I_2\,\frac{\partial W}{\partial I_2}+I_3\,\frac{\partial W}{\partial I_3}\right) ,\qquad \beta _1=\frac{2}{\sqrt{I_3}}\,\frac{\partial W}{\partial I_1},\qquad \beta _{-1}=-2\sqrt{I_3}\,\frac{\partial W}{\partial I_2}. \end{aligned}$$

(34)

Lemma A.1

Let $\varphi {:}\varOmega \rightarrow \mathbb {R}$, $\varphi (x)=(x_1+\gamma \,x_2,x_2,x_3)$ be a simple shear deformation, with $\gamma \in \mathbb {R}$ denoting the amount of shear. Then the Cauchy shear stress $\sigma _{12}$ 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 $\gamma$if and only ifW is APS-convex.

Proof

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

$$\begin{aligned} \sigma =\beta _0\,{\mathbb {1}}+\beta _1\,B+\beta _{-1}\,B^{-1}. \end{aligned}$$

(35)

In the case of simple shear we compute [36, p.41]

$$\begin{aligned} \nabla \varphi&=\left( \begin{matrix}1&{}\quad \gamma &{}\quad 0\\ 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\end{matrix}\right) ,\quad B=FF^T=\left( \begin{matrix}{ll} 1+\gamma ^2&{}\quad \gamma &{}\quad 0\\ \gamma &{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{matrix}\right) ,\nonumber \\&B^{-1}=\left( \begin{matrix}1&{}\quad -\gamma &{}\quad 0\\ -\gamma &{}\quad 1+\gamma ^2&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\end{matrix}\right) ,\nonumber \\ I_1&={{\mathrm{tr}}}B=3+\gamma ^2,\quad I_2={{\mathrm{tr}}}({{\mathrm{Cof}}}B)={{\mathrm{tr}}}\left( \begin{matrix}1&{}\quad -\gamma &{}\quad 0\\ -\gamma &{}\quad 1+\gamma ^2&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\end{matrix}\right) =3+\gamma ^2,\nonumber \\&\quad I_3=\det B=1,\nonumber \\ \implies \,\sigma&=(\beta _0+\beta _1+\beta _{-1})\,{\mathbb {1}}+\left( \begin{matrix}\beta _1\,\gamma ^2&{}\quad (\beta _1-\beta _{-1})\,\gamma &{}\quad 0\\ (\beta _1-\beta _{-1})\,\gamma &{}\quad \beta _{-1}\,\gamma ^2&{}\quad 0\\ 0&{}\quad 0&{}\quad 0\end{matrix}\right) . \end{aligned}$$

(36)

Therefore, the Cauchy shear stress component $\sigma _{12}$ is a scalar-valued function depending on the amount of shear $\gamma$, given by

$$\begin{aligned} \sigma _{12}(\gamma )&=(\beta _1-\beta _{-1})\,\gamma =\gamma \left. \frac{2}{\sqrt{I_3}}\left( \frac{\partial W}{\partial I_1}+I_3\,\frac{\partial W}{\partial I_2}\right) \right| _{I_1=I_2=3+\gamma ^2,I_3=1} \end{aligned}$$

(37)

$$\begin{aligned}&=2\gamma \left. \left( \frac{\partial W}{\partial I_1}+\frac{\partial W}{\partial I_2}\right) \right| _{I_1=I_2=3+\gamma ^2,I_3=1}=\frac{d}{d\gamma }\,W\left( 3+\gamma ^2,3+\gamma ^2,1\right) . \end{aligned}$$

(38)

The positivity of the Cauchy shear stress is already implied by the (weak) empirical inequalities $\beta _1>0,\;\beta _{-1}\le 0$. The condition for shear-monotonicity is given by

$$\begin{aligned} \frac{d}{d\gamma }\,\sigma _{12}(\gamma )=\frac{d^2}{(d\gamma )^2}\,W\left( 3+\gamma ^2,3+\gamma ^2,1\right) >0\qquad \qquad \forall \,\gamma \ge 0, \end{aligned}$$

(39)

which is equivalent to APS-convexity condition (APS2) of the energy function $W(I_1,I_2,I_3).$$\square$

Remark A.1

The empirical inequalities (24) state that $\beta _0\le 0,\;\beta _1>0,\;\beta _{-1}\le 0$. In the case of APS-deformations ($I_1=I_2=3+\gamma ^2,I_3=1$), Pucci et al. [13, eq.(4.3)] obtain the inequality

$$\begin{aligned} (I_1-3)^p(h^*)'+q^2h^*>0,\qquad \forall \,I_1\ge 3\qquad \text {with }\; h^*(I_1)=\left. \beta _1-\beta _{-1}\right| _{I_1=I_2,I_3=1},\nonumber \\ \end{aligned}$$

(40)

“where p, q are real numbers such that $p>0$ and $q\ne 0$”, by a “simple manipulation of the empirical inequalities [(24)]” and [the stress-free reference configuration]. In [13, 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

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

(41)

is equivalent to APS-convexity by equation (APS3) with

$$\begin{aligned} \left. \left( \frac{\partial W}{\partial I_1}+\frac{\partial W}{\partial I_2}\right) \right| _{I_1=I_2=3+\gamma ^2,I_3=1}\overset{{}(34)}{=}\left. \beta _1-\beta _{-1}\right| _{I_1=I_2=3+\gamma ^2,I_3=1}=h^*(3+\gamma ^2). \end{aligned}$$

(42)

We are, however, not able to reproduce a proof of inequality (40), see also the counterexample in Remark 4.3.

Lemma A.2

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 ${\mathbb {1}}$), the Cauchy shear stress is a monotone function of the amount of shear.

Proof

In Lemma A.1, we already computed the Cauchy shear stress corresponding to a simple shear to be $\sigma _{12}(\gamma )=(\beta _1-\beta _{-1})\,\gamma$, with $\gamma \in \mathbb {R}$ denoting the amount of shear. The monotonicity of this mapping is equivalent to

$$\begin{aligned} 0&<\frac{d}{d\gamma }\sigma _{12}(\gamma )=\frac{d}{d\gamma }\left[ (\beta _1(3+\gamma ^2)-\beta _{-1}(3+\gamma ^2))\,\gamma \right] \nonumber \\&=\left( \beta _1'(3+\gamma ^2)-\beta _{-1}'(3+\gamma ^2)\right) 2\,\gamma ^2+\beta _1(3+\gamma ^2)-\beta _{-1}(3+\gamma ^2). \end{aligned}$$

(43)

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

$$\begin{aligned} \frac{d}{d\gamma }\sigma _{12}(\gamma ) {=} \underbrace{\beta _1(3+\gamma ^2)-\beta _{-1}(3+\gamma ^2)}_{\ge \varepsilon } \,+\, \underbrace{\left( \beta _1'(3+\gamma ^2)-\beta _{-1}'(3+\gamma ^2)\right) }_{\le \;\text {const.}} \,\cdot \,2\,\gamma ^2{>}0 \end{aligned}$$

for every sufficiently small shear deformation, i.e., sufficiently small $\gamma$. $\square$