Incompressible Isotropic Material Research Articles

Modelling the rate-dependent mechanical behaviour of the brain tissue

A new modelling approach is employed in this work for application to the rate-dependent mechanical behaviour of the brain tissue, as an incompressible isotropic material. Extant datasets encompassing single- and multi-mode compression, tension and simple shear deformation(s) are considered, across a wide range of deformation rates from quasi-static to rates akin to blast loading conditions, in the order of 1000 s−1 . With a simple functional form and a reduced number of parameters, the model is shown to capture the considered rate-dependent behaviours favourably, including in both single- and multi-mode deformation fits, and over all range of deformation rates. The provided modelling results here are obtained from either first fitting the model to the quasi-static data, or/and predicting the behaviour at a different rate than those used for calibrating the model parameters. Given its simplicity, versatility, predictive capability and accuracy, the application of the utilised modelling framework in this work to the rate-dependent mechanical behaviour of the brain tissue is proposed.

Accretion–ablation mechanics

In this paper, we formulate a geometric nonlinear theory of the mechanics of accreting–ablating bodies. This is a generalization of the theory of accretion mechanics of Sozio & Yavari (Sozio & Yavari 2019 J. Nonlinear Sci. 29 , 1813–1863 (doi:10.1007/s00332-019-09531-w)). More specifically, we are interested in large deformation analysis of bodies that undergo a continuous and simultaneous accretion and ablation on their boundaries while under external loads. In this formulation, the natural configuration of an accreting–ablating body is a time-dependent Riemannian 3 -manifold with a metric that is an unknown a priori and is determined after solving the accretion–ablation initial-boundary-value problem. In addition to the time of attachment map, we introduce a time of detachment map that along with the time of attachment map, and the accretion and ablation velocities, describes the time-dependent reference configuration of the body. The kinematics, material manifold, material metric, constitutive equations and the balance laws are discussed in detail. As a concrete example and application of the geometric theory, we analyse a thick hollow circular cylinder made of an arbitrary incompressible isotropic material that is under a finite time-dependent extension while undergoing continuous ablation on its inner cylinder boundary and accretion on its outer cylinder boundary. The state of deformation and stress during the accretion–ablation process, and the residual stretch and stress after the completion of the accretion–ablation process, are computed. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

A generalisation of the Pucci–Saccomandi model of rubber elasticity

The Pucci–Saccomandi model of rubber elasticity, also known as the Gent–Gent model, is generalised in this manuscript for a more enhanced applicability to a wider range of finite deformation behaviours, rubber-like material types, and an increased modelling accuracy. The generalising of interest is developed by devising a higher order rational approximant, i.e., of order [1/1], of the response functions. The proposed model is shown to be a parent to some of the celebrated limiting chain extensibility, classical WI1,I2 and generalised neo-Hookean models. The application of the proposed generalised model to the finite deformation of various isotropic incompressible rubber-like materials is demonstrated, including natural unfilled and filled rubbers, commercial polymers and biomaterials. The considered datasets exhibit various deformation behaviours from the classical large deformation of natural rubbers to the downward concavity of filled rubbers, strain/shear softening behaviours of polymers and the softening behaviour of biomaterials up to the onset of failure. The modelling results are directly compared with those of the Gent–Gent model, where the higher accuracy of the proposed model and its capability to favourably capture the aforementioned challenging and varied behaviours is established. In particular, the Gent–Gent model is shown unsuitable for capturing such downward concavity, shear/strain softening and continuous softening to failure behaviours. Given the improved modelling predictions, simplicity of the functional form of the devised model, and its relatively low number of model parameters, the proposed model is presented as a more general, comprehensive and versatile modelling tool for application to the finite deformation of various isotropic incompressible soft materials.

Simple tension three ways

The consequences of using a principal Cauchy stress formulation of simple tension of cuboid specimens are explored within the context of incompressible isotropic materials, with the prototypical neo-Hookean material used to illustrate the main ideas. When simple tension is defined in this way it is shown that there are three natural Cartesian forms for the Cauchy stress: the conventional form, a plane stress form and a uniform stress form. A deformation consistent with each of these forms is obtained for the neo-Hookean material and it is shown that there are some counter-intuitive effects. It is also shown that the plane stress form of simple tension could alternatively describe the effect of applying equal and opposite forces to diagonally opposite edges of a cuboid specimen, with the uniform stress form describing the effect of applying these forces to diagonally opposite corners of a cuboid.

A generalised WI1,I2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{W}}\\left({{\\varvec{I}}}_{1},{{\\varvec{I}}}_{2}\\right)$$\\end{document} strain energy function of binomial form with unified applicability across various isotropic incompressible soft solids

A generalised WI1,I2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W\\left({I}_{1},{I}_{2}\\right)$$\\end{document} strain energy function, a generalisation of previously devised response functions W1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${W}_{1}$$\\end{document} and W2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${W}_{2}$$\\end{document}, of binomial form is presented in this work for application to the finite deformation of isotropic incompressible soft solids. It is shown that the proposed model is the parent to many of the well-known existing invariants-based models in the literature. The first-order expansion of the model, with six model parameters, is then applied to extant multiaxial deformation of a wide range of materials, from filled and unfilled rubbers to hydrogels, liquid crystal elastomers and biomaterials. The model captures the experimental data accurately, with typical relative errors below 4%, while favourably modelling various challenging mechanical behaviours such as the asymmetry of compression—tension, high nonlinearity of the simple shear response, deformation softening effects, pronounced Payne effect, the soft elasticity phenomenon, and the reverse Poynting effect. The predictive capabilities of the model are also demonstrated and verified against experimental data. Given the analyses and results presented here, the devised model is proposed to serve as a standard choice for a priori selection for application to the finite deformation of isotropic incompressible soft materials.

Continuous softening up to the onset of failure: A hyperelastic modelling approach with intrinsic softening for isotropic incompressible soft solids

This paper seeks to put forward an alternative notion for modelling the continuous softening in the finite deformation of incompressible isotropic rubber-like materials, up to the onset of failure. Based on the proposition that ‘there is no fundamental reason as to why the basic constitutive parameters of the (hyper)elastic state of a material cannot determine the natural evolution of softening and failure in that material’, the concept of hyperelasticity with intrinsic softening is posited. In contrast to the currently available theories of continuum damage mechanics and energy limiters, which superpose external parameters onto the basic hyperelastic model, the alternative approach presented here postulates that if a hyperelastic strain energy function accommodates a comprehensive set of constitutive parameters and has an appropriate functional form, it will intrinsically capture the observed continuous softening in the stress–deformation curves of soft solids, up to the onset of failure. Examples of this notion are presented using a specific hyperelastic model previously proposed by the author, applied here to extant datasets of a wide variety of isotropic incompressible soft solids, ranging from soft tissues to protein gels and 3D printed biomaterials, that include the softening behaviour. The favourable modelling results obtained via this approach are demonstrated. It is conferred that these results portend a versatile modelling tool for application to the whole-range deformation of soft solids, up to the onset of failure, and make the case for devising a unified theory for modelling both continuous and discontinuous softening in the finite deformation of rubber-like materials.

Some models for initially stressed and initially strained structurally anisotropic incompressible materials

In this paper, we determine the constitutive relations for structurally anisotropic materials from a reference with initial strain or initial stress. Using reference independence of strain energy density, we first deduce the relationship of various initial strain measures with material symmetry and determine the resulting structural tensors. We use these structural tensors to derive the necessary invariants for initially strained materials. We also Examine the polyconvexity of the derived invariants and determine constitutive relations for the initially strained Ogden model. Next, we determine the constitutive relation for a combination of initial stress-texture symmetry which may be a more popular approach of recent interest. We deduce four reference-independent constitutive relations for initially stressed transversely isotropic incompressible materials. The first two strain energy functions are linear in the left Cauchy stretch and its cofactor, respectively, in the stress-free state. In the absence of initial stress, the third strain energy function involves all standard invariants for transversely isotropy. To deduce this strain energy in the presence of initial stress, we invert a fourth-order tensor and perform several involved tensor calculations. Finally, we obtain the constitutive relation for an initially stressed transversely isotropic limited chain extensible (LCE) model. To determine the initial stretch in the preferred texture direction in terms of initial stress, we utilize the geometrical interpretation of the Almansi strain measure.

Large Isotropic Elastic Deformations: On a Comprehensive Model to Correlate the Theory and Experiments for Incompressible Rubber-Like Materials

A comprehensive model, i.e., a model that: (i) suitably captures the mechanical behaviour of various types of rubber-like materials; (ii) describes the constitutive behaviour of a subject rubber-like specimen under different deformation modes via a single set of model parameter values, and (iii) is parent to many of the existing models of distinct types, is presented in this paper for application to the finite deformation of incompressible isotropic rubber-like materials. The model breaks away from Rivlin’s principal invariants I_{i} and Valanis-Landel’s separable function of principal stretches lambda _{i} representations, and instead adopts a general non-separable functional form W( lambda _{1}, lambda _{2}, lambda _{3} ), subject to the incompressibility constraint. By way of example, the application of the model to extant datasets from four types of rubber-like materials that exhibit discernible mechanical behaviours, namely natural unfilled and filled rubbers, hydrogels and (extremely) soft tissue specimens, will be considered. The favourable correlation between the model predictions and the considered experimental datasets, obtained via simultaneous fitting of the model to the data of various deformations, will be demonstrated. It will be shown that most of the landmark models in the literature including the W( I_{1}, I_{2} ) form Mooney-Rivlin, the Gent-like limiting chain extensibility, the nonaffine tube and the separable principal stretches-based type Ogden models are all a special sub-set of the presented parent model, and are all recovered from this model. An important implication of the non-separable functional form of the model will be conferred, through a specific dataset, where the predictions of the separable functions prove inherently inadequate. The consequences of these improvements for a more accurate modelling of the finite deformation of incompressible isotropic rubber-like materials will also be discussed.

A non-linear complementary energy-based constitutive model for incompressible isotropic materials

Inverse non-linear stress–strain relations between the logarithmic Hencky strain tensor and the Cauchy stress tensor are derived for incompressible isotropic materials. The constitutive model is based on a new type of complementary energy potential that considered to be the function of the second and third invariants of the deviatoric Cauchy stress tensor in a power-law form. A proper scaling of the third deviatoric stress invariant allows separate fitting to uniaxial tension and equibiaxial tension (or uniaxial compression) data. The predictive capabilities of a four- and a six-parameter model are presented through a parameter fitting procedure using Treloar’s experimental data for rubber. The performance of the new constitutive model is also demonstrated by solving the inflation problem of a spherical shell.

Hyperelasticity models extending Hooke’s law from small to moderate strains and experimental verification of their scope of application

Lagrangian formulations of Hill’s linear isotropic hyperelastic material models based on the one-parameter (r) Itskov family of strain tensors (including the Hencky, Pelzer, and Mooney strain tensors generating the H, P, and M material models, respectively) were developed. Explicit basis-free expressions for fourth-order elasticity tensors were obtained for these material models. These expressions use the right Cauchy–Green deformation tensor. The developed formulations of isotropic hyperelastic material models were implemented in the commercial finite element MSC.Marc code. Simple tension stress–strain diagrams were obtained for dog-bone shaped flat specimens made of Duothane QA965 polyurethane. The experimental data were used to fit the parameters E and ν (having the meaning of Young’s modulus and Poisson’s ratio for specimens at small strains) for the H, P, and M material models and the parameters C1 and C2 for the Mooney–Rivlin (M–R) isotropic incompressible material model included in the library of standard material models of the MSC.Marc code. For the investigated material, the H and M material models were shown to adequately reproduce stress–strain diagrams up to 50% elongation, and the P and M–R models up to 150% elongation. In addition, cylindrical specimens were made of the same polyurethane and tested for simple torsion (with constrained edges) and generalized torsion (with edges free to move in the longitudinal direction). In both problems, the resultant moments obtained by computer simulations using the P and M–R material models agree well with the experimental data on rod torsion up to buckling; computer simulations using the H and M material models give less satisfactory agreement with the experimental data. Similar qualitative agreement is obtained between the experimental and simulated resultant axial force and axial elongation; however, in this case, the quantitative agreement is less satisfactory than for the resultant moment.

A three-parameter structurally motivated robust constitutive model for isotropic incompressible unfilled and filled rubber-like materials

A structurally motivated three-parameter strain energy function W is presented in this paper for application to the constitutive modelling of various types of incompressible isotropic rubber-like materials, from unfilled to filled rubbers, polymers and soft tissues. A particular objective of the current work is to achieve another step towards devising a comprehensive model for application to an assortment of incompressible isotropic hyperelastic materials, not limited only to a specific subset type of rubbers or polymers. To this end, the application of the model to extant datasets from various types of rubber-like materials is presented, via simultaneous fitting of the model to the considered deformation modes. The modelling results are then compared with those of a selected set of established three-parameter models in the literature, including a generalised neo-Hookean strain energy function and another two models which include an I2 term. It will be shown that the proposed model most favourably captures the datasets for all unfilled, filled rubber and polymer specimens considered. Finally, the model is applied to the deformation datasets of a soft tissue, namely the human brain, for which the (one-term) Ogden model is currently known to provide the best fit. It will be demonstrated that the application of the Ogden model will lead to loss of convexity, while our proposed model provides a better fit, with lower relative errors, and remains convex over the deformation domain. By way of the presented examples and applications, it is concluded that the proposed model provides a reliable degree of robustness for a suitable application to constitutive modelling of various types of rubber-like materials.

A novel class of universal relation for incompressible isotropic electro-viscoelastic materials

The present paper proposes a novel class of universal relation for incompressible isotropic electro-viscoelastic materials. A new universal relation valid for all nonlinear electro-viscoelastic isotropic materials is established. The proposed relation connects the stress and deformation components for any electro-viscoelastic material in an assigned constitutive family. A continuum mechanics-based electro-viscoelasticity theory is utilized to develop the relation, following the second law of thermodynamics. A new class of controllable deformation family in shearing motions based on the proposed universal relation is also obtained. In general, the proposed universal relation may help the modelers and materialist community in characterizing the material behavior, which was often ignored.

Cylindrical bending of transversely isotropic composite panels with finite strains

sThe article is devoted to solving the problem of cylindrical bending of a flat panel made of a transversely isotropic incompressible composite material with finite deformations. The focus of the article is that in this problem the so-called universal model of constitutive relations is considered, which allows one to obtain solutions to problems simultaneously for several classes of models related to different conjugate pairs of stress-strain tensors. This method was proposed earlier in the works of Yu.I. Dimitrienko. Using this method, the difference in the calculation results, which are obtained using various models of composites with finite strains, is shown.

Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i.e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

Adiabatic heating effect in elastic-plastic contraction / expansion of spherical cavity in isotropic incompressible material

The paper presents an analytical solution to the coupled problem of spherically symmetric elastic-plastic deformation accompanied by heating of the material due to the plastic dissipation, which in turn changes the mechanical properties of the material. The dependence of both elastic and plastic mechanical properties of the material on temperature can be arbitrary. Both elastic and plastic deformations are assumed to be finite. The thermal expansion of material is neglected. Heating is assumed to be adiabatic. Instead of a spatial coordinate, temperature is considered as an independent monotonic variable. This made it possible to reduce the problem to solving the first order ODE. The obtained solution is valid for any incompressible hyperelastic solid with arbitrary pressure-independent non-singular yield condition, perfectly-plastic or isotropic strain-hardening/softening. Example of solution for the linear thermal-softening and strain hardening material with tension-compression asymmetry in yielding is given.

Implementation of Particle Swarm Optimization Algorithm in Matlab Code for Hyperelastic Characterization

The purpose of this paper is to demonstrate the applicability of Particle Swarm Optimization algorithm to determine material parameters in incompressible isotropic elastic strain-energy functions using combined tension and torsion loading. Simulation of rubber behavior was conducted from the governing equations of the deformation of a cylinder composed of isotropic hyperelastic incompressible materials. Four different forms of strain-energy function were considered based respectively on polynomial, exponential and logarithmic terms to reproduce load force (N) and torque (M) trends using natural rubber experimental data. After highlighting the minimization of the objective function generated in the fitting process, the study revealed that a particle swarm optimization algorithm could be successfully used to identify the best material parameters and characterize the behavior of rubber-like hyperelastic materials.

Material tailoring in three-dimensional flexural deformations of functionally graded material beams

Soft materials such as rubberlike and biological tissues are usually modeled as incompressible. Mechanical properties of polymeric materials can be controlled either by exposing them to ultraviolet light for different time durations or by changing their molecular structure. A challenge is to find the spatial variation of the moduli to fully utilize the material. One way to achieve this is to have a uniform distribution of the stress component likely to cause their failure. To achieve this, we analytically analyze 3-dimensional infinitesimal flexural deformations of a functionally graded (FG) and linearly elastic beam of rectangular cross-section with Young’s modulus a continuous function of the thickness coordinate. The problem is first studied for an incompressible material and then for a compressible material for which Poisson’s ratio is assumed to be a constant. It is found that when Young’s modulus at a point is inversely proportional to its distance from the neutral axis, then the magnitude of the bending stress is uniform over beam’s cross-section, the beam is the lightest possible and its deflections are 2/3rd of that of the corresponding beam of a homogeneous material that has the same maximum bending stress as the FG beam. Noting that Young’s modulus cannot be infinity at the neutral axis, we avoid this by assuming that it is a constant over a small region around the neutral axis. For a transversely isotropic incompressible material beam, it is shown that (2μ1 + μ2) determines beam’s flexural stiffness where μ1 and μ2 are, respectively, the shear moduli along and perpendicular to the axis of transverse isotropy. Beams of homogeneous materials having the same geometry as the FG beam have different shear moduli depending upon whether the FG and the homogeneous material beams have the same maximum deflection, the same maximum bending stress or the same total strain energy of deformation.

Universal Solutions of Nonlinear Dislocation Theory for Elastic Cylinder

For an elastic isotropic incompressible material of general form, a number of exact solutions have been found about large torsional and tensile–compressive deformations of a solid circular cylinder, taking into account distributed dislocations. Explicit formulas are obtained that determine the effect of dislocations on the dependences of the torque and longitudinal force on the twist angle and axial elongation. The main results are formulated in a form that allows experimental verification.

The Poisson Ratio of the Cellular Actin Cortex Is Frequency Dependent

Cell shape changes are vital for many physiological processes such as cell proliferation, cell migration, and morphogenesis. They emerge from an orchestrated interplay of active cellular force generation and passive cellular force response, both crucially influenced by the actin cytoskeleton. To model cellular force response and deformation, cell mechanical models commonly describe the actin cytoskeleton as a contractile isotropic incompressible material. However, in particular at slow frequencies, there is no compelling reason to assume incompressibility because the water content of the cytoskeleton may change. Here, we challenge the assumption of incompressibility by comparing computer simulations of an isotropic actin cortex with tunable Poisson ratio to measured cellular force response. Comparing simulation results and experimental data, we determine the Poisson ratio of the cortex in a frequency-dependent manner. We find that the Poisson ratio of the cortex decreases in the measured frequency regime analogous to trends reported for the Poisson ratio of glassy materials. Our results therefore indicate that actin cortex compression or dilation is possible in response to acting forces at sufficiently fast timescales. This finding has important implications for the parameterization in active gel theories that describe actin cytoskeletal dynamics.

The multiscale simulation of apical myocardial infarction and shape variation of the left ventricle of the heart

A multiscale model of the cardiovascular system (CVS) in which the left ventricle (LV) of the heart was approximated by an axisymmetrical thick-wall body made of transversely isotropic incompressible material was used to simulate the performance of the heart with apical myocardial infarction (MI). The material model reproduced mechanical properties and calcium regulation of active tension in cardiac muscle. The changes in the LV strain and the reduction of the LV stroke volume and arterial blood pressure obtained in the MI simulations were similar to those observed in patients with the apical MI. In contrast to the decrease in heart performance in the MI simulations, the simulation of changes in the LV shape from “normal” to a spherical or conical one revealed only slight changes in haemodynamics provided that the LV preload and the mass of the LV wall were kept constant.