Incompressible Hyperelasticity Research Articles

A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models

We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist several different Hamiltonians introduced for quasi-incompressible elastodynamics based on different multifield variational formulations, there is not much study on the fully incompressible materials in the literature. The adopted mixed formulation can be viewed as a finite-strain generalization of Herrmann variational formulation, and it naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient and scaled mid-point formulas, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. Our analysis and numerical evidence also reveal that the scaled mid-point formula is non-robust numerically. The generalized Taylor–Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano’s formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.

Stabilized formulation for phase‐field fracture in nearly incompressible hyperelasticity

AbstractThis work presents a stabilized formulation for phase‐field fracture of hyperelastic materials near the limit of incompressibility. At this limit, traditional mixed displacement and pressure formulations must satisfy the inf‐sup condition for solution stability. The mixed formulation coupled with the damage field can lead to an inhibition of crack opening as volumetric changes are severely penalized effectively creating a pressure‐bubble. To overcome this bottleneck, we utilize a mixed formulation with a perturbed Lagrangian formulation which enforces the incompressibility constraint in the undamaged material and reduces the pressure effect in the damaged material. A mesh‐dependent stabilization technique based on the residuals of the Euler–Lagrange equations multiplied with a differential operator acting on the weight space is used, allowing for linear interpolation of all field variables of the elastic subproblem. This formulation was validated with three examples at finite deformations: a plane‐stress pure‐shear test, a two‐dimensional geometry in plane‐stress, and a three‐dimensional notched sample. In the last example, we incorporate a hybrid formulation with an additive strain energy decomposition to account for different behaviors in tension and compression. The results show close agreement with analytical solutions for crack tip opening displacements and performs well at the limit of incompressibility.

Bayesian model uncertainty quantification for hyperelastic soft tissue models

Abstract Patient-specific surgical simulations require the patient-specific identification of the constitutive parameters. The sparsity of the experimental data and the substantial noise in the data (e.g., recovered during surgery) cause considerable uncertainty in the identification. In this exploratory work, parameter uncertainty for incompressible hyperelasticity, often used for soft tissues, is addressed by a probabilistic identification approach based on Bayesian inference. Our study particularly focuses on the uncertainty of the model: we investigate how the identified uncertainties of the constitutive parameters behave when different forms of model uncertainty are considered. The model uncertainty formulations range from uninformative ones to more accurate ones that incorporate more detailed extensions of incompressible hyperelasticity. The study shows that incorporating model uncertainty may improve the results, but this is not guaranteed.

Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).

A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity

In Parts I (Bonet et al., 2015) and II (Gil et al., 2016) of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie–Grüneisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. Fourth, and in line with the previous papers of the series, an explicit stabilised Petrov–Galerkin framework is presented for the numerical solution of the thermo-elastic system of conservation laws when considering the entropy as an unknown of the system. Finally, a series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulation.

An efficient and stabilised SPH method for large strain metal plastic deformations

Due to its simplicity and robustness, smooth particle hydrodynamics (SPH) has been widely used in the modelling of solid and fluid mechanics problems. Through the years, various formulations and stabilisation techniques have been adopted to enhance it. Recently, the authors developed JST–SPH, a mixed formulation based on the SPH method. Originally devised for modelling (nearly) incompressible hyperelasticity, the JST–SPH formulation is mixed in the sense that linear momentum and a number of strain definitions, instead of the displacements, act as main unknowns of the problem. The resulting governing system of conservation laws conveniently enables the application of the Jameson–Schmidt–Turkel (JST) artificial dissipation term, commonly employed in computational fluid dynamics, to solid mechanics. Coupled with meshless SPH discretisation, this novel scheme eliminates the shortcomings encountered when implementing fast dynamics explicit codes using traditional mesh-based methods. This paper focuses on the applicability of the JST–SPH mixed formulation to the simulation of high-rate, large metal elastic–plastic deformations. Three applications—including the simulation of an industry-relevant metal forming process—are examined under different loading conditions, in order to demonstrate the reliability of the method. Results compare favourably with both data from the previous literature, and simulations performed with a commercial finite elements package. Most noticeably, these results demonstrate that the total Lagrangian framework of JST–SPH, fundamental to reduce the computational effort associated with the scheme, retains its accuracy in the presence of large distortions. Moreover, an algorithmic flow chart is included at the end of this document, to facilitate the computer implementation of the scheme.

Magic angles and fibre stretch in arterial tissue: Insights from the linear theory

This work is motivated by the current widespread interest in modelling the mechanical response of arterial tissue. A widely used approach within the context of anisotropic nonlinear elasticity is to use an orthotropic incompressible hyperelasticity model which, in general, involves a strain-energy density that depends on seven independent invariants. The complexity of such an approach in its full generality is daunting and so a number of simplifications have been introduced in the literature to facilitate analytical tractability. An extremely popular model of this type is where the strain energy involves only three invariants. While such models and their generalisations have been remarkably successful in capturing the main features of the mechanical response of arterial tissue, it is generally acknowledged that such simplified models must also have some drawbacks. In particular, it is intuitively clear that the correlation of such models with experiment will suffer limitations due to the restricted number of invariants considered. Our purpose here is to use the linearised theory for infinitesimal deformations to provide some guidelines for the development of a more robust nonlinear theory. The linearised theory for incompressible orthotropic materials is developed and involves six independent elastic constants. The general stress-strain law is inverted to provide an expression for the fibre stretch in terms of the stress. We examine the linearised response for simple tension in two mutually perpendicular directions corresponding to the axial and circumferential directions in the artery, obtaining an explicit expression for the fibre stretch in terms of the applied tension, fibre angle and linear elastic constants. The focus is then on determining the range of fibre orientation angles that ensure that the fibres are in tension in these simple tension tests. It is shown that the fibre stretch is positive for both simple tension tests if and only if the fibre angle is restricted to lie between two special angles called generalised magic angles. For the special case where the strain-energy function for the nonlinear model depends only on the three invariants I1,I4,I6, it is shown that the corresponding linearised model, called the standard linear model (SLM), depends on three elastic constants and the fibre stretch is positive only in the small range of fibre angles between the classic magic angles 35.26° and 54.74°. However, when the two additional invariants I5,I7 are included in the nonlinear strain energy so that the corresponding linear model involves four elastic constants, it is shown that the domain of fibre angle for which the stretch is positive is much larger and that the fibre stretch is monotonic with respect to the fibre angle in this range.

Method of Nearby Problems for Nearly Incompressible Hyperelasticity

Method of Nearby Problems for Nearly Incompressible Hyperelasticity

A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity

A novel Hu–Washizu type 5-field virtual work principle for nearly inextensible and almost incompressible finite hyperelasticity is developed and implemented. The formulation is implemented in an hp-adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation. The novel implementation includes residual based error estimation and mesh adaptivity. In the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraints of inextensibility and incompressibility. A study using a semi-inverse analytical solution and h-refinements and p-enrichments corroborates the convergence characteristics of the 5-field implementation. A new closed form solution for pure torsion of a circular cylindrical tube with inextensible fibres is derived and used for verification. As typical applications are found in e.g. soft tissue biomechanics, passive pressurisation of an ellipsoidal geometry resembling the left ventricle of a rabbit heart is analysed to demonstrate the capability of modelling transversely isotropic materials.

2-Dimensional Homogenization FEM Analysis of Hyperelastic Foamed Rubber

Foamed rubbers are widely used for the shoe-soles, shin-guards, protectors and so on. The complex microstructure of foamed rubber consists of rubber matrix and pores. Therefore, foamed rubbers show good shock-absorbing properties in addition to good formability and lightweight. The mechanical characteristics of foamed rubber are determined by the mechanical characteristics of rubber matrix and its microstructure.In this study, a homogenization numerical analysis of mechanical characteristics of foamed rubber considering the rubber matrix and microstructure was shown. In order to develop the evaluation method, the microstructure of the foamed rubber was assumed to have the periodical holes, and a homogenization FEM code for formed rubber was developed based on homogenization theory with hyperelasticity. To verify the applicability of our developed code, rectangular rubber specimens with periodically holes were prepared to observe stress distributions of periodical inner structure. Incompressible hyperelasticity was applied to the rubber matrix. The material parameters of hyperelasticity for rubber matrix were identified by the biaxial tensile test results. Compression test of the rubber specimen reproducing FEM model was conducted to verify the applicability of the analysis code. The analysis result showed good agreement with the compression test result in the low strain region.

A comparison of conservation law construction approaches for the two-dimensional incompressible Mooney-Rivlin hyperelasticity model

An extended Kovalevskaya form is derived for the two-dimensional incompressible Mooney-Rivlin nonlinear hyperelasticity equations and is used to compute a complete set of local conservation laws of the model through the direct method. Conserved densities and fluxes of the conservation laws are derived, and their physical interpretation is discussed. Since the model admits a variational formulation, the equations are rewritten in the self-adjoint form. Computation of local conservation laws through the direct method applied to the self-adjoint form, as well as a conservation law computation through the local symmetry analysis and the Noether’s first theorem, is performed. A correspondence between local variational symmetries and conservation law multipliers is illustrated. It is argued that even though it leads to more complicated forms of multipliers, the direct conservation law construction method applied to the Kovalevskaya form of the equations is a preferred systematic way of conservation law computations for complicated physical models of the type considered in this work, since it yields complete results, and naturally avoids singular multipliers.

One-dimensional nonlinear elastodynamic models and their local conservation laws with applications to biological membranes

The framework of incompressible nonlinear hyperelasticity and viscoelasticity is applied to the derivation of one-dimensional models of nonlinear wave propagation in fiber-reinforced elastic solids. Equivalence transformations are used to simplify the resulting wave equations and to reduce the number of parameters. Local conservation laws and global conserved quantities of the models are systematically computed and discussed, along with other related mathematical properties. Sample numerical solutions are presented. The models considered in the paper are appropriate for the mathematical description of certain aspects of the behavior of biological membranes and similar structures.

A viscoelastic poromechanical model of the knee joint in large compression

The elastic response of the knee joint in various loading and pathological conditions has been investigated using anatomically accurate geometry. However, it is still challenging to predict the poromechanical response of the knee in realistic loading conditions. In the present study, a viscoelastic, poromechanical model of the knee joint was developed for soft tissues undergoing large deformation. Cartilages and menisci were modeled as fibril-reinforced porous materials and ligaments were considered as fibril-reinforced hyperelastic solids. Quasi-linear viscoelasticty was formulated for the collagen network of these tissues and nearly incompressible Neo-Hookean hyperelasticity was used for the non-fibrillar matrix. The constitutive model was coded with a user defined FORTRAN subroutine, in order to use ABAQUS for the finite element analysis. Creep and stress relaxation were investigated with large compression of the knee in full extension. The contact pressure distributions were found similar in creep and stress relaxation. However, the load transfer in the joint was completely different in these two loading scenarios. During creep, the contact pressure between cartilages decreased but the pressure between cartilage and meniscus increased with time. This led to a gradual transfer of some loading from the central part of cartilages to menisci. During stress relaxation, however, both contact pressures decreased monotonically.

Distortion of Anisotropic Hyperelastic Solids Under Pure Pressure Loading: Compressibility, Incompressibility and Near-Incompressibility

The purpose of this note is to examine distortion during pure pressure loading for anisotropic hyperelastic solids. We contrast the corresponding issues in compressible and incompressible hyperelasticity, and then use these results to examine nearly incompressible materials. An anisotropic compressible hyperelastic solid will generally exhibit both volume change and distortion under hydrostatic pressure loading. In contrast, an incompressible hyperelastic solid—both isotropic and anisotropic—exhibits no change to its current state of deformation as the hydrostatic pressure is varied. Nearly incompressible hyperelastic materials are compressible, but approach an incompressible response in an appropriate limit. We examine this limiting process in the context of transverse isotropy. The issue arises as to how to implement a nearly incompressible version of a given truly incompressible material model. Here we examine how certain implementations eliminate distortion under pure pressure loading and why alternative implementations do not eliminate the distortion.

20213 均質化理論を用いた発泡材料の特性評価手法の開発(OS3-6【高分子系複合材料の成形と力学的特性(6)】)

In this paper, development of evaluation method for foam materials under large deformation using homogenization theory was described to analyze the mechanical characteristics of foam materials. The foam materials are applied to packing material, automobile seat and sports equipment The mechanical uniaxialensile tests using foam materials were conducted to investigate the effect of the microscopic structure on the mechanical characteristics. The porosity of the specimens was 26%, 61%, 66%, 76%, and the pore shape was nearly spherical. In the simulation the deformation of microscopic structure was uniform, and microscopic periodicity was kept under large deformation. The polymer matrix was supposed to have incompressible hyperelasticity, and the hyperelasticity was represented by Mooney-Rivlin model. The applicability of the proposed model for the foam materials was shown by comparing the numerical simulation with the mechanical loading test results. The effect of the microscopic structure on the mechanical characteristics was shown.

108 微細構造を考慮した軟質発泡材料の均質化解析(OS9 オーガナイズドセッション《計算力学と数値シミュレーション》)

In this study, a homogenization analysis of foam materials to consider the microscopic structure was conducted. A foam material was defined as a material which has a eriodical microscopic structure. Deformation of the microscopic structure was uniform, and microscopic periodicity was kept under finite deformation. The polymer matrix was assumed to have incompressible hyperelasticity, which was represented by the Mooney-Rivlin model. An original 2-dimensional simulation program was developed. Uniaxial tensile tests using foam materials were conducted to investigate the effect of porosity on mechanical characteristics. Specimen porosities were 26%, 61%, 66%, and 76%, and pore shape was nearly spherical under microscope observation. The simulation model also had periodic and equally-sized pores. Uniform deformation and periodic boundary condition were applied to the unit cell. The applicability of the proposed model for foam materials was shown by comparing the numerical simulation with the mechanical loading test results.

Homogenization Analysis of Porous Polymer Considering Microscopic Structure

In this study, a homogenization analysis of porous polymer to consider the microscopic structure was conducted. This analysis allows the establishment of a design method for porous polymer for sports equipment. The mechanical characteristics of porous polymers are determined by mechanical characteristics of the polymer's matrix and microscopic structure. In this study, a porous polymer was defined as a material which has a periodical microscopic structure. Deformation of the microscopic structure was uniform, and microscopic periodicity was kept under finite deformation. The polymer matrix was assumed to have incompressible hyperelasticity, which was represented by the Mooney-Rivlin model. An original 2-dimensional simulation program was also developed. Mechanical uniaxial tensile tests using porous polymers for sports equipment were conducted. The effect of porosity on mechanical characteristics was investigated. Specimen porosities were 26%, 61%, 66%, and 76%, and pore shape was nearly spherical under microscope observation. Results revealed a reduction in stiffness with increase porosity. Also, a linear relationship between porosity and initial stiffness was revealed. The simulation model also had periodic and equally-sized pores. Porosity was reproduced by adjusting the diameter of the pores. Uniform deformation and a periodic boundary condition were applied to the unit cell. Results revealed a reduction in the simulated stiffness. Stress concentration on the microscopic structure was confirmed. The stiffness reduction can serve as a design index of porous polymer.

Uniqueness of the elastography inverse problem for incompressible nonlinear planar hyperelasticity

The uniqueness of several 2D inverse problems for incompressible nonlinear hyperelasticity is studied. These problems are motivated by elastography, in which one is given a measured deformation field in a 2D domain Ω and seeks to reconstruct the pointwise distribution of material parameters within Ω. Two classes of models are considered. The simpler class is material models characterized by a single material parameter exemplified by the Neo–Hookean model. The second class of material models considered is characterized by two material parameters, and includes a simplified Veronda–Westmann model, a Blatz model and a modified Blatz model. Consistent with the results in linear elasticity, we find that significantly fewer data are required to determine the material properties under plane stress conditions than under plane strain conditions. The results show that, roughly speaking, one needs one measured deformation for each material parameter sought under plane stress conditions, and twice as much data for plane strain conditions.

J031035 スポーツ用発泡材料の均質化解析手法の開発

To analyze the mechanical properties of porous polymers under large deformation, the homogenization analysis of hyperelasticy and mechanical loading test results of porous polymers for sports equipment were described in this paper. The porous polymers are applied to sports equipment like as shoe sole, shock protector and grip tape of racquets. In this research, the microstructures of porous polymer were assumed to deform uniformly and the microscopic periodicity remains under large deformation. To evaluate the applicability of the simulation program to the porous polymers, homogenization analysis of microscopic skeletal wall of which was assumed to be represented by nearly incompressible hyperelasticity is demonstrated. In the mechanical loading tests, fundamental properties of porous polymers for sports equipment were shown to investigate the effect of microscopic pore ratio on the macroscopic mechanical characteristics. By the comparing the numerical simulation with experimental results, applicability of the proposed model to the porous polymer is shown.

A finite strain Eulerian formulation for compressible and nearly incompressible hyperelasticity using high‐order B‐spline finite elements

SUMMARYWe present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions. This finite element formulation is based on the Eulerian description of motion and the transport of the deformation gradient. When modeling a nearly incompressible solid, the transport of the deformation gradient is decomposed into its isochoric part and the Jacobian determinant as independent fields. A homogeneous isotropic hyperelastic solid is assumed and B‐splines‐based finite elements are used for the spatial discretization. A variational multiscale residual‐based approach is employed to stabilize the transport equations. The performance of the scheme is explored for both compressible and nearly incompressible applications. The numerical results are in good agreement with theory illustrating the viability of the computational scheme. Copyright © 2011 John Wiley & Sons, Ltd.