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.     

A micro-mechanically based continuum damage model for fatigue life prediction of filled rubbers

This paper presents a continuum damage model based on two mechanisms: decohesion between fillers and matrix at a micro-scale followed by a crack nucleation at a macro-scale. That scenario was developed considering SEM observations and an original experimental procedure based on simple shear and tension specimens. Damage accumulation is related to fatigue life using the continuum damage mechanics (CDM). The material behavior is investigated using the statistical framework introduced by Martinez et al. (2011). A Finite Element implementation is proposed and some numerical examples are provided.     

A Micromechanics Finite-Strain Constitutive Model of Fibrous Tissue

Biological tissues have unique mechanical properties due to the wavy fibrous collagen and elastin microstructure. In inflation, a vessel easily distends under low pressure but becomes stiffer when the fibers are straightened to take up the load. The current microstructural models of blood vessels assume affine deformation, i.e., the deformation of each fiber is assumed to be identical to the macroscopic deformation of the tissue. This uniform-field (UF) assumption leads to the macroscopic (or effective) strain energy of the tissue that is the volumetric sum of the contributions of the tissue components. Here, a micromechanics-based constitutive model of fibrous tissue is developed to remove the affine assumption and to take into consideration the heterogeneous interactions between the fibers and the ground substance. The development is based on the framework of a recently developed second-order homogenization theory, and takes into account the waviness, orientations and spatial distribution of the fibers, as well as the material nonlinearity at finite-strain deformation. In an illustrative simulation, the predictions of the macroscopic stress-strain relation and the statistical deformation of the fibers are compared to the UF model, as well as finite-element (FE) simulation. Our predictions agree well with the FE results, while the UF predictions significantly overestimate. The effects of fiber distribution and waviness on the macroscopic stress-strain relation are also investigated. The present mathematical model may serves as a foundation for native as well as for engineered tissues and biomaterials.     

Homogenization of long fiber reinforced composites including fiber bending effects

This paper presents a homogenization method, which accounts for intrinsic size effects related to the fiber diameter in long fiber reinforced composite materials with two independent constitutive models for the matrix and fiber materials. A new choice of internal kinematic variables allows to maintain the kinematics of the two material phases independent from the assumed constitutive models, so that stress–deformation relationships, can be expressed in the framework of hyper-elasticity and hyper-elastoplasticity for the fiber and the matrix materials respectively. The bending stiffness of the reinforcing fibers is captured by higher order strain terms, resulting in an accurate representation of the micro-mechanical behavior of the composite. Numerical examples show that the accuracy of the proposed model is very close to a non-homogenized finite-element model with an explicit discretization of the matrix and the fibers.     

Mechanical response of neo-Hookean fiber reinforced incompressible nonlinearly elastic solids

The mechanical behavior of an incompressible neo-Hookean material, directionally reinforced by neo-Hookean fibers, is examined under homogeneous deformations. A composite model for this transversely isotropic material is developed based on a multiplicative decomposition of the deformation gradient which considers interaction between the fiber and the matrix. The so-called standard reinforcing model exhibits non-monotonic behavior in compression. The present composites-based approach leads to a modification of the standard reinforcing model in which monotonic behavior in compression is observed. This stems from the micromechanical basis of the model in which the fiber is treated as a neo-Hookean material. The conditions for loss of monotonicity and positivity in the stress-shear behavior in off-axis simple 2D shear are also obtained.     

A constitutive model for fibrous tissues considering collagen fiber crimp

A micromechanically based constitutive model for fibrous tissues is presented. The model considers the randomly crimped morphology of individual collagen fibers, a morphology typically seen in photomicrographs of tissue samples. It describes the relationship between the fiber endpoints and its arc-length in terms of a measurable quantity, which can be estimated from image data. The collective mechanical behavior of collagen fibers is presented in terms of an explicit expression for the strain-energy function, where a fiber-specific random variable is approximated by a Beta distribution. The model-related stress and elasticity tensors are provided. Two representative numerical examples are analyzed with the aim of demonstrating the peculiar mechanism of the constitutive model and quantifying the effect of parameter changes on the mechanical response. In particular, a fibrous tissue, assumed to be (nearly) incompressible, is subject to a uniaxial extension along the fiber direction, and, separately, to pure shear. It is shown that the fiber crimp model can reproduce several of the expected characteristics of fibrous tissues.     

Anisotropic damage–plasticity model for concrete

A material model for concrete is proposed here within the framework of a thermodynamically consistent elasto-plasticity–damage theory. Two anisotropic damage tensors and two damage criteria are adopted to describe the distinctive degradation of the mechanical properties of concrete under tensile and compressive loadings. The total stress tensor is decomposed into tensile and compressive components in order to accommodate the need for the above mentioned damage tensors. The plasticity yield criterion presented in this work accounts for the spectral decomposition of the stress tensor and allows multiple hardening rules to be used. This plastic yield criterion is used simultaneously with the damage criteria to simulate the physical behavior of concrete. Non-associative flow rule for the plastic strains is used to account for the dilatancy of concrete as a frictional material. The thermodynamic Helmholtz free energy concept is used to consistently derive dissipation potentials for damage and plasticity and to allow evolution laws for different hardening parameters. The evolution of the two damage tensors is accounted for through the use of fracture-energy-based continuum damage mechanics. An expression is derived for the damage–elasto-plastic tangent operator. The theoretical framework of the model is described here while the implementation of this model will be discussed in a subsequent paper.     

Growth and remodeling of load-bearing biological soft tissues

The past two decades reveal a growing role of continuum biomechanics in understanding homeostasis, adaptation, and disease progression in soft tissues. In this paper, we briefly review the two primary theoretical approaches for describing mechano-regulated soft tissue growth and remodeling on the continuum level as well as hybrid approaches that attempt to combine the advantages of these two approaches while avoiding their disadvantages. We also discuss emerging concepts, including that of mechanobiological stability. Moreover, to motivate and put into context the different theoretical approaches, we briefly review findings from mechanobiology that show the importance of mass turnover and the prestressing of both extant and new extracellular matrix in most cases of growth and remodeling. For illustrative purposes, these concepts and findings are discussed, in large part, within the context of two load-bearing, collagen dominated soft tissues—tendons/ligaments and blood vessels. We conclude by emphasizing further examples, needs, and opportunities in this exciting field of modeling soft tissues.     

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.     

Rate-dependent transition from tensile damage to discrete fracture in dynamic brittle failure

Based on the work for a combined damage/plasticity model of geologic materials and the bifurcation analysis of material failure, an analytical framework is established to study the rate-dependent transition from continuum damage to discrete fracture in dynamic brittle failure. Because of the simple formulation, a vectorized constitutive model solver can be designed for large-scale computer simulation. A continuum tangent stiffness tensor is invoked for the tensile damage evolution such that the bifurcation analysis can be performed to identify the initiation and orientation of tensile failure. It is shown that the orientation of tensile failure is rate-independent although the limit state is rate-dependent for the rate-dependent tensile damage model. Sample problems are considered to demonstrate the features of the proposed approach.     

Numerical study of strength properties for a composite material with short reinforcing fibers

A numerical approach to the determination of strength properties for concrete with short reinforcing fibers on the basis of the finite element method is proposed. The mathematical model takes into account various processes of nonlinear deformation of the concrete matrix under compressive and tensile loads, the possibility of developing the inelastic strains in the concrete matrix and reinforcing fibers and the nonlinear interaction between them. The effect of fiber concentration, various loading surfaces for the material matrix, and the bonding type on the deformation of a composite material is analyzed. Numerical examples of strength analysis are given.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

An elastic–plastic crack bridging model for brittle-matrix fibrous composite beams under cyclic loading

A fibrous composite beam with an edge crack is submitted to a cyclic bending moment and the crack bridging actions due to the fibers. Assuming a general elastic-linearly hardening crack bridging model for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging actions are obtained from compatibility conditions. The elastic and plastic shake-down phenomena are examined in terms of generalised cross-sectional quantities and, by employing a fatigue crack growth law, the mechanical behaviour up to failure is captured. Within the framework of the proposed fracture mechanics-based model, the cyclic crack bridging due to debonding at fiber–matrix interface of short fibers is analysed in depth. By means of some simplifying assumptions, such a phenomenon can be described by a linear isotropic tensile softening/compressive hardening law. Finally, numerical examples are presented for fibrous composite beams with randomly distributed short fibers.     

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.     

钢纤维增强砂浆的轴拉损伤力学模型

基于Li Fa Ming的平行棒模型,对短钢纤维增强砂浆的平板试件,在直接拉伸条件下的破坏行为进行了分析,假设试件由N根相互平行的复合棒组成,每根复合棒又由一根纤维棒和S根砂浆棒组成,考虑纤维在基体中分布的方向因子和长度因子.砂浆的损伤可按连续损伤力学进行处理,将Loland模型和Mazars模型加以改进来描述.依据多根纤维的拉拨模型,假定纤维与基体间界面的损伤由纤维脱粘长度与纤维插入长度的比值来描述,复合材料的损伤包括基体的损伤和纤维的损伤,借助已有的试验数据和文献资料来确定本构模型中的各种参数,成功建立了短钢纤维增强砂浆直接拉伸应力一应变全曲线模型.所建模型与试件在直接拉伸试验下的应力-应变全曲线进行了对比,结果较为吻合.     

A constitutive framework of plastically deformed damaged continuum and a formulation using theendochronic concept

The concepts of continuum damage mechanics (CDM) are discussed and aconstitutive framework of CDM is proposed for infinitesimal deformation based on the internalvariables approach. The framework involves transforming the actual damaged continuum into anequivalent fictitious undamaged continuum. A distinction is made between the state of damageand the damage measure. The development makes use of the concept of damage force. Thenegative of the damage force is related to the energy required to restore the fictitious undamagedcontinuum to its undamaged state after each step of deformation and damage. A set of equationand constraint governs the deformation of the fictitious continuum, while another set of equationand constraint governs the damage behavior. The coupling between the deformation and damageprocesses is provided for by the damage restoring force concept. Within the proposedconstitutive framework, the endochronic concept has been used to derive explicit constitutiveequations. The proposed model has been shown to describe the three-dimensional state ofdeformation of a cylindrical concrete specimen subjected to uniaxial compression.     

A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.