Journal of Elasticity最新文献

In isotropic and incompressible elastodynamics, Carroll’s solutions for plane, circularly polarized shear waves represent an important class of controllable, non-universal solutions. It has already been shown that these solutions can be obtained in a compact way by rewriting the equations governing transverse waves as a single complex differential wave equation. In this work, we show that a complex wave equation formally identical to Carroll’s one can be used to construct new classes of solutions, still in the isotropic and incompressible setting, but in a more general case involving plane wave superimposed to homogeneous deformations. We analyze these new solutions and observe that they also exist in an interesting asymptotic regime frequently encountered in non-linear acoustics. Moreover, we demonstrate that these solutions are closely linked to symmetry properties of the complex wave equation and satisfy a nonlinear universal relation—a noteworthy and rare result.

We investigate universal deformations in compressible isotropic Cauchy elastic solids with residual stress, without assuming any specific source for the residual stress. We show that universal deformations must be homogeneous, and the associated residual stresses must also be homogeneous. Since a non-trivial residual stress cannot be homogeneous, it follows that residual stress must vanish. Thus, a compressible Cauchy elastic solid with a non-trivial distribution of residual stress cannot admit universal deformations. These findings are consistent with the results of Yavari and Goriely (Proc. R. Soc. A 472(2196):20160690, 2016), who showed that in the presence of eigenstrains, universal deformations are covariantly homogeneous and in the case of simply-connected bodies the universal eigenstrains are zero-stress (impotent).

A new proof is presented for the plane version of Ericksen’s theorem.

This work proposes a cono-spherical indentation method for characterizing the parameters of Mooney-Rivlin and Arruda-Boyce constitutive models in rubber-like hyperelastic materials. The cono-spherical indentation model (CSIM) is formulated based on the principle of equivalent energy to establish a relationship between load-depth responses and constitutive model parameters. This model enables the efficient, in-situ and non-destructive properties characterization of hyperelastic materials during service conditions. Validation of CSIM is performed through extensive finite element simulations covering a broad spectrum of hyperelastic constitutive parameters, encompassing the Mooney-Rivlin and Arruda-Boyce models. The constitutive model parameters for four rubber-like materials are inversely identified from load-depth curves obtained through cono-spherical indentation using CSIM, and stress-stretch curves are derived from the inversely identified parameters. The accuracy of the reverse-predicted results is confirmed by comparing them with results from uniaxial tensile tests conducted over a wide range of deformations. These results highlight the efficacy of CSIM, utilizing the Mooney-Rivlin and Arruda-Boyce constitutive models, as a precise and dependable approach for predicting constitutive parameters of rubber-like materials.

Steady shock wave shapes are studied in a size-dependent Cosserat continuum. The Cosserat continuum enriches a standard continuum by introducing a deformable triad of linearly independent director vectors that are determined by additional balances of director momentum, which introduce size-dependence. Limiting attention to purely mechanical nonlinear elastic response, it is shown that steady shock waves in uniaxial strain admit non-trivial trailing wave shapes after the jump at the shock front. The dependence of the wave shapes on the Cosserat material parameters are investigated for weak and strong shocks.

It is shown that some of the assumptions and further restrictions imposed in Reference (Muki and Sternberg in Z. Angew. Math. Phys. 16:611–648, 1965) in relation to development of the plane strain version of the conventional couple-stress theory are invalid. This finding comes a result a new development presented in the first part of this study (Soldatos in J. Elast. 153:185–206, 2023), according to which a refined couple-stress theory of isotropic linear elasticity enables determination of the trace of the couple-stress tensor and, henceforth, of the antisymmetric part of the stress and, thus, the total stress tensors. The present communication reconsiders those assumptions and restrictions (Muki and Sternberg in Z. Angew. Math. Phys. 16:611–648, 1965), separates the ones that are still valid from their invalid counterpart, and, to the extent that this is possible and acceptable, rebuilds the foundation that the plane strain concept can be based upon in polar isotropic linear elasticity. It is accordingly found, and also demonstrated with a couple of relevant boundary value problem applications, that, while the assumption of plane strain naturally guarantees that displacements and strains depend only on in-plane co-ordinates parameters, the couple-stress and the stress tensors generally depend on the out-of-plane co-ordinate parameter as well.

A general semi-analytical solution is developed for the time-dependent, axisymmetric problem of elastoplastic stress concentration around expanding cavities in the presence of heat conduction. The formulation is based on the incremental theory of plasticity within a Lagrangian framework of thermo-elastoplastic constitutive relations, while incorporating the von Mises yield criterion, strain hardening, and the associated flow rule. The transient heat conduction problem is treated through application of the Laplace integral transform. It is shown that the problem constitutive equations reduce to a system of three nonlinear ordinary differential equations describing the path-dependent evolution of the elastoplastic stress components with time. The long-time asymptotic solution provides the distribution of residual stresses. The proposed general solution for the thermal cavity expansion problem offers a rigorous benchmark for verification of relevant numerical solvers. The application of the proposed solution to the stress analysis of shrink-fit assemblies is demonstrated by examining the mechanical interaction between a solid shaft and a hollow hub, wherein the hub undergoes elastoplastic deformation due to the thermal expansion of the inner shaft. The results show that the strain-hardening parameter plays a critical role in controlling the extent of plastic deformation in the hub. Furthermore, a case study highlights the influence of constitutive behavior and stress-path dependency on the development of residual stresses in shrink-fit assemblies.

In this paper the nonlinear elasticity theory of volumetric growth based on residual stress that was introduced in previous contribution (Huang et al. in J. Elast. 145:223–241, 2021) is developed further, and is then focused on an applications of the theory with computational examples. The main idea here is to use residual stress in an intact unloaded configuration, or the deformation from a fixed and intact reference configuration (which may itself be residually stressed), as a means to assess the growth in a soft solid, the developing unloaded configuration and the accompanying developing residual stress. The general theory is presented in terms of the free energy per unit mass and the associated energy functions relative to the reference configuration and the unloaded configuration. Growth of a thick-walled spherical shell is examined in order to illustrate the theory using simple prototype energy functions. A general programme for obtaining the developing deformed configuration is outlined and several possible growth laws are discussed for the growth of a spherical shell under internal pressure. This study shows that growth modelling based on the unloaded configurations may provide insights into the development of residual stress and morphology, both of which are, in principle, accessible to experimental observation. For several possible growth laws detailed numerical results are provided to illustrate the evolution of growth and the associated residual stress.

The representation theory of tensor functions is essential to constitutive modeling of materials including both mechanical and physical behaviors. Generally, material symmetry is incorporated in the tensor functions through a structural or anisotropic tensor that characterizes the corresponding point group. The general mathematical framework was well-established in the 1990s. Nevertheless, the traditional theory suffers from a grand challenge that many point groups involve fourth or sixth order structural tensors that hinder its practical applications in engineering. Recently, researchers have reformulated the representation theory and opened up opportunities to model anisotropic materials using lower-order (i.e., 2nd- order and lower) structural tensors only, although the theory was not fully established. This work aims to fully establish the reformulated representation theory of tensor functions for all two-dimensional point groups. It was found that each point group needs a structural tensor set to characterize the symmetry. For each two-dimensional point group, the structural tensor set is proposed and the general tensor functions are derived. Only lower-order structural tensors are introduced so researchers can readily apply these tensor functions for their modeling applications. The theory presented here is useful for constitutive modeling of materials in general, especially for composites, nanomaterials, soft tissues, etc.