A continuum mechanics framework and a constitutive model for remodelling of collagen gels and collagenous tissues

Collagen is a very important protein of the human body and is responsible for the structural stability of many body components. Furthermore, collagen fibre networks are able to grow and remodel themselves, which enables them to adjust to varying physiological conditions. This remodelling is accomplished by fibre-producing cells, such as fibroblasts. The ability to adjust to new physiological conditions is very important, for example in wound healing. In the present paper, a theoretical framework for modelling collagenous tissues and collagen gels is proposed. Continuum mechanics is employed to describe the kinematics of the collagen, and affine deformations of fibres are assumed. Biological soft tissues can be approximated as being hyperelastic, and the constitutive model for the collagen fabric is therefore formulated in terms of a strain energy function. This strain energy function includes a density function that describes the distribution of the collagen fibre orientation. The density function evolves according to an evolution law, where fibres tend to reorient towards the direction of maximum Cauchy stress. The remodelling of the collagen network is also assumed to include a pre-stretching of collagen fibres, accomplished by fibroblasts. The theoretical framework is applied to experiments performed on collagen gels, where gels were exposed to remodelling under both biaxial and uniaxial constraints. The proposed model was able to predict both the resulting collagen distribution and the resulting stress-strain relationships obtained for the remodelled collagen gels. The influence of the most important model parameters is demonstrated, and it appears that there is a fairly unique set of model parameters that gives an optimal fit to the experimental data.     

Phenomenological constitutive model for a CNT turf

Carbon nanotubes (CNT), grown on a substrate, form a turf – a complex structure of intertwined, mostly nominally vertical tubes, cross-linked by adhesive contact and few bracing tubes. The turfs are compliant and good thermal and electrical conductors. In this paper, we consider the micromechanical analysis of the turf deformation reported earlier, and develop a phenomenological constitutive model of the turf. We benchmark the developed model using a finite element implementation and compare the model predictions to the results two different nanoindentation tests.The model includes: nonlinear elastic deformation, small Kelvin–Voigt type relaxation, caused by the thermally activated sliding of contacts, and adhesive contact between the turf and the indenter. The pre-existing (locked-in) strain energy of bent nanotubes produces a high initial tangent modulus, followed by an order of magnitude decrease in the tangent modulus with increasing deformation. The strong adhesion between the turf and indenter tip is due to the van der Waals interactions.The finite element simulations capture the results from the nanoindentation experiments, including the loading, unloading, viscoelastic relaxation during hold, and adhesive pull-off.     

Rate dependent finite strain constitutive model of polyurea

Continuous loading and unloading experiments are performed at different strain rates to characterize the large deformation behavior of polyurea under compressive loading. In addition, uniaxial compression tests are carried out with multistep strain history profiles. The analysis of the experimental data shows that the concept of equilibrium path may not be applied to polyurea. This finding implies that viscoelastic constitutive models of the Zener type are no suitable for the modeling of the rate dependent behavior of polyurea. A new constitutive model is developed based on a rheological model composed of two Maxwell elements. The soft rubbery response is represented by a Gent spring while nonlinear viscous evolution equations are proposed to describe the time-dependent material response. The eight material model parameters are identified for polyurea and used to predict the experimentally-measured stress-strain curves for various loading and unloading histories. The model provides a good prediction of the response under monotonic loading over wide range of strain rates, while it overestimates the stiffness during unloading. Furthermore, the model predictions of the material relaxation and viscous dissipation during a loading-unloading cycle agree well with the experiments.     

A continuum constitutive model for the active behaviour of skeletal muscle

In the present paper we propose a continuum constitutive model for the passive and active mechanical behaviour of skeletal muscle. Unlike most works in this field, the model is not based on an additive split between passive and active components but considers muscle tissue as one continuous biological material, which alters its properties when activated. This alteration also allows for a kinematic interpretation on the muscle fibre level and is described by a single activation-dependent model parameter. This as well as the other material parameters are obtained from standard experiments on resting and activated muscle or from microstructural information such as fibre type and twitch characteristics. In the passive state, the constitutive equations are governed by a transversely isotropic polyconvex and coercive strain-energy function. The model shows excellent agreement with experimental stress-stretch data of a passive and activated rat tibialis anterior muscle.     

An efficient constitutive model for room-temperature,low-rate plasticity of annealed Mg AZ31B sheet

Metals and alloys with hexagonal close packed (HCP) crystal structures can undergo twinning in addition to dislocation slip when loaded mechanically. The complexity of the plastic response and the limited extent of twinning are impediments to their room-temperature formability and thus their widespread adoption. In order to exploit the unusual deformation characteristics of twinning sheet materials in designing novel forming operations, a practical plane stress material model for finite element implementation was sought. Such a model, TWINLAW, has been constructed based on three phenomenological deformation modes for Mg AZ31B: S (slip), T (twinning), and U (untwinning). The modes correspond to three testing regimes: initial in-plane tension (from the annealed state), initial in-plane compression, and in-plane tension following compression, respectively. A von Mises yield surface with initial non-zero back stress was employed to account for plastic yielding asymmetry, with evolution according to a novel isotropic and nonlinear kinematic hardening model. Texture and its evolution were represented throughout deformation using a weighted discrete probability density function of c-axis orientations. The orientation of c-axes evolves with twinning or untwinning using explicit rules incorporated in the model.     

Influence of isotropic and anisotropic material models on the mechanical response in arterial walls as a result of supra-physiological loadings

As accepted in the literature, arterial tissues have in principle anisotropic material properties. Although some very special situations in arteries exist where isotropic constitutive models may approximate the real material behavior with sufficient accuracy, the larger part of analyses requires an anisotropic model. In particular for overstretched arteries, as e.g. a result of a balloon angioplasty, an accurate representation of the complex softening phenomena is important and then the consideration of anisotropy may be necessary. However, a variety of publications found in the literature, where such supra-physiological loading situations are analyzed to optimize e.g. stent designs, consider isotropic models. Therefore, in this contribution, the response of an isotropic and an anisotropic material model is compared in numerical calculations where arteries are subjected to supra-physiological loading. The constitutive formulations include the typical nonlinear stiffening of the fiber response as well as softening due to microscopic damage. In detail, the isotropic and the anisotropic model are adjusted to the same experimental stress–stretch curves of different arterial layers and then both models are applied to finite element simulations of overstretched arterial walls. As it turns out a significant difference is obtained for both calculations showing the importance of anisotropic models for these loading situations.     

A constitutive model of aluminum foam for crash simulations

A new constitutive model for metallic foams is developed to overcome the deficiency of existing models in commercial finite element codes. The proposed constitutive model accounts for volume changes under hydrostatic compression and combines the hydrostatic pressure and von Mises stress into one yield function. The change of the compressibility of the metallic foam is handled in the constitutive model by allowing for shape changes of the yield surface in the hydrostatic pressure-von Mises stress space. The backward Euler method is adopted to integrate the constitutive equations to achieve numerical accuracy and stability. The model is implemented into LS-DYNA as a user-defined subroutine, verified with existing solutions, and validated with foam testing data. The verified and validated model is then utilized in the crushing simulations of foam-filled columns with square and hexagonal cross-sections. Two constitutive models are studied: the first using an exponential function to describe the relationship of plastic Poisson's ratio with respect to true strain and the second using linear interpolation function as an alternative approximation. The new foam model provides satisfactory prediction of crushing forces and deformed shapes compared to experimental results. Additionally, the new foam model was shown to have better numerical stability and accuracy than existing LS-DYNA built-in material models.     

The study of equivalent material parameters in a hyperelastic model

We study a hyperelastic model of some biological soft tissues with emphasis on the problem of its matching with the material parameters acquired by experimental mechanical tests. First, we study the polyconvexity property of the hyperelastic model. Then, we explore the notion of equivalent sets of material parameters. We perform a numerical study of the regions of equivalent material parameters characterizing the curves predicted by the hyperelastic model that are close, within a prefixed tolerance, to those given by the experimental data. In the numerical study we use the quadratic variation and the Hausdorff distance. The study suggests that a qualitative knowledge of shape and volume of the regions of equivalent material parameters can provide both a criterion for the optimal match between the model with the experimental data and an indication on the reducibility of the number of parameters used in the model.     

A novel constitutive model for semiconductors: The case of silicon

The photovoltaic industry relies heavily on solar-grade silicon multicrystals. Understanding their mechanical behavior requires the development of adequate constitutive models accounting for the effects of both high dislocation densities and complex loading situations in a wide range of temperature, strain rate, and impurity contents. The traditional model of Alexander and Haasen poses several limitations. We introduce in this work a novel constitutive model for covalent single crystals and its implementation into a rate-dependent crystal plasticity framework. It is entirely physically based on the dislocation generation, storage and annihilation processes taking place during plastic flow. The total dislocation density is segmented according to the dislocation mobility potential and their character. A dislocation multiplication law for the yield region more accurate than the one of Alexander and Haasen is proposed. The influence of additional dislocation sources created on forest trees, usually disregarded in models for semiconductors, is assessed. The dislocation velocity law combines three potentially rate-limiting mechanisms: the standard double kink mechanism, jog dragging and the influence of localized obstacles. The model is valid at finite strains, in multiple slip conditions and captures accurately the high temperature- and strain rate sensitivity of semiconductors. The experimental stress overshoot is qualitatively reproduced only when jog dragging is accounted for. Localized obstacles are shown not to have any significant effect on dislocation motion in silicon. The broader case of extrinsic semiconductors is discussed and the influence of dissolved oxygen on the upper yield stress of silicon monocrystals is successfully reproduced.     

A constitutive model for the formation of martensite in austenitic steels under large strain plasticity

A constitutive model for diffusionless phase transitions in elastoplastic materials undergoing large deformations is developed. The model takes basic thermodynamic relations as its starting point and the phase transition is treated through an internal variable (the phase fractions) approach. The usual yield potential is used together with a transformation potential to describe the evolution of the new phase. A numerical implementation of the model is presented, along with the derivation of a consistent algorithmic tangent modulus. Simulations based on the presented model are shown to agree well with experimental findings. The proposed model provides a robust tool suitable for large-scale simulations of phase transformations in austenitic steels undergoing extensive deformations, as is demonstrated in simulations of the necking of a bar under tensile loading and also in simulations of a cup deep-drawing process.     

A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation

A constitutive model to describe macroscopic elastic and transformation behaviors of polycrystalline shape-memory alloys is formulated using an internal variable thermodynamic framework. In a departure from prior phenomenological models, the proposed model treats initiation, growth kinetics, and saturation of transformation distinctly, consistent with physics revealed by recent multi-scale experiments and theoretical studies. Specifically, the proposed approach captures the macroscopic manifestations of three micromechanial facts, even though microstructures are not explicitly modeled: (1) Individual grains with favorable orientations and stresses for transformation are the first to nucleate martensite, and the local nucleation strain is relatively large. (2) Then, transformation interfaces propagate according to growth kinetics to traverse networks of grains, while previously formed martensite may reorient. (3) Ultimately, transformation saturates prior to 100% completion as some unfavorably-oriented grains do not transform; thus the total transformation strain of a polycrystal is modest relative to the initial, local nucleation strain. The proposed formulation also accounts for tension–compression asymmetry, processing anisotropy, and the distinction between stress-induced and temperature-induced transformations. Consequently, the model describes thermoelastic responses of shape-memory alloys subject to complex, multi-axial thermo-mechanical loadings. These abilities are demonstrated through detailed comparisons of simulations with experiments.     

Formulation of a damage internal state variable model for amorphous glassy polymers

The following article proposes a damage model that is implemented into a glassy, amorphous thermoplastic thermomechanical inelastic internal state variable framework. Internal state variable evolution equations are defined through thermodynamics, kinematics, and kinetics for isotropic damage arising from two different inclusion types: pores and particles. The damage arising from the particles and crazing is accounted for by three processes of damage: nucleation, growth, and coalescence. Nucleation is defined as the number density of voids/crazes with an associated internal state variable rate equation and is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing as an internal state variable rate equation that is a function of stress state, rate sensitivity, and strain rate. The coalescence internal state variable rate equation is an interactive term between voids and crazes and is a function of the nearest neighbor distance of voids/crazes and size of voids/crazes, temperature, and strain rate. The damage arising from the pre-existing voids employs the Cocks–Ashby void growth rule. The total damage progression is a summation of the damage volume fraction arising from particles and pores and subsequent crazing. The modeling results compare well to experimental findings garnered from the literature. Finally, this formulation can be readily implemented into a finite element analysis.     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

A unified constitutive theory for polymeric liquids I. Basic considerations and a simplified model

By combining a continuum mechanical approach with considerations of network theory and thermodynamics of irreversible processes, a set of differentialtype constitutive equations for polymeric liquids are obtained which provide expressions for the stress tensor, evolution equations of the effective Finger strain and Cauchy strain for the network deformation, and a first order differential equation governing the rigidity modulus. Unlike Giesekus' recent unified approach that starts from the bead-spring model, the theory lends itself more readily to a better understanding of most of the current theories based on continuum mechanics and molecular network concepts. Different recent models such as those due to Leonov, Dashner—Van Arsdale, Phan Thien—Tanner, and Acierno et al. (or Marrucci) can be unambiguously interpreted as resulting from specific approximations or additional assumptions.     

各向同性超弹性本构模型数值计算及验证

现有多种形式的橡胶本构模型试图预测橡胶力学性质,其中部分模型已写入有限元软件中用于仿真计算,还存在较多拟合性较好的模型无法在有限元材料库中直接获得。本文详述了由不变量和主伸长率描写的各向同性超弹性本构模型的数值实现方法,并结合最新的本构模型开发了UHYPER和UMAT子程序。将UHYPER用于有限元实现对多孔橡胶板的拉伸仿真,对比仿真和试验结果,验证子程序的正确性以及评估本构模型预测复杂应变场的准确性;将UMAT用于单轴、等双轴和剪切拉伸的有限元仿真,对比仿真和本构模型理论结果,验证子程序的可靠性。结果表明,有限元仿真结果与理论结果拟合较好,子程序能够契合本构模型的力学描述,所述方法可以用于超弹性材料的数值计算。     

A new constitutive model for monodispersed suspensions of spheres at high concentrations

We describe a constitutive model for monodispersed concentrated suspensions of spheres at high volume fractions. In this model the motion of a pair of generic spheres is represented by that of a pair of force-free and torque-free spheres tumbling along with the flow. The interaction with the surrounding spheres is modelled by an anisotropic diffusion process. Viscometric and elongational properties of the model are obtained by a Brownian dynamic simulation. Received: 24 September 1998 Accepted: 1 December 1998     

Towards a constitutive law for the unsteady contact stress in granular media

In order to build a unified modelling for granular media by means of Eulerian averaged equations, it is necessary to study two contributions in the effective Cauchy stress tensor: the first one concerns solid and fluid matter, including contact and collisions between grains; the second one focuses on the random movements of grains and fluid, similar to Reynolds stress for turbulent flows. It is shown that the point of view of piecewise continuous media already used for two phase flows allows one to derive a constitutive equation for the first contribution, under the form of a partial differential equation. Similarly as for the Reynolds stress in turbulent flows, this equation cannot be written only in terms of averaged quantities without adequate approximations. The structure of the closed equation is discussed with respect to irreversible thermodynamics, and in connection with some already existing models. It is emphasised that numerical simulations by the discrete elements method can be used in order to validate these approximations, through numerical experiments statistically considered. Finally an extension of this approach to the second contribution of the effective Cauchy stress tensor is briefly discussed, showing how the modelling of both contributions have to be coupled.