# Again anti-plane shear

@article{Voss2018AgainAS, title={Again anti-plane shear}, author={Jendrik Voss and Herbert Baaser and Robert J. Martin and Patrizio Neff}, journal={arXiv: Analysis of PDEs}, year={2018} }

We reconsider anti-plane shear deformations of the form $\varphi(x)=(x_1,\,x_2,\,x_3+u(x_1,x_2))$ 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.

## 3 Citations

Global Bifurcation of Anti-plane Shear Fronts

- Mathematics, Computer ScienceJ. Nonlinear Sci.
- 2021

For a class of generalized neo-Hookean strain energy densities and live body forces, this work constructs unbounded curves of front-type solutions using global bifurcation theory and finds some of these curves contain solutions with deformations of arbitrarily large magnitude.

A P ] 1 2 O ct 2 01 8 Shear , pure and simple

- 2018

In a 2012 article in the International Journal of Non-Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell’s so-called empirical inequalities, the…

## References

SHOWING 1-10 OF 20 REFERENCES

Remarks on Analytic Solutions and Ellipticity in Anti-plane Shear Problems of Nonlinear Elasticity

- Physics
- 2017

This paper revisits a well-studied anti-plane shear deformation problem formulated by Knowles in 1976. It shows that a homogenous hyper-elasticity for general anti-plane shear deformation must be…

Injectivity of the Cauchy-stress tensor along rank-one connected lines under strict rank-one convexity condition

- Mathematics
- 2016

In this note, we show that the Cauchy stress tensor σ$\sigma$ in nonlinear elasticity is injective along rank-one connected lines provided that the constitutive law is strictly rank-one convex. This…

Duality in G. Saccomandi’s Challenge on Analytical Solutions to Anti-plane Shear Problem in Finite Elasticity

- Physics, Mathematics
- 2015

This note is a response to G. Saccomandi’s recent challenge by showing basic mistakes in his conclusions. The proof is elementary, but leads to some fundamental results in correctly understanding an…

Geometry of Logarithmic Strain Measures in Solid Mechanics

- Mathematics, Physics
- 2015

AbstractWe consider the two logarithmic strain measures
$$\begin{array}{ll} {\omega_{\mathrm{iso}}} = ||{{\mathrm dev}_n {\mathrm log} U} || = ||{{\mathrm dev}_n {\mathrm log} \sqrt{F^TF}}|| \quad…

On the determination of semi-inverse solutions of nonlinear Cauchy elasticity: The not so simple case of anti-plane shear

- Mathematics
- 2015

Remarks on Analytic Solutions in Nonlinear Elasticity and Anti-Plane Shear Problem

- Mathematics, Physics
- 2015

This paper revisits a well-studied anti-plane shear deformation problem formulated by Knowles in 1976 and analytical solutions in general nonlinear elasticity proposed by Gao since 1998. Based on…

Analytical solutions to general anti-plane shear problems in finite elasticity

- Mathematics
- 2014

This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical duality–triality theory developed by the…

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

- Mathematics, Physics
- 2014

We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$\begin{aligned} F\mapsto W_{\mathrm{eH}}(F):=\widehat{W}_{\mathrm{eH}}(U):=\left \{ \begin{array}{l@{\quad}l}…

The Anti-Plane Shear Problem in Nonlinear Elasticity Revisited

- Mathematics
- 2013

A classical problem in the framework of nonlinear elasticity theory is the characterization of the materials that may sustain a pure state of anti-plane shear in the absence of body forces. This…

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility.

- Materials Science, MedicinePhysics in medicine and biology
- 2009

The feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound using ultrasound is established and it is conjectured that these properties may be used to discern malignant tumors from benign tumors.