Incompressible Hyperelastic Materials Research Articles

Finite anti-plane shearing of centrally-cracked solids of incompressible hyperelastic material

Closed-form solutions are derived for two problems of nonlinear elastic fracture mechanics. The cases considered deal with the out-of-plane deformation of a centrally-cracked cylinder of elliptic cross section involving hyperelastic materials of either Neo-Hookean or Mooney-Rivlin type. Each solid is loaded by a self-equilibrating anti-plane shear traction applied to the faces of its crack and has its remote boundary either free or fixed. It is shown that, as in the case of small strains, the equation governing the out-of-plane response decouples from those defining in-plane behavior. It is also found that a finite state of pure out-of-plane deformation can only be sustained in the presence of Poynting stresses. In particular, nonlinear solids are seen to require out-of-plane direct axial stresses to satisfy the prescribed kinematics of deformation. Additionally, the non-regular shear stresses at the crack tips retain the same power of singularity as would exist in their linear elastic counterparts. Interestingly enough, however, the direct stresses, which are regular in linear materials, exhibit a singularity of higher order than that of the shear stresses.

Construction of two-phase equilibria in a non-elliptic hyperelastic material

This work focuses on the construction of equilibrated two-phase antiplane shear deformations of a non-elliptic isotropic and incompressible hyperelastic material. It is shown that this material can sustain metastable, two-phase equilibria which are neither piecewise homogeneous nor axisymmetric, but, rather, involve non-planar interfaces which completely segregate inhomogeneously deformed material in distinct elliptic phases. These results are obtained by studying a constrained boundary value problem involving an interface across which the deformation gradient jumps. The boundary value problem is recast as an integral equation and conditions on the interface sufficient to guarantee the existence of a solution to this equation are obtained. The contraints, which enforce the segregation of material in the two elliptic phases, are then studied. Sufficient conditions for their satisfaction are also secured. These involve additional restrictions on the interface across which the deformation gradient jumps — which, with all restrictions satisfied, constitutes a phase boundary. An uncountably infinite number of such phase boundaries are shown to exist. It is demonstrated that, for each of these, there exists a solution — unique up to an additive constant — for the constrained boundary value problem. As an illustration, approximate solutions which correspond to a particular class of phase boundaries are then constructed. Finally, the kinetics and stability of an arbitrary element within this class of phase boundaries are analyzed in the context of a quasistatic motion.

An arbitrary Lagrangian-Eulerian finite element method for incompressible hyperelasticity

In this paper, we present an arbitrary Lagrangian-Eulerian (ALE) formulation for large deformation analysis of incompressible hyperelastic materials, which are often employed for modeling rubber-like materials. Conventional ALE formulations, which have been successfully applied to not only fluids but also solid materials with rate constitutive equations, are usually described in velocity-based variables. Thus, in the application of such formulations to hyperelasticity, the velocity needs to be integrated in the time variable to obtain the deformation gradient necessarily appearing in hyperelastic material modeling. However, such integration is expensive and difficult to evaluate precisely. From such aspects, we propose a new ALE finite element formulation which directly incorporates the total deformation to avoid integration of velocity-based variables and can deal with arbitrary mesh motion by considering the variation of geometric mappings. Some basic numerical results are also given to illustrate the validity of our formulation.

Alternative methods for the solution of hyperelastic problems with incompressibility

A new approach for analysing incompressible hyperelastic and rubber-like material behaviour is discussed. The approach is based on extending the linear analysis bubble function concept into nonlinear analysis. To eliminate external or residual forces on the bubble function or nodeless degrees of freedom, two iterative methods are discussed. The new approach is compared to other approaches available in the literature including multified principle, reduced integration penalty and average constraint procedures. A unified numerical treatment for handling various constitutive relations of hyperelastic and rubber-like materials is presented. Numerical examples and comparisons include thick-walled cylinder subjected to internal pressure, inflation of a circular rubber plate and inflation of a hollow sphere.

Passive Material Properties of Intact Ventricular Myocardium Determined From a Cylindrical Model

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.

Derivation of hyperelastic incompressible material constitutive tensor within a total lagrangian framework

A tridimensional non-linear elastic constitutive relationship of the hyperelastic incompressible materials is derived from the Mooney-Rivlin strain energy density function. This works within large displacement theory, in the total Lagrangian formulation. Several applications, referring to the implementation in a non-linear computer code, are also described.

Efficient non‐linear finite element shell formulation involving large strains

A previously presented finite element shell formulation is extended to the application of large strains. The finite elements are those based on the concept of ‘the degenerated solids’, which are widely used in non‐linear finite element programs. The constitutive equation of hyperelastic incompressible material is adopted and specialized to the Mooney‐Rivlin law. The additional state variable, the hydrostatic pressure, which occurs for incompressible materials, is eliminated on element level using the plane stress condition. Attention is drawn to the efficient calculation of the element matrices by applying a layer concept. The effectiveness of the proposed total Lagrangian formulation is demonstrated in a number of example problems.

Large deformation finite element calculations for slightly compressible hyperelastic materials

Finite element calculations have been made previously for large deformation problems using incompressible hyperelastic material models. However, there is an important class of problems in which the finite compressibility of high elongation rubbery materials influences the stress distribution and must be included in the calculation. To address these problems we have formulated a new rubber elasticity model with finite compressibility and improved material representation. We have incorporated this model into the finite element code ADINA. Test problems demonstrate that good results can be obtained using the usual displacement-based Lagrangian formulation.

Shape of a polytropic primary

A phenomenological energy-based model for stress-softening of isotropic, incompressible hyperelastic rubberlike materials is derived here. In this model, the microstructural damage is characterized by an exponential softening function that depends on the current magnitude of the strain–energy function and its maximum previous value in a deformation of the virgin material. Theoretical models are presented for uniaxial, equibiaxial and pure shear deformations by using Gaussian and non-Gaussian material molecular network models. The accuracy of the resulting constitutive equations is demonstrated on uniaxial, equibiaxial and pure shear experimental data provided in the literature. Comparisons between the energy-based model and the strain intensity based phenomenological model described in [Elías-Zúñiga A, Beatty MF. ZAMP 2002;53:794–814. [1]] show that the model developed here is slightly superior in following experimental data.

Two problems with mixed boundary conditions for an incompressible isotropic hyperelastic material

Within the framework of nonlinear elasticity theory there is considered the equilibrium of a layer of incompressible Isotropie hyperelastic material under plane strain under the effect of gravity and forces P applied at infinity. The linearized equations generated by this state of stress and strain are investigated. It is shown that under for relationships between the material parameters, the layer thickness and the force P the equilibrium position can become unstable. Two problems are considered: the contact problem for a strip and the problem of a vertical crack of finite length emerging on the half-plane boundary. The action of the stamp and the crack is considered a small perturbation of the state of stress and strain caused by the action of the intrinsic weight and the force P.

On finite plane strain of an isotropic incompressible hyperelastic material

Abstract The purpose of this paper is to derive a new formulation of the governing equations for the complex variable in the deformed configuration of an isotropic incompressible hyperelastic material, assuming that the material undergoes a constant finite axial stretch and then a finite plane strain. As an application we consider the circular shear of a tube of such a material and, if we do not take into account torsion, it is proved that the plane deformation is given by an ordinary differential equation of second order, the integration of which requires the determination of the first derivative of the unknown function at the outer wall of the tube and the knowledge of the strain‐energy function. The axial stretch of the tube can be determined through two equations only when the strain‐energy function is specified.

Generalized shear deformations for isotropic incompressible hyperelastic materials

AbstractFor isotropic incompressible hyperelastic materials the single function characterizing generalized shear deformations or as they are sometimes called anti-plane strain deformations must satisfy two distinct partial differential equations. Knowles [5] has recently given a necessary and sufficient condition for the strain–energy function of the material which if satisfied ensures that the two equations have consistent solutions. It is shown here for the general material not satisfying Knowles' criterion that the only possible consistent solution of the two partial differential equations are those which are already known to exist for all strain–energy functions. More general types of generalized shear deformations for such meterials are shown to exist only for special or restricted form ot the strain-energy function. In derving these results we also obtain an alternative derivation of Knowles' criterion.

Radial deflections of thin precompressed cylindrical rubber bush mountings

For an isotropic incompressible hyperelastic Varga material the plane stress (membrane) theory of thin sheets is employed to formulate the load-deflection relation for small superimposed radial deflections of a cylindrical rubber bush which is precompressed by a large uniform radial inflation. The Varga material is a prototype for rubber over a limited range of deformation and the load-deflection relation obtained provides an extreme lower bound to the practical situation. Moreover this relation complements existing results for cylindrical rubber bushes so that now at least some assessment can be made of the effect of precompression on the radial mode of deflection for bushes of finite length. Typical numerical values are given and are contrasted with corresponding values obtained from existing plane strain radial load-deflection relations for long precompressed cylindrical rubber bush mountings.

Self-adjoint differential equations arising in finite elasticity for small superimposed deformations

For a number of problems involving small deformations superimposed upon large non-homogeneous deformations of isotropic incompressible hyperelastic materials the governing fourth order linear ordinary differential equations are shown to be self-adjoint. Moreover it is shown that at least in principle every fourth order linear self-adjoint differential equation can be factorized in terms of a single second order self-adjoint operator. These two results unify the derivation of a number of solutions of these equations which have been derived previously by a variety of ad-hoc procedures. Two problems are posed for the interested reader. Firstly the problem of obtaining these self-adjoint differential equations directly from the underlying variational principle and secondly the problem of obtaining the explicit factorizations for the differential equations given.

Closed form solutions for small deformations superimposed upon the symmetrical expansion of a spherical shell

For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are super-imposed upon the symmetrical expansion of a spherical shell, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the n=1 buckling criterion for thick-walled spherical shells which are subjected to uniform external pressure. They are also used to deduce an upper bound to the force deflection relation for small superimposed translational deflections of bonded pre-compressed spherical rubber bush mountings.

Closed form solutions for small deformations superimposed upon the simultaneous inflation and extension of a cylindrical tube

For small deformations of isotropic incompressible hyperelastic materials which are superimposed upon the simultaneous inflation and extension of a cylindrical tube, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the load-deflection relation for small radial deformations of pre-compressed long bonded cylindrical rubber bush mountings. They are also used to formulate the n=1 buckling criterion for long cylindrical tubes which are subjected to uniform external pressure.

Symmetric stiffness matrix for incompressible hyperelastic materials

Symmetric structure matrices are derived for solving plane strain and axisymmetric problems involving incompressible hyperelastic materials. An infinite hollow cylinder subjected to internal pressure is considered as an example. Displacement and hydrostatic pressure profiles are calculated using the Newton-Raphson iteration technique. The results are in good agreement with the exact curves.

CRITICAL PRESSURES FOR THE BUCKLING OF THICK-WALLED SPHERICAL SHELLS UNDER UNIFORM EXTERNAL PRESSURE

For isotropic incompressible hyperelastic materials the problem of determining the critical pressure at which a thick-walled spherical shell buckles when subjected to a uniform external pressure involves solving a fourth-order system of highly non-homogeneous ordinary differential equations. Closed-form solutions of this system are derived here for a particular hyperelastic material which has not been studied previously. These solutions are used to derive the buckling criterion, and numerical values are obtained for the resulting critical pressures. For thin shells these values are in agreement with classical thin-shell theory while for thicker shells close agreement is obtained with existing results for the neo-Hookean material. However for thin shells the predicted buckling mode is different from that given by previous authors. It can be shown from previously published experimental results on simple extension and simple compression that the new strain-energy function is a valid prototype for vulcanized rubber over a limited range of deformation and moreover that it agrees with the neo-Hookean theory for small strains.

Buckling of long thick-walled circular cylindrical shells of isotropic incompressible hyperelastic materials under uniform external pressure

For isotropic incompressible hyperelastic materials, the problem of determining the critical external pressure at which a long thick-walled circular cylindrical shell will buckle involves solving a fourth-order system of highly non-homogeneous, ordinary differential equations. Closed-form solutions of this system are derived here for plane-strain conditions and for the particular case of the Varga material. These solutions are used to derive the buckling criterion and numerical values are obtained for the resulting critical pressures. They are found to be in good agreement with existing theoretical and experimental results for the neo-Hookean material.

Formation of shock waves in hyperelastic solids

This paper studies the mechanism of shock formation from generalised simple waves in the case of an isotropic incompressible hyperelastic material. Fast and slow waves are identified and it is established that slow simple waves cannot tend to shocks on the wavefront. The time and position of occurrence of shocks when they do form is investigated with the aid of generalised Riemann invariants and a circularly polarised slow wave is identified. The special case of plane shear waves is commented upon.