A structural multi-mechanism constitutive equation for cerebral arterial tissue

A structural multi-mechanism constitutive equation is developed to describe the nonlinear, anisotropic, inelastic mechanical behavior of cerebral arterial tissue. Elastin and collagen fibers are treated as separate components (mechanisms) of the artery. Elastin is responsible for load bearing at low strain levels while the collagen mechanism is recruited for load bearing at higher strain levels. This work builds on an earlier model in which both the elastin and collagen mechanisms are represented by isotropic response functions [Wulandana, R., Robertson, A.M., 2005. An inelastic multi-mechanism constitutive equation for cerebral arterial tissue. Biomech. Model. Mechan. 4 (4), 235–248]. Here, the anisotropic material response of the wall is introduced through the collagen mechanism which is composed of helically distributed families of fibers. The orientation of these families is described using either a finite number of fiber orientations or a fiber distribution function. The fiber orientation or dispersion function can be prescribed directly from arterial histology data, or, taking a phenomenological approach, based on data fitting from bi-axial measurements. The activation of the collagen mechanism is specified using a new fiber strain based activation criterion. The multi-mechanism constitutive equation is applied to the simple case of cylindrical inflation and material constants are determined based on available inelastic experimental data for cerebral arteries. While the proposed model captures all features of this inelastic data, there is a pressing need for further experiments to refine the model.     

短纤维增强EPDM包覆薄膜超弹性本构模型

短纤维增强三元乙丙橡胶(EPDM)包覆薄膜用于一种新型缠绕包覆工艺,主要解决复杂构型自由装填药柱外表面可靠性包覆问题.为了描述其在固体火箭发动机工作过程中产生的大变形、非线性和各向异性等力学行为,根据纤维增强复合材料连续介质力学理论,提出了各向异性超弹性本构模型.该模型中单位体积的应变能函数被解耦成两部分:表征各向同性的橡胶基体应变能和表征各向异性的纤维拉伸应变能,通过引入纤维方向对纤维应变能进行修正,给出了通过单轴拉伸、偏轴拉伸实验数据获取模型参数的具体方法.研究结果表明,该模型能够很好地预测材料在纤维方向0°~45°时的各向异性力学特性,并将预测结果与实验数据对比,误差在5%以下.所建立的各向异性超弹性本构模型准确性高、易于实现数值开发,在一定程度上能够为固体火箭发动机的装药结构完整性分析提供理论依据.     

Modeling of anisotropic wound healing

Biological soft tissues exhibit non-linear complex properties, the quantification of which presents a challenge. Nevertheless, these properties, such as skin anisotropy, highly influence different processes that occur in soft tissues, for instance wound healing, and thus its correct identification and quantification is crucial to understand them. Experimental and computational works are required in order to find the most precise model to replicate the tissues' properties. In this work, we present a wound healing model focused on the proliferative stage that includes angiogenesis and wound contraction in three dimensions and which relies on the accurate representation of the mechanical behavior of the skin. Thus, an anisotropic hyperelastic model has been considered to analyze the effect of collagen fibers on the healing evolution of an ellipsoidal wound. The implemented model accounts for the contribution of the ground matrix and two mechanically equivalent families of fibers. Simulation results show the evolution of the cellular and chemical species in the wound and the wound volume evolution. Moreover, the local strain directions depend on the relative wound orientation with respect to the fibers.     

Mechanical response of a short fiber-reinforced thermoplastic: Experimental investigation and continuum mechanical modeling

The present work can be regarded as a first step toward an integrated modeling of mold filling during injection molding process of polymer composites and the resulting material behavior under service loading conditions. More precisely, the emphasis of the present paper is laid on how to account for local fiber orientation in the ground matrix on the prediction of the mechanical response of the composite at its final solid state. To this end, a set of experiments which captures the mechanical behavior of an injection molded short fiber-reinforced thermoplastic under different strain histories is described. It is shown that the material exhibits complex response mainly due to non-linearity, anisotropy, time/rate-dependence, hysteresis and permanent strain. Furthermore, the relaxed state of the material is characterized by the existence of an equilibrium hysteresis independently of the applied strain rate. A three-dimensional phenomenological model to represent experimentally observed response is developed. The microstructure configuration of the material is simplified and assumed to be entirely represented by a distributed fiber orientation in the ground matrix. In order to account for distributed short fiber orientations in a continuum sense, a concept of (symmetric) generalized structural tensor (tensor of orientation) of second order is adopted. The proposed model is based on assumption that the strain energy function of the composite is given by a linear mixture of the strain energy of each constituent: an isotropic part representing Phase 1 which is essentially related to the ground matrix and an anisotropic part describing Phase 2 which is mainly related to the fibers and the interphase as a whole. Hence, taking into account the fiber content and orientation, the efficiency of the model is assessed and perspectives are drawn.     

帘线/橡胶复合材料各向异性黏-超弹性本构模型

帘线/橡胶复合材料广泛应用于轮胎等重要工程领域,为了描述其在服役条件下的大变形、非线性、各向异性和高应变率等材料力学行为,基于纤维增强复合材料连续介质力学理论,提出了一种考虑应变率效应的帘线/橡胶复合材料各向异性黏-超弹性本构模型. 该模型中单位体积的应变能被解耦为便于参数识别的基体等容变形能、帘线拉伸变形能、剪切应变能和黏性应变能四部分. 给出了模型参数的确定方法,并通过拟合文献中单轴拉伸、偏轴拉伸实验数据,得到了模型参数. 利用该模型预测了不同加载和变形条件下的力学行为,并将预测结果与实验结果对比分析. 结果表明, 考虑黏性模型和不考虑黏性模型对不同应变率变形条件下的预测结果相差很大,且考虑黏性模型的预测结果与实验结果吻合很好. 因此,与不考虑黏性模型相比,所提出的各向异性黏-超弹性本构模型能更好地表征帘线/橡胶复合材料在大变形、高应变率条件下的力学特性.      

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.     

Visco-electro-elastic models of fiber-distributed active tissues

We present a constitutive model for stochastically distributed fiber reinforced visco-active tissues, where the behavior of the reinforcement depends on the relative orientation of the electric field. Following our previous works, for the passive behaviors we adopt a second order approximation of the strain energy density associated to the parameters of the fiber distribution. Consistently, we also assume that the active behavior accounts for the stochastic distribution of the fibers. The ensuing mechanical quantities result to be dependent on two average structure tensors. We introduce an extended Helmholtz free energy density characterized by the inclusion of a directional active potential, dependent on a stochastic anisotropic permittivity tensor. The permittivity tensor is expanded in Taylor series up to the second order, allowing to obtain an approximated active potential with the same structure of the passive Helmholtz free energy density. In particular, the explicit expression of active stress and stiffness are dependent on the two average structure tensors that characterize the passive response. Anisotropy follows from the fiber distribution and inherits its stochastic nature through statistics parameters. The active fiber distributed model is extended here to viscous materials by including the contribution of a dual dissipation potential in the variational formulation of the constitutive updates. Additionally, we present a computational example of application of the electro-viscous-mechanical material model by simulating peristaltic contractions on a portion of human intestine.     

A mixed Von Mises distribution for modeling soft biological tissues with two distributed fiber properties

Constitutive modeling of biological tissues plays an important role in the understanding of tissue behavior and the development of synthetic materials for medical and bio-inspired applications. A structural continuum model that incorporates principal structural features of the tissue can potentially provide the link between microstructure and the macroscopic mechanical response of biological tissues. For most soft biological tissues, including arterial walls and skin tissue, the main load-carrying constituent is presumed to be the distributed collagen fibers embedded in a base matrix. It is believed that the organization of the collagen fibers gives rise to the anisotropy of the material. In this paper, a semi-structural constitutive model is proposed to account for planar fiber distributions with more than one distributed planar fiber property. Motivated by histology information of the wing membrane of the bat, a statistical treatment is formulated in this paper to capture the overall effect of the distribution of fiber cross-sectional area and the distribution of the number of fibers. This formulation is suitable for general cases when more than one fiber property varies spatially. Furthermore, this model is a two-dimensional specialization within the framework of a three-dimensional theory, which is different the formulation based on a fundamentally two-dimensional theory.     

Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites

This paper presents constitutive models for the anisotropic, finite-deformation viscoelastic behavior of soft fiber-reinforced composites. An essential assumption of the models is that both the fiber reinforcements and matrix can exhibit distinct time-dependent behavior. As such, the constitutive formulation attributes a different viscous stretch measure and free energy density to the matrix and fiber phases. Separate flow rules are specified for the matrix and the individual fiber families. The flow rules for the fiber families then are combined to give an anisotropic flow rule for the fiber phase. This is in contrast to many current inelastic models for soft fiber-reinforced composites which specify evolution equations directly at the composite level. The approach presented here allows key model parameters of the composite to be related to the properties of the matrix and fiber constituents and to the fiber arrangement. An efficient algorithm is developed for the implementation of the constitutive models in a finite-element framework, and examples are presented examining the effects of the viscoelastic behavior of the matrix and fiber phases on the time-dependent response of the composite.     

短纤维增强三元乙丙橡胶横观各向同性黏-超弹性本构模型

短纤维增强三元乙丙橡胶包覆薄膜,是一种应用于固体火箭发动机缠绕包覆装药的新型复合材料.为了描述其在工作过程中受振动、冲击等载荷作用时的力学行为,基于黏弹性理论和纤维增强连续介质力学理论,提出了一种考虑应变率强化效应的横观各向同性黏-超弹本构模型.模型中应变能函数被分解为超弹性应变能和黏性应变能,其中超弹性应变能包括表征各向同性的橡胶基体应变能和表征各向异性的纤维拉伸应变能,黏性应变能采用表征橡胶和纤维黏性响应的宏观唯象模型.选取表征各应变能的函数形式,经过数学变换、替代、叠加,求解确定最终的应力应变形式,明确模型参数获取的具体步骤,将预测结果与实验结果对比分析,准确性较高.研究表明:该模型能有效预测材料在低应变率下纤维方向为0?～45?的非线性率相关力学特性;模型形式易于实现有限元开发,对固体火箭发动机装药结构完整性分析具有参考价值.     

Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling

A fully three-dimensional anisotropic elastic model for vascular tissue modelling is presented here. The underlying strain energy density function is assumed to additively decouple into volumetric and deviatoric contributions. A straightforward isotropic neo-Hooke-type law is used to model the deviatoric response of the ground substance, whereas a micro-structurally or rather micro-sphere-based approach will be employed to model the contribution and distribution of fibres within the biological tissue of interest. Anisotropy was introduced by means of the use of von Mises orientation distribution functions. Two different micro-mechanical approaches—a, say phenomenological, exponential ansatz, and a worm-like-chain-based formulation—are applied to the micro-fibres and illustratively compared. The passage from micro-structural contributions to the macroscopic response is obtained by a computational homogenisation scheme, namely numerical integration over the surface of the individual micro-spheres. The algorithmic treatment of this integration is discussed in detail for the anisotropic problem at hand, so that several cubatures of the micro-sphere are tested in order to optimise the accuracy at reasonable computational cost. Moreover, the introduced material parameters are identified from simple tension tests on human coronary arterial tissue for the two micro-mechanical models investigated. Both approaches are able to recapture the experimental data. Based on the identified sets of parameters, we first discuss a homogeneous deformation in simple shear to evaluate the models’ response at the micro-structural level. Later on, an artery-like two-layered tube subjected to internal pressure is simulated by making use of a non-linear finite element setting. This enables to obtain the micro- and macroscopic responses in an inhomogeneous deformation problem, namely a blood vessel representative boundary value problem. The effect of residual stresses is additionally included in the model by means of a multiplicative decomposition of the deformation gradient tensor which turns out to crucially affect the simulation results.     

A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus

This paper presents a composites-based hyperelastic constitutive model for soft tissue. Well organized soft tissue is treated as a composite in which the matrix material is embedded with a single family of aligned fibers. The fiber is modeled as a generalized neo-Hookean material in which the stiffness depends on fiber stretch. The deformation gradient is decomposed multiplicatively into two parts: a uniaxial deformation along the fiber direction and a subsequent shear deformation. This permits the fiber-matrix interaction caused by inhomogeneous deformation to be estimated by using effective properties from conventional composites theory based on small strain linear elasticity and suitably generalized to the present large deformation case. A transversely isotropic hyperelastic model is proposed to describe the mechanical behavior of fiber-reinforced soft tissue. This model is then applied to the human annulus fibrosus. Because of the layered anatomical structure of the annulus fibrosus, an orthotropic hyperelastic model of the annulus fibrosus is developed. Simulations show that the model reproduces the stress-strain response of the human annulus fibrosus accurately. We also show that the expression for the fiber-matrix shear interaction energy used in a previous phenomenological model is compatible with that derived in the present paper.     

Hyperelastic modeling of location-dependent human distal femoral cartilage mechanics

Knee articular cartilage exhibits complex mechanical behavior, even under high strain rates, which poses a challenge to developing accurate and efficient cartilage models. In particular, the tissue׳s stress–strain response is non-linear and the stiffness of the response is location-dependent. Hyperelastic models such as those of Alan Gent and others have increasingly found use in soft tissue biomechanics. Recently, a hyperelastic statistical chain network model representing the transverse isotropy of the collagen matrix in the superficial tangential zone has been developed. The model successfully simulated the 100% strain/s unconfined compression response of human proximal tibial cartilage. Moreover, spatial variations in the tangent modulus to the nominal stress–strain curve taken at 10% strain were reflected in the variability of a single parameter of the model. Given the success of the model, we desired to determine whether these outcomes are equally applicable to healthy human distal femoral cartilage so that a complete model of tibiofemoral joint cartilage can be developed. The transversely isotropic model was employed along with two other hyperelastic chain network models to determine which model best simulated unconfined compression data for healthy distal femoral cartilage. The transversely isotropic model fit the data excellently (R²=0.999). The model was subsequently simplified to depend on a single parameter and reapplied to the dataset. The modified model maintained an excellent fit to the data (R²=0.999), and its single parameter varied in a statistically similar regional pattern (p<0.05) to the experimentally-obtained elastic modulus of the tissue. Outcomes suggest that this model is suitable for modeling the spatially-varying, non-linear mechanics of healthy human distal femoral cartilage. Implementation of this constitutive relation within computational models of the knee will provide novel insight into the relationship between joint mechanics, cartilage loading, and knee osteoarthritis development.     

Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part II: Concrete material application

On the basis of the strong discontinuity analysis, a discrete model expressed in terms of traction vector-displacement jump has been constructed from a continuous model expressed in terms of stress–strain law. In the first part of the paper, this approach has been extended to a class of anisotropic continuum damage constitutive models [1]. In this second part of the paper, the proposed class of discrete anisotropic damage constitutive models is particularized. More precisely, a micromechanical-based anisotropic damage constitutive model is derived. This model accounts in a natural manner for particular crack families orientation. The aims of this paper are (i) to illustrate the capabilities of the proposed discrete enhanced model in reproducing the induced anisotropy appearing in quasi-brittle materials when cracking and (ii) to assess the numerical robustness of the time integration scheme. For this purpose, two numerical examples at the material point level are exposed after a brief presentation of the time integration scheme. The correspondence between the continuous and the discrete model as well as the induced anisotropy features are emphasized.     

A variational approach for materially stable anisotropic hyperelasticity

In this work we propose an anisotropic stored energy function which satisfies a priori the Legendre–Hadamard condition, which is strongly related to the material stability of the constitutive equations. In the linearized case this condition implies positive wave speeds. The Legendre–Hadamard condition plays also an important role for the (local) existence of solutions in the neighborhood of stationary points. We apply the proposed hyperelastic energies to soft tissues and compare the formulation with existing models which have been used for the calculation of medial collateral ligament and arterial walls. In our numerical and analytical investigations we discuss the distribution of wave speeds for a sequence of deformation states containing some essential stress–strain characteristics of the compared models.     

Optimal material orientation of linear and non-linear elastic 3D anisotropic materials

Optimal material orientation problems of linear and non-linear elastic three-dimensional anisotropic materials are studied. Most commonly, the energy based formulation is applied for solving orientational design problems of anisotropic materials, considering elastic energy density as a measure of the stress strain state. The same approach is used in the current study, but the strength criteria based approaches are also discussed. A simple relation between the stationary conditions in terms of Euler angles and the optimality conditions in terms of strains is pointed out. The complexity analysis of the different existing optimality conditions has been performed. The solution of the posed optimization problem is decomposed into the strain level solution, search for global extremes and evaluation of Euler angles (parameters). The results obtained are extended to some nonlinear elastic material models.     

Evolution of plastic anisotropy in amorphous polymers during finite straining

The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce, and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran, and Boyce [in press].     

On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre–Hadamard condition automatically. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results.