Class Of Nonlinear Materials Research Articles

Remarks on pure and simple shear

AbstractIn a 2012 article in the International Journal of Non‐Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell's so‐called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear. Similar results can be found in a 2011 article of L. A. Mihai and A. Goriely. We confirm their results under weakened assumptions and consider the case of a shear load, i.e. a Biot pure shear stress. In addition, conditions under which Cauchy pure shear stresses correspond to (idealized) pure shear stretch tensors are stated and a new notion of idealized finite simple shear is introduced, showing that for certain classes of nonlinear materials, the results by Destrade et al. can be simplified considerably.

Shear, pure and simple

In a 2012 article in the International Journal of Non-Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell’s so-called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear. Similar results can be found in a 2011 article of L. A. Mihai and A. Goriely. We confirm their results under weakened assumptions and consider the case of a shear load, i.e. a Biot pure shear stress. In addition, conditions under which Cauchy pure shear stresses correspond to (idealized) pure shear stretch tensors are stated and a new notion of idealized finite simple shear is introduced, showing that for certain classes of nonlinear materials, the results by Destrade et al. can be simplified considerably.

Asymptotic fields ahead a crack for a class of non linear materials under mode III

The paper considers the mechanical fields near the tip of a crack deformed by an anti-plane shear at infinity for a class of non linear elastic materials. For brittle material rupture occurs when a maximal stretch is reached. Taking account of this critical value, the crack is replaced by a totally damaged zone of finite thickness named a quasicrack. Inside this domain, the stress is identically zero and the shape of the boundary between damaged and undamaged body is found analytically.

Breaking the diffraction limit in nonlinear materials

Optical writing of small structures is usually governed by the diffraction limit. However, if the substrate exhibits nonlinear properties, it is possible to break this limit with conventional optical focusing systems. We estimate here theoretically the fundamental limits of resolution achievable in two different classes of nonlinear materials using a scalar approximation. Within this approximation our calculations suggest that the best possible resolution is given by 0.45λ0/(NASNR), where λ0 is the wavelength in vacuum, NA is the numerical aperture of the focusing system, and SNR is the signal to noise ratio of the system. Thus, we find that nanometer resolution is only feasible if the SNR is sufficiently large.

An overview of interface cracks

In many cases cracks leading to fracture occur at interfaces between two different constituents, e.g. a fiber and the matrix in a composite. The early solutions of such problems in the context of linear elastic fracture mechanics (LEFM) revealed the presence of an unsatisfactory behavior: rapid oscillations in the stress and displacement fields, implying the physically impossible phenomenon of interpenetration. In the late seventies two major modifications were proposed: one by Atkinson and the other by Comninou. The Atkinson modification recognizes that the interface between two different materials is almost never sharp, and provides a gradual transition which avoids the oscillatory behavior. The Comninou approach simply resolves the interface crack problem by accepting its inherently unilateral nature (presence of inequalities) and allowing for partial closure at the tips. Both solutions have received high praise and severe criticism, especially since the oscillatory behavior is absent in some, at least, classes of nonlinear materials (Knowles and Sternberg). Current emphasis is placed on numerical models of the elastoplastic behavior of interface cracks. An additional complication is the apparent presence of mixed mode crack tip fields regardless of the type of the applied loading. Valuable as these theoretical efforts may be, it is becoming increasingly imperative to perform experiments to determine the mode of propagation and critical parameters governing interface fracture. Even then, the results must be viewed with caution, because the quantities of interest can only be determined indirectly. The present paper presents an overview of the interface crack problem and describes some preliminary experimental results in the fatigue and fracture of interface cracks.

Characteristic Forms of Differential Equations for Wave Propagation in Nonlinear Media

Characteristic forms of differential equations for wave propagation in nonlinear media are derived directly from equations of motion and equations which combine the constitutive equations and the equations of continuity. Both Lagrangian coordinates and Eulerian coordinates are considered. The constitutive equations considered here apply to a large class of nonlinear materials. The characteristic forms derived here clearly show which components of the stress and velocity are involved in the differentiation along the bicharacteristics. Moreover, the reduction to one-dimensional cases from three-dimensional problems is obvious for the characteristic forms obtained here. Examples are given and compared with the known solution in the literature for wave propagation in linear isotropic elastic solids and isentropic compressible fluids.

An Analysis for Plane Strain Plastic Deformation in Metal-Working Process

The two-dimensional plane strain equation of large plastic flow, expressed in terms of the stream function gradients, is modified using complex variables. The resulting governing equation is solved analytically for a class of nonlinear materials whose stress-strain rate behavior can be expressed by σ¯ = ce¯˙m. A one-to-one correspondence between the plastic flow equation using the Levy-Mises constitutive relations and the Navier-Stokes equation of fluid flow with zero inertia term is established for constant λ˙. This correspondence allows the existing fluid dynamic solution to be used for the plasticity analysis. An analytical solution of plane strain extrusion of linear material through square-cornered die is presented to illustrate the procedure.

An incremental theory of large strain and large displacement problems and its finite element formulation

This paper is concerned with the development of an incremental finite'element theory for the large strain and the large displacement problems, referred to the current configuration of the body. Using the convected coordinate system which is embedded in the body, the incremental representations of strain and stress tensors and the energy relations are presented, and then the general procedure to construct the so-called element stiffness matrix in incremental form is considered. The finite element formulation is developed for a typical constitutive relation and it is shown that some correction matrices, some of which have been omitted in the previous works, are to be added to the element stiffness matrix. Finally the method to assemble the equations of the element to the global system is discussed and a simple finite element model satisfying the compatibility condition is presented. The finite element theory developed in this paper is able to be extended to the problems for the general thermodynamical process of a broad class of nonlinear materials.