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Keywords Two-scale model, propagating cracks, partition-of-unity method, finite strain. This research focuses on the fracture that may occur in intervertebral discs. The intervertebral disc, which lies between adjacent vertebrae in the spine, serves as a shock absorber, load distributor, spacer and flexibility provider to the spine. The presence of damage in the discs, e.g. cracks, faults and shear bands, can obviously change their physical behaviour of the disc which may affect its capacity of absorbing and transmitting load. This, in turn may cause severe lower back pain on human body. An extension to a finite strain framework of a two-scale numerical model for propagating cracks in porous media (1, 2) is proposed to model the fracture on intervertebral discs with in combination with a hyper-elastic material response. The crack is described in discrete manner by exploiting the partition-of-unity property of finite element shape functions (3). At the microscopic scale, the flow in the cohesive crack is modeled as viscous fluid using Stokes equations and averaged over the cross section of the cavity. The governing equations are derived for the momentum and the mass for a fluid saturated porous medium, which are assumed to hold on the macroscopic scale. The weak formulations are derived for this model and calculated under total Lagrangian formulation. The resulting discrete equations are non- linear due to the the cohesive crack model, the geometrical non-linear effect and non-linearity of the coupling terms. A Crank-Nicholson scheme takes care of the time integration of the system. In this contribution the derived model and numerical implementation are used to analyze a quasi-static crack growth in confined compression under shear band loading.

A two-scale approach for propagating cracks in a fluid-saturated porous material