General Strain-energy Function Research Articles

Elasto-capillary necking, bulging and Maxwell states in soft compressible cylinders

Localized pattern formations and “two-phase” deformations are studied theoretically in soft compressible cylinders subject to surface tension and axial loading through several force-controlled loading scenarios. By drawing upon known results for separate yet mathematically similar elastic localization problems, a concise family of analytical bifurcation conditions for localized bulging or necking are derived in terms of a general compressible strain energy function. The effect of material compressibility and strain-stiffening behaviour on the bifurcation point is analysed, and comparisons between our theoretical bifurcation conditions and the corresponding numerical simulation results of Dortdivanlioglu and Javili (Extreme Mech. Lett. 55, 2022) are made. It is then explained how the fully developed “two-phase” (Maxwell) state which evolves from the initial localized bifurcation solution can be comprehensively understood using the simple analytical expressions for the force parameters corresponding to the primary axial tension deformation. The power of this simple analytical approach in validating numerical simulation results for elastic localization and phase-separation-like problems of this nature is highlighted.

A nonlinear constitutive model for a two preferred direction electro-elastic body with residual stresses

A nonlinear spectral formulation, which is more general than the traditional classical-invariant formulation, is used to describe the mechanical behaviour of a residually stressed electro-elastic body with two preferred directions (ESTPD); the generality of the spectral formulation could facilitate the quest for good constitutive equations for ESTPDs. The strain energy is a function of spectral invariants (each with a clear physical meaning) that depend on the right stretch tensor, the residual stress tensor, two preferred-direction-structural tensors and one of the electric variables; clear meaningful physical invariants are useful in aiding the design of a rigorous experiment to construct a specific form of constitutive equation. Separable finite strain constitutive equations containing general single-variable functions, which depend only on a principal stretch or the electric field, are proposed and, in view of this, the corresponding infinitesimal strain energy functions can be easily constructed. A specific form for the strain energy function is generally easier to obtain from the general strain energy function via experiment if it is expressed in terms of general single-variable functions. The proposed constitutive equations can be easily converted to allow the mechanical influence of compressed fibres to be excluded or partially excluded and to model fibre dispersion in collagenous soft tissues. With the aid of spectral invariants, we easily prove that at most 15 of the 98 classical invariants in the corresponding minimal integrity basis are independent; this proof cannot be found in the literature. A simple tension boundary value problem with cylindrical symmetry is studied, where the residual stress is assumed to depend only on the radial position.

Large acoustoelastic effect for Lamb waves propagating in an incompressible elastic plate.

In this paper, the effect of a large pre-stress on the propagation of small amplitude Lamb waves in an incompressible elastic plate is investigated. Using the theory of incremental elasticity, the dispersion equations, which give the phase velocity of the symmetric and anti-symmetric wave modes as a function of the wavenumber, plate thickness, and pre-stress state, are derived for a general strain energy function. By considering the fourth-order strain energy function of incompressible isotropic elasticity, the correction to the phase velocity due to the pre-stress is obtained implicitly to the second order in the pre-strain/stress, and depends on the second, third, and fourth-order elastic constants. Numerical results are presented to show the dependence of the phase velocity of the Lamb wave modes upon the applied stress. These are compared to the first-order correction, and agree well with the limiting and asymptotic values obtained previously. It is envisaged that the present results may well find important practical applications in various guided wave based ultrasonic techniques utilising gels and rubber-like materials.

Finite element implementation of a new model of slight compressibility for transversely isotropic materials

Modelling transversely isotropic materials in finite strain problems is a complex task in biomechanics, and is usually addressed by using finite element (FE) simulations. The standard method developed to account for the quasi-incompressible nature of soft tissues is to decompose the strain energy function (SEF) into volumetric and deviatoric parts. However, this decomposition is only valid for fully incompressible materials, and its use for slightly compressible materials yields an unphysical response during the simulation of hydrostatic tension/compression of a transversely isotropic material. This paper presents the FE implementation as subroutines of a new volumetric model solving this deficiency in two FE codes: Abaqus and FEBio. This model also has the specificity of restoring the compatibility with small strain theory. The stress and elasticity tensors are first derived for a general SEF. This is followed by a successful convergence check using a particular SEF and a suite of single-element tests showing that this new model does not only correct the hydrostatic deficiency but may also affect stresses during shear tests (Poynting effect) and lateral stretches during uniaxial tests (Poisson's effect). These FE subroutines have numerous applications including the modelling of tendons, ligaments, heart tissue, etc. The biomechanics community should be aware of specificities of the standard model, and the new model should be used when accurate FE results are desired in the case of compressible materials.

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.

Bending of an Isotropic Compressible Circular Plate into an Ellipsoidal Shell

The problem of bending of an isotropic compressible circular plate into an ellipsoidal shell has been solved and the results obtained in terms of the general strain energy function w.

Finite elastic deformations of transversely isotropic circular cylindrical tubes

We consider the finite radially symmetric deformation of a circular cylindrical tube of a homogeneous transversely isotropic elastic material subject to axial stretch, radial deformation and torsion, supported by axial load, internal pressure and end moment. Two different directions of transverse isotropy are considered: the radial direction and an arbitrary direction in planes normal locally to the radial direction, the only directions for which the considered deformation is admissible in general. In the absence of body forces, formulas are obtained for the internal pressure, and the resultant axial load and torsional moment on the ends of the tube in respect of a general strain-energy function. For a specific material model of transversely isotropic elasticity, and material and geometrical parameters, numerical results are used to illustrate the dependence of the pressure, (reduced) axial load and moment on the radial stretch and a measure of the torsional deformation for a fixed value of the axial stretch.

An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids

The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic half-space coated with a thin pre-stressed elastic layer. The half-space and the layer are assumed to be compressible and in welded contact with each other. By using the effective boundary condition method, an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function. When the pre-strains are absent, the secular equation obtained coincides with the one for the isotropic case. Numerical investigation shows that the approximate secular equation obtained is a good approximation. Since explicit dispersion relations are employed as theoretical bases for extracting pre-stresses from experimental data, the secular equation obtained will be useful in practical applications.

A Micromechanically Based Anisotropic Constitutive Model for the Microtubule Wall

Microtubules serve as one of the structural components of the cell and govern several important cellular functions including mitosis and vesicular transport. Microtubules are comprised of tubulin subunits formed by α and β tubulin dimers arranged in a cylindrical hollow tube with diameter ∼20 nm. The tube is typically comprised of 13 or 14 protofilaments extending axially and staggered to give a spiral configuration. The longitudinal bonds between the tubulin dimers are much stiffer and stronger than the lateral bonds. This gives a highly anisotropic structure and mechanical properties of the microtubule. In this work, the aim is to define a complete set of effective anisotropic elastic properties of the tube wall that capture the atomistic interactions. A seamless microtubule wall is represented as a two dimensional triangulated lattice of dimers from which a representative volume element is defined. A harmonic potential is adapted for the dimer–dimer interactions. Estimating the lattice elastic constants and following the methodology from the analysis of the mechanical behavior of the triangulated spectrin network of the red blood cell membrane (Arslan and Boyce, 2006, “Constitutive Modeling of the Finite Deformation Behavior of Membranes Possessing a Triangulated Network Microstructure,” ASME J. Appl. Mech., 73, pp. 536–543), a general anisotropic hyperelastic strain energy function is formulated and used to define the effective anisotropic continuum level constitutive model of the mechanical behavior of the microtubule wall. In particular, the role of the anisotropic microstructure resulting from the different lattice bond lengths and bond stiffnesses is examined to explain nature’s optimization of microstructural orientation in providing a high axial stiffness combined with low shear stiffness.

On formulas for the Rayleigh wave velocity in pre-stressed compressible solids

In this paper, formulas for the velocity of Rayleigh waves in compressible isotropic solids subject to uniform initial deformations are derived using the theory of cubic equation. They are explicit, have simple algebraic forms, and hold for a general strain energy function. Unlike the previous investigations where the derived formulas for Rayleigh wave velocity are approximate and valid for only small enough values of pre-strains, this paper establishes exact formulas for Rayleigh wave velocity being valid for any range of pre-strains. When the prestresses are absent, the obtained formulas recover the Rayleigh wave velocity formula for compressible elastic solids. Since obtained formulas are explicit, exact and hold for any range of pre-strains, they are good tools for evaluating nondestructively prestresses of structures.

Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer

The dispersive behavior of small amplitude waves propagating along a non-principal direction in a pre-stressed, compressible elastic layer is considered. One of the principal axes of stretch is normal to the elastic layer and the direction of propagation makes an angle θ with one of the in-plane principal axes. The dispersion relations which relate wave speed and wavenumber are obtained for both symmetric and anti-symmetric motions by formulating the incremental boundary value problem for a general strain energy function. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of the anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress and propagation angle, it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds, while other higher modes have an infinite phase speed. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the Rayleigh surface wave speed and the limiting wave speeds of the layer, respectively. Numerical results are presented for a Blatz–Ko material and the effect of the propagation angle is clearly illustrated.

Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.

On Formulas for the Velocity of Rayleigh Waves in Prestrained Incompressible Elastic Solids

In the present paper, formulas for the velocity of Rayleigh waves in incompressible isotropic solids subject to a general pure homogeneous prestrain are derived using the theory of cubic equation. They have simple algebraic form and hold for a general strain-energy function. The formulas are concretized for some specific forms of strain-energy function. They then become totally explicit in terms of parameters characterizing the material and the prestrains. These formulas recover the (exact) value of the dimensionless speed of Rayleigh wave in incompressible isotropic elastic materials (without prestrain). Interestingly that, for the case of hydrostatic stress, the formula for the Rayleigh wave velocity does not depend on the type of strain-energy function.

A generalization of Hencky’s strain-energy density to model the large deformations of slightly compressible solid rubbers

A power-law relationship between the stretches has been shown to describe the inherent slight compressibility of isotropic solid rubbers in simple tension. It is well known that the Hencky strain-energy function predicts such a kinematic power-law relationship. Although the resulting constitutive relation adequately models the stress response for moderate deformations, it is not capable of predicting the stress response for large deformations. A strain-energy function for slightly compressible rubber is derived that both incorporates the kinematic power-law relation and reduces to the Hencky form on using a simple linearization process. Special forms of this new general strain-energy function are shown to be consistent with the stress–stretch experimental data in simple tension over a large range of axial stretches. This general form is also shown to be consistent with the observed behavior of rubbers in compression tests involving moderate volume changes.

Nonlinear transversely isotropic elastic solids: an alternative representation

A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress-strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis-Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.

Acoustic waves at the interface of a pre-stressed incompressible elastic solid and a viscous fluid

We analyse the influence of pre-stress on the propagation of interfacial waves along the boundary of an incompressible hyperelastic half-space that is in contact with a viscous fluid extending to infinity in the adjoining half-space. One aim is to derive rigorously the incremental boundary conditions at the interface; this derivation is delicate because of the interplay between the Lagrangian and the Eulerian descriptions but is crucial for numerous problems concerned with the interaction between a compliant wall and a viscous fluid. A second aim of this work is to model the ultrasonic waves used in the assessment of aortic aneurysms, and here we find that for this purpose the half-space idealization is justified at high frequencies. A third goal is to shed some light on the stability behaviour in compression of the solid half-space, as compared with the situation in the absence of fluid; we find that the usual technique of seeking standing waves solutions is not appropriate when the half-space is in contact with a fluid; in fact, a correct analysis reveals that the presence of a viscous fluid makes a compressed neo-Hookean half-space slightly more stable. For a wave travelling in a direction of principal strain, we obtain results for the case of a general (incompressible isotropic) strain-energy function. For a wave travelling parallel to the interface and in an arbitrary direction in a plane of principal strain, we specialize the analysis to the neo-Hookean strain-energy function.

Homogenization and Global Responses of Inhomogeneous Spherical Nonlinear Elastic Shells

Homogenization of radially inhomogeneous spherical nonlinear elastic shells subject to internal pressure is studied. The equivalent homogeneous material is defined in such a way that it gives rise to exactly the same global response to the pressure load as that of the inhomogeneous shell. For a shell with general strain–energy function and inhomogeniety, the strain–energy function of the equivalent homogeneous material is determined explicitly. The resulting formula is used to study layered composite shells. The equivalent homogeneous material for an infinitely fine layered composite shell is examined, and is found to give not only the same global response, but also the same average stress field as the composite shell does.

The Valanis–Landel Strain-Energy Function

A constitutive equation is derived for the Cauchy stress matrix for arbitrary deformations of an isotropic elastic solid characterized by a Valanis-Landel strain-energy function. A simple example is given of the way in which results for controllable deformations of an incompressible elastic solid, with a Valanis-Landel strain-energy function, can be obtained from the known results for the more general strain-energy function employed by Rivlin.

Some comments on the dynamic properties of anisotropic and strongly anisotropic pre-stressed elastic solids

The effect of pre-stress on the propagation of small amplitude waves in an incompressible, transversely isotropic elastic solid is discussed in respect of the most general appropriate strain energy function. A simple set of sufficient conditions is noted which ensures that two real wave speeds exist for all directions of propagation. In the case of bi-axial primary deformations, and for propagation within each principal plane of the homogeneous primary deformation, the propagation condition is factorised and conditions which are both necessary and sufficient to ensure the existence of two real wave speeds are established. The paper also includes some graphical illustrations of the associated slowness and wave surfaces and discussion of the strongly anisotropic case, for which the extensional modulus along the preferred fibre direction is much larger than other material parameters.

An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

An asymptotically consistent two–dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long–wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading–order and refined higher–order equations for the mid–surface deflection are derived. In the case of zero normal initial static stress and in–plane tension, the leading–order equation reduces to the classical membrane equation, with its refined counterpart also being obtained. The theory is applied to a one–dimensional edge loading problem for a semi–infinite plate. In doing so, the leading– and higher–order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi–front. A solution is derived by using the method of matched asymptotic expansions.