Bifurcation behavior of compressible elastic half-space under plane deformations

• Published in: International journal of non-linear mechanics Vol. 126; p. 103553
• Main Authors: Bakiler, A. Derya, Javili, Ali
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.11.2020; Elsevier BV
• Subjects: Bifurcation of half-space; Bifurcations; Compressibility; Compressible elasticity; Compressive properties; Computational fluid dynamics; Eigenvalues; Elastic deformation; Elastic half spaces; Elastic limit; Elastic recovery; Exact solutions; Finite element method; Incompressibility; Instabilities; Non-linear Poisson’s ratio; Poisson's ratio; Stability analysis; Bifurcation of half-space; Compressible elasticity; Instabilities; Non-linear Poisson’s ratio
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2020.103553

A finitely deformed elastic half-space subject to compressive stresses will experience a geometric instability at a critical level and exhibit bifurcation. While the bifurcation of an incompressible elastic half-space is commonly studied, the bifurcation behavior of a compressible elastic half-space remains elusive and poorly understood to date. The main objective of this manuscript is to study the bifurcation of a neo-Hookean compressible elastic half-space against the well-established incompressible case. The formulation of the problem requires a novel description for a non-linear Poisson’s ratio, since the commonly accepted definitions prove insufficient for the current analysis. To investigate the stability of the domain and the possibility of bifurcation, an incremental analysis is carried out. The incremental analysis describes a small departure from an equilibrium configuration at a finite deformation. It is shown that at the incompressibility limit, our results obtained for a compressible elastic half-space recover their incompressible counterparts. Another key feature of this contribution is that we verify the analytical solution of this problem with computational simulations using the finite element method via an eigenvalue analysis. The main outcome of this work is an analytical expression for the critical stretch where bifurcation arises. We demonstrate the utility of our model and its excellent agreement with the numerical results ranging from fully compressible to incompressible elasticity. Moving forward, this approach can be used to comprehend and harness the instabilities in bilayer systems, particularly for compressible ones. •A new class of nonlinear Poisson’s ratio is introduced, simplifying the derivations.•Our Poisson’s ratio matches Hencky’s only at compressible and incompressible limits.•Our solution for critical stretch is verified by FEM enhanced by eigenvalue analysis.•At the fully compressible limit, critical stretch λcr is found to be 0.4859.•At the incompressible limit λcr=0.5437, which is in agreement with the literature.•An explicit estimate λcr=0.0467ν2+0.1045ν+0.4859 is provided, relating critical stretch λcr to Poisson’s ratio ν.