In addition, the tissue responses captured in the experiments are passive, i.e. the effect of muscle cells is neglected. The inelastic deformation behavior of soft tissue depends on the stress state and extent of the damage. Damage models phenomenologically capture the damage in the soft tissue as a result of pathological or supra-physiological conditions.
In the HGO model, the isotropic and anisotropic parts of the strain energy function are decomposed as: Experimental studies have reported four phenomena related to the damage in soft tissues: (a) Mullins effect, (b) hysteresis, (c) permanent set and (d) fracture [6,35]. Furthermore, they cause the stable breakdown of the collagen fibers and the ground matrix of the soft tissue.
The role of collagen fibers has also been studied with the injury model and found to be predominant. The mechanical response of soft tissues varies with fiber position and direction. In the reported study [71], matrix damage was neglected and only the contribution of fiber damage was considered.
The elastic damage model postulated in terms of the strain energy density is given as:. where Φf i is the smooth damage function of the fibers and ηf i ∈ [0, 1] represents the damage variables for the two fiber families given as:.
Rupture Modelling
The strain energy remains constant (Wf =ψif) and prevents healing in the material and allows energy dissipation. In CZM, a coherent surface is placed in the intact region of the material where the crack propagates, as shown in Figure 7. In accordance with XFEM, a damage initiation criterion is needed for the crack propagation, i.e. the phase-field evolution of the crack begins.
Furthermore, numerical aspects related to aortic dissections were investigated in the studies by Raina and Miehe [114] and Gultekin et al. It is defined as the variation in the total potential energy per unit propagation of the crack. 116] modeled the initiation and propagation of a crack in the coronary artery to study the relationship between rupture in the coronary artery and atherosclerosis.
To achieve that, crack initiation and propagation in the walls of healthy and atherosclerotic human coronary arteries were simulated with cracks placed circularly along the luminal and in the radial direction. The study of Jayendran and Ruimi [118] aims to investigate the state of stresses in the artery during crack propagation, which would help in studying the mechanics of aortic dissections. The initial tear was placed circularly in the middle of the medium loaded with internal pressure.
In CZM, a layer of the surface is placed between two bulk materials where the crack propagates. This layer of the surface is modeled with special elements that disappear when the crack propagates based on the traction separation and evolution. The simulation uses a two-dimensional model of the artery, that is, partially embedded in the myocardium and epicardium.
The overall damage is denoted by D, a scalar damage variable that is determined at the onset of damage at the interface. It is a monotonically increasing variable from zero for no damage to the variable where the crack propagates. In one study, the delamination process of a fibrous cap was simulated to investigate its underlying process that causes delamination. Mainly two field variables are used in CPFM viz. strain map (ϕ) and crack phase field (d) as shown in Fig-.
The crack phase field (d) is solved with the crack evolution equation (102), while the strain field (ϕ) is solved with linear momentum balance (103). 34] extended the approach by considering the fiber distribution to incorporate the anisotropy in the crack phase field.
Discussion
The fundamental behavior of the damage model in pseudoelasticity makes it a good fit for continuous and discontinuous damping, and it is also extended for the permanent set. In particular, the Holzapfel and Ogden damage model [86] phenomenologically captures damage with an additional physical parameter, i.e., crosslinks. The pseudoelastic damage in the material is controlled by the maximum stress reached, making it numerically simple.
Finally, in hyperelastic softening, the damage is included in the constitutive model and does not include any damage variables and their evolution equations. The authors have validated the reviewed models with basic mechanical tests, which may differ from the physiological state. For example, damage models for the rectus sheath are validated by the uniaxial tests of Martins et al.
However, the biaxial tension will better represent the physiological load condition in the rectus sheath. Damage models developed for arterial tissues can be validated with internal pressure tests by Perez et al. The CZM by Maiti and Geubelle [124] was particularly used to study arterial rupture under uniaxial tension by Fortunato et al.
125], and the same model was applied by Ferrer et al. to study tendon rupture. In the discussed damage models, the occurrence of damage is considered based on the loading condition, where the damage occurs after reaching a certain load or a certain stretch in the tissue caused by the load. The damage model for diseased tissue requires knowledge of both supraphysiological stresses and pathological effects.
So far, the reported studies are based on experiments with healthy tissue and diseased tissue, which give the constitutive response of the tissue under different conditions. However, the effect of disease progression on the constitutive response is still large. The advances in tissue engineering and developing in vitro disease modeling [143] may enable the experiments to study the effect of disease progression [144] in the tissues.
Conclusions
Damage parameter modeling considers the representation of damage in an inactive tissue neglecting all biological aspects of the tissue [6,32]. Therefore, the numerical simulation of damage due to pathological conditions requires a model where its parameters represent the mechanical and physiological changes due to the disease [6,139]. However, the active soft tissue response in models of injury and rupture can be developed by inducing growth and remodeling [140–142].
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