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.     

基于横观各向同性超弹性理论的短纤维增强橡胶本构模型的建立与应用Establishment and application of short fiber reinforced rubber constitutive model based on transversely isotropic hyperelastic theory

提出了一种能够表征短纤维增强橡胶的横观各向同性超弹性本构模型,并结合试验体系,对其在数值分析中的应用方法和效果进行了研究。基于连续介质力学理论,建立了横观各向同性材料的应变能函数,推导得到不同变形形式下的应力应变关系,给出材料参数辨识试验方法,并成功应用于某短纤维增强橡胶测试中,得到表征其超弹性特性的相关材料参数。利用有限元软件ANSYS对不同纤维排布方向的单轴拉伸和平行纤维方向的平面拉伸进行仿真计算,并对比相应试验数据,以验证材料参数的可靠性。最后基于已验证的本构模型,建立了某铣槽装备减振环仿真模型,并对其进行了校核计算。研究结果表明,本文提出的本构模型能够有效表征短纤维增强橡胶的静态力学特性并且方便嵌入现有的有限元软件中,具有材料参数少、测试简便和结果准确等特点,工程实用性强。     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

A general hyperelastic model for incompressible fiber-reinforced elastomers

This work presents a new constitutive model for the effective response of fiber-reinforced elastomers at finite strains. The matrix and fiber phases are assumed to be incompressible, isotropic, hyperelastic solids. Furthermore, the fibers are taken to be perfectly aligned and distributed randomly and isotropically in the transverse plane, leading to overall transversely isotropic behavior for the composite. The model is derived by means of the “second-order” homogenization theory, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” Compared to other constitutive models that have been proposed thus far for this class of materials, the present model has the distinguishing feature that it allows consideration of behaviors for the constituent phases that are more general than Neo-Hookean, while still being able to account directly for the shape, orientation, and distribution of the fibers. In addition, the proposed model has the merit that it recovers a known exact solution for the special case of incompressible Neo-Hookean phases, as well as some other known exact solutions for more general constituents under special loading conditions.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

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.     

Torsional Swelling of a Hyperelastic Tube with Helically Wound Reinforcement

This paper examines the combination of radial deformation with torsion for a circular cylindrical tube composed of a transversely isotropic hyperelastic material subject to finite deformation swelling. The stored energy function involves separate matrix and fiber contributions such that the fiber contribution is minimized when the fiber direction is at a natural length. This natural length is not affected by the swelling. Hence swelling preferentially expands directions that are orthogonal to the fiber. The swelling itself is described via a swelling field that prescribes the local free volume at each location in the body. Such a treatment is a relatively simple generalization of the conventional incompressible theory. The direction of transverse isotropy associated with the fiber reinforcement is described by a helical winding about the tube axis. The swelling induced preferential expansion orthogonal to this direction induces the torsional aspect of the deformation. For a specific class of strain energy functions we find that the twist increases with swelling and approaches a limiting asymptotic value as the swelling becomes large. The fibers reorient such that fibers at the inner portion of the tube assume a more circumferential orientation whereas, at least for small and moderate swelling, the fibers in the outer portion of the tube assume a more axial orientation. For large swelling the fibers in the outer portion of the tube reorient beyond the axial orientation, and so are described by helices with orientation in the opposite sense to that in the reference configuration.      

On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

The paper compares two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic materials with different elastic moduli in tension and compression. The approaches use different nonlinear constitutive equations describing the elastic deformation of the transversally isotropic bimodulus materials. Both approaches employ the theory of deformation along paths of small curvature and the Kirchhoff-Love hypothesis for the whole laminate to describe the deformation of the isotropic materials. The problem is solved by the method of successive approximations. The solutions of specific problems obtained by the two approaches are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 59–69, June 2008.     

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

The Boussinesq problem, that is, determining the deformation in a hyperelastic half-space due to a point force normal to the boundary, is an important problem of engineering, geomechanics, and other fields to which elasticity theory is often applied. While linear solutions produce useful Greens functions, they also predict infinite displacements and other physically inconsistent results nearby and at the point of application of the load where the most critical and interesting material behavior occurs. To illuminate the deformation due to such a load in the region of interest, asymptotic analysis of the nonlinear Boussinesq problem has been considered in the context of isotropic hyperelasticity. Studies considering transversely isotropic materials have also been broadly used in the linear theory, but have not been treated within the nonlinear framework. In this paper we extend the nonlinearly elastic isotropic analysis to transverse isotropy, producing a more general theory which also better encompasses applications involving layered media. The governing equations for nonlinearly elastic, transversely isotropic solids are derived, conservation laws of elastostatics are invoked, asymptotic forms of the deformation solutions are hypothesized, and the differential equations governing deformation near the point load are determined. The analysis also develops sequences of simple tests to determine if a transversely isotropic material can possibly sustain a finite deflection under the point load. The results are applied to a variety of transversely isotropic materials, and the effects of the anisotropy considered is demonstrated by comparison of the resulting deformation with similar asymptotic solutions in the isotropic theory. Mathematics Subject Classifications (2000) 74B20, 74E10, 74G10, 74G15, 74G70.     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Thermo-Chemo-Electro-Mechanical Formulation of Saturated Charged Porous Solids

A thermo-chemo-electro-mechanical formulation of quasi-static finite deformation of swelling incompressible porous media is derived from a mixture theory including the volume fraction concept. The model consists of an electrically charged porous solid saturated with an ionic solution. Incompressible deformation is assumed. The mixture as a whole is assumed locally electroneutral. Different constituents following different kinematic paths are defined: solid, fluid, anions, cations and neutral solutes. Balance laws are derived for each constituent and for the mixture as a whole. A Lagrangian form of the second law of thermodynamics for incompressible porous media is used to derive the constitutive restrictions of the medium. The material properties are shown to be contained in one strain energy function and a matrix of frictional tensors. A principle of reversibility results from the constitutive restrictions. Existing theories of swelling media should be evaluated with respect to this principle.     

Material symmetry and the evolution of anisotropies in a simple material—I. Change of reference configuration

Noll's rule is used to determine the structure of a material symmetry group written with respect to one reference configuration when the representation of the symmetry with respect to another configuration is the traditional material symmetry group associated with isotropy, transverse isotropy or orthotropy, and for an arbitrary deformation gradient relating the two configurations. It is shown that the former symmetry group can contain an orthogonal subgroup. It is determined whether this subgroup is that for isotropic, transversely isotropic, orthotropic, monoclinic, or triclinic response, and the preferred directions of the symmetry are determined.     

PREDICTION OF THE VISCOELASTIC PROPERTIES OF THE EQUIVALENT PARTICLE FOR THE INTERCALATED MULTI-LAYER STACK OF NANOPLASTICS

The aim of this paper is to apply the asymptotic homogenization method to deter- mining analytically and numerically the transversely isotropic viscoelastic relaxation moduli of the equivalent particle for the intercalated multi-layer stack of intercalated type nanoplastics.A two-phase multilayered material containing n layers is considered.The matrix is assumed to be an isotropic viscoelastic standard linear body and the reinforcement is assumed to be an isotropic elastic body.Final explicit analytical formulae for the effective elastic moduli of the multilay- ered material are derived first;and then the correspondence principle is employed to obtain the homogenized relaxation moduli of the equivalent intercalated particle.A numerical example is given.Final explicit analytical formulae in the time domain derived here make it convenient to estimate the influence of all the particle parameters of micro-structural details on the effective properties of the equivalent intercalated particle.The results of this paper can also be applied to multi-layer composites.     

A finite-deformation theory of plasticity

Based on a multiplicative decomposition of local deformation into elastic and plastic deformations general constitutive equations of elastic-plastic materials are proposed. Two alternative approaches are discussed: one in which the elastic deformation is used as an independent variable, and the other in which the stress is one of the independent variables. The appropriate material symmetries are defined, and it is shown that the plastic spin is absent in the theory of isotropic materials. Analysis of a simple extension is given as an example.     

气泡在液体中运动过程的数值模拟

本文用数值方法预测气泡在液体中的百定常运动。运用位标函数进行界面的隐含跟踪并且与有限体积法相结合构成一种可行的计算方法。本文方法允许在界面处存在很大的物性差，而且较容易将表面张力引入控制方程。我们对气液两相流中单个气泡的运动进行了计算，得到了与实验结果符合很好的数值结果。     

On thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic solids with application to biomechanics

In this paper, we are interested in developing thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids, in isothermal processes. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation: 10 of them are isotropic invariants and 8 of them are associated with the deformation and the orientation of the fiber. Among the 8 anisotropic invariants just 6 are related to the viscoelastic response. The terms in the Cauchy stress tensor associated to these 6 invariants are analyzed with respect to thermodynamical consistency, and we obtain restrictions for the corresponding constitutive coefficients. This framework is applied to viscoelastic potentials within the context of biomaterials.     

Deformation and Microdamage of a Discrete-Fibrous Composite with Transversely Isotropic Components

The theory of microdamage for materials with a transversely isotropic matrix and unidirectional ellipsoid-like fibers is set forth. Microdamage is modeled by empty pores. The failure criterion for a microvolume is assumed to have the Huber–Mises form where the ultimate strength is a random function of coordinates with a power or Weibull distribution. The stress–strain state and effective properties of the material are determined from the theory of elasticity for materials with a transversely isotropic matrix and unidirectional fibers. The deformation and microdamage equations are closed by the porosity-balance equations. The nonlinear dependences of the coupled processes of deformation and microdamage on macrodeformations are constructed. The effect of physical and geometrical parameters on the processes is studied