Compressible Hyperelastic Materials Research Articles

Development of the variation-of-elastic-energy-based virtual fields method for parameter identification of incompressible and compressible hyperelastic materials

The virtual fields method (VFM) has played a pivotal role in parameter identification of hyperelastic materials, spanning from rubbers to soft tissues. A key challenge in VFM lies in calculating the internal virtual work, which necessitates the computation of the stress and its conjugate virtual strain. In this study, we introduce a novel theoretical framework for VFM, which directly computes the internal virtual work based on the variation of elastic energy (VEE). The proposed method, namely VEE-VFM, eliminates the calculation of stress and virtual strain, making it remarkably user-friendly compared with the conventional approach. We extend VEE-VFM to both incompressible and compressible hyperelastic materials, including invariant-based and principal stretch-based models. In the end, we demonstrate the accuracy of VEE-VFM and its robustness against noise with numerical experiments.

Stabilization-free virtual element method for finite strain applications

In this paper, a novel higher stabilization-free virtual element method is proposed for compressible hyper-elastic materials in 2D. Different from the most traditional virtual element formulation, the method does not need any stabilization. The main idea is to modify the virtual element space to allow the computation of a higher-order polynomial L2 projection of the gradient. Based on that the stiffness matrix can be obtained directly which greatly simplifies the analysis process, especially for nonlinear problems. Hyper-elastic materials are considered and some benchmark nonlinear problems are solved to verify the capability and accuracy of the stabilization-free virtual element method.

Two-dimensional growth of incompressible and compressible soft biological tissues

Soft tissues have the ability of growth as a response to some especial changes of their environment. The aim of this work is to develop a non-linear finite element formulation for two-dimensional growth of soft tissues. The formulation is derived in Total Lagrangian (TL) framework. Moreover, the growth variable is considered as a single-valued scalar parameter known as growth multiplier. To deal with evolution equation resulting from the growth parameter, an implicit midpoint scheme is employed. Furthermore, a method is presented for compressible hyperelastic material models, and the formulas can be applied to incompressible and compressible, isotropic and anisotropic tissues. Finally, to investigate the performance of the proposed model, some examples are provided and compared to those available in the literature.

Calculation of the macroscopic mechanical properties of polyamide 6.6 underwater influence using molecular dynamics simulations, hyperelastic material modeling of the amorphous fraction, and homogenization schemes

AbstractThe macroscopic mechanical properties of water‐influenced polyamide 6.6 is analyzed by combining molecular dynamics simulation and experimental measurements of the media concentration and the degree of crystallization. The mechanical properties of α‐crystalline and amorphous polyamide 6.6 are analyzed separately by using molecular dynamics simulations. The anisotropic linear‐elastic stiffness tensor of the crystalline material is determined by unidirectional tensile and shear stress simulations. The deformations of amorphous simulation cells with different water concentrations are fitted in a compressible hyperelastic Yeoh material model. The media concentration of tensile specimens conditioned in air with 50% relative humidity at 23°C and in water is determined after different conditioning times. The degree of crystallization of the material is determined using Differential Scanning Calorimetry. Both parameters are used to estimate the macroscopic mechanical properties of polyamide 6.6. The homogenization schemes by Reuss and Ahzi are compared to calculate the resulting macroscopic properties. Tensile tests are used to validate the calculated properties: The weakening of the material due to water molecules in the amorphous phase can be shown, but it is underestimated in the methods used.

New classes of linearizable hyperelastic compressible materials

The governing equations for plane deformations of isotropic compressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. At present, there are only two classes of material in which the governing equations are known to be linearizable. The first is the harmonic material, and the second is a class of material recently introduced by the authors. In this paper, we show new materials exist where the governing equations are linearizable.

Internally balanced hyperelastic constitutive model in terms of principal stretches

Internal variables are included in material constitutive models to capture realistic mechanical effects such as viscosity. Recent work for compressible hyperelastic material is developed based on applying the argument of calculus variation to two-factor multiplicative decomposition of the deformation gradient. Strain energy function is expressed in terms of principal invariants of the deformation gradient decomposed counterparts. The present work permits the use of strain energy function in terms principal stretches of the deformation gradient multiplicatively decomposed counterparts directly. This simplifies significantly the expression of second Piola–Kirchhoff stress and internal balance tensor.

Symmetry classification of plane deformations of hyperelastic compressible materials

The governing equations for plane deformations of isotropic compressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. For materials known as harmonic materials, only one solution of any generality is known. It is natural to ask whether the governing equations possess some special property for this particular material. In this paper, we show that the governing equations admit an extensive class of symmetries for these so-called harmonic materials and further show that there exists a second class of strain-energy functions, where the governing equations admit a general class of symmetries. We further show that for this strain-energy function, the governing equations are linearizable.

Updated Lagrangian for Compressible Hyperelastic Material with Frictionless Contact

The Updated Lagrangian method for nonlinear elasticity with contact is presented. First, we describe the Total Lagrangian for a compressible Neo-Hookean material. Next, we introduce the Updated Lagrangian formulation for Neo-Hookean and Ogden compressible materials with contact. An advantage of this approach is that at each iteration only a linear system is solved. The linear problem to be solved is written in the updated domain. Numerical results are presented: compression of a Hertz half ball and of a hyperelastic ring against a flat rigid foundation, and contact of an elastic cube and a ball.

Investigating the impact behavior of wrestling mats via finite element simulation and falling weight impact tests

In this paper, a novel approach is used to represent the viscoelastic behavior of cross-linked polyethylene foams at high strain rate: a compressible hyperelastic material model is combined with the Rayleigh damping method. We developed finite element models to simulate the impact tests described in the international assessment protocol of wrestling mats. The accuracy of the finite element analysis was validated with experimental data. We showed the nonlinear viscoelastic nature of the foam material by determining the relationship between the damping ratio and maximum compressive strain. We demonstrated the applicability of the method through the quality assessment of wrestling mats, and determined the minimum required thickness to prevent traumatic head injuries. The modeling and testing method can be implemented in the study of sandwich structures, thus the design process can be simplified. With further development, our approach can be used to design multilayered wrestling mats with better energy absorbing characteristics.

Interfacial wave propagation in initially stressed compressible hyperelastic materials

In this paper, the propagation of interfacial waves at the joint boundary of two initially stressed compressible half-spaces is discussed. The materials are subject to pure homogeneous strain and the wave is assumed to travel along one of the principle axes. For mathematical formulation of the problem, non-linear theory of elasticity and theory of invariants are used. Boundary conditions at the interface are applied which lead to an implicit secular equation governing the wave speed. A prototype strain-energy function is used for specialized theoretical results to understand the physical behavior of waves at the interface. A special case of biaxial initial stress is considered and the results are presented theoretically and graphically for representative numerical values of parameters. It is observed that the wave speed is considerably affected by intrinsic properties, i.e., material parameters as well as the amount of initial stress.

Stability of Classical Shock Fronts for Compressible Hyperelastic Materials of Hadamard Type

This paper studies the uniform and weak Lopatinski\u{\i} conditions associated to classical (Lax) shock fronts of arbitrary amplitude for compressible hyperelastic materials of Hadamard type in several space dimensions. Thanks to the seminal works of Majda (Mem. Amer. Math. Soc. 43 (1983), no. 281; Mem. Amer. Math. Soc. 41 (1983), no. 275) and M\'etivier (Trans. Amer. Math. Soc. 296 (1986), pp. 431-479; Comm. Partial Diff. Eqs. 15 (1990), no. 7, pp. 983-1028), the uniform Lopatinski\u{\i} condition ensures the local-in-time, multidimensional, nonlinear stability of such fronts. The stability function (also called Lopatinski\u{\i} determinant) for shocks of arbitrary amplitude in this large class of hyperelastic materials is computed explicitly. This information is used to establish the conditions for uniform and weak shock stability in terms of the parameters of the shock and of the elastic moduli of the material.

Finite deformation of a blunt crack represented by a parabolic cavity in a harmonic solid

We study the finite plane deformation of a blunt crack represented by a parabolic cavity in a particular class of compressible hyperelastic materials of harmonic type. A complete solution to the blunt crack problem is derived using complex variable techniques. Elementary expressions of the stresses along the real axis ahead of the vertex of the parabola and the finite deformation of the parabola are obtained. When the nominal stress intensity factors approach zero, the stress distributions along the real axis are identical to those in linear elasticity. When the nominal stress intensity factors approach infinity, the tractions along the real axis are inversely proportional to the square root of the horizontal coordinate. A finite and non-zero mode II nominal stress intensity factor will induce both shear and normal stresses along the real axis, a phenomenon quite different from that for linear elasticity.

Swelling and inflation of a toroidal gel balloon

This work mainly focuses on the free swelling and inflation mechanics of a hyperelastic homogeneous and isotropic gas-filled toroidal gel balloon of an initially circular cross-section. Two compressible hyperelastic material models, namely neo-Hookean and Gent, are considered with Flory–Huggins mixing energy. Mechanics of free swelling configurations are studied using equilibrium equations obtained for incompressible and compressible axisymmetric inflation. The two-point boundary value problem of the balloon was solved using the Nelder–Meads search technique by constructing an optimization problem. Non-uniform solvent concentration is resulted in the meridional direction for the equilibrium swelling and inflation. For the neo-Hookean material, the instant/delayed burst phenomenon is concluded for almost all the geometric and material parametric ranges due to the softening type nonlinearity. On the other hand, for the Gent material model, instant/delayed short burst phenomenon is found for a specific range of dimensionless material constants. The inflation pressure range for the burst phenomenon and equilibrium swelling is identified for particular material parametric ranges. The critical pressure for the delayed short burst is roughly estimated and found to be highly dependent on the nonlinear nature of the swollen toroidal gel balloon inflation pressure-stretch curve. Also, deswelling is observed at the larger inflation stretch values of the gel.

A study on deformation-induced degradation of the compressible polymeric pressurized vessel through a non-equilibrium thermodynamic framework

The present paper investigates the degradation of compressible polymers based on the proposed model on strain-induced degradation of incompressible polymers. In a non-equilibrium thermodynamic framework, constitutive equations and evolution laws are derived using the principle of maximum energy dissipation rate and specifying how energy can be stored and dissipated. As a computational model, the governing equations are applied to the pressurized polymeric vessel subjected to the Ogden–Hill compressible hyperelastic material model. To analyze the axisymmetric plane-strain degradable vessel, programming in ANSYS Parametric Design Language (APDL) and the Standard Galerkin Finite Element Method (SGFEM) are applied. The results show that the degradable compressible Ogden–Hill model can also predict the degradation of incompressible polymers subjected to the neo-Hookean model. Results also reveal that the highest dissipation rate and material softening occur at the inner radius of the inflated degradable vessel. Creep-like and stress-relaxation-like responses of the polymeric vessel with time-position-dependent material properties are examined. ANSYS coding indicates good accuracy and efficiency in studying the compressible vessel subjected to inhomogeneous degradation.

PH-sensitive hydrogel-based valves: A transient fully-coupled fluid-solid interaction study

pH-sensitive hydrogels are promising materials to be employed in microfluidic devices, especially microvalves. In this paper, a theory of transient swelling of pH-sensitive micro-valve is presented. A transient constitutive model that captures electrical, chemical, and mechanical fields is considered to model the swelling phenomenon. The diffusion of ions into the hydrogel, the electromigration, and convection are described by implementing the Nernst-Planck equation. Assuming Gent model, hydrogel is considered as a compressible hyperelastic material and osmotic pressure is assumed as an external loading. Due to benefits of in-plane valves, a design of the micro-valve is studied. Design simplicity and great sealing are vital factors which can be considered as an advantage of this valve for fabrication. This design and modeling approach has not been used for pH-sensitive hydrogels in earlier works. Thus, we have studied the transient swelling of pH-sensitive hydrogel microvalve, when effects of fluid-structure-interaction are examined on valve performance. It is noted that in most previous studies, equilibrium conditions have been assumed. While considering transient fully-coupled fluid-structure-interaction is necessary to capture a more realistic modeling. The results illustrate that the microvalve blocks the channel much earlier than reaching the equilibrium-state, which implies importance of the transient behavior of hydrogels.

Role of fluid phase in compression of nonlinear elastic fluid-saturated porous medium

This paper is about the role of the fluid in supporting the overall compressive stress applied to nonlinear elastic fluid-saturated porous material. The case of large deformation and compressible hyperelastic matrix material is considered. Due to complexity of the problem, a simple model of the porous medium consisting of an assemblage of fluid-filled cylinders is studied in more detail. The procedure for computing the stress partitioning between the solid phase of the cylinder and the fluid in withstanding the overall stress consists of two steps. In the first step, the radial stress σ of given magnitude is applied to outer radius of the fluid-filled cylinder and the radial displacement is determined. In the second step, the displacement of the outer surface of the cylinder is equal to that of the first step but the cavity of the cylinder is empty. The radial stress acting on the outer boundary of the cylinder in the second step is defined as an effective stress in the solid skeleton of the porous medium, σsk. For the given overall stress σ and the computed value of the effective stress σsk, the effective stress supported by the fluid Peff is computed as the difference σ−σsk. Each of the two steps requires a solution to a boundary value problem which is solved numerically with the help of Python bvp solver.

A slightly compressible hyperelastic material model with the Mullins effect

The paper presents a hyperelastic material model which is intended to describe slightly compressible materials with encountered the Mullins effect. It is assumed the volumetric-isochoric stored energy function split regarding the primary response of a material. Constitutive equations based on the concept of pseudo-elasticity are derived. The problems of a simple uniaxial compression/tension involving one finite element and a compression of a ball are solved using ABAQUS. The material model is defined with the help of user subroutine UHYPER in terms of the primary material response. Its main advantage is a physically correct description of volumetric changes while the polynomial model in ABAQUS does not meet basic growth conditions. In order to include the Mullins effect, it is combined with available in the software library the Mullins effect material model which is based on Ogden-Roxburgh proposition of stress softening function. Experimental data on synthetic rubber neoprene published in the literature is utilized.

Propagation of Rayleigh wave in initially-stressed compressible hyperelastic materials

In this paper, propagation of Rayleigh surface wave in a compressible half-space solid with initial stress is discussed. The basic formulation of the problem includes the non linear theory of elasticity and theory of invariants. The equations governing infinitesimal motions superimposed on a finite deformation are used to study the combined effect of initial stress and finite deformation on wave speed. Secular equation governing the wave speed of a surface wave is presented and analyzed in detail. Various conditions are found for the existence and uniqueness of the wave speed. Furthermore, to understand the physical behavior of surface wave propagation, the theoretical results are plotted and analyzed for various numerical values of parameters involved in a prototype model. It is concluded that the surface wave speed is considerably affected by compression and tension as well as due to the stretch ratios of deformation. Moreover, for various choices of numerical values of governing parameters, the admissible numerical values of initial stress components and stretch ratios are also illustrated graphically.

A general higher‐order shell theory for compressible isotropic hyperelastic materials using orthonormal moving frame

SummaryThe primary objective of this study is threefold: (1) to present a general higher‐order shell theory to analyze large deformations of thin or thick shell structures made of general compressible hyperelastic materials; (2) to formulate an efficient shell theory using the orthonormal moving frame, and (3) to develop and apply the nonlinear weak‐form Galerkin finite element model for the proposed shell theory. The displacement field of the line normal to the shell reference surface is approximated by the Taylor series/Legendre polynomials in the thickness coordinate of the shell. The use of an orthonormal moving frame makes it possible to represent kinematic quantities (e.g., the determinant of the deformation gradient) in a far more efficient manner compared with the nonorthogonal covariant bases. Kinematic quantities for the shell deformation are obtained in a novel way in the surface coordinate described in the appendix of this study with the help of exterior calculus. Furthermore, the governing equation of the shell deformation has been derived in the general surface coordinates. To obtain the nonlinear solution in the quasi‐static cases, we develop the weak‐form finite element model in which the reference surface of the shell is modeled exactly. The general invariant based compressible hyperelastic material model is considered. The formulation presented herein can be specialized for various other nonlinear compressible hyperelastic constitutive models, for example, in biomechanics and other soft‐material problems (e.g., compressible neo‐Hookean material, compressible Mooney–Rivlin material, Saint Venant–Kirchhoff model, and others). A number of numerical examples are presented to verify and validate the formulation presented in this study. The scope of potential extensions are outlined in the final section of this study.

Zona pellucida shear modulus, a possible novel non-invasive method to assist in embryo selection during in-vitro fertilization treatment

The present study investigated the association between oocyte zona pellucida shear modulus (ZPSM) and implantation rate (IR). Ninety-three oocytes collected from 38 in-vitro fertilization patients who underwent intracytoplasmic sperm injection were included in this case–control study. The ZP was modeled as an isotropic compressible hyperelastic material with parameter C_{10}, which represents the ZPSM. Computational methodology was used to calculate the mechanical parameters that govern ZP deformation. Fifty-one developed embryos were transferred and divided into two groups—implanted and not implanted. Multivariate logistic regression analysis was performed to identify the association between ZPSM and IR while controlling for confounders. Maternal age and number of embryos per transfer were significantly associated with implantation. The IR of embryos characterized by C_{10} values in the range of 0.20–0.40 kPa was 66.75%, while outside this range it was 6.70%. This range was significantly associated with implantation (p < 0.001). Geometric properties were not associated with implantation. Multivariate logistic regression analysis that controlled for relevant confounders indicated that this range was independently associated with implantation (adjusted OR 38.03, 95% confidence interval 4.67–309.36, p = 0.001). The present study suggests that ZPSM may improve the classic embryo selection process with the aim of increasing IR.