Pure azimuthal shear of isotropic, incompressible hyperelastic materials with limiting chain extensibility (original) (raw)
Related papers
1999
The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy-Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model.
1999
The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy-Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney-Rivlin model remains an open question which can be resolved only by large strain experiments.
Azimuthal Shear of a Transversely Isotropic Elastic Solid
Mathematics and Mechanics of Solids, 2008
In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either “with” or “against” the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ell...
Polymer Testing, 2013
The concepts of simple and pure shear are well known in continuum mechanics. For small deformations, these states differ only by a rotation. However, correlations between them are not well defined in the case of large deformations. The main goal of this study is to compare these two states of deformation by means of experimental and theoretical approaches. An incompressible isotropic hyperelastic material was used. The experimental procedures were performed using digital image correlation (DIC). The simple shear deformation was obtained by single lap joint testing, while the pure shear was achieved by means of planar tension testing. Classical hyperelastic constitutive equations available in the literature were used. As a consequence, the results indicate that simple shear cannot be considered as pure shear combined with a rotation when large deformation is assumed, as widely considered in literature.
We find the strain energy function for isotropic incompressible solids exhibiting a linear relationship between shear stress and amount of shear, and between torque and amount of twist, when subject to large simple shear or torsion deformations. It is inclusive of the well-known neo-Hookean and the Mooney–Rivlin models, but also can accommodate other terms, as certain arbitrary functions of the principal strain invariants. Effectively, the extra terms can be used to account for several non-linear effects observed experimentally but not captured by the neo-Hookean and Mooney–Rivlin models, such as strain stiffening effects due to limiting chain extensibility.
Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials
Journal of Elasticity, 2010
The classical problem of simple shear in nonlinear elasticity has played an important role as a basic pilot problem involving a homogeneous deformation that is rich enough to illustrate several key features of the nonlinear theory, most notably the presence of normal stress effects. Here our focus is on certain ambiguities in the formulation of simple shear arising from the determination of the arbitrary hydrostatic pressure term in the normal stresses for the case of an incompressible isotropic hyperelastic material. A new formulation in terms of the principal stretches is given. An alternative approach to the determination of the hydrostatic pressure is proposed here: it will be required that the stress distribution for a perfectly incompressible material be the same as that for a slightly compressible counterpart. The form of slight compressibility adopted here is that usually assumed in the finite element simulation of rubbers. For the particular case of a neo-Hookean material, the different stress distributions are compared and contrasted.
Simple shearing and azimuthal shearing of an internally balanced compressible elastic material
International Journal of Non-Linear Mechanics, 2016
The finite deformation response of a compressible internally balanced elastic material is studied for deformations that involve progressive shearing. The internally balanced material theory requires that an equation of internal balance is satisfied at each material point. This arises from the constitutive theory which makes use of a multiplicative decomposition of the deformation gradient. Satisfaction of the internal balance requirement then yields the most energetically favorable decomposition. Here we consider a particular compressible internally balanced material model that is motivated by a Blatz-Ko type energy from the conventional hyperelastic theory. The conventional hyperelastic theory occurs as a special limiting case of the internally balanced constitutive theory. More generally, the internally balanced material exhibits softer mechanical behavior. This gives rise to a stress-plateau in the simple shearing response whereas such plateaus do not occur in the corresponding hyperelastic treatment. The boundary value problem for azimuthal shearing with a possible radial stretching is then studied. The internally balanced material response is again found to be softer than that of the hyperelastic limiting case. This is manifest in terms of an upper bound to the applied twisting moment for the existence of solutions to the boundary value problem. In contrast, the hyperelastic limiting case has solutions for all values of applied moment.
On small azimuthal shear deformation of fibre-reinforced cylindrical tubes
Journal of Mechanics of Materials and Structures, 2011
The problem of azimuthal shear deformation of a transversely isotropic elastic circular cylindrical tube is considered and studied in the small deformation regime. The preferred direction of the transverse isotropy is assumed to lie on the plane of the tube cross-section and is due to the existence of a single family of plane spiral fibres. Consideration of the manner that either the tube material or the fibres may be constrained gives rise to four different versions of the problem which are all susceptible to an exact closed form solution when fibres are perfectly flexible. Particular attention is paid to the special case of straight fibres aligned along the radial direction of the tube cross-section, where comparisons are made between the aforementioned solution obtained when fibres are perfectly flexible and a corresponding solution obtained when fibres posses bending stiffness. It is found that the conventional linear elasticity considerations associated with the perfectly flexible fibre assumption cannot adequately account for the effects of material anisotropy. In contrast, effects of material anisotropy can be accounted for when fibres posses bending stiffness, by taking into consideration the action of couple-stress and therefore asymmetric stress. Moreover, an intrinsic material length parameter which appears naturally in the associated governing equations may be chosen as a representative of the fibre thickness in this case. It is also seen that deformation patterns of fibres possessing bending stiffness as well as corresponding stress distributions developed within the tube cross-section fit physical expectation much closer than their perfectly flexible fibre counterparts.
A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling
International Journal of Plasticity, 2006
A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure Ce. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.
Incompressible Transversely Isotropic Hyperelastic Materials and Their Linearized Counterparts
Journal of Elasticity
The strain-energy density W WWW for incompressible transversely isotropic hyperelastic materials depends on four independent invariants of the strain tensor. For consistency with the infinitesimal theory, it is well known that there are three necessary conditions on the derivatives of W WWW (evaluated in the undeformed state) that have to be to be satisfied in terms of the three independent elastic moduli of the linear theory for incompressible transversely isotropic materials. We consider three different sets of these linear elastic moduli and express them in terms of the relevant derivatives of W WW W evaluated in the undeformed state. Necessary and sufficient conditions on the linearized strain-energy to ensure positive-definiteness are given and thus we derive such conditions expressed in terms of derivatives of W WWW evaluated in the undeformed state. These conditions are proposed to be fundamental constitutive inequalities which must be satisfied by any physically realistic strain-energy for incompressible transversely isotropic hyperelastic material, in particular those models proposed for fiber-reinforced soft tissues. Some particular strain-energies that have been proposed in the literature are used as illustrations of the results.