Modeling rate-independent hysteresis in large deformations of preconditioned soft tissues

A phenomenological model is proposed for characterizing rate-independent hysteresis exhibited by preconditioned soft tissues. The preconditioned tissue is modeled as an isotropic composite of a hyperelastic component and a dissipative (inelastic) component. Specifically, the constitutive equations are hyperelastic in the sense that the stress is determined by derivatives of a strain energy function. Inelasticity of the dissipative component is controlled by a yield function with different functional forms for the hardening variable during deformation loading and unloading. The constitutive equations proposed in this paper are simple. In particular, they depend on only seven material constants: three controlling the response of the elastic component and the remainder controlling the response of the dissipative component. More importantly, the material constants can be determined to match rather general loading and unloading behavior. It is observed that the hysteretic response of the model compares well with experimental data for passive uniaxial loading/unloading of Manduca muscle. Moreover, the present model treats partial loading and reloading of preconditioned tissue as elastic–plastic response, which is different from the treatment of pseudo-elastic models used in the literature.     

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.     

Simulation of damage evolution in ductile metals undergoing dynamic loading conditions

A set of constitutive equations for large rate-dependent elastic-plastic-damage materials at elevated temperatures is presented to be able to analyze adiabatic high strain rate deformation processes for a wide range of stress triaxialities. The model is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative approach is proposed. It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. In addition, the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure is discussed.Furthermore, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor-elastic corrector technique. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. Numerical simulations of dynamic experiments with different specimens are performed and good correlation of numerical results and published experimental data is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson-Bar experiments.     

A structural constitutive model for smooth muscle contraction in biological tissues

A structural constitutive model that characterizes the active and passive responses of biological tissues with smooth muscle cells (SMCs) is proposed. The model is formulated under the assumption that the contractile units in SMCs and the connected collagen fibers are the active tissue component, while the collagen fibers not connected to the SMCs are the passive tissue component. An evolution law describing the deformation of the active tissue component over time is developed based on the sliding filament theory. In this evolution law the contraction force is the sum of a motor force that initiates contraction, a viscous force that describes the actin–myosin filament sliding, and an elastic force that accounts for the deformation of the cross-bridges. The mechanical response of the collagen fibers is governed by the fiber recruitment process: collagen fibers support load and behave as a linear elastic material only after becoming taut. The proposed structural constitutive model is tested with published active and passive, uniaxial and biaxial experimental data on pig arteries.     

Derivation of strain measures from strand convection models for polymer melts

Constitutive equations for polymer melts often fail to describe realistically experimental data because they do not describe adequately the melt's elastic behaviour; i.e., the instantaneous response of the melt to the imposed deformation. Elastic behaviour is represented in differential constitutive equations by the convected time derivative and in BKZ integral models by the elastic potential function or strain measure. Here we propose a general molecular model to describe strand convection; that is, the instantaneous response of a polymer strand to an imposed deformation. The model includes the affine, Gordon-Schowalter, and apparently the Seth models as special cases. It is found that by requiring the convection to be such that the principal axes of stress and of strain rotate together, the strain measure becomes a function of strain history, and the Lodge-Meissner relationship is satisfied, in accord with experimental observations.     

Microcanonical Entropy and Mesoscale Dislocation Mechanics and Plasticity

A methodology is devised to utilize the statistical mechanical entropy of an isolated, constrained atomistic system to define constitutive response functions for the dissipative driving-force and energetic fields in continuum thermomechanics. A thermodynamic model of dislocation mechanics is discussed as an example. Primary outcomes are constitutive relations for the back-stress tensor and the Cauchy stress tensor in terms of the elastic distortion, mass density, polar dislocation density, and the scalar statistical density.     

A crystal plasticity model that incorporates stresses and strains due to slip gradients

This work is concerned with incorporating the kinematic and stress effects of excess dislocations in a constitutive model for the elastoplastic behavior of crystalline materials. The foundation of the model is a three term multiplicative decomposition of the deformation gradient in which the two classical terms of plastic and elastic deformation are included along with an additional term for long range strain due to the collective effects of excess dislocations. The long range strain is obtained from an assumed density of Volterra edge dislocations and is directly related to gradients in slip. A new material parameter emerges which is the size the region about a continuum point that contributes to long range strains.Using Hookean elasticity, the stress at a point is linearly related to the sum of the elastic plus the long range strain fields. However, the driving force for slip is postulated to be due only to the elastic stress so that the long range stress is a back stress in the constitutive relationship for plastic deformation. A consistent balance of the total deformation rate with the three proposed mechanisms of deformation leads to a set of differential equations that can be solved for the elastic stress, rotation and pressure which then implicitly defines the material state and equilibrium stress. Results from the simulation of a tapered tensile specimen demonstrate that the constitutive model exhibits isotropic and kinematic type hardening effects as well as changes in the pattern of plastic deformation and necking when compared to a material without slip gradient effects.     

Large amplitude oscillatory shear of pseudoplastic and elastoviscoplastic materials

We explore the utility of strain-controlled large amplitude oscillatory shear (LAOS) deformation for identifying and characterizing apparent yield stress responses in elastoviscoplastic materials. Our approach emphasizes the visual representation of the LAOS stress response within the framework of Lissajous curves with strain, strain rate, and stress as the coordinate axes, in conjunction with quantitative analysis of the corresponding limit cycle behavior. This approach enables us to explore how the material properties characterizing the yielding response depend on both strain amplitude and frequency of deformation. Canonical constitutive models (including the purely viscous Carreau model and the elastic Bingham model) are used to illustrate the characteristic features of pseudoplastic and elastoplastic material responses under large amplitude oscillatory shear. A new parameter, the perfect plastic dissipation ratio, is introduced for uniquely identifying plastic behavior. Experimental results are presented for two complex fluids, a pseudoplastic shear-thinning xanthan gum solution and an elastoviscoplastic invert-emulsion drilling fluid. The LAOS test protocols and the associated material measures provide a rheological fingerprint of the yielding behavior of a complex fluid that can be compactly represented within the domain of a Pipkin diagram defined by the amplitude and timescale of deformation.     

Plane strain bending of strain-stiffening rubber-like rectangular beams

This paper is concerned with investigation of the effects of strain-stiffening for the classical problem of plane strain bending by an end moment of a rectangular beam composed of an incompressible isotropic nonlinearly elastic material. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress–stretch response, the stresses and resultant moments are obtained explicitly. While such results are well known for classical constitutive models such as the Mooney-Rivlin and neo-Hookean models, our primary focus is on materials that undergo severe strain-stiffening in the stress–stretch response. In particular, we consider in detail two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of bending that beams composed of such materials can sustain is limited by the constraint. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Numerical analysis and elastic–plastic deformation behavior of anisotropically damaged solids

The present paper is concerned with the numerical modelling of the large elastic–plastic deformation behavior and localization prediction of ductile metals which are sensitive to hydrostatic stress and anisotropically damaged. The model is based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. The formulation relies on a multiplicative decomposition of the metric transformation tensor into elastic and damaged-plastic parts. Furthermore, undamaged configurations are introduced which are related to the damaged configurations via associated metric transformations which allow for the interpretation as damage tensors. Strain rates are shown to be additively decomposed into elastic, plastic and damage strain rate tensors. Moreover, based on the standard dissipative material approach the constitutive framework is completed by different stress tensors, a yield criterion and a separate damage condition as well as corresponding potential functions. The evolution laws for plastic and damage strain rates are discussed in some detail. Estimates of the stress and strain histories are obtained via an explicit integration procedure which employs an inelastic (damage-plastic) predictor followed by an elastic corrector step. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. A variety of large strain elastic–plastic-damage problems including severe localization is presented, and the influence of different model parameters on the deformation and localization prediction of ductile metals is discussed.     

On the mechanics of solids with a growing mass

A general constitutive theory of the stress-modulated growth of biomaterials is presented with a particular accent given to pseudo-elastic soft living tissues. The governing equations of the mechanics of solids with a growing mass are revisited within the framework of finite deformation continuum thermodynamics. The multiplicative decomposition of the deformation gradient into its elastic and growth parts is employed to study the growth of isotropic, transversely isotropic, and orthotropic biomaterials. An explicit representation of the growth part of the deformation gradient is given in each case, which leads to an effective incremental formulation in the analysis of the stress-modulated growth process. The rectangular components of the instantaneous elastic moduli tensor are derived corresponding to selected forms of the elastic strain energy function. Physically appealing structures of the stress-dependent evolution equations for the growth induced stretch ratios are proposed.     

考虑应变率的广义压电热弹理论及其应用

工程中大量材料的形变介于弹性与黏性之间,既具有弹性固体特性,又具有黏性流体特点,即为黏弹性.黏弹性使得材料出现很多力学松弛现象,如应变松弛、滞后损耗等行为.在研究受热载荷作用的多场耦合问题的瞬态响应时,考虑此类问题中的热松弛和应变松弛现象,对准确描述其瞬态响应尤为重要.针对广义压电热弹问题的瞬态响应,尽管已有学者建立了考虑热松弛的广义压电热弹模型,但迄今,尚未计入应变松弛.本文中,考虑到材料变形时的应变松弛,通过引入应变率,在Chandrasekharaiah广义压电热弹理论的基础之上,经拓展,建立了考虑应变率的广义压电热弹理论.借助热力学定律,给出了理论的建立过程并得到了相应的状态方程及控制方程.在本构方程中,引入了应变松弛时间与应变率的乘积项,同时,分别在本构方程和能量方程中引入了热松弛时间因子.其后,该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题.采用拉普拉斯变换及其数值反变换,对问题进行了求解,得到了不同应变松弛时间和热源移动速度下的瞬态响应,即无量纲温度、位移、应力和电势的分布规律,并重点考察了应变率对各物理量的影响效应,将结果以图形形式进行了表示.结果表明:...     

A GENERALIZED PIEZOELECTRIC-THERMOELASTIC THEORY WITH STRAIN RATE AND ITS APPLICATION 1)

工程中大量材料的形变介于弹性与黏性之间, 既具有弹性固体特性, 又具有黏性流体特点, 即为黏弹性. 黏弹性使得材料出现很多力学松弛现象, 如应变松弛、滞后损耗等行为. 在研究受热载荷作用的多场耦合问题的瞬态响应时, 考虑此类问题中的热松弛和应变松弛现象, 对准确描述其瞬态响应尤为重要. 针对广义压电热弹问题的瞬态响应, 尽管已有学者建立了考虑热松弛的广义压电热弹模型, 但迄今, 尚未计入应变松弛. 本文中, 考虑到材料变形时的应变松弛, 通过引入应变率, 在Chandrasekharaiah广义压电热弹理论的基础之上, 经拓展, 建立了考虑应变率的广义压电热弹理论. 借助热力学定律, 给出了理论的建立过程并得到了相应的状态方程及控制方程. 在本构方程中, 引入了应变松弛时间与应变率的乘积项, 同时, 分别在本构方程和能量方程中引入了热松弛时间因子. 其后, 该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题. 采用拉普拉斯变换及其数值反变换, 对问题进行了求解, 得到了不同应变松弛时间和热源移动速度下的瞬态响应, 即无量纲温度、位移、应力和电势的分布规律, 并重点考察了应变率对各物理量的影响效应, 将结果以图形形式进行了表示. 结果表明: 应变率对温度、位移、应力和电势的分布规律有显著影响.     

On some connections between equivalent single material and mixture theory models for fiber reinforced hyperelastic materials

There are two approaches that can be used to model the large strain mechanical response of material systems in which elastic fibers are embedded in an elastic matrix. In the first approach, a fiber reinforced material undergoing large deformation is homogenized in the sense that it is assumed to act as an equivalent single material that is transversely isotropic and hyperelastic. Both constituents then share a common reference configuration, which is typically assumed to be a natural or stress-free configuration for the equivalent single material. The stress depends on a single deformation gradient defined with respect to the natural configuration.In the second approach, the fiber/matrix system is treated as a mixture, with the matrix and the fibrous constituents having their own reference configurations and material symmetries. The total stress depends on the deformation gradients and material symmetries for both constituents, defined with respect to their reference configurations.Under appropriate assumptions, the constitutive theory developed using mixture theory can coincide with the constitutive theory assuming an equivalent single material that is transversely isotropic and hyperelastic. This paper explores the connection between the two approaches by considering the various reference configurations and material symmetries.     

A theory of amorphous solids undergoing large deformations,with application to polymeric glasses

This paper develops a continuum theory for the elastic–viscoplastic deformation of amorphous solids such as polymeric and metallic glasses. Introducing an internal-state variable that represents the local free-volume associated with certain metastable states, we are able to capture the highly non-linear stress–strain behavior that precedes the yield-peak and gives rise to post-yield strain softening. Our theory explicitly accounts for the dependence of the Helmholtz free energy on the plastic deformation in a thermodynamically consistent manner. This dependence leads directly to a backstress in the underlying flow rule, and allows us to model the rapid strain-hardening response after the initial yield-drop in monotonic deformations, as well as the Bauschinger-type reverse-yielding phenomena typically observed in amorphous polymeric solids upon unloading after large plastic deformations. We have implemented a special set of constitutive equations resulting from the general theory in a finite-element computer program. Using this finite-element program, we apply the specialized equations to model the large-deformation response of the amorphous polymeric solid polycarbonate, at ambient temperature and pressure. We show numerical results to some representative problems, and compare them against corresponding results from physical experiments.     

Thermomechanical constitutive equations for the dynamic response of ceramics

The behavior and failure of brittle materials is significantly influenced by the existence of inhomogeneities such as pores and cracks. The proposed constitutive equations model the coupled micro-mechanical response of these inhomogeneities through evolution equations for scalar measures of porosity, and a “density” function of randomly oriented penny-shaped cracks. A specific form for the Helmholtz free energy is proposed which incorporates the known Mie–Grüneisen constitutive equation for the nonporous solid. The resulting thermomechanical constitutive equations are valid for large deformations and the elastic response is hyperelastic in the sense that the stress is related to a derivative of the Helmholtz free energy. These equations allow for the simulation of the following physical phenomena exhibited by brittle materials: (1) high compressive strength compared with much lower tensile strength; (2) inelastic deformation due to growth and nucleation of cracks and pores instead of due to dislocation dynamics associated with metal plasticity; and (3) loss of integrity (degradation of elastic moduli) due to damage accumulation. The main features of the model are demonstrated by examples of cyclic loading in homogeneous deformation and by a simulation of a dynamic plate-impact experiment on AD85 ceramic. The theoretical predictions of the model are in excellent agreement with the dynamic experimental data.     

A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys

The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Compared with other strain measures, e.g., the commonly used Green-Lagrange measure, logarithmic strain is a more physical measure of strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. The derivation is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, and on the use of the isotropic property of the Helmholtz strain energy function. We also use the fact that the corotational rate of the Eulerian Hencky strain associated with the so-called logarithmic spin is equal to the strain rate tensor (symmetric part of the velocity gradient tensor). Satisfying the second law of thermodynamics in the Clausius-Duhem inequality form, we derive a thermodynamically-consistent constitutive model in a Lagrangian form. In comparison with the available finite strain models in which the unsymmetric Mandel stress appears in the equations, the proposed constitutive model includes only symmetric variables. Introducing a logarithmic mapping, we also present an appropriate form of the proposed constitutive equations in the time-discrete frame. We then apply the developed constitutive model to shape memory alloys and propose a well-defined, non-singular definition for model variables. In addition, we present a nucleation-completion condition in constructing the solution algorithm. We finally solve several boundary value problems to demonstrate the proposed model features as well as the numerical counterpart capabilities.     

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.     

An implicit algorithm for hypoelasto-plastic and hypoelasto-viscoplastic endochronic theory in finite strain isotropic–kinematic-hardening model

This paper is concerned with objective stress update algorithm for elasto-plastic and elasto-viscoplastic endochronic theory within the framework of additive plasticity. The elastic response is stated in terms of hypoelastic model and endochronic constitutive equations are stated in unrotated frame of reference. A trivially incrementally objective integration scheme for rate constitutive equations is established. Algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. The implementation is validated by means of a set of simple deformation paths (simple shear, extension and rotation), two benchmark test in nonlinear mechanics (the necking of a circular bar and expansion of a thick-walled cylinder), a test which demonstrates the capabilities of the proposed model in simulation of cyclic loading and ratcheting in finite strain case (cyclically loaded notched bar) and finally, the analysis of a tensile test, which presents a shear band with a finite thickness independent of the finite element mesh using endochronic viscoplastic constitutive model.