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A perturbation expansion is offered for the micromechanical prediction of the bifurcation buckling of soft viscoelastic composites with imperfections (e.g. wavy fibers). The composites of periodic microstructure are subjected to compressive loading and are undergoing large deformations. The perturbation expansion applied on the imperfect composites results in a zero and first order problems of perfect composites. In the former problem, loading exists and interfacial and periodicity conditions are imposed. In the latter one, however, loading is absent, the interfacial conditions possess complicated terms that have been already established by the zero order problem, and Bloch-Floquet boundary conditions are imposed. Both problems are solved by the high-fidelity generalized method of cells (HFGMC) micromechanical analysis. The ideal critical bifurcation stress can be readily predicted from the asymptotic values of the form of waviness growth with applied loading. This form enables also the estimation of the actual critical stress. The occurrence of the corresponding critical deformation and time is obtained by generating the stress-deformation response of the composite. The offered approach is illustrated for the prediction of bifurcation buckling of viscoelastic bi-layered and polymer matrix composites as well as porous materials. Finally, bifurcation buckling stresses of unidirectional composites in which the matrix is represented by the quasi-linear viscoelasticity theory are predicted. This quasi-linear viscoelasticity model exhibits constant damping which is observed by the actual viscoelastic behavior of biological materials. [ABSTRACT FROM AUTHOR]

In contemporary elasticity theory, the strain–energy function predominantly relies on the first invariant I 1 of the deformation tensor; a practice that has been influenced by models derived from rubber elasticity. However, this approach may not fully capture the complexities of materials exhibiting pronounced shear deformations, such as very soft biological tissues. Here, we explore the implications and potential benefits of constitutive models where the strain–energy function is exclusively a function of the second invariant, I 2. By shifting the focus towards I 2 , we aim to address the limitations of current models in accurately describing shear-dominated behaviors and to provide a more comprehensive understanding of material responses, particularly for materials that do not conform to the assumptions underlying I 1 -centric theories. Through theoretical musings, data analysis, and automated model discovery, we investigate the feasibility of this approach and its consequences for predicting material behavior under various loading conditions. We show that the so-called "second-invariant materials" conforming to I 2 -only have interesting properties that are found in biological tissues and are fundamentally different from the traditional "first-invariant materials". [ABSTRACT FROM AUTHOR]

Abstract The Lamina Cribrosa is a part of the optic nerve head acting as a scaffold for collecting the retinal ganglion cell axons. It can be modeled as a poroelastic material where the saturated porosity stands for the capillary network running inside the collagen beams. Our aim is to study the interaction between tissue porosity, deformation and hemodynamics. To this end we first focus on the derivation of a poroelastic model in a rather general case, using as a prototype a model of species diffusion in an elastic material. Then we outline the clinical significance of the mechanical behavior of the Lamina Cribrosa and show, through numerical simulations, how an increased intraocular pressure results in a deformation affecting porosity and blood perfusion. We emphasize how the model behavior relies on the free energy expression. [ABSTRACT FROM AUTHOR]

In this work, a continuum constitutive framework for the mechanical modelling of soft tissues that incorporates strain rate and temperature dependencies as well as the transverse isotropy arising from fibres embedded into a soft matrix is developed. The constitutive formulation is based on a Helmholtz free energy function decoupled into the contribution of a viscous-hyperelastic matrix and the contribution of fibres introducing dispersion dependent transverse isotropy. The proposed framework considers finite deformation kinematics, is thermodynamically consistent and allows for the particularisation of the energy potentials and flow equations of each constitutive branch. In this regard, the approach developed herein provides the basis on which specific constitutive models can be potentially formulated for a wide variety of soft tissues. To illustrate this versatility, the constitutive framework is particularised here for animal and human white matter and skin, for which constitutive models are provided. In both cases, different energy functions are considered: Neo-Hookean, Gent and Ogden. Finally, the ability of the approach at capturing the experimental behaviour of the two soft tissues is confirmed. [ABSTRACT FROM AUTHOR]

Abstract: This paper reports the construction of viscoelasticity models for simulation of wave propagation in soft tissues. Aided by the Carson transform, Fung’s model and Iatridis’s model are transformed into the frequency domain with the imaginary part of the transformed relaxation function shown to be related to wave absorption effect. Based on measured wave absorption coefficients, the viscoelasticity models in the time domain are determined. Results show that Fung’s model does not support the frequency-dependent damping effect, but Iatridis’ model does. To validate the approach, numerical simulation of propagating wavelets in a skeletal pig muscle is performed. The constructed relaxation function is discretized by parallelly connected Standard Linear Solid (SLS) models. The hereditary integration is then transformed into a set of partial differential equations by introducing the internal variables. Together, the governing equations include the equation of motion, the constitutive equation, and the equations of internal variables. Velocity, stress, and internal variables are the primary unknowns. The model equations are solved by using the space–time CESE method. The calculated wave absorption effect compares well with the measured data. [Copyright &y& Elsevier]

Abstract: Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical response of soft biological tissues. Inelastic phenomena which occur in mechanical testing of biological tissues are very likely to be associated with alterations in the internal structure of these materials. In this study, a novel structural constitutive model is formulated to describe the inelastic effects in soft biological tissues such as Mullins type behavior, damage and permanent set as a result of residual strains after unloading. Anisotropic softening is considered by evolution of internal variables governing the anisotropic properties of the material. We consider two weight factors w i (softening) and s k (discontinuous damage) as internal variables characterizing the structural state of the material. Numerical simulations of several soft tissues are used to demonstrate the performance of the model in reproducing the inelastic behavior of soft biological tissues. [Copyright &y& Elsevier]

Abstract: The purpose of this work is to provide a theoretical analysis of the mechanical behavior of the growth of soft materials under geometrical constraints. In particular, we focus on the swelling of a gel layer clamped to a substrate, which is still the subject of many experimental tests. Because the constrained swelling process induces compressive stresses, all these experiments exhibit surface instabilities, which ultimately lead to cusp formation. Our model is based on fixing a neo-Hookean constitutive energy together with the incompressibility requirement for a volumetric, homogeneous mass addition. Our approach is developed mostly, but not uniquely, in the plane strain configuration. We show how the standard equilibrium equations from continuum mechanics have a similarity with the two-dimensional Stokes flows, and we use a nonlinear stream function for the exact treatment of the incompressibility constraint. A free energy approach allows the extension both to arbitrary hyperelastic strain energies and to additional interactions, such as surface energies. We find that, at constant volumetric growth, the threshold for a wavy instability is completely governed by the amount of growth. Nevertheless, the determination of the wavelength at threshold, which scales with the initial thickness of the gel layer, requires the coupling with a surface effect. Our findings, which are valid in proximity of the threshold, are compared to experimental results. The proposed treatment can be extended to weakly nonlinearities within the aim of the theory of bifurcations. [Copyright &y& Elsevier]

Abstract: A technique suitable for the modelling of large deforming biological tissues with a nearly periodic microstructure is presented in this work. The proposed approach takes into account the heterogeneous material constitution and geometrical arrangement of the tissues at the microstructural level. The global material properties are described in terms of the homogenized (effective) parameters. Numerical simulations are focused on the mechanical behaviour of an arterial wall. [Copyright &y& Elsevier]

Abstract: A fully three dimensional finite-strain damage model for fibrous soft tissue is developed. The model assumes uncoupled contributions for the matrix and collagen fibers, and uncoupled bulk and deviatoric response over any range of deformations. A simple isotropic damage mechanism within the framework of continuum damage mechanics has been used to describe the softening behavior under deformation for the matrix. On the other hand, statistical aspects related to the length distribution of the reinforcing fibers lead to a damage model for the reinforcing material. As a result, a general theoretical framework for constitutive modeling of biological soft tissue with continuum damage is obtained. A theoretical example consisting of a biaxial test of a soft tissue reinforced with two families of collagen fibers has been considered to demonstrate the capabilities of the proposed model and to study the sensitivity to changes in the statistical parameters associated with the reinforcing material. Also, a preliminary numerical example is included to demonstrate the model on a inhomogeneous boundary value problem. Results show that the model is able to capture the typical stress–strain behavior observed in fibrous soft tissue and seems to confirm the soundness of the proposed formulation. [Copyright &y& Elsevier]

Abstract: The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading. [Copyright &y& Elsevier]

• A porohyperviscoelastic model is presented to address the shear wave elastography of soft tissues with emphasis on the liver; • An analytical solution has been derived to reveal the dependence of the wave velocity on the constitutive parameters and deformation of the solid matrix generated by the internal pressure; • The model and solution help understand the dispersion of the shear waves in porohyperviscoelastic soft tissues; • The analysis reveals that both the hyperelastic and viscoelastic deformation behaviors of the solid matrix are responsible for the pronounced effect of the internal pressure on the shear wave velocity. Shear wave elastography (SWE) enables the probing of the mechanical properties of soft tissues in vivo and has been demonstrated to be a promising technique for the diagnosis of diseases such as staging liver fibrosis, evaluating artery stiffening, and detecting cancers. In this study, a porohyperviscoelastic model is presented to address the shear wave elastography of soft tissues with emphasis on the liver. Based on this constitutive model, an analytical solution has been derived to reveal the dependence of the wave velocity on the constitutive parameters and deformation of the solid matrix generated by the internal pressure. The model and solution not only help understand the dispersion of the shear waves in porohyperviscoelastic soft tissues such as the liver, but also reveal that both the hyperelastic and viscoelastic deformation behaviors of the solid matrix may play an important role in the pronounced effect of the internal pressure on the shear wave velocity, as observed in experiments. The analysis results are expected to advance our understanding of SWE and facilitate its use in clinics. [ABSTRACT FROM AUTHOR]