Modelling the time-dependent, damage and fracture mechanical properties of load-bearing soft biological tissues

Abstract

Load-carrying soft biological tissues exhibit a wide range of complex time-dependent, damage, and fracture mechanical properties. An effective comprehension of such behaviour is advantageous in characterising tolerance to injury and can provide valuable insights regarding the incidence and progression of certain diseases. Improving knowledge in such areas directly benefits the successful integration of engineering concepts within the clinical workflow. Moreover, it can aid in the advancement of preventative measures, patient treatment strategies, and the optimisation of medical device design. The macroscopic material response of soft tissue is inextricably linked to the deformation of mechanically significant extracellular matrix (ECM) components such as fibrous collagen and associated constituents like proteoglycans. There is, however, a fundamental lack of understanding concerning the tensile properties of the collagenous ECM at different length scales. The inherent challenges associated with the experimental discernment of such processes have motivated the deployment of modelling strategies as an effective investigative tool. This thesis has dealt with the design of generalised constitutive descriptions that propose salient microstructural deformation, damage, and failure-related mechanisms.

In Paper A, A multiscale constitutive framework based on a novel description of collagen is introduced. The description accounts for the gradual recruitment of undulated collagen fibrils and introduces proteoglycan-mediated time-dependent fibrillar sliding. Crucially, the proteoglycan deformation allows for the reduction of overstressed fibrils towards a preferential homeostatic stress. An implicit Finite Element implementation of the model uses an interpolation strategy towards collagen fibre stress determination and results in a memory-efficient representation of the model. A number of test cases, including patient-specific geometries, establish the efficiency of the description and demonstrate its ability to explain qualitative properties reported from macroscopic experimental studies of tendon and vascular tissue.

In Paper B, the aforementioned description is extended such that it additionally incorporates an interfibrillar failure (fibril pull-out) mechanism. The resulting damage-induced mechanical behaviour across several length scales is showcased for the microstructurally motivated continuum damage model. Notably, a bottom-up approach is further demonstrated, whereby the model is employed in a single-element representation of the modes of fracture. A qualitative description of soft tissue rupture is accordingly attained, to which an appropriate cohesive zone model for the equivalent fracture surface is then calibrated. In doing so, a surface-based discontinuous characterisation of failure is directly derived from the upscaling of irreversible and dissipative damage mechanisms from the microscale.

In Paper C, we present the novel coupling of the above continuum damage model with an embedded phenomenological representation of the fracture surface. Tissue separation is therefore accounted for through the integration of the cohesive crack concept within the partition of unity finite element method. A transversely isotropic cohesive potential per unit undeformed area is introduced that comprises rate-dependent damage evolution and accounts for mixed-mode failure. Furthermore, a novel crack initialisation procedure is detailed that identifies the occurrence of localised deformations in the continuum material and the orientation of the inserted discontinuity. Proof of principle is demonstrated via the application of the computational framework to two representative numerical simulations, illustrating the robustness and versatility of the formulation.

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