The rectilinear shear of fiber-reinforced incompressible non-linearly elastic solids

In this paper we study the problem of rectilinear shear of a slab of transversely isotropic incompressible non-linearly elastic material. In particular, the material under consideration is a base neo-Hookean model augmented with a function that accounts for the existence of a unidirectional reinforcement. The slab is of infinite length in two dimensions and finite thickness in the other one and is clamped to two rigid plates. Closed form analytic solutions are found for this problem. It is shown that, depending on the reinforcement strength and the fiber orientation in the undeformed configuration, weak solutions, i.e. solutions for which the smoothness required by the differential equations is relaxed, are to be expected. These solutions give rise to fiber kinking. It is shown that: (i) both sides of the kink involve fiber contraction; (ii) a suitable intermediate deformation between the two conjoined kink deformation states is non-elliptic.     

A phenomenological framework of constitutive modelling for incompressible and compressible elasto-plastic solids

A unified framework of constructing phenomenological constitutive models for a broad class of elasto-plastic materials exhibiting either plastical incompressibility (e.g., grey cast iron) or plastical compressibility (e.g., metal foams) is proposed. The constitutive framework also enables the different yielding behaviours under tension and compression as well as differential hardening along different loading paths to be accounted for in a relatively simple manner. The resulting plasticity model does not require the difficult task of experimentally probing the initial yield surface and its subsequent evolution; it is completely determined from a set of as few as two distinctive stress–strain curves measured along the characteristic loading paths for isotropic materials. The predicted yielding behaviours for grey cast iron and metal foams compare favourably with those measured.     

A general hyperelastic model for incompressible fiber-reinforced elastomers

This work presents a new constitutive model for the effective response of fiber-reinforced elastomers at finite strains. The matrix and fiber phases are assumed to be incompressible, isotropic, hyperelastic solids. Furthermore, the fibers are taken to be perfectly aligned and distributed randomly and isotropically in the transverse plane, leading to overall transversely isotropic behavior for the composite. The model is derived by means of the “second-order” homogenization theory, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” Compared to other constitutive models that have been proposed thus far for this class of materials, the present model has the distinguishing feature that it allows consideration of behaviors for the constituent phases that are more general than Neo-Hookean, while still being able to account directly for the shape, orientation, and distribution of the fibers. In addition, the proposed model has the merit that it recovers a known exact solution for the special case of incompressible Neo-Hookean phases, as well as some other known exact solutions for more general constituents under special loading conditions.     

Dynamical analysis of cavitation for a transversely isotropic incompressible hyper-elastic medium: Periodic motion of a pre-existing micro-void

In this paper, the dynamical cavitation behavior is analyzed for a sphere composed of a class of transversely isotropic incompressible hyper-elastic materials, where there is a pre-existing micro-void in the interior of the sphere. A second-order non-linear ordinary differential equation that governs the motion of the initial micro-void is obtained by using the boundary conditions. On analyzing the qualitative properties of the solutions of the differential equation, some interesting conclusions are proposed. It is proved that the number of equilibrium points of the differential equation depends on the values of the material parameters, and that the phase diagrams of the equation are closed, smooth and convex trajectories. For any prescribed surface tensile dead-loads, the motion of the initial micro-void undergoes a non-linear periodic oscillation. The dependence of the periodic motion of the initial micro-void on material parameters and the radius of the initial micro-void is examined, and numerical results are also provided. It is worth pointing out that the conclusions in this paper can be used to describe approximately the physical implications of the dynamical formation of a cavity in the sphere.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.