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This paper computes stationary profiles of an isotropic, homogeneous, linearly elastic rod with its endpoint locations and tangents specified. One end of the rod is clamped and the other end makes contact with a flat, rigid, impenetrable surface, which is displaced towards the clamped end. This boundary value problem has applications to biomechanical sensory devices such as mammal whiskers. The paper gives exact analytical solutions to the boundary value problem, embracing the planar equilibrium configurations for both point contact and line contact with the wall. Plots of loading paths for different inclinations of the wall provide an insight into the force-displacement relationship appertaining to real world slender rods under this type of loading. This report is complemented by data obtained from corresponding experimental studies which shed light on the differences between the model, which is based on the mathematical theory of elasticity, and the mechanics of real world long slender bodies, such as mammalian vibrissal systems. [ABSTRACT FROM AUTHOR]

Abstract: Numerical modeling of localization phenomena shows that constitutive equations with internal length scale are necessary to properly model the post-localization behavior. Moreover, these models allow an accurate description of the scale effects observed in some phenomena like micro-indentation. This paper proposes some analytical results concerning a boundary value problem in a medium with microstructure. In addition to their own usefulness, such analytical solutions can be used in benchmark exercises for the validation of numerical codes. The paper focuses on the thick-walled cylinder problem, using a general small strain isotropic elastic second gradient model. The most general isotropic elastic model involving seven different constants is used and the expression of the analytical solutions is explicitly given. The influence of the microstructure is controlled by the internal length scale parameter. The classical macrostress is no more in equilibrium with the classical forces at the boundary. Double stresses are indeed also generated by the classical boundary conditions and, as far as the microstructure effects become predominant (i.e. the internal length scale is much larger than the thickness of the cylinder), the macrostresses become negligible. This leads to solutions completely different from classical elastic ones. [Copyright &y& Elsevier]

Abstract: This paper compares and evaluates strain-gradient extensions of the conventional plasticity theory. Attention is focused on the ability of individual formulations to act as localization limiters, i.e., to regularize the boundary value problem in the presence of softening and to prevent localization of plastic strain increments into a set of zero measure. To keep the presentation simple and to highlight the essential properties of the investigated models, only the static, rate-independent response in the small-strain range and in the one-dimensional setting is considered. These restrictions permit an analytical or semi-analytical treatment of the problem, while the basic characteristics of the solutions remain valid in the general, multi-dimensional case. The onset of localization is characterized as a bifurcation from a uniform state. The subsequent evolution of the localized process zone and of the shape of the strain profile is studied numerically. It is shown that certain pathologies, e.g., expansion of the plastic region accompanied by stress locking, may arise at later stages of localization. A similar analysis of models with gradients of internal variables is presented in a companion paper. [Copyright &y& Elsevier]

Abstract: The wave motion in micromorphic microstructured solids is studied. The mathematical model is based on ideas of Mindlin and governing equations are derived by making use of the Euler–Lagrange formalism. The same result is obtained by means of the internal variables approach. Actually such a model describes internal fields in microstructured solids under external loading and the interaction of these fields results in various physical effects. The emphasis of the paper is on dispersion analysis and wave profiles generated by initial or boundary conditions in a one-dimensional case. [Copyright &y& Elsevier]

Abstract: The paper presents analytical or semi-analytical solutions for the formation and evolution of localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. A variationally based formulation of explicit gradient plasticity with linear softening is used, and the ensuing jump conditions and boundary conditions are discussed. Three cases with different regularity of the stress distribution are considered, and the problem is converted to a dimensionless form. Relations linking the load level, size of the plastic zone, distribution of plastic strain and plastic elongation of the bar are derived and compared to another, previously analyzed gradient formulation. [Copyright &y& Elsevier]

Abstract: Various macroscopic models to describe instability pattern formation are discussed in this paper. They are similar to the Ginzburg–Landau envelope equation, but they can remain valid away from the bifurcation and are based on the technique of Fourier series with slowly varying coefficients. We focus on two questions: the need to take phase changes into account and the boundary conditions to be associated with macroscopic models. The analysis is carried out on the basis of numerical simulations for the problem of a compressed beam on a nonlinear foundation that is quite similar to the well known Swift–Hohenberg equation. The first macroscopic model involves a real envelope so that the phase is assumed to be constant. The second model is also macroscopic and it is a sort of Ginzburg–Landau equation with a complex envelope. The third one follows from a multi-scaled approach with a numerical bridging between the full model near the boundary and a macroscopic model in the bulk. [Copyright &y& Elsevier]

Abstract: In this work, we have assessed the results of the local and nonlocal versions of Rousselier’s damage model, which have been used here for simulation of ductile crack growth. There are several issues regarding the accuracy of the results which has been addressed in this paper, e.g., accuracy in simulation of crack path, extent and width of the damaged region, fracture resistance behaviour in situations such as symmetric vs. non-symmetric boundary-value problems, mixed-mode loading vs. mode-I loading of the crack-tip, etc. It was also observed that the shape and orientation of the elements at the crack-tip, in addition to their size, influence the results of the local damage model. In this work, it was shown that the above issues can be resolved through the use of nonlocal damage models. The predictions of the nonlocal model are also consistent with the experimental observations unlike its local counterpart. Several examples were presented, where the results as obtained by both the local and nonlocal models were compared. From this experience, it is recommended that the local damage models should not be used blindly by the analysts for all kinds of mesh design, loading, boundary conditions, etc. [Copyright &y& Elsevier]