In this paper we consider a nonlinear poroelasticity model that describes the quasi-static mechanical behaviour of a fluid-saturated porous medium whose permeability depends on the divergence of the displacement. Such nonlinear models are typically used to study biological structures like tissues, organs, cartilage and bones, which are known for a nonlinear dependence of their permeability/hydraulic conductivity on solid dilation. We formulate (extend to the present situation) one of the most popular splitting schemes, namely the fixed-stress split method for the iterative solution of the coupled problem. The method is proven to converge linearly for sufficiently small time steps under standard assumptions. The error contraction factor then is strictly less than one, independent of the Lam\'{e} parameters, Biot and storage coefficients if the hydraulic conductivity is a strictly positive, bounded and Lipschitz-continuous function.

翻译：本文研究描述流体饱和多孔介质准静态力学行为的非线性孔隙弹性模型，其中介质的渗透率依赖于位移场的散度。这类非线性模型通常用于研究生物组织、器官、软骨和骨骼等生物结构——这些结构的渗透率/水力传导率对固体膨胀具有已知的非线性依赖性。我们针对当前情形构建（推广）了最流行的分裂格式之一——固定应力分裂法，用于求解该耦合问题的迭代解。在标准假设下，该方法被证明对足够小的时间步长具有线性收敛性。若水力传导率是严格正、有界且满足Lipschitz连续的函数，则误差收缩因子严格小于1，且该收敛性独立于Lamé参数、Biot系数及储集系数。