Antiplane Shear Deformation Research Articles

On the refined boundary condition at the edge of a thin elastic strip supported by a Winkler-type foundation under antiplane shear deformation

The derivation of the boundary conditions is the most challenging part of the asymptotic techniques underlying low-dimensional models for thin elastic structures. At the moment, these techniques do not take into consideration the effect of the environment, e.g., a Winkler foundation, when tackling boundary conditions, and have to be amended. In this paper as an example we consider an antiplane problem for a thin elastic strip contacting with a relatively compliant Winkler foundation. Refined boundary conditions at an edge loaded by prescribed stresses are established using a properly adjusted Saint-Venant’s principle. They appear to be useful for advanced structure modelling including analysis of the static equilibrium under self-equilibrated loading.

Weak solvability of elliptic variational inequalities coupled with a nonlinear differential equation

In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.

Neutral inhomogeneity in circular cylinder subjected to prescribed boundary displacement

In this paper a single circular inhomogeneity embedded within a solid circular cylinder whose curved boundary surface is subjected to a given boundary displacement in axial direction is considered. The displacement neutrality of the coupled system of host body and inclusion is studied. The neutral inhomogeneity (inclusion) does not disturb the displacement, strain and stress fields in the host body. In this paper the deformation of the considered inhomogeneous cylinder is a linear antiplane shear deformation.

On dynamic crack propagation in a lattice Boltzmann method for elastodynamics in 2D

AbstractIn recent years, the development of lattice Boltzmann methods (LBMs) for solids has gained attention. Fracture mechanics as a viable application for these methods has been presented before, albeit for mode III cracks in the context of an LBM for antiplane shear deformation. The performance of the LBM itself is promising, while the usage of a regular lattice simplifies the modelling of fractures significantly. Recent advancements in LBMs for solids, especially the description of Dirichlet‐ and Neumann‐type boundary conditions, now make it possible to extend the LBM simulation of crack propagation to the plane strain case with modes I and II crack opening, including growth with non‐uniform speed in arbitrary directions. For this, the configurational force acting on a crack tip is utilised. The definition of the moments of the LBM, which are based on the balance laws of continuum mechanics, render the evaluation of macroscopic fields in the configuration straightforward. In this work, the general in‐plane case of dynamic crack propagation is shown and necessary considerations for the implementation are discussed. Lastly, numerical examples showcase the capabilities of the proposed method to model dynamic fractures and establish a proof‐of‐concept.

Influence of Creep Compaction and Dilatancy on Earthquake Sequences and Slow Slip

AbstractFluids influence fault zone strength and the occurrence of earthquakes, slow slip events, and aseismic slip. We introduce an earthquake sequence model with fault zone fluid transport, accounting for elastic, viscous, and plastic porosity evolution, with permeability having a power‐law dependence on porosity. Fluids, sourced at a constant rate below the seismogenic zone, ascend along the fault. While the modeling is done for a vertical strike‐slip fault with 2D antiplane shear deformation, the general behavior and processes are anticipated to apply also to subduction zones. The model produces large earthquakes in the seismogenic zone, whose recurrence interval is controlled in part by compaction‐driven pressurization and weakening. The model also produces a complex sequence of slow slip events (SSEs) beneath the seismogenic zone. The SSEs are initiated by compaction‐driven pressurization and weakening and stalled by dilatant suctions. Modeled SSE sequences include long‐term events lasting from a few months to years and very rapid short‐term events lasting for only a few days; slip is ∼1–10 cm. Despite ∼1–10 MPa pore pressure changes, porosity and permeability changes are small and hence fluid flux is relatively constant except in the immediate vicinity of slip fronts. This contrasts with alternative fault valving models that feature much larger changes in permeability from the evolution of pore connectivity. Our model demonstrates the important role that compaction and dilatancy have on fluid pressure and fault slip, with possible relevance to slow slip events in subduction zones and elsewhere.

Decay of solutions for strain gradient mixtures

AbstractWe study antiplane shear deformations for isotropic and homogeneous strain gradient mixtures of the Kelvin‐Voigt type in a cylinder. Our aim is to analyze the behaviour of the solutions with respect to the time variable when a dissipative structural mechanism is considered. We study three different cases, each at a time. For each case we prove existence and uniqueness of solutions. We obtain the exponential decay of the solutions in the hyperviscosity and viscosity cases. Exponential decay is also expected when the dissipation is generated by the relative velocity (in the generic case, although a particular combination of the constitutive parameters leads to slow decay). These results are proved with the help of the theory of semigroups.

Dynamic propagation of mode III cracks in a Lattice Boltzmann method for solids

This work presents concepts and algorithms for the simulation of dynamic fractures with a Lattice Boltzmann method (LBM) for linear elastic solids. This LBM has been presented previously and solves the wave equation, which is interpreted as the governing equation for antiplane shear deformation. Besides the steady growth of a crack at a prescribed crack velocity, a fracture criterion based on stress intensity factors has been implemented. This is the first time that crack propagation with a mechanically relevant criterion is regarded in the context of LBMs. Numerical results are examined to validate the proposed method. The concepts of crack propagation introduced here are not limited to mode III cracks or the simplified deformation assumption of antiplane shear. By introducing a rather simple processing step into the existing LBM at the level of individual lattice sites, the overall performance of the LBM is maintained. Our findings underline the validity of the LBM as a numerical tool to simulate solids in general as well as dynamic fractures in particular.

Superposing plane strain on anti-plane shear deformations in a special class of fiber-reinforced incompressible hyperelastic materials

The purpose of the present research is to investigate how the nonlinearity and the boundary conditions have the power to influence the coupling of the various modes of deformation when a plane strain deformation is superimposed on anti-plane shear deformation for a class of fiber-reinforced incompressible hyperelastic materials described by a strain energy density W. Attention is confined when W depends only on the first invariant of the strain tensor and on the square of the stretch in the direction of the fibers.We are able to write down the governing equations of equilibrium as a coupled system of three nonlinear partial differential equations for three displacement fields. Two displacements are the in-plane components and one displacement is the anti-plane state. The system that we are able to deduce in a compact form is always compatible at variance with the case in which the anti-plane shear problem is analyzed. As explicit example of our findings we study the problem of the helical shear and we investigate into details the coupling of its axial and azimuthal components.

Lattice Boltzmann method for antiplane shear deformation: non-lattice-conforming boundary conditions

In this work, two different approaches to treat boundary conditions in a lattice Boltzmann method (LBM) for the wave equation are presented. We interpret the wave equation as the governing equation of the displacement field of a solid under simplified deformation assumptions, but the algorithms are not limited to this interpretation. A feature of both algorithms is that the boundary does not need to conform with the discretization, i.e., the regular lattice. This allows for a larger flexibility regarding the geometries that can be handled by the LBM. The first algorithm aims at determining the missing distribution functions at boundary lattice points in such a way that a desired macroscopic boundary condition is fulfilled. The second algorithm is only available for Neumann-type boundary conditions and considers a balance of momentum for control volumes on the mesoscopic scale, i.e., at the scale of the lattice spacing. Numerical examples demonstrate that the new algorithms indeed improve the accuracy of the LBM compared to previous results and that they are able to model boundary conditions for complex geometries that do not conform with the lattice.

Approximate controllability of nonsimple elastic plate with memory

<p style='text-indent:20px;'>In this paper, we give some qualitative results on the behavior of a nonsimple elastic plate with memory corresponding to anti-plane shear deformations. First we describe briefly the equations of the considered plate and then we study the well-posedness of the resulting problem. Secondly, we perform the spectral analysis, in particular, we establish conditions on the physical constants of the plate to guarantee the simplicity and the monotonicity of the roots of the characteristic equation. The spectral results are used to prove the exponential stability of the solutions at an optimal decay rate given by the physical constants. Then we present an approximate controllability result of the considered control problem. Finally, we give some numerical experiments based on the spectral method developed with implementation in MATLAB for one and two-dimensional problems.</p>

Lattice Boltzmann Method for Antiplane Shear with Non‐Mesh Conforming Boundary Conditions

AbstractLattice Boltzmann methods [1] have been extended beyond their initial usage in transport problems, and can be used to solve a broader range of partial differential equations, e.g. the wave equation [2]. Thereby they can be utilized for fracture mechanics [3]. In the context of antiplane shear deformation we previously examined a stationary crack [4, 5] with a finite width. In this work we present two implementation strategies for non‐mesh conforming boundary conditions, for which the bounding geometry does not need to adhere to the underlying lattice. This rectifies problems in modeling the crack. A numerical example shows the improvement compared to the previous results.

Shape of a single inclusion enclosing uniform stress for a non-uniform far-field loading in anti-plane shear

The problem as to whether an inclusion when embedded in an elastic infinite matrix with a perfectly bonded interface could achieve uniform internal stress has been studied extensively in the literature. It has been shown that in almost all cases ellipsoidal shape (including elliptical shape for two-dimensional deformation) is the only inclusion shape ensuring the uniformity of the stress inside the inclusion if the external loading exerted remotely on the edge of the matrix is uniform. From a practical point of view, this result remains valid for a non-uniform remote loading imposed on the matrix provided the product of the inclusion's size and the gradient of the non-uniform loading is small enough (as compared with the overall level of the stress field induced by the non-uniform loading at the inclusion's position). In this paper, we explore the possibility of designing an inclusion enclosing uniform internal stress in an elastic infinite plane under anti-plane shear deformation for a non-uniform remote loading with arbitrary gradients. We derive the equations with respect to the parameters describing the configuration of the inclusion leading to a uniform internal stress inside the inclusion for a general polynomial remote loading, and establish the analytic and numerical solutions for the shape of the inclusion which are illustrated via several examples.

Effective moduli of multiferroic fibrous composites with strain gradient and electromagnetic field gradient effects

This paper studies the effective moduli of multiferroic fibrous composites with strain gradient and electromagnetic field gradient effects subjected to generalized anti-plane shear deformation. Using the energy variational formulation and the generalized self-consistent method, we can determine the effective moduli of the heterogeneous material. The derived solutions are then applied to a BaTiO3–CoFe2O4 composite to demonstrate the effect of strain gradient, electric field gradient, and magnetic field gradient. Numerical results show that the gradient effects have significant influence on the effective moduli of the multiferroic composite. We show that the magnetoelectric voltage coefficient, the figure of merit of the multiferroic material, can be enhanced many-fold than that of the classical solution. In addition, most of the effective moduli increase as the radius of the fiber decreases (or equivalently the characteristic length increases).

Decay and numerical results in nonsimple viscoelasticity

In this paper, we give some mathematical and numerical results on the behavior of a nonsimple viscoelastic plate corresponding to anti-plane shear deformations. First, we derive briefly the equations of the considered plate, and then, we study the well-posedness problem. Second, we prove that the solutions decay exponentially at a rate proportional to the total mass of the second order. Finally, we give some numerical experiments based on the spectral method developed for multi-dimensional problems with implementation in MATLAB for one and two-dimensional spaces.

Global Bifurcation of Anti-plane Shear Fronts

We consider anti-plane shear deformations of an incompressible elastic solid whose reference configuration is an infinite cylinder with a cross section that is unbounded in one direction. For a class of generalized neo-Hookean strain energy densities and live body forces, we construct unbounded curves of front-type solutions using global bifurcation theory. Some of these curves contain solutions with deformations of arbitrarily large magnitude.

Effects of strain gradient and electromagnetic field gradient on potential and field distributions of multiferroic fibrous composites

The objective of this work is to study the effect of strain gradient, electric field gradient, and magnetic field gradient on the potential and field distributions of multiferroic fibrous composites subjected to generalized anti-plane shear deformation. A detailed energy variational formulation with strain, strain gradient, electric field, electric field gradient, magnetic field, and magnetic field gradient as independent variables is provided, and the equilibrium equations with the complete boundary conditions are simultaneously determined. Three internal characteristic lengths of the underlying microstructure are introduced in the constitutive equations. The general solutions in cylindrical polar coordinates consisting of the modified Bessel functions of first and second kind are derived. One inclusion problem’s solution is also obtained. The derived solutions are applied to a $$\hbox {CoFe}_{2}\hbox {O}_{4}$$ -PZT-4 composite to demonstrate the effect of strain gradient, electric field gradient, and magnetic field gradient. Numerical results provide insights into the role of characteristic lengths on the potential and field distributions of a multiferroic fibrous composite.

A boundary value problem of orthotropic electroelastic circular cylinder

A boundary value problem of orthotropic piezoelectric solid circular cylinder which is in the state of antiplane shear deformation is studied. The whole boundary surface is loaded by an equilibrium axial traction. This paper gives an analytical solution of the considered antiplane shear deformation.

The symmetry and loading-independency of multiple inclusions enclosing uniform stresses in an infinite elastic plane

The identification of multiple interacting inclusions with uniform internal stresses in an infinite elastic matrix subjected to a uniform remote loading is of fundamental importance in the mechanics and design of particulate composite materials. In anti-plane shear and plane deformations, certain sufficient conditions have been established in the literature which guarantee uniform internal stresses inside multiple interacting inclusions displaying various symmetries when the matrix is subjected to specific uniform remote loading. Correspondingly, sufficient conditions which allow for the design of multiple interacting inclusions independent of any specific form of (uniform) remote loading have also been established. In this paper, we demonstrate rigorously that, in all cases, these sufficient conditions are also necessary conditions and indeed allow for the identification of all possible collections of such inclusions.

Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.

Boundary optimal control for antiplane contact problems with power-law friction

• A variational inequality models the antiplane shear deformation of a body in frictional contact with a foundation. • The distance between the solution of the variational inequality and a given target can be minimized by acting with a boundary control force. • The boundary control can be approximated by using linearization and fixed point method. We consider a contact model with power-law friction in the antiplane context. Our study focuses on the boundary optimal control, paying special attention to optimality conditions and computational methods. Depending on the exponent of the power-law friction, we are able to deduce an optimality condition for the original problem or for a regularized version of it. Furthermore, we introduce and analyze a computational technique based on linearization, saddle point theory and a fixed point method. For a slightly modified optimal control problem, some numerical experiments are presented.