Anisotropic Hyperelastic Material Research Articles

A new invariant-based method for building biomechanical behavior laws – Application to an anisotropic hyperelastic material with two fiber families

In this article, we present a general constructive and original approach that allows us to calculate the invariants associated with an anisotropic hyperelastic material made of two families of collagen fibers. This approach is based on mathematical techniques from the theory of invariants:•Definition of the material symmetry group.•Analytical calculation of a set of generators using the Noether’s theorem.•Analytical calculation of an integrity basis.•Comparison between the proposed invariants and the classical ones.

The Effect of Ligament Modeling Technique on Knee Joint Kinematics: A Finite Element Study.

Finite element (FE) analysis has become an increasingly popular technique in the study of human joint biomechanics, as it allows for detailed analysis of the joint/tissue behavior under complex, clinically relevant loading conditions. A wide variety of modeling techniques have been utilized to model knee joint ligaments. However, the effect of a selected constitutive model to simulate the ligaments on knee kinematics remains unclear. The purpose of the current study was to determine the effect of two most common techniques utilized to model knee ligaments on joint kinematics under functional loading conditions. We hypothesized that anatomic representations of the knee ligaments with anisotropic hyperelastic properties will result in more realistic kinematics. A previously developed, extensively validated anatomic FE model of the knee developed from a healthy, young female athlete was used. FE models with 3D anatomic and simplified uniaxial representations of main knee ligaments were used to simulate four functional loading conditions. Model predictions of tibiofemoral joint kinematics were compared to experimental measures. Results demonstrated the ability of the anatomic representation of the knee ligaments (3D geometry along with anisotropic hyperelastic material) in more physiologic prediction of the human knee motion with strong correlation (r ≥ 0.9 for all comparisons) and minimum deviation (0.9º ≤ RMSE ≤ 2.29°) from experimental findings. In contrast, non-physiologic uniaxial elastic representation of the ligaments resulted in lower correlations (r ≤ 0.6 for all comparisons) and substantially higher deviation (2.6° ≤ RMSE ≤ 4.2°) from experimental results. Findings of the current study support our hypothesis and highlight the critical role of soft tissue modeling technique on the resultant FE predicted joint kinematics.

902 応力軟化を考慮した異方性超弾性モデルの開発(OS9 オーガナイズドセッション《発電機器の材料・構造解析と評価》)

902 応力軟化を考慮した異方性超弾性モデルの開発(OS9 オーガナイズドセッション《発電機器の材料・構造解析と評価》)

Finite Element Modeling of Mitral Valve Dynamic Deformation Using Patient-Specific Multi-Slices Computed Tomography Scans

The objective of this study was to develop a patient-specific finite element (FE) model of a human mitral valve. The geometry of the mitral valve was reconstructed from multi-slice computed tomography (MSCT) scans at middle diastole with distinguishable mitral leaflet thickness, chordal origins, chordal insertion points, and papillary muscle locations. Mitral annulus and papillary muscle dynamic motions were also quantified from MSCT scans and prescribed as boundary conditions for the FE simulation. Material properties of the human mitral leaflet tissues were obtained from biaxial tests and characterized by an anisotropic hyperelastic material model. In vivo dynamic closing of the mitral valve was simulated. The closed shape of the mitral valve output from the simulation was similar to the mitral valve geometry reconstructed from MSCT images at middle systole. Forces from the anterolateral and posteromedial papillary muscle groups at middle systole were 4.51N and 5.17N, respectively. The average maximum principal stress of the midsection of the anterior mitral leaflet was approximately 160kPa at the systolic peak. Results demonstrated that the developed FE model could closely replicate in vivo mitral valve dynamic motion during middle diastole and systole. This model may serve as a basis for utilizing computational simulations to obtain a better understanding of mitral valve mechanics, disease and surgical repair.

A comparison between inverse form finding and shape optimization methods for anisotropic hyperelasticity in logarithmic strain space

AbstractIn this paper an inverse mechanical formulation and a Limited‐Broyden‐Flechter‐Goldfarb‐Shanno method for shape optimization are compared. Both methods deal with the determination of the undeformed shape of an hyperelastic part knowing its deformed configuration and the applied loads. We consider anisotropic hyperelastic materials that are formulated in the logarithmic strain space. Beside the theoretical aspects, we present a numerical example. We established that no difference could be found between the node coordinates on the undeformed sheets computed with both methods. However the convergence to the solution is faster for the inverse mechanical formulation compared to the L‐BFGS algorithm. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Effects of intima stiffness and plaque morphology on peak cap stress

BackgroundRupture of the cap of a vulnerable plaque present in a coronary vessel may cause myocardial infarction and death. Cap rupture occurs when the peak cap stress exceeds the cap strength. The mechanical stress within a cap depends on the plaque morphology and the material characteristics of the plaque components. A parametric study was conducted to assess the effect of intima stiffness and plaque morphology on peak cap stress.MethodsModels with idealized geometries based on histology images of human coronary arteries were generated by varying geometric plaque features. The constructed multi-layer models contained adventitia, media, intima, and necrotic core sections. For adventitia and media layers, anisotropic hyperelastic material models were used. For necrotic core and intima sections, isotropic hyperelastic material models were employed. Three different intima stiffness values were used to cover the wide range reported in literature. According to the intima stiffness, the models were classified as stiff, intermediate and soft intima models. Finite element method was used to compute peak cap stress.ResultsThe intima stiffness was an essential determinant of cap stresses. The computed peak cap stresses for the soft intima models were much lower than for stiff and intermediate intima models. Intima stiffness also affected the influence of morphological parameters on cap stresses. For the stiff and intermediate intima models, the cap thickness and necrotic core thickness were the most important determinants of cap stresses. The peak cap stress increased three-fold when the cap thickness was reduced from 0.25 mm to 0.05 mm for both stiff and intermediate intima models. Doubling the thickness of the necrotic core elevated the peak cap stress by 60% for the stiff intima models and by 90% for the intermediate intima models. Two-fold increase in the intima thickness behind the necrotic core reduced the peak cap stress by approximately 25% for both intima models. For the soft intima models, cap thickness was less critical and changed the peak cap stress by 55%. However, the necrotic core thickness was more influential and changed the peak cap stress by 100%. The necrotic core angle emerged as a critical determinant of cap stresses where a larger angle lowered the cap stresses. Contrary to the stiff and intermediate intima models, a thicker intima behind the necrotic core increased the peak cap stress by approximately 25% for the soft intima models. Adventitia thickness and local media regression had limited effects for all three intima models.ConclusionsFor the stiff and intermediate intima models, the cap thickness was the most important morphological risk factor. However for soft intima models, the necrotic core thickness and necrotic core angle had a bigger impact on the peak cap stress. We therefore need to enhance our knowledge of intima material properties if we want to derive critical morphological plaque features for risk evaluation.

Mechanics of formation and rupture of human aneurysm

The mechanical response of the human arterial wall under the combined loading of inflation, axial extension, and torsion is examined within the framework of the large deformation hyper-elastic theory. The probability of the aneurysm formation is explained with the instability theory of structure, and the probability of its rupture is explained with the strength theory of material. Taking account of the residual stress and the smooth muscle activities, a two layer thick-walled circular cylindrical tube model with fiber-reinforced composite-based incompressible anisotropic hyper-elastic materials is employed to model the mechanical behavior of the arterial wall. The deformation curves and the stress distributions of the arterial wall are given under normal and abnormal conditions. The results of the deformation and the structure instability analysis show that the model can describe the uniform inflation deformation of the arterial wall under normal conditions, as well as formation and growth of an aneurysm under abnormal conditions such as the decreased stiffness of the elastic and collagen fibers. From the analysis of the stresses and the material strength, the rupture of an aneurysm may also be described by this model if the wall stress is larger than its strength.

Structural analysis of the natural aortic valve in dynamics: From unpressurized to physiologically loaded

A novel finite element model of the natural aortic valve was developed implementing anisotropic hyperelastic material properties for the leaflets and aortic tissues, and starting from the unpressurized geometry. Static pressurization of the aortic root, silicone rubber moulds and published data helped to establish the model parameters, while high-speed video recording of the leaflet motion in a left-heart simulator allowed for comparisons with simulations. The model was discretized with brick elements and loaded with time-varying pressure using an explicit commercial solver. The aortic valve model produced a competent valve whose dynamic behavior (geometric orifice area vs. time) closely matched that observed in the experiment. In both cases, the aortic valve took approximately 30 ms to open to an 800 mm 2 orifice and remained completely or more than half open for almost 200 ms, after which it closed within 30–50 ms. The highest values of stress were along the leaflet attachment line and near the commissure during diastole. Von Mises stress in the leaflet belly reached 600–750 kPa from early to mid-diastole. While the model using the unpressurized geometry as initial configuration was specially designed to satisfy the requirements of continuum mechanics for large deformations of hyperelastic materials, it also clearly demonstrated that dry models can be adequate to analyze valve dynamics. Although improvements are still needed, the advanced modeling and validation techniques used herein contribute toward improved and quantified accuracy over earlier simplified models.

多凸性ひずみエネルギー関数を用いた異方性超弾性シェル要素の開発と衣服圧シミュレーションへの応用

The present study addresses the constitutive model of cloth using anisotropic hyperelastic materials and its implementation into a nonlinear finite shell element in order to simulate large deformation behavior of cloth. Present work focuses on a macroscopic continuum constitutive model able to capture the appropriate mechanical behavior of cloth, which is characterized by two families of yarns, i.e., warp and weft. Hyperelasticity allows the formulation of large deformation including the anisotropic effect by using a structural tensor which consists of fiber-directional unit vectors. In order to avoid non-physical behavior, the related strain-energy function must be polyconvex. The polyconvexity of a strain-energy function insures the existence of the global minimizers for the total elastic energy. In this paper, interaction effect of each fiber is accounted for using weight factors of the principal material directions in order for the accurate and robust modeling of clothing behavior. The present material model is implemented into a four-node shell element in Abaqus/Standard (S4R Type) via user-subroutine UMAT, and it is used to predict the outcome of uniaxial tensile tests and compute the clothing pressure applied to human body to design a form of cloth which better fits the human body.

On universal relations in orthotropic material subclasses

Universal relations that hold true for specific classes or subclasses of elastic materials form an integral part of the characterisation process for any material that can deform elastically. On the other hand, it has become understood that any class of hyper-elastic materials possessing anisotropic properties more complicated to those of transverse isotropy obeys no universal relations. However, there are met in practice special cases (subclasses) of anisotropic hyper-elastic materials that, under certain circumstances, may still obey a certain number of universal relations. Material characterisation purposes suggest therefore that there is a practical purpose in identifying and classifying different subclasses of anisotropic hyper-elastic materials that may obey any kind of universal relations and, hence, to derive and gradually build up catalogues or tables of different universal relations that such material subclasses obey when subjected to different deformation patterns. This paper heads towards this research direction by initially considering the orthotropic material class which is alternatively identified as the hyper-elastic crystal class of rhombic symmetry [A.E. Green, J.E. Adkins, Large Elastic Deformations, Clarendon Press, Oxford, 1960]. Then, for a certain set of appropriately chosen orthotropic and transverse isotropic hyper-elastic material subclasses, it develops and presents a considerable number of relevant universal relations, most of which are new in the literature.

Large‐scale simulation of arterial walls: mechanical modeling

AbstractBiological 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. When arteries are overstretched, discontinuous damage is observed. For the modeling of this effect we apply a damage model, which basically assumes that the damage occurs mainly in fiber direction. For the numerical simulation we consider an atherosclerotic artery and apply a high internal pressure which is comparable to the pressure applied during a balloon‐angioplasty. The 3D‐discretization results in a large system of equations, therefore, a parallel algorithm using FETI‐DP is applied to solve the boundary value problem. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An energetic approach to the analysis of anisotropic hyperelastic materials

We demonstrate that for the class of anisotropic hyperelastic materials with stiffening behaviour (i.e., the stiffness increases for increasing strain), it is possible to find an approximation by means of the linear superposition of an anisotropic quadratic potential, generated by the true linear elasticity tensor of the target material, and a suitable correction potential that is isotropic and hyperelastic. The proposed method can be implemented into commercially available Finite Element software by use of featured options only. This approach is intended to provide the solution to a stress–strain problem, based entirely on energetic considerations, which ensures the convexity of the potentials, and provides a simple material characterisation procedure.

Inflation of an artery leading to aneurysm formation and rupture.

Formation and rupture of aneurysms due to the inflation of an artery with collagen fibers distributed in two preferred directions, subjected to internal pressure and axial stretch are examined within the framework of nonlinear elasticity. A two layer tube model with a fiber-reinforced composite based incompressible anisotropic hyperelastic constitutive material is employed to model the stress-strain behavior of the artery wall with distributed collagen fibers. The artery wall takes up a uniform inflation deformation, and there are no aneurysms in the artery under the normal condition. But an aneurysm may be formed in arteries when the stiffness of the fibers is decreased to a certain value or the direction of the fibers is changed to a certain degree towards the circumferential direction. The aneurysm may expand to much large extent and become complex in shape. One portion of the aneurysm becomes highly distended as a bubble while the rest remains lightly inflated. The rupture of the aneurysm is discussed along with the distribution of stresses. Critical pressures and the rupture pressures are given for different collagen fiber orientations or stiffness. Furthermore, the stability of the solutions is discussed to explain the formation of aneurysm.

Large deformation analysis of inflated air-spring shell made of rubber-textile cord composite

This paper deals with the mechanical behaviour of the thin-walled cylindrical air-spring shell (CAS) made of rubber-textile cord composite (RCC) subjected to different types of loading. An orthotropic hyperelastic constitutive model is presented which can be applied to numerical simulation for the response of biological soft tissue and of the nonlinear anisotropic hyperelastic material of the CAS used in vibroisolation of driver`s seat. The parameters of strain energy function of the constitutive model are fitted to the experimental results by the nonlinear least squares method. The deformation of the inflated CAS is calculated by solving the system of five first-order ordinary differential equations with the material constitutive law and proper boundary conditions. Nonlinear hyperelastic constitutive equations of orthotropic composite material are incorporated into the finite strain analysis by finite element method (FEM). The results for the deformation analysis of the inflated CAS made of RCC are given. Numerical results of principal stretches and deformed profiles of the inflated CAS obtained by numerical deformation analysis are compared with experimental ones.

On Multi-Scale Modeling of Elastomeric Laminated Composites for Structural Analysis

Abstract Analysis of complex structures is often based on multi-scale modeling, where effective properties of certain substructures are used instead of actual properties of smaller components. Laminated composites are usually considered as such sub-structures in components with laminated design. In case of elastomeric composites, well-known classical laminate theories could hardly be used due to significant non-linearity and incompressibility of material deformation. A convenient engineering variant of laminated model is proposed in this study for composites where neither physical nor geometrical non-linearity may be ignored. Possible material incompressibility is also taken into account. The model is primarily based on a previously developed constitutive approach to describe effective properties of anisotropic hyperelastic materials. Analytical and computational implementation of the model is considered in detail. Numerical examples illustrate accuracy and convenience of the model for representative cord/rubber composites.

Using FEM for large deformation analysis of inflated air-spring cylindrical shell made of rubber-textile cord composite

An orthotropic hyperelastic constitutive model is presented for large deformation analysis of the nonlinear anisotropic hyperelastic material of the cylindrical air-spring shell used in vibroisolation of driver's seat. Nonlinear hyperelastic constitutive equations of orthotropic composite material are incorporated into the finite strain analysis by finite element method (FEM). The results of deformation analysis of the inflated air-spring shell made of composite with rubber matrix reinforced by textile cords are given. Obtained numerical results of deformation corresponding to the experimentally measured deformation of the inflated cylindrical air-spring.

Identification parameters of material model and large deformation analysis of inflated air-spring shell made of rubber-textile cord composite

In the paper an orthotropic hyperelastic constitutive model is presented which can be applied to numerical simulation for the response of biological soft tissue and of the nonlinear anisotropic hyperelastic material of the cylindrical air-spring shell used in vibroisolation of driver's seat. The parameters of strain energy function of the proposed constitutive model are fitted to the experimental results by the nonlinear least squares method. The deformation of the inflated cylindrical air-spring shell is calculated by solving the system of five first-order ordinary differential equations with the material constitutive law and proper boundary conditions. Numerical results of principal stretches and deformed profiles of the inflated cylindrical air-spring shell obtained by numerical deformation analysis are compared with experimental ones.

Directional model for isotropic and anisotropic hyperelastic rubber-like materials

A material direction-dependent constitutive model has been formulated for large deformations for isotropic and anisotropic rubber-like materials. Although such materials are usually isotropic, anisotropic behavior has been observed in calendered plates of filled rubbers. Strain energy density function characterizing rubber-like materials is usually dependent on principal stretch ratios and thus is unable to account for anisotropy, whereas the proposed strain energy density depends on material directions and accounts for anisotropy. The material directions have simply been chosen using regular solid geometry. The strain energy density is given as the sum, over all material directions, of elementary directional strain energy densities. Then the elementary strain energy form is phenomenologically determined to account for the state of strain dependence of the material response. The model response is compared to uniaxial tension experimental data for anisotropic hyperelastic rubber-like materials and to uniaxial and biaxial tension for isotropic rubber-like materials.

Directional model for anisotropic hyperelastic rubber-like materials

A material direction-dependent constitutive model has been formulated for large deformation of anisotropic rubber-like materials. Anisotropic behavior has been observed in calendered plates of filled elastomers. Strain energy density functions characterizing rubber-like material behavior are usually dependent on the principal stretch ratios and are unable to take into account anisotropy. The proposed strain energy density depends on material directions and accounts for anisotropy. Model material directions have simply been chosen using existing macromolecular model chains geometry. The material strain energy density is given as the sum, over all material directions, of the elementary directional strain energy density. This elementary strain energy is determined by analogy with chain entropy of macromolecular models using the Langevin statistics. To evaluate the effectiveness of the proposed model, it is compared to uniaxial tension experimental data of anisotropic hyperelastic rubber-like materials.

Unified constitutive modeling of anisotropic hyperelastic materials at large strains

In the present contribution we propose a generalized hyperelastic model for fiber-reinforced structures. Along with the earlier generalized orthotropic hyperelastic formulation [1] this constitutive model forms a reliable basis for the adequate numerical simulation of various anisotropic materials at large elastic strains.