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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.

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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

Mathematics Subject Classification

  • 74B20
  • 74A20
  • 74A10