# 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.

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## Notes

1. 1.

Possible boundary conditions are Dirichlet or Neumann boundary conditions, which permit an APS-deformation of the surface $$\partial \varOmega$$ of $$\varOmega$$. Here, we restrict our attention to Dirichlet boundary condition for simplicity of exposition.

2. 2.

For the homogeneous deformation $$u(x_1,x_2)=c_1\,x_1+c_2\,x_2+c_3$$ with constants $$c_1,c_2,c_3\in \mathbb {R},$$ follows directly from the linearity of u that $$\alpha =u_{,x_1}=c_1$$ and $$\beta =u_{,x_2}=c_2.$$ This implies $$I_1=I_2=3+\alpha ^2+\beta ^2=\text {const.}$$, which shows that $$G(I_1,I_2),H(I_1,I_2),p(I_1,I_2),q(I_1,I_2)=$$ const. Thus, all three Euler–Lagrange equations are trivially fulfilled.

3. 3.

Convexity is clearly not necessary for the existence of a minimizer; see, e.g., [10]; however it will turn out later that this convexity condition is not a particularly limiting property for most elastic energy functions.

4. 4.

For the necessity of (K1), see Knowles [1, eq.(3.22)].

5. 5.

With the notation from (9), we can restate (K1) as $$b\,H(I_1,I_2)=G(I_1,I_2)$$ with constant b. Therefore, the relationship $$\mathrm{div}(H\,\nabla u)=0$$ together with $$b\,H(I_1,I_2)=G(I_1,I_2)$$ yields $$\mathrm{div}(G\,\nabla u)=0.$$

6. 6.

Note again that $$I_1=I_2=3+\gamma ^2=3+\Vert \nabla u\Vert ^2$$.

7. 7.

For detailed calculations, see [20].

8. 8.

Gao [34]: “ [...] the equilibrium equation [...] has just one non-trivial component [namely equation (III)].” Gao claims that Knowles’ condition (K1) is automatically satisfied for every elastic energy function with $$b=0$$, which is clearly not the case (Table 1).

9. 9.

For APS-deformations, $$I_1=I_2=3+\Vert \nabla u\Vert ^2$$ and $$I_3=1$$.

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## Acknowledgements

We thank Giuseppe Saccomandi (University of Perugia) and Roger Fosdick (University of Minnesota) for helpful discussions.

## Author information

Authors

### Corresponding author

Correspondence to Jendrik Voss.

## Appendix A

### 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 pq 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 $$\gamma$$. $$\square$$

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Voss, J., Baaser, H., Martin, R.J. et al. More on Anti-plane Shear. J Optim Theory Appl 184, 226–249 (2020). https://doi.org/10.1007/s10957-018-1358-6

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### Keywords

• Isotropic nonlinear elasticity
• Constitutive inequalities
• Convexity
• Constitutive law
• Anti-plane shear deformations
• Ellipticity
• Empirical inequalities

• 74B20
• 74A20
• 74A10