Framework Of Mixture Theory Research Articles

Numerical schemes for mixture theory models with filling constraint: application to biofilm ecosystems

The mixture theory framework is a powerful way to describe multi-phasic systems at an intermediary scale between microscopic and macroscopic scales. In particular, mixture theory reveals a powerful approach to represent microbial biofilms where a consortium of cells is embedded in a polymeric structure. To simulate a model of microalgal biofilm, we propose an upgraded numerical scheme, consolidating the one proposed by Berthelin et al. (2016) to enforce the volume-filling constraint in mixture models including mass exchanges. The strategy consists in deducing the discrete version of the incompressibility constraint from the discretized mass balance equations. Numerical simulations show that this method constrains the total volume filling constraint, even at the discrete level. Moreover, we add viscous terms in the biofilm model to properly represent biofilms interactions with its fluidic environment. It turns out that a well-balanced numerical scheme becomes of outmost importance to capture the biofilm dynamic when including the viscosity. This modelling upgrade also involves recalibrating model parameters. In particular, the elastic tensors to recover realistic front features. With the new parameters, the numerical set-up becomes more demanding to reach convergence.

Modeling inelastic responses using constrained reactive mixtures

This study reviews the progression of our research, from modeling growth theories for cartilage tissue engineering, to the formulation of constrained reactive mixture theories to model inelastic responses in any solid material, such as theories for damage mechanics, viscoelasticity, plasticity, and elasto-plastic damage. In this framework, multiple solid generations α can co-exist at any given time in the mixture. The oldest generation is denoted by α=s and is called the master generation, whose reference configuration Xs is observable. The solid generations α are all constrained to share the same velocity vs, but may have distinct reference configurations Xα. An important element of this formulation is that the time-invariant mapping Fαs=∂Xα/∂Xs between these reference configurations is a function of state, whose mathematical formulation is postulated by constitutive assumption. Thus, reference configurations Xα are not observable (α≠s). This formulation employs only observable state variables, such as the deformation gradient Fs of the master generation and the referential mass concentrations ρrα of each generation, in contrast to classical formulations of inelastic responses which rely on internal state variable theory, requiring evolution equations for those hidden variables. In constrained reactive mixtures, the evolution of the mass concentrations is governed by the axiom of mass balance, using constitutive models for the mass supply densities ρˆrα. Classical and constrained reactive mixture approaches share considerable mathematical analogies, as they both introduce a multiplicative decomposition of the deformation gradient, also requiring evolution equations to track some of the state variables. However, they also differ at a fundamental level, since one adopts only observable state variables while the other introduces hidden state variables. In summary, this review presents an alternative foundational approach to the modeling of inelastic responses in solids, grounded in the classical framework of mixture theory.

A modified mixture theory for one-dimensional melting of pure PCM and PCM/metal foam composite: Numerical analysis and experiment validation

Phase change materials (PCMs) have been attracting researchers’ attention in the last few decades for their great potential in energy storage. Nevertheless, the precise modeling of phase change problem of PCMs and PCM composites remains a problem due to the extra complexity generated by interactions between several constituents. In our study, we propose a modified mixture theory framework for one-dimensional pure PCM and PCM/metal foam composite melting problems, which allows us to study the thermal interactions between each constituent. In this modified mixture theory model, the local thermal non-equilibrium effect is considered, and the velocity correlated to the density change during paraffin melting is introduced. A new constituential heat flux term is presented to explain the heat supply from different constituents, and a new interpretation for internal energy supply term is introduced. The governing equations are numerically solved by using finite difference method. Temperature field, mushy zone evolution, and velocity profile at the boundary are predicted. Experiments are conducted to validate the numerical analysis. By comparing the temperatures at 4 positions, it is shown that the numerical results obtained by the mixture theory model have a satisfying agreement with the real situation.

Inhibition of toll-like receptor 4 protects against inflammation-induced mechanobiological alterations to intervertebral disc cells

Intervertebral disc (IVD) degeneration is associated with elevated levels of inflammatory cytokines implicated in disease aetiology and matrix degradation. Toll-like receptor-4 (TLR4) has been shown to participate in the inflammatory responses of the nucleus pulposus (NP) and its levels are upregulated in disc degeneration. Activation of TLR4 in NP cells leads to significant, persistent changes in cell biophysical properties, including hydraulic permeability and osmotically active water content, as well as alterations to the actin cytoskeleton. The study hypothesis was that inflammation-induced changes to cellular biomechanical properties and actin cytoskeleton of NP cells could be prevented by inhibiting TLR4 signalling. Isolated NP cells from bovine discs were treated with lipopolysaccharide (LPS), the best studied TLR4 agonist, with or without treatment with the TLR4 inhibitor TAK-242. Cellular volume regulation responses to step osmotic loading were measured and the transient volume-response was captured by time-lapse microscopy. Volume-responses were analysed using mixture theory framework to investigate hydraulic permeability and osmotically active intracellular water content. Hydraulic permeability and cell radius were significantly increased with LPS treatment and these changes were blocked in cells treated with TAK-242. LPS-induced remodelling of cortical actin and IL-6 upregulation were also mitigated by TAK-242 treatment. These findings indicated that TLR4 signalling participated in NP cell biophysical regulation and may be an important target for mitigating altered cell responses observed in IVD inflammation and degeneration.

A mixture theory for the moisture transport in polyamide

Polyamide exhibits hygroscopic nature and can absorb up to 10% of moisture relative to its dry weight. The absorbed moisture increases the mobility of the molecular chains and causes a reduction in the glass transition temperature. Thus, depending on the moisture distribution, a polyamide component can show different stiffness and relaxation times. Moreover, the moisture distribution also depends on the mechanical loading of the material as the volumetric deformation results in a change of the available free volume for the moisture. Thus, a strongly coupled model is required to describe the material behaviour. In this work, a thermodynamically consistent coupled model within the framework of mixture theory is developed. The mechanical deformation of polyamide 6 (PA6) is based on a linear viscoelastic material model, and the moisture transport is based on a nonlinear diffusion model. The stiffness and the relaxation time of the viscoelastic model change with the moisture concentration. Furthermore, the moisture transport is affected by the pressure gradient generated by the mechanical loading of the material. This strongly coupled model has been implemented using the finite element method, and simulation results are presented for a three-point bending experiment.

Coupled heat, mass and momentum transport in swelling cellulose based materials with application to retorting of paperboard packages

Understanding the complex interplay of mass transport, heat transport and deformation in swelling cellulose based materials is a challenging task. To aid in this endeavour a multiphysics model is developed within the mixture theory framework. The model is macroscopic and accounts for swelling, non-equilibrium mass exchange of water, elastic deformation and multiphase transport phenomena, where non-isothermal conditions are considered. The investigated material is viewed as an immiscible mixture of cellulose fibers, pore gas and inter-fiber pore water separated by interfaces. Fiber and gas are considered as miscible mixtures of dry fiber and fiber water, and dry air and water vapor, respectively. The constitutive equations are obtained in a coupled and thermodynamically consistent manner by exploitation of the dissipation inequality. As an application of the model, conditions representative of the retorting process for liquid packaging board are simulated using the proposed constitutive model. The influence of the process control parameters during the heating, cooking and cooling steps of the retorting procedure are analysed. It was found that through-plane temperature gradients and the presence of dry air in the retort had a significant impact on the moisture uptake and the moisture distribution. Spatial distribution of hydrophobicity had a great effect on the distribution of fiber water and inter-fiber water.

Size segregation in compressible granular shear flows of binary particle systems

This paper deals with the modelling and simulation of segregation in granular materials. The basis is a hydrodynamic model for granular material flows, which is extended to capture the dynamic process of segregation in shear flows of systems with small and large particles. The granular flow equations consist of a set of compressible Navier–Stokes-like equations as well as an equation for the granular temperature. With the help of the granular temperature equation, the granular flow equations are able to cover a wide range of regimes, starting from dilute to arresting flows. However, this paper focuses on dry granular shear flows. It extends this hydrodynamic system in a dense shear flow regime by a segregation equation using the framework of mixture theory. Special focus is lain on the segregation direction. A procedure from mechanics is adapted to obtain the segregation direction from the granular flow system independent of the choice of the coordinate system. In particular, this is done in three-dimensional space. Due to the compressibility of the granular flow system and the structure of the derived segregation equation, solving the segregation equation requires special numerical treatment. Therefore, a suitable numerical scheme is presented which prevents the system from reaching unphysical states.

The Effect of Oxidation on the Mechanical Response of Isolated Elastin and Aorta.

Oxidation of aorta by hydroxyl radicals produces structural changes in arterial proteins like elastin and collagen. This in turn results in change in the mechanical response of aorta. In this paper, a thermodynamically consistent constitutive model is developed within the framework of mixture theory, to describe the changes in aorta and isolated elastin with oxidation. The model is then studied under uniaxial extension using experimental data from literature.

The paradifferential approach to the local well-posedness of some problems in mixture theory in two space dimensions

In this paper, we consider a class of models describing multiphase fluids in the framework of mixture theory. The considered systems, in their more general form, contain both the gradient of a hydrostatic pressure, generated by an incompressibility constraint, and a compressible pressure depending on the volume fractions of some of the different phases. To approach these systems, we propose an approximation based on the Leray projection, which involves the use of a symbolic symmetrizer for quasi-linear hyperbolic systems and related paradifferential techniques. In two space dimensions, we prove the well-posedness of this approximation and its convergence to the unique classical solution to the original system. In the last part, we shortly discuss the three dimensional case.

Modeling diffusion and reaction of sulfates with cement concrete using mixture theory

Here mixture theory is used to capture the changes in cement concrete exposed to sodium sulfate till cracks develop. Toward this, the mixture is assumed to be made of eleven constituents of which the sodium sulfate and water move relative to themselves and the remaining nine solid constituents. The nine solid constituents constrained to move together are the eight relevant chemical constituents in concrete that react with sodium sulfate and all the other remaining chemical constituents of concrete that do not react with sulfates. Constitutive assumptions needed to be made within this mixture theory framework are the same as those reported by Gouder and Saravanan (Acta Mech 227(11):3123–3146, 2016). Within this framework of mixture theory, the radial ingress and reaction of sodium sulfate solution with the concrete cylinder sealed at top and bottom, exposed to a constant concentration of sodium sulfate at its outer surface, are formulated. The resulting nonlinear governing differential equations are converted into a system of nonlinear algebraic equations using a forward finite difference scheme in space and a backward difference in time. The nonlinear algebraic equations are solved simultaneously using constrained minimization technique till the water reaches the center of the cylinder. The results obtained for ingress without chemical reactions agree with those predicted by Fick’s equation. The axial expansion of the cylinder and the increase in the value of Young’s modulus of the part of concrete which reacted with sulfates agree qualitatively with the experiments.

A thermodynamically consistent model accounting for viscoplastic creep and anisotropic damage in unsaturated rocks

This paper presents a new poro-viscoplastic damageable model for partially saturated rocks. The hydromechanical modelling is based on the framework of mixture theory for multiphase porous media combined with Continuum Damage Mechanics. The proposed dissipative evolution laws are governed by an effective stress variable, which is a function of an equivalent pore pressure. The viscoplastic creep law is an extension of Lemaitre's model incorporating pressure-dependency and a non-associated flow rule. A new anisotropic damage evolution law for unsaturated rocks under compressive stresses is suggested. Despite a large number of coupled phenomena involved, the model remains relatively simple with a quite moderate number of parameters, while still ensuring the thermodynamic consistency. The model is capable to describe the main features of the time-dependent behaviour of rocks such as strain hardening, stress-dependent creep, moisture effects, damage-induced anisotropy and failure. Its efficiency is evidenced by simulating a number of laboratory tests performed on different rocks.

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi–Douglas–Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two- and three-dimensional porous media are presented to demonstrate the accuracy of the method (displaying an optimal convergence rate), the physics-preserving properties of the mixed–primal scheme, as well as the robustness of the method (which is successfully used to simulate diverse physical phenomena such as density fingering, Terzaghi's consolidation, deformation of a cantilever bracket, and Boycott effects). The applicability of the method is not limited to flow in porous media, but can also be employed to describe many other physical systems governed by a similar set of equations, including e.g. multi-component materials.

Anti-Inflammatory Agents Prevent Intervertebral Disc Cell Mechanobiological Alterations

Introduction Intervertebral disc (IVD) degeneration (DD) is characterized by increased catabolic activity, ECM breakdown, and elevated levels of pro-inflammatory cytokines, particularly TNF-α and IL-1β. DD also alters hydrostatic pressure and biomechanical loading in the nucleus pulposus (NP), potentially leading to an altered biomechanical microenvironment around the NP cells. In a previous study we have shown that stimulation of isolated NP cells with the inflammatory cytokine TNFα, or activation of innate immune signaling via toll-like-receptor-4 (TLR4) with the ligand LPS, causes significant increases in both cell volume and hydraulic permeability of the cell. The goal of the current study is to examine the effects of anti-inflammatory drugs on mitigation of inflammatory induced biophysical changes of NP cells. Material and Methods NP tissue was isolated from bovine lumbar discs. Inflammatory stimulation was performed with either TNFα (10ng/ml) or LPS (0.1 µg/ml) for 24 hours. To inhibit the effects of TNFα, cells were pre-treated with dexamethasone (DEX, 1 µg/mL) for 90 minutes prior to inflammatory stimulation. For TLR4 inhibition groups, cells were pre-treated with TAK-242 (1µM), an intracellular inhibitor of TLR4, for 1 hour prior to addition of LPS. Supernatants were collected and nitrite (NO) concentration was measured. Cell osmotic properties were also measured using a custom Y-shaped microfluidic channel. Cells were equilibrated in a 333 mOsm/L NaCl solution after which a single hyper-osmotic loading followed by a hypo-osmotic loading step was applied with NaCl solutions at 466mOsm/L followed by 333 mOsm/L. During osmotic loading, cells were imaged using DIC and volume response was computed over time for each cell. Volume response was curve fitted to a mixture theory framework to determine biophysical properties for each cell including membrane hydraulic permeability (Lp), and reference intracellular water content (Φir). Data was analyzed with ANOVA and Fisher LSD post-hoc test ( p < 0.05 considered significant). Results DEX and TAK-242 significantly inhibited TNFα and LPS induced nitrite release, respectively. Both TNFα and LPS stimulation significantly increased cell size at 24 hours post treatment. Hydraulic permeability (Lp) of cells significantly increased for both osmotic steps. Treatment of cells with anti-inflammatory drugs significantly reduced Lp back to baseline levels in both TNFα and LPS groups. No significant changes in Φir were observed due to inflammatory simulation. Conclusion The goal of this study was to investigate the effects of anti-inflammatory drugs on the biophysical properties of isolated NP cells. DEX, an anti-inflammatory glucocorticoid, and TAK-242, a small molecule TLR4 inhibitor, were both found to decrease TNFα and LPS induced NO release respectively. TNFα and LPS each increased Lp in NP cells, consistent with our previous findings. Interestingly, treatment of cells with anti-inflammatory drugs inhibited TNFα and LPS induced changes in Lp, where Lp was found to be comparable to baseline control levels. These findings indicate that anti-inflammatory drugs may prevent alterations in cell biomechanical properties and protect the mechanobiological function of NP cells from inflammatory changes. Our findings identify potential targets and therapeutic biologics agents for further consideration in the treatment of disc degeneration.

Continuum theory of fibrous tissue damage mechanics using bond kinetics: application to cartilage tissue engineering.

This study presents a damage mechanics framework that employs observable state variables to describe damage in isotropic or anisotropic fibrous tissues. In this mixture theory framework, damage is tracked by the mass fraction of bonds that have broken. Anisotropic damage is subsumed in the assumption that multiple bond species may coexist in a material, each having its own damage behaviour. This approach recovers the classical damage mechanics formulation for isotropic materials, but does not appeal to a tensorial damage measure for anisotropic materials. In contrast with the classical approach, the use of observable state variables for damage allows direct comparison of model predictions to experimental damage measures, such as biochemical assays or Raman spectroscopy. Investigations of damage in discrete fibre distributions demonstrate that the resilience to damage increases with the number of fibre bundles; idealizing fibrous tissues using continuous fibre distribution models precludes the modelling of damage. This damage framework was used to test and validate the hypothesis that growth of cartilage constructs can lead to damage of the synthesized collagen matrix due to excessive swelling caused by synthesized glycosaminoglycans. Therefore, alternative strategies must be implemented in tissue engineering studies to prevent collagen damage during the growth process.

Modeling diffusion of sulfate through concrete using mixture theory

Concrete exposed to sulfates deteriorates. Here, an attempt is made to see whether the framework of mixture theory can be used to model the changes that occur in concrete exposed to sodium sulfate. Toward this, diffusion and reaction of sulfates with concrete is modeled within the framework of mixture theory. Appropriate choices are made for the Helmholtz free energy and interaction momentum so that the process of diffusion of sulfates in concrete can be captured. As expected in mixture theory, diffusion causes deformation of the solid. The parameters in the mixture theory model are determined by comparing the steady-state concentration profile of the diffusing sodium sulfate solution and inlet velocity of the fluid with that of Fick’s solution for the same boundary conditions. An assumption is made for the mass production term to capture the reaction of sulfates with concrete. The rate constant and order of the reaction are estimated using the concentration of gypsum, calcium hydroxide and ettringite reported in the literature, for various duration of exposure of cement pastes to known concentration of sodium sulfate solution. Finally, the governing equations for the combined problem of steady-state diffusion and reaction of sulfates with concrete are presented. A numerical scheme to solve the governing equations is outlined. Long-term concentration profiles of sodium sulfate predicted by the framework agree qualitatively with the experimentally observed profile reported in the literature.

Study of blood flow in several benchmark micro-channels using a two-fluid approach

It is known that in a vessel whose characteristic dimension (e.g., its diameter) is in the range of 20–500μm, blood behaves as a non-Newtonian fluid, exhibiting complex phenomena, such as shear-thinning, stress relaxation, and also multi-component behaviors, such as the Fahraeus effect, plasma-skimming, etc. For describing these non-Newtonian and multi-component characteristics of blood, using the framework of mixture theory, a two-fluid model is applied, where the plasma is treated as a Newtonian fluid and the red blood cells (RBCs) are treated as shear-thinning fluid. A computational fluid dynamic (CFD) simulation incorporating the constitutive model was implemented using OpenFOAM® in which benchmark problems including a sudden expansion and various driven slots and crevices were studied numerically. The numerical results exhibited good agreement with the experimental observations with respect to both the velocity field and the volume fraction distribution of RBCs.

Inflammation induces irreversible biophysical changes in isolated nucleus pulposus cells.

Intervertebral disc degeneration is accompanied by elevated levels of inflammatory cytokines that have been implicated in disease etiology and matrix degradation. While the effects of inflammatory stimulation on disc cell metabolism have been well-studied, their effects on cell biophysical properties have not been investigated. The hypothesis of this study is that inflammatory stimulation alters the biomechanical properties of isolated disc cells and volume responses to step osmotic loading. Cells from the nucleus pulposus (NP) of bovine discs were isolated and treated with either lipopolysaccharide (LPS), an inflammatory ligand, or with the recombinant cytokine TNF-α for 24 hours. We measured cellular volume regulation responses to osmotic loading either immediately after stimulation or after a 1 week recovery period from the inflammatory stimuli. Cells from each group were tested under step osmotic loading and the transient volume-response was captured via time-lapse microscopy. Volume-responses were analyzed using mixture theory framework to investigate two biomechanical properties of the cell, the intracellular water content and the hydraulic permeability. Intracellular water content did not vary between treatment groups, but hydraulic permeability increased significantly with inflammatory treatment. In the 1 week recovery group, hydraulic permeability remained elevated relative to the untreated recovery control. Cell radius was also significantly increased both after 24 hours of treatment and after 1 week recovery. A significant linear correlation was observed between hydraulic permeability and cell radius in untreated cells at 24 hours and at 1-week recovery, though not in the inflammatory stimulated groups at either time point. This loss of correlation between cell size and hydraulic permeability suggests that regulation of volume change is disrupted irreversibly due to inflammatory stimulation. Inflammatory treated cells exhibited altered F-actin cytoskeleton expression relative to untreated cells. We also found a significant decrease in the expression of aquaporin-1, the predominant water channel in disc NP cells, with inflammatory stimulation. To our knowledge, this is the first study providing evidence that inflammatory stimulation directly alters the mechanobiology of NP cells. The cellular biophysical changes observed in this study are coincident with documented changes in the extracellular matrix induced by inflammation, and may be important in disease etiology.

Interstitial growth and remodeling of biological tissues: Tissue composition as state variables

Growth and remodeling of biological tissues involves mass exchanges between soluble building blocks in the tissue's interstitial fluid and the various constituents of cells and the extracellular matrix. As the content of these various constituents evolves with growth, associated material properties, such as the elastic modulus of the extracellular matrix, may similarly evolve. Therefore, growth theories may be formulated by accounting for the evolution of tissue composition over time in response to various biological and mechanical triggers. This approach has been the foundation of classical bone remodeling theories that successfully describe Wolff's law by establishing a dependence between Young's modulus and bone apparent density and by formulating a constitutive relation between bone mass supply and the state of strain. The goal of this study is to demonstrate that adding tissue composition as state variables in the constitutive relations governing the stress–strain response and the mass supply represents a very general and straightforward method to model interstitial growth and remodeling in a wide variety of biological tissues. The foundation for this approach is rooted in the framework of mixture theory, which models the tissue as a mixture of multiple solid and fluid constituents. A further generalization is to allow each solid constituent in a constrained solid mixture to have its own reference (stress-free) configuration. Several illustrations are provided, ranging from bone remodeling to cartilage tissue engineering and cervical remodeling during pregnancy.

Revisiting total, matric, and osmotic suction in partially saturated geomaterials

An accurate quantification of negative pore pressure (commonly referred to as ‘suction’) in the pore network is necessary for modeling the mechanical response of unsaturated geomaterials. Traditional definitions and formulations of total, matric, and osmotic suction suggest incorrect pore fluid pressures under certain conditions. In this paper, the notion of suction is revisited by deriving an expression for pore fluid pressure in a simple osmotic, capillary tube using the framework of mixture theory in conjunction with the fundamental laws of thermodynamics. Based on the derived expression for the tube, expressions are derived for total, matric, and osmotic suction for partially saturated geomaterials. Particular attention is given to osmotic suction since confusion regarding its mechanisms has apparently contributed to its misapplication in geomechanics. The new expressions derived herein adequately explain behavior that is incorrectly explained by the traditional formulations and unifies two approaches to modeling osmotic suction previously considered to be in contradiction.

A MULTIPHASE MODEL OF TUMOR AND TISSUE GROWTH INCLUDING CELL ADHESION AND PLASTIC REORGANIZATION

The main aim of the paper is to embed the experimental results recently obtained studying the detachment force of single adhesion bonds in a multiphase model developed in the framework of mixture theory. In order to do that the microscopic information is upscaled to the macroscopic level to describe the dependence of some crucial terms appearing in the PDE model on the sub-cellular dynamics involving, for instance, the density of bonds on the membrane, the probability of bond rupture and the rate of bond formation. In fact, adhesion phenomena influence both the interaction forces among the constituents of the mixtures and the constitutive equation for the stress of the cellular components. Studying the former terms a relationship between interaction forces and relative velocity is found. The dynamics presents a behavior resembling the transition from epithelial to mesenchymal cells or from mesenchymal to ameboid motion, though the chemical cues triggering such transitions are not considered here. The latter terms are dealt with using the concept of evolving natural configurations consisting in decomposing in a multiplicative way the deformation gradient of the cellular constituent distinguishing the contributions due to growth, to cell rearrangement and to elastic deformation. This allows the description of situations in which if in some points the ensemble of cells is subject to a stress above a threshold, then locally some bonds may break and some others may form, giving rise to an internal reorganization of the tissue that allows to relax exceedingly high stresses.