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.

受到压缩应力的有限变形的弹性半空间将在临界水平经历几何不稳定，并出现分叉。虽然通常研究不可压缩的弹性半空间的分叉，但是迄今为止，可压缩的弹性半空间的分叉行为仍然难以捉摸，并且了解甚少。该手稿的主要目的是研究新霍克可压缩弹性半空间的分叉与公认的不可压缩案件。问题的表述需要对非线性泊松比进行新颖的描述，因为公认的定义不足以进行当前的分析。为了研究域的稳定性和分叉的可能性，进行了增量分析。增量分析描述了在有限变形下偏离平衡构型的微小变化。结果表明，在不可压缩极限下，我们对可压缩弹性半空间所获得的结果恢复了其不可压缩的对应物。这种贡献的另一个关键特征是，我们通过特征值分析，使用有限元方法，通过计算仿真验证了该问题的解析解。这项工作的主要成果是对出现分叉的临界拉伸的分析表达式。我们用从完全可压缩到不可压缩弹性的数值结果证明了该模型的实用性及其出色的一致性。向前发展，这种方法可用于理解和利用双层系统中的不稳定性，尤其是对于可压缩的系统。