A viscoelastic anisotropic hyperelastic constitutive model of the human cornea

A constitutive model based on the continuum mechanics theory has been developed which represents interlamellar cohesion, regional variation of collagen fibril density, 3D anisotropy and both age-related viscoelastic and hyperelastic stiffening behaviour of the human cornea. Experimental data gathered from a number of previous studies on 48 ex vivo human cornea (inflation and shear tests) enabled calibration of the constitutive model by numerical analysis. Wide-angle X-ray scattering and electron microscopy provided measured data which quantify microstructural arrangements associated with stiffness. The present study measures stiffness parallel to the lamellae of the cornea which approximately doubles with an increase in strain rate from 0.5 to 5%/min, while the underlying stromal matrix provides a stiffness 2–3 orders of magnitude lower than the lamellae. The model has been simultaneously calibrated to within 3% error across three age groups ranging from 50 to 95 years and three strain rates across the two loading scenarios. Age and strain-rate-dependent material coefficients allow numerical simulation under varying loading scenarios for an individual patient with material stiffness approximated by their age. This present study addresses a significant gap in numerical representation of the cornea and has great potential in daily clinical practice for the planning and optimisation of corrective procedures and in preclinical optimisation of diagnostic procedures.     

The anisotropic material constitutive models for the human cornea

This paper presents an anisotropic analysis model for the human cornea. The model is based on the assumption that the fibrils in the cornea are organised into lamellae, which may have preferential orientation along the superior-inferior and nasal-temporal directions, while the alignment of lamellae with different orientations is assumed to be random. Hence, the cornea can be regarded as a laminated composite shell. The constitutive equation describing the relationships between membrane forces, bending moments, and membrane strains, bending curvatures are derived. The influences of lamella orientations and the random alignment of lamellae on the stiffness coefficients of the constitutive equation are discussed.     

A general framework for the numerical implementation of anisotropic hyperelastic material models including non-local damage

The highly nonlinear mechanical behaviour of soft tissues solicited within the physiological range usually involves degradation of the material properties. Mechanically, having these biostructures undergoing such stretch patterns may bring about pathological conditions related to the steady deterioration of both collagen fibres and material’s ground substance. Tissue and subject variability observed in the phenomenological mechanical characterisation of soft tissues often hinder the choice of the computational constitutive model. Therefore, this contribution brings forth a detailed overview of the constitutive implementation in a computational framework of anisotropic hyperelastic materials with damage. Surmounting the challenge posed by the mesh dependency pathology requires the incorporation of an integral-type non-local averaging, which seeks to include the effects of the microstructure in order to limit the localisation phenomena of the damage variables. By adopting this approach, one can make use of multiple developed material models available in the literature, a combination of those, or even propose new models within the same numerical framework. The numerical examples of three-dimensional displacement and force-driven boundary value problems highlight the possibility of using multiple material models within the same numerical framework. Particularities concerning the considered material models and the damage effect implications to represent the Mullins effect, induced anisotropy, hysteresis, and mesh dependency are discussed.     

An inhomogeneous and anisotropic constitutive model of human dentin

Dentin constitutes the major part of human tooth. It is composed of a large number of tubules with both variational radii and radially parallel pattern. In addition, peritubular dentin surrounds each tubule lumen and has a higher elastic modulus than the matrix of dentin, i.e. intertubular dentin. Considering the above microstructural characteristics, a micromechanics model is used in this paper to evaluate the overall elastic properties of dentin. Five independent effective elastic parameters in transverse isotropic elasticity matrix can be expressed analytically by the material parameters of peri- and intertubular dentin and the volume fraction of tubules. To determine the effectivity of this theoretical model, a finite element (FE) model simulating a longitudinal tooth slice in moire fringe testing of Wang and Weiner (J. Biomech. 31 (1998) 135) was performed. Furthermore, the FE model was developed incorporate modeling of variation of tubule's diameter and softer characteristic of intertubular dentin near the dentin-enamel junction and around the pulp chamber. It turned out that the isoline figure of longitudinal displacement by FE calculation has very similar patterns to the moire fringe results. However, the FE results of displacement by traditional stress-strain models which regard dentin as a homogeneous and isotropic material show an obviously different strain distributions as compared to published moire fringes results. Thus the inhomogeneous and anisotropic model developed in this paper more accurately reflects the true physical nature of human dentin.     

Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials

A comprehensive study on the spherical indentation of hyperelastic soft materials is carried out through combined theoretical, computational, and experimental efforts. Four widely used hyperelastic constitutive models are studied, including neo-Hookean, Mooney–Rivlin, Fung, and Arruda–Boyce models. Through dimensional analysis and finite element simulations, we establish the explicit relations between the indentation loads at given indentation depths and the constitutive parameters of materials. Based on the obtained results, the applicability of Hertzian solution to the measurement of the initial shear modulus of hyperelastic materials is examined. Furthermore, from the viewpoint of inverse problems, the possibility to measure some other properties of a hyperelastic material using spherical indentation tests, e.g., locking stretch, is addressed by considering the existence, uniqueness, and stability of the solution. Experiments have been performed on polydimethylsiloxane to validate the conclusions drawn from our theoretical analysis. The results reported in this study should help identify the extent to which the mechanical properties of hyperelastic materials could be measured from spherical indentation tests.     

Hyperelastic anisotropic microplane constitutive model for annulus fibrosus

In a recent paper, Peng et al. (2006, "An Anisotropic Hyperelastic Constitutive Model With Fiber-Matrix Interaction for the Human Annulus Fibrosis," ASME J. Appl. Mech., 73(5), pp. 815-824) developed an anisotropic hyperelastic constitutive model for the human annulus fibrosus in which fiber-matrix interaction plays a crucial role in simulating experimental observations reported in the literature. Later, Guo et al. (2006, "A Composites-Based Hyperelastic Constitutive Model for Soft Tissue With Application to the Human Fibrosis," J. Mech. Phys. Solids, 54(9), pp. 1952-1971) used fiber reinforced continuum mechanics theory to formulate a model in which the fiber-matrix interaction was simulated using only composite effect. It was shown in these studies that the classical anisotropic hyperelastic constitutive models for soft tissue, which do not account for this shear interaction, cannot accurately simulate the test data on human annulus fibrosus. In this study, we show that the microplane model for soft tissue developed by Caner and Carol (2006, "Microplane Constitutive Model and Computational Framework for Blood Vessel Tissue," ASME J. Biomech. Eng., 128(3), pp. 419-427) can be adjusted for human annulus fibrosus and the resulting model can accurately simulate the experimental observations without explicit fiber-matrix interaction because, in microplane model, the shear interaction between the individual fibers distributed in the tissue provides the required additional rigidity to explain these experimental facts. The intensity of the shear interaction between the fibers can be adjusted by adjusting the spread in the distribution while keeping the total amount of the fiber constant. A comparison of results obtained from (i) a fiber-matrix parallel coupling model, which does not account for the fiber-matrix interaction, (ii) the same model but enriched with fiber-matrix interaction, and (iii) microplane model for soft tissue adapted to annulus fibrosus with two families of fiber distributions is presented. The conclusions are (i) that varying degrees of fiber-fiber and fiber-matrix shear interaction must be taking place in the human annulus fibrosus, (ii) that this shear interaction is essential to be able to explain the mechanical behavior of human annulus fibrosus, and (iii) that microplane model can be fortified with fiber-matrix interaction in a straightforward manner provided that there are new experimental data on distribution of fibers, which indicate a spread so small that it requires an explicit fiber-matrix interaction to be able to simulate the experimental data.     

Structural three-dimensional constitutive law for the passive myocardium

A three-dimensional constitutive law is proposed for the myocardium. Its formulation is based on a structural approach in which the total strain energy of the tissue is the sum of the strain energies of its constituents: the muscle fibers, the collagen fibers and the fluid matrix which embeds them. The ensuing material law expresses the specific structural and mechanical properties of the tissue, namely, the spatial orientation of the comprising fibers, their waviness in the unstressed state and their stress-strain behavior when stretched. Having assumed specific functional forms for the distribution of the fibers spatial orientation and waviness, the results of biaxial mechanical tests serve for the estimation of the material constants appearing in the constitutive equations. A very good fit is obtained between the measured and the calculated stresses, indicating the suitability of the proposed model for describing the mechanical behavior of the passive myocardium. Moreover, the results provide general conclusions concerning the structural basis for the tissue overall mechanical properties, the main of which is that the collagen matrix, though comprising a relatively small fraction of the whole tissue volume, is the dominant component accounting for its stiffness.     

Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration

The annulus fibrosus (AF) of the intervertebral disk undergoes large and multidirectional stresses and strains. Uniaxial tensile tests are limited for measuring AF material properties, because freely contracting edges can prevent fiber stretch and are not representative of in situ boundary conditions. The objectives of this study were to measure human AF biaxial tensile mechanics and to apply and validate a constitutive model to determine material properties. Biaxial tensile tests were performed on samples oriented along the circumferential–axial and the radial–axial directions. Data were fit to a structurally motivated anisotropic hyperelastic model composed of isotropic extra-fibrillar matrix, nonlinear fibers, and fiber–matrix interactions (FMI) normal to the fibers. The validated model was used to simulate shear and uniaxial tensile behavior, to investigate AF structure–function, and to quantify the effect of degeneration. The biaxial stress–strain response was described well by the model (R ²?>?0.9). The model showed that the parameters for fiber nonlinearity and the normal FMI correlated with degeneration, resulting in an elongated toe-region and lower stiffness with degeneration. The model simulations in shear and uniaxial tension successfully matched previously published circumferential direction Young’s modulus, provided an explanation for the low values in previously published axial direction Young’s modulus, and was able to simulate shear mechanics. The normal FMI were important contributors to stress and changed with degeneration, therefore, their microstructural and compositional source should be investigated. Finally, the biaxial mechanical data and constitutive model can be incorporated into a disk finite element model to provide improved quantification of disk mechanics.     

A three-dimensional elastic plastic damage constitutive law for bone tissue

Motivated by mechanical analysis of bones and bone-implant systems, a 3D constitutive law describing the macroscopic mechanical behaviour of both cortical and trabecular bone in cyclic (not fatigue) overloads is developed. The proposed model which mathematical formulation is established within the framework of generalized standard materials accounts for three distinct material evolution modes where elastic, plastic and damage aspects are closely related. The anisotropic elasticity of bone is described by a morphology-based model and distinct damage behaviour in tension and compression by a halfspacewise generalized Hill criterion. The plastic criterion is based on the intact elastic compliance tensor. The algorithm applies three distinct projections based on the relationship between the internal variables and criteria. Their respective consistent tangent operators are presented. Numerical resolutions of several boundary value problems and a biomechanical application are presented to illustrate the potential of the constitutive model and demonstrate the expected quadratic convergence of the algorithm.     

A visco-hyperelastic constitutive approach for modeling polyvinyl alcohol sponge

This study proposes the quasi-linear viscoelastic (QLV) model to characterize the time dependent mechanical behavior of poly(vinyl alcohol) (PVA) sponges. The PVA sponges have implications in many viscoelastic soft tissues, including cartilage, liver, and kidney as an implant. However, a critical barrier to the use of the PVA sponge as tissue replacement material is a lack of sufficient study on its viscoelastic mechanical properties. In this study, the nonlinear mechanical behavior of a fabricated PVA sponge is investigated experimentally and computationally using relaxation and stress failure tests as well as finite element (FE) modeling. Hyperelastic strain energy density functions, such as Yeoh and Neo-Hookean, are used to capture the mechanical behavior of PVA sponge at ramp part, and viscoelastic model is used to describe the viscose behavior at hold part. Hyperelastic material constants are obtained and their general prediction ability is verified using FE simulations of PVA tensile experiments. The results of relaxation and stress failure tests revealed that Yeoh material model can define the mechanical behavior of PVA sponge properly compared with Neo-Hookean one. FE modeling results are also affirmed the appropriateness of Yeoh model to characterize the mechanical behavior of PVA sponge. Thus, the Yeoh model can be used in future biomechanical simulations of the spongy biomaterials. These results can be utilized to understand the viscoelastic behavior of PVA sponges and has implications for tissue engineering as scaffold.     

Analysis of prolapse in cardiovascular stents: a constitutive equation for vascular tissue and finite-element modelling

The effectiveness of a cardiovascular stent depends on many factors, such as its ability to sustain the compression applied by the vessel wall, minimal longitudinal contraction when it is expanded, and its ability to flex when navigating tortuous blood vessels. The long-term reaction of the tissue to the stent is also device dependant; in particular some designs provoke in-stent restenosis (i.e., regrowth of the occlusion around the stent). The mechanism of restenosis is thought to involve injury or damage to the vessel wall due to the high stresses generated around the stent when it expands. Because of this, the deflection of the tissue between the struts of the stent (called prolapse or "draping") has been used as a measure of the potential of a stent to cause restenosis. In this paper, uniaxial and biaxial experiments on human femoral artery and porcine aortic vascular tissue are used to develop a hyperelastic constitive model of vascular tissue suitable for implementation in finite-element analysis. To analyze prolapse, four stent designs (BeStent 2, Medtronic AVE; NIROYAL, Boston Scientific; VELOCITY, Cordis; TETRA, Guidant) were expanded in vitro to determine their repeating-unit dimensions. This geometric data was used to generate a finite element model of the vascular tissue supported within a repeating-unit of the stent. Under a pressure of 450 mm Hg (representing the radial compression of the vessel wall), maximum radial deflection of 0.253 mm, 0.279 mm, 0.348 mm and 0.48 mm were calculated for each of the four stents. Stresses in the vascular wall were highest for the VELOCITY stent. The method is proposed as a way to compare stents relative to their potential for restenosis and as a basis for a biomechanical design of a stent repeating-unit that would minimize restenosis.     

Experimentally tractable,pseudo-elastic constitutive law for biomembranes: II. Application

This is the second paper of a 2 part series. In the first, a theoretical framework is developed that provides 3 enhancements for the constitutive theory of biomembranes. 1) Covariance amongst response terms is minimized, 2) the pseudo-strain-energy potential is redefined so as to be measurable, and 3) the stress response is decomposed into 2 parts--a hyperelastic part plus a hypoelastic-stress-residual. This paper illustrates the experimental advantages of this novel constitutive theory via analysis of biaxial test data obtained from chemically treated bovine pericardium.     

Experimentally tractable,pseudo-elastic constitutive law for biomembranes: I. Theory

Although visco-elastic in general, the stress-strain relation of biomembranes is one-to-one or pseudo-elastic when being loaded after preconditioning. This pseudo-elastic relation is hypoelastic (i.e., it is not hyperelastic), yet much of the stress response can be characterized by a scalar function omega that represents the work done (per unit reference volume) on the specimen during loading. (Since a pseudo-strain-energy function W is optimized to fit the test data and not the work done, omega is not equal to W in general.) The remaining part tR of the stress response does no work during loading. With biaxial testing, omega can be definitively determined from data. Moreover, for tests with the stretch directions coaxial to the axes of anisotropy, tR can be accurately characterized by a scalar function omega that depends on the strain. This paper is part 1 of 2 with "I. Theory" and "II. Application."     

An anisotropic interaction model for simulating fingerprints

Journal of Mathematical Biology - Evidence suggests that both the interaction of so-called Merkel cells and the epidermal stress distribution play an important role in the formation of fingerprint...     

Estimating material parameters of a structurally based constitutive relation for skin mechanics

This paper presents a structurally based modeling framework to characterize the structure–function relation in skin tissues, based upon biaxial tensile experiments performed in vitro on porcine skin. Equi-axial deformations were imposed by stretching circular skin specimens uniformly along twelve directions, and the resultant loads at the membrane attachment points were measured. Displacement fields at each deformation step were tracked using an image 2D cross-correlation technique. A modeling framework was developed to simulate the experiments, whereby measured forces were applied to finite element models that were created to represent the geometry and structure of the tissue samples. Parameters of a structurally based constitutive relation were then identified using nonlinear optimization. Results showed that the ground matrix stiffness ranged from 5 to 32 kPa, fiber orientation mean from 2 to 13^◦ from the torso midline, fiber undulation mean from 1.04 to 1.34 and collagen fiber stiffness from 48 to 366 MPa. It was concluded that the objective function was highly sensitive to the mean orientation and that a priori information about fiber orientation mean was important for the reliable identification of constitutive parameters.     

Fitted hyperelastic parameters for Human brain tissue from reported tension,compression, and shear tests

The mechanical properties of human brain tissue are the subject of interest because of their use in understanding brain trauma and in developing therapeutic treatments and procedures. To represent the behavior of the tissue, we have developed hyperelastic mechanical models whose parameters are fitted in accordance with experimental test results. However, most studies available in the literature have fitted parameters with data of a single type of loading, such as tension, compression, or shear. Recently, Jin et al. (Journal of Biomechanics 46:2795−2801, 2013) reported data from ex vivo tests of human brain tissue under tension, compression, and shear loading using four strain rates and four different brain regions. However, they do not report parameters of energy functions that can be readily used in finite element simulations. To represent the tissue behavior for the quasi-static loading conditions, we aimed to determine the best fit of the hyperelastic parameters of the hyperfoam, Ogden, and polynomial strain energy functions available in ABAQUS for the low strain rate data, while simultaneously considering all three loading modes. We used an optimization process conducted in MATLAB, calling iteratively three finite element models developed in ABAQUS that represent the three loadings. Results showed a relatively good fit to experimental data in all loading modes using two terms in the energy functions. Values for the shear modulus obtained in this analysis (897−1653 Pa) are in the range of those presented in other studies. These energy-function parameters can be used in brain tissue simulations using finite element models.     

Computational efficiency of numerical approximations of tangent moduli for finite element implementation of a fiber-reinforced hyperelastic material model

In this study, we evaluated computational efficiency of finite element (FE) simulations when a numerical approximation method was used to obtain the tangent moduli. A fiber-reinforced hyperelastic material model for nearly incompressible soft tissues was implemented for 3D solid elements using both the approximation method and the closed-form analytical method, and validated by comparing the components of the tangent modulus tensor (also referred to as the material Jacobian) between the two methods. The computational efficiency of the approximation method was evaluated with different perturbation parameters and approximation schemes, and quantified by the number of iteration steps and CPU time required to complete these simulations. From the simulation results, it can be seen that the overall accuracy of the approximation method is improved by adopting the central difference approximation scheme compared to the forward Euler approximation scheme. For small-scale simulations with about 10,000 DOFs, the approximation schemes could reduce the CPU time substantially compared to the closed-form solution, due to the fact that fewer calculation steps are needed at each integration point. However, for a large-scale simulation with about 300,000 DOFs, the advantages of the approximation schemes diminish because the factorization of the stiffness matrix will dominate the solution time. Overall, as it is material model independent, the approximation method simplifies the FE implementation of a complex constitutive model with comparable accuracy and computational efficiency to the closed-form solution, which makes it attractive in FE simulations with complex material models.     

An anisotropic elastic-viscoplastic damage model for bone tissue

A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain ${D(\kappa)}$ , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.