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One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to $$ H ^1$$ H 1 , such that standard nodal $$ H ^1$$ H 1 -finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces $$ H ^1$$ H 1 and $$ H (\mathrm {curl})$$ H ( curl ) , demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.

Abstract The equilibrium problem for an elastic body having an inhomogeneous inclusion with curvilinear boundaries is considered within the framework of antiplane shear. We assume that there is a power-law dependence of the shear modulus of the inclusion on a small parameter characterizing its width. We justify passage to the limit as the parameter vanishes and construct an asymptotic model of an elastic body containing a thin inclusion. We also show that, depending on the exponent of the parameter, there are the five types of thin inclusions: crack, rigid inclusion, ideal contact, elastic inclusion, and a crack with adhesive interaction of the faces. The strong convergence is established of the family of solutions of the original problem to the solution of the limiting one.

Homogenization theories provide models that simplify the constitutive description of heterogeneous media while retaining their macroscopic features. These theories have shown how the governing fields can be macroscopically coupled, even if they are microscopically independent. A prominent example is the Willis theory which predicted the strain–momentum coupling in elastodynamic metamaterials. Recently, a theory that is based on the Green’s function method predicted analogous electro–momentum coupling in piezoelectric metamaterials. Here, we develop a simpler scheme for fibrous piezoelectric composites undergoing antiplane shear waves. We employ a source-driven approach that delivers a unique set of effective properties for arbitrary frequency–wavevector pairs. We numerically show how the resultant homogenized model recovers exactly the dispersion of free waves in the composite. We also compute the effective properties in the long-wavelength limit and off the dispersion curves, and show that the resultant model satisfy causality, reciprocity and energy conservation. By contrast, we show how equivalent models that neglect the electromomentum coupling violate these physical laws.

We study the problem of аntiplane shear of an elastic body containing a finite array of arbitrarily located and oriented elliptic inclusions under the conditions of imperfect mechanical contact on the interfaces. The analytic solution of the problem is obtained by the method of multipole expansions with the use of the technique of complex potentials. By expanding the disturbances of the field of displacements caused by inclusions in a series in the system of elliptic harmonics and using the formulas for their reexpansion and exact validity of all contact conditions, we reduce the boundary-value problem of the theory of elasticity to an infinite system of linear algebraic equations. It is also proved that the reduction method is applicable to the indicated system, the rate of convergence of the solution is investigated, and the accumulated results are compared with the data available from the literature. The presented numerical results of parametric investigations reveal the presence of a strong dependence of stress concentration on the conditions of contact on the interfaces, as well as on the sizes, shapes, and relative positions of the inclusions.

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals is considered. The Fourier transform technique is used to solve a moving crack problem under the action of antiplane shear and inplane electric field. Full elastic stresses of phonon and phason fields and electric fields are derived for a crack running with constant speed in the periodic plane. Obtained results show that the coupled elastic fields inside piezoelectric quasicrystals depend on the speed of crack propagation, and exhibit the usual square-root singularity at the moving crack tip. Electric field and phason stresses do not have singularity and electric displacement and phonon stresses have the inverse square-root singularity at the crack tip for a permeable crack. The field intensity factors and energy release rates are obtained in closed form. The crack velocity does not affect the field intensity factors, but alters the dynamic energy release rate. Bifurcation angle of a moving crack in a 1D hexagonal piezoelectric quasicrystal is evaluated from the viewpoint of energy balance. Obtained results are helpful to better understanding crack advance in piezoelectric quasicrystals.

The transient response of an edge interfacial crack in bonded media with a functionally graded interlayer is investigated under the condition of an antiplane shear impact. The graded interlayer is assumed to follow power-law variations of the shear modulus and mass density between two dissimilar, homogeneous semi-infinite strips. Based on the use of Laplace and Fourier integral transforms, the crack problem is formulated in terms of a singular integral equation with a generalized Cauchy kernel, which is solved by the expansion-collocation technique in the Laplace domain. The time-dependent crack-tip behavior is determined through an inverse Laplace transform and the values of the mode III stress intensity factors are obtained as a function of time. The numerical results include the variations of such dynamic stress intensity factors for various combinations of the material and geometric parameters of the bonded system; more specifically, the effects of shear modulus, mass density, layer thickness, and their interactions on the dynamic overshoot characteristics of the transient crack-tip behavior in the presence of the graded interlayer are examined.

In this work, the lattice Boltzmann method is applied to study the dynamic behaviour of linear elastic solids under antiplane shear deformation. In this case, the governing set of partial differential equations reduces to a scalar wave equation for the out of plane displacement in a two dimensional domain. The lattice Boltzmann approach developed by Guangwu (J Comput Phys 161(1):61–69, 2000) in 2006 is used to solve the problem numerically. Some aspects of the scheme are highlighted, including the treatment of the boundary conditions. Subsequently, the performance of the lattice Boltzmann scheme is tested for a stationary crack problem for which an analytic solution exists. The treatment of cracks is new compared to the examples that are discussed in Guangwu’s work. Furthermore, the lattice Boltzmann simulations are compared to finite element computations. Finally, the influence of the lattice Boltzmann relaxation parameter on the stability of the scheme is illustrated.

In this paper, the generalized finite difference method is adopted for solving antiplane shear problems with multiple inclusions. This method is applied to domains in matrix and inclusions respectively in which the local Taylor expansion and moving least square method are applied to form the local system. And the final sparse system is obtained through satisfying the continuity and equilibrium conditions along the interface between matrix and inclusions. Two numerical examples containing several number of inclusions with various locations, shapes, sizes as well as different shear moduli are provided to assess the applicability and accuracy of the proposed method. It is verified that the proposed method can be used for problems when the ratio of shear moduli is extremely large. The comparison is made between the proposed method and reference solutions.

This paper investigates the elastic responses of fibrous nano-composites with imperfectly bonded interface under longitudinal shear. The proposed imperfect interface model is the shear lag (or the spring layer) model; the presented nano interfacial stress model is the Gurtin–Murdoch surface/interface model; and the three-phase confocal elliptical cylinder model is the geometry model accounting for the fiber section shape. By virtue of the complex variable method, a generalized self-consistent method is employed to derive the closed from solution of the effective antiplane shear modulus of the fibrous nano-composites with imperfect interface. Five existing solutions can be regarded as the limit form the present analytic expression. The influences of the interface elastic constant, the interfacial imperfection parameter, the size of the elliptic section fiber, the fiber section aspect ratio, the fiber volume fraction and the fiber elastic property on the effective antiplane shear modulus of the nano-composites are discussed. Particularly, numerical results demonstrate that the interfacial elastic imperfection will always cause a significant reduction in the effective antiplane shear modulus; and the fiber interface stress effect on the effective modulus of the fibrous nano-composites will weaken with the interfacial imperfection increases.

Calculating the yield limit $Y_c$ (the critical ratio of the yield stress to the driving stress), of a viscoplastic fluid flow is a challenging problem, often needing iteration in the rheological parameters to approach this limit, as well as accurate computations that account properly for the yield stress and potentially adaptive meshing. For particle settling flows, in recent years calculating $Y_c$ has been accomplished analytically for many antiplane shear flow configurations and also computationally for many geometries, under either two dimensional (2D) or axisymmetric flow restrictions. Here we approach the problem of 3D particle settling and how to compute the yield limit directly, i.e. without iteratively changing the rheology to approach the yield limit. The presented approach develops tools from optimization theory, taking advantage of the fact that $Y_c$ is defined via a minimization problem. We recast this minimization in terms of primal and dual variational problems, develop the necessary theory and finally implement a basic but workable algorithm. We benchmark results against accurate axisymmetric flow computations for cylinders and ellipsoids, computed using adaptive meshing. We also make comparisons of accuracy in calculating $Y_c$ on comparable fixed meshes. This demonstrates the feasibility and benefits of directly computing $Y_c$ in multiple dimensions. Lastly, we present some sample computations for complex 3D particle shapes., Comment: 30 pages, 6 figures, 3 tables

© 2019 Elsevier Ltd This article models the elastodynamic transient contact between two elastically similar half planes under antiplane loading and in the presence of friction. Contact is maintained along the positive real line under the presence of a certain remote contact pressure. An antiplane shear load is applied, which entails interfacial shear traction that opposes the frictional force entailed by the contact pressure. In order to balance the surface tractions, the surface must be allowed to slip. We derive the closed form solution of the interfacial traction due to a general antiplanar displacement distribution using a variant of the Wiener-Hopf technique. We also find closed-form expressions for the interfacial shear traction due to this remote antiplane load. In combination with the frictional force, this leads to an integral equation the solution to which is the distribution of relative slip. We quantify both this and the magnitude of the interfacial shear tractions under diverse loading, showing that transient loading leads to partial reverse slip of the contact surfaces. We show that the reverse slip tends to vanish over time, and that it is ameliorated if the friction coefficient is reduced.

In linearised continuum elasticity, the elastic strain due to a straight dislocation line decays as $O(r^{-1})$, where $r$ denotes the distance to the defect core. It is shown in Ehrlacher, Ortner, Shapeev (2016) that the core correction due to nonlinear and discrete (atomistic) effects decays like $O(r^{-2})$. In the present work, we focus on screw dislocations under pure anti-plane shear kinematics. In this setting we demonstrate that an improved decay $O(r^{-p})$, $p > 2$, of the core correction is obtained when crystalline symmetries are fully exploited and possibly a simple and explicit correction of the continuum far-field prediction is made. This result is interesting in its own right as it demonstrates that, in some cases, continuum elasticity gives a much better prediction of the elastic field surrounding a dislocation than expected, and moreover has practical implications for atomistic simulation of dislocations cores, which we discuss as well.

In this work, a unified solution approach is proposed for the analytical evaluation of the stress fields close to notches under antiplane shear and torsion loadings, which allows a large variety of notch problems to be tackled. The method is based on the complex potential approach for antiplane elasticity combined with the use of proper conformal mappings. In particular, it is shown that a well defined analytical link does exist between the complex potential to be used to determine stresses and the first derivative of the conformal mapping used to mathematically describe the notch profile. This makes some methodologies such as Schwarz-Christoffel transformation, which allows describing any polygonal domain automatically, very attractive for the direct solution of notch problems. A bulk of solutions are provided to support this finding, from cracks and pointed notches, to radiused notches. In addition, the accuracy of each proposed solution is discussed in detail taking advantage of a bulk of results from FE analyses.

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, and the friction is modeled with Tresca’s law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field with a time-dependent variational equation for the potential field. Then we prove the existence of a unique weak solution to the model. Moreover, the proof is based on arguments of evolution equations and on the Banach fixedpoint theorem.

Transient stress intensification due to the interaction of two parallel, edge interfacial cracks in bonded media with a functionally graded interlayer is investigated under the condition of antiplane shear impact. The graded interlayer is assumed to exhibit power-law variations in shear modulus and mass density between dissimilar, homogeneous semi-infinite strips. Laplace and Fourier integral transforms are employed to derive a system of singular integral equations with generalized Cauchy kernels, which is solved in the Laplace domain. Mode III stress intensity factors are defined and evaluated as a function of time. Numerical results include variations of dynamic stress intensity factors, addressing the impact-induced interaction between two nearby edge interfacial cracks affected by geometric and material parameters of the bonded system. More specifically, this study elaborates on the effects that layer thickness, crack length, shear modulus, and mass density have on the dynamic overshoot characteristics of transient behavior in the near-tip regions.

An investigation is presented of an antiplane shear strip crack subjected to representative general non-constant crack-face loading conditions which is moving uniformly in one-dimensional hexagonal quasicrystals, using an extended technique of continuous dislocation layers. Readily-calculable explicit closed-form representations are determined and discussed for the components of the phonon and phason stress fields created throughout the media. Illustrative numerical results are presented graphically. As a special case, the corresponding expressions are deduced explicitly for a non-uniformly loaded stationary crack.

This paper reconsiders the problem of determining the elastostatics fields near the tip of a crack in a body deformed by an antiplane shear for a class of incompressible, homogeneous, isotropic materials. The study is generalized to the formation of a quasicrack under the same conditions of loading for brittle material that cannot support any further loading when a critical strength is reached. The crack is then replaced by a totally damaged zone where the stress is identically zero. The shape of the boundary between the damaged and undamaged body is found analytically. A numerical approach is proposed to address the problem for more general constitutive law. The analytical solution is recovered by a process of shape optimization.

A rod or beam is one of the most widely used members in engineering construction. Such members must be properly designed to resist the applied loads. When subjected to antiplane (longitudinal) shear and torsional loading, homogeneous, isotropic, and elastic materials are governed by the Laplace equation in two dimensions under the assumptions of classical continuum mechanics, and are considerably easier to solve than their three-dimensional counterparts. However, when using the finite element method in conjunction with linear elastic fracture mechanics, crack nucleation and its growth still pose computational challenges, even under such simple loading conditions. This difficulty is mainly due to the mathematical structure of its governing equations, which are based on the local classical continuum theory. However, the nonlocal peridynamic theory is free of these challenges because its governing equations do not contain any spatial derivatives of the displacement components, and thus are valid everywhere in the material. This study presents the peridynamic equation of motion for antiplane shear and torsional deformations, as well as the peridynamic material parameters. After establishing the validity of this equation, solutions for specific components that are weakened by deep edge cracks and internal cracks are presented.

In this paper, a doubly periodic interfacial cracking problem in a layered periodic composite is analyzed under antiplane shear loads. To further understand the interaction effect among the periodic interfacial cracks, four crack configurations are considered, namely a bilayered composite containing a single crack and a periodic array of collinear cracks, a layered periodic composite with a periodic array of parallel cracks and a doubly periodic rectangular array of cracks. In view of the periodic symmetry, the three periodic problems are transformed to a mixed boundary value problem for a single crack problem in an appropriate cell with the suitable periodic boundary conditions on its boundaries. The treatment skills for the periodic array of collinear cracks and the disposal techniques for the periodic array of parallel cracks are used together to solve the doubly periodic rectangular array of cracking problem. To demonstrate the computational accuracy of the present method, detailed comparative analyses are carried out. Numerical results are presented to show the interaction effect among the periodic interfacial cracks. For the doubly periodic rectangular array of cracking problem, parametric studies on the stress intensity factors leads out the following rule: the shielding effect of multiple parallel cracks and the amplifying effect of multiple collinear cracks exist simultaneously, and a coupled effect between geometrical and physical parameters on the interfacial fracture behavior exists clearly.

In the current study, we rigorously analyze an arbitrarily shaped piezoelectric inclusion surrounded by an infinite isotropic piezoelectric matrix subject to antiplane shear and in plane electric field loadings. The inclusion and matrix are separated by a homogeneously imperfect interface that characterizes a spring type interaction between the elastic and electric interfacial boundary conditions. Furthermore, the boundary conditions for a mechanically compliant, weakly conducting and mechanically compliant, highly conducting interface are incorporated into the analysis. Using complex variable techniques the potential function inside the inclusion is formulated as a Faber Series expansion and a system of linear algebraic equations for a closed form solution is developed for the corresponding Faber coefficients under a finite number of terms. Under this approach, expressions for both the elastic and electric fields are developed for the inclusion and matrix. The results are presented in exact form for an elliptic inclusion and numerically simulated for a finite number of terms for purposes of verification. Additionally, the cases of a square and star inclusion geometry are analyzed and results are presented numerically. The results clearly demonstrate that not only is the stress distribution inside the inclusion interface non-uniform, but that the magnitude of the peak stresses are highly dependent on the inclusion shape and imperfect interface condition.

We study the effect of surface elasticity on an arc-shaped crack in a linearly elastic isotropic homogeneous material under antiplane shear deformation. The surface mechanics is incorporated by using a continuum-based surface/interface model of Gurtin and Murdoch. We obtain a complete solution by reducing the problem to two decoupled first-order Cauchy-type singular integro-differential equations. It is shown that different from the case of a straight crack, the stresses exhibit both the weak logarithmic and the strong square root singularities at the tips of the arc crack.

We study the phenomenon of a spatial localization of elastic waves in periodic composite materials with local defects. The wave spectrum in heterogeneous composite solids includes pass and stop frequency bands. If the frequency of the signal falls within a stop band, the group velocity vanishes and the wave attenuates exponentially. In such a case, a local perturbation of the microstructure may lead to the localization of the wave energy in the vicinity of the defect. Longitudinal tension-compression waves in a layered composite and transverse antiplane shear waves in a unidirectional fibrous composite are considered. Local perturbations of the density and of the volume fractions of the components are taken into account. The analysis is based on the transfer-matrix method and on the plane-wave expansions method. As the result, the frequencies of the wave localization and the corresponding attenuation factors are determined.

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking.

Mode I and mode III solutions for sharp notch roots are derived and compared with representative full-field solutions. The example geometry studied is a circumferentially notched shaft, loaded in both tension and torsion. The calibration for the notch root intensity factors has been carried out using the finite element method. The combined effects of tension and torsion have been briefly considered as an example.

An infinite class of nonuniform antiplane shear fields is considered for a linear elastic isotropic space and (non-intersecting) isotoxal star-shaped polygonal voids and rigid inclusions perturbing these fields are solved. Through the use of the complex potential technique together with the generalized binomial and the multinomial theorems, full-field closed-form solutions are obtained in the conformal plane. The particular (and important) cases of star-shaped cracks and rigid-line inclusions (stiffeners) are also derived. Except for special cases (addressed in Part II), the obtained solutions show singularities at the inclusion corners and at the crack and stiffener ends, where the stress blows-up to infinity, and is therefore detrimental to strength. It is for this reason that the closed-form determination of the stress field near a sharp inclusion or void is crucial for the design of ultra-resistant composites., 19 pages, 6 figures, 1 table

Notch stress intensity factors and stress intensity factors are obtained analytically for isotoxal star-shaped polygonal voids and rigid inclusions (and also for the corresponding limit cases of star-shaped cracks and stiffeners), when loaded through remote inhomogeneous (self-equilibrated, polynomial) antiplane shear stress in an infinite linear elastic matrix. Usually these solutions show stress singularities at the inclusion corners. It is shown that an infinite set of geometries and loading conditions exist for which not only the singularity is absent, but the stress vanishes ('annihilates') at the corners. Thus the material, which even without the inclusion corners would have a finite stress, remains unstressed at these points in spite of the applied remote load. Moreover, similar conditions are determined in which a star-shaped crack or stiffener leaves the ambient stress completely unperturbed, thus reaching a condition of 'quasi-static invisibility'. Stress annihilation and invisibility define optimal loading modes for the overall strength of a composite and are useful for designing ultra-resistant materials., 21 pages, 11 figures

We consider fiber reinforced composites where both the matrix and the fibers are ideally plastic materials. We restrict our attention to microstructures and applied stresses that lead to both microscopic and macroscopic antiplane shear deformations. We describe a bound we have recently obtained on the yield set of the composite in terms of the shape of the fibers, their volume fraction, and the yield set of the matrix. We construct examples of composites showing that our bound is essentially optimal.

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive support. We model the material's behavior with an electro-elastic constitutive law; the frictional contact is described with a boundary condition involving Clarke's generalized gradient and the electrical condition on the contact surface is modelled using the subdifferential of a proper, convex and lower semicontinuous function. We derive a variational formulation of the model and then, using a fixed point theorem for set valued mappings, we prove the existence of at least one weak solution. Finally, the uniqueness of the solution is discussed; the investigation is based on arguments in the theory of variational-hemivariational inequalities.

Multimaterial wedges are frequently observed in composite materials. They consist of two or more sectors of dissimilar materials joined together, whose interfaces converge at the same vertex. Due to the mismatch in material properties such as Young’s modulus, thermal conductivity, dielectric permittivity, or magnetic permeability, these geometrical configurations can lead to singular fields at the junction vertex. This paper discusses mathematical analogies, focused on singular harmonic problems, between antiplane shear problem in elasticity due to mode III loading or torsion, the steady-state heat transfer problem, and the diffraction of waves in electromagnetism. In the case of a single material wedge, a mathematical analogy between elasticity and fluid dynamics is also outlined. The proposed unified mathematical formulation is particularly convenient for the identification of common types of singularities (power-law or logarithmic type), the definition of a standardized method to solve nonlinear eigenvalue problems, and the determination of common geometrical and material configurations allowing the relief or removal of different singularities.

The interaction of time harmonic antiplane shear waves and multiple cracks embedded in a plate is studied by using the dislocation model and images method. The effect of the wave number, incident angle and relative positions of the cracks and free surface are presented. Resonance vibrations and the relaxation phenomenon of the layer between the cracks and the free surface are discussed in detail. With the approach, the strain energy density factors (SEDFs) of edge cracks can also be derived by assuming one of the crack tips to be nearly in contact with the free surface.

The problem of diffraction of antiplane shear waves by a crack of finite length under the permeable electric boundary conditions is investigated analytically. Using Fourier transforms the mixed boundary value problem is reduced to two pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The iterative solutions of the Fredholm integral equations have been obtained for small values of the wave number. And analytical expressions for the dynamic stress intensity factor and electric displacement intensity factor are obtained. Finally the numerical results for dynamic stress intensity factor are displayed graphically.

We consider fiber reinforced composites where both the matrix and the fibers are made of ideally plastic materials with the fibers being much stronger than the matrix. We restrict our attention to microstructures and applied stresses that lead to both microscopic and macroscopic antiplane shear deformations. We obtain a bound on the yield set of the composite in terms of the shapes of the fibers, their volume fraction and the yield set of the matrix.

We have ascertained the limits of reasonable application of the classical boundary element method for the solution of the antiplane problem of the theory of elasticity in the study of bodies with thin-walled elements of structure and geometry. We have proposed an approach for the regularization of singular and quasisingular integrals, which appear inevitably in analyzing thin structures. We give also numerical examples demonstrating the reliability and efficiency of the proposed approach.

We analyze the antiplane shear deformation of an elastic cylinder in frictional contact with a rigid foundation, for static processes, under the small deformation hypothesis. Based on the Knaster–Kuratowski–Mazurkiewicz technique in the theory of the hemivariational inequalities, we prove that the model has at least one weak solution. Moreover, we present several examples of constitutive laws and friction laws for which our theoretical results are valid. Finally, we comment on the conditions which guarantee the uniqueness of the solution.

We study the antiplane shear deformation of a cylindrical body in frictional contact with a rigid foundation, under the hypothesis of the small deformations. The envisaged material is assumed to be elastic, physically nonlinear and nonhomogeneous, such that the Lame coefficient μ satisfies \({{\rm inf}_{{\bf x}\in\Omega}\,\mu(\bf x)=0}\), where Ω denotes the cross section of the cylinder. We establish the existence of a unique weak solution for this model on an appropriate weighted functional space. The proof is based on arguments of variational inequalities with strongly monotone operators.

A cracked orthotropic elastic strip of finite width is analyzed under antiplane shear loading. For four frequently encountered constraint edges, i.e. free-free, clamped-clamped, free-clamped, or clamped-free edges, the Fourier series method is used to reduce triple series equations, and then to a mixed boundary value problem associated with a mode-III crack for each case to a singular integral equation. Closed-form solutions are derived for the resulting equations, and formulae for calculating stress intensity factors are obtained for an internal crack and edge cracks subjected to arbitrarily varying loading. Numerical results of the stress intensity factors for an eccentric, central, and edge crack are shown graphically for the case of uniform loading at the crack faces. Obtained formulae for determining stress intensity factors can be taken as a benchmark of numerical evaluations. The derived results are also applicable to an isotropic elastic strip with an anti-plane shear crack normal to the edges.

The paper deals with nonlinear stress and strain distributions at the root of sharp and rounded notches with different opening angles under antiplane shear loading and small scale yielding. In order to make an easier comparison with the Neuber rule, the material is thought of as obeying the particular nonlinear law used in the past just by Neuber. By solving the linear differential equation resulting from the use of the hodograph transformation, a new relationship linking linear and nonlinear stress and strain concentrations is found. The relationship is written also in terms of the relevant notch stress intensity factors. In contrast with the Neuber rule, this relationship strictly depends on the notch opening angle. Even when the notch opening angle is zero, it does not match the Neuber Rule, but results in an additional factor 2 which is in agreement with Hult and McClintock’s solution when the notch tip radius tends to zero and the notch becomes a crack.

We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period $\varepsilon$ of the grid and the ratio $\delta$ between the thickness of the fibers and the period $\varepsilon$. We show that the asymptotic behavior as $\varepsilon\to0^+$ and $\delta\to0^+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\varepsilon\ll\delta$ and $\varepsilon\gg\delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.

Closed-form solutions are developed for the stress fields induced by circumferential hyperbolic and parabolic notches in axisymmetric shafts under torsion and uniform antiplane shear loading. The boundary value problem is formulated by using complex potential functions and two different coordinate systems, providing two classes of solutions. It is also demonstrated that some solutions of linear elastic fracture and notch mechanics reported in the literature can be derived as special cases of the general solutions proposed herein. Finally the analytical frame is used to link the Mode III notch stress intensity factor to the maximum shear stress at the notch tip, as well as to give closed-form expressions for the strain energy averaged over a finite size volume surrounding the notch root.

Based on the Lekhnitskii-Eshelby approach of two-dimensional anisotropic elasticity, a semi-analytical solution is derived for the problem associated with an anisotropic elliptical inhomogeneity embedded in an infinite anisotropic matrix subjected to remote uniform antiplane shear stresses. In this research, the linear spring type imperfect bonding conditions are imposed on the inhomogeneity-matrix interface. We use a different approach than that developed by Shen et al. (2000) to expand the function encountered during the analysis for an imperfectly bonded interface. Our expansion method is in principle based on Isaac Newton's generalized binomial theorem. The solution is verified, both theoretically and numerically, by comparison with existing solution for a perfect interface. It is observed that the stress field inside an anisotropic elliptical inhomogeneity with a homogeneously imperfect interface is intrinsically nonuniform. The explicit expression of the nonuniform stress field within the inhomogeneity is presented. The nonuniform stress field inside the inhomogeneity is also graphically illustrated. A difference in internal stress distribution between a composite composed of anisotropic constituents and a composite composed of isotropic constituents is also observed. We finally extend the solution derived for a linear spring type imperfect interface to address an elliptical inhomogeneity with a viscous interface described by the linear law of rheology. It is observed that the stress field inside an elliptical inhomogeneity with a viscous interface is nonuniform and time-dependent.

In this paper, we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions. The solution is represented by a distribution of double-layer potentials. The singularities of kernels are regularized by using a subtracting and adding-back technique. Therefore, the troublesome singularity in the method of fundamental solutions (MFS) is avoided and the diagonal terms of influence matrices are determined. An inclusion problem is decomposed into two parts: one is the exterior problem for a matrix with holes subjected to remote shear, the other is the interior problem for each inclusion. The two boundary densities, essential and natural data, along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions. A linear algebraic system is obtained by matching boundary conditions and interface conditions. Finally, numerical results demonstrate the accuracy of the present solution. Good agreements are obtained and compare well with analytical solutions and Gong's results.

We study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $$\varepsilon $$ , which forces damage concentration on regions of codimension one. We determine the $$\Gamma $$ -limit as $$\varepsilon $$ tends to zero and show that it contains an energy term involving the crack opening.

The interaction of a generalized screw dislocation with circular arc interfacial cracks under remote antiplane shear stresses, in-plane electric and magnetic loads in transversely isotropic magnetoelectroelastic solids is dealt with. By using the complex variable method, the general solutions to the problem are presented. The closed-form expressions of complex potentials in both the inhomogeneity and the matrix are derived for a single circular-arc interfacial crack. The intensity factors of stress, electric displacement and magnetic induction are provided explicitly. The image forces acting on the dislocation are also calculated by using the generalized Peach–Koehler formula. For the case of piezoelectric matrix and piezomagnetic inclusion, the shielding and anti-shielding effect of the dislocation upon the stress intensity factors is evaluated in detail. The results indicate that if the distance between the dislocation and the crack tip remains constant, the dislocation in the interface will have a largest shielding effect which retards the crack propagation. In addition, the influence of the interfacial crack geometry and materials magnetoelectroelastic mismatch upon the image force is discussed. Numerical computations show that the perturbation effect of the above parameters upon the image force is significant. The main result shows that a stable or unstable equilibrium point may be found when a screw dislocation approaches the surface of the crack from infinity which differs from the perfect bonded case under the same conditions. The present solutions contain a number of previously known results which can be shown to be special cases.

The piezoelectric composites with a doubly periodic parallelogrammic array of piezoelectric fibers are dealt with under antiplane shear coupled with inplane electrical load. A rigorous analytical method is developed by using the doubly quasi-periodic Riemann boundary value problem theory integrated with the eigenstrain and eigen-electrical-field concepts. The numerical results are presented and a comparison with finite element calculations, experimental data and micromechanical analysis is made to demonstrate the efficiency and accuracy of the present method. Subsequently, the present solutions are used to study two important topics in piezoelectric fiber composites, i.e., (1) stress and electrical field fluctuations in the microstructure, (2) the macroscopic effective electroelastic moduli. The relation between the macroscopic properties of the composites and their microstructural parameters is discussed and many interesting electroelastic interaction phenomena are revealed, which are useful to estimate and optimize the performance of piezoelectric composites.

Antiplane shear deformation of finite wedges is considered under different boundary conditions. First, the assertions and results of a recent paper, namely Chue and Liu [C.H. Chue, W.J. Liu, Comments on “Analysis of an isotropic finite wedge under antiplane deformation”, Int. J. Solids Struct. 41 (2004) 5023–5034] are invalidated. Then, closed form solutions are extracted for the stress distribution in the wedge. These closed forms have the advantages of showing the possible geometric stress singularity as well as the load singularity explicitly, in addition to the continuity or discontinuity as well as the convergence of the results in the entire region. Finally, the stress intensity factors are extracted in the special case of a circular shaft containing an edge crack under different boundary conditions.

It may be well known in practice that high stress concentrations occur in fiber‐reinforced composites. There have been several works by analysis to estimate for the stresses between closed spaced fibers. However, the known results on stiff fibers have until now been restricted to the particular case of circular cross‐sections. Thus, we extend the blow‐up results on the stresses specialized only for disks to the general case of arbitrary shapes. Moreover, we prove that the blow‐up rate of the general case is exactly the same as that of disks. Nevertheless, from the viewpoint of methodology, the technique we use is significantly different from the previous one restricted to the case of disks. Referring to antiplane shear problems, these works are reduced to the gradient estimates for the solution to the conductivity problem containing two closely spaced conductors which represent the cross‐sections of fibers. We establish a novel representation for the solution on conductors by a probability function. Based...

In this paper, a nonconforming discretization method for the frictional contact between two bodies subjected to antiplane shear deformation is considered. The method is based on a mixed variational formulation where dual basis functions are used for the discretization of the Lagrange multiplier. Under some regularity assumptions on the solution, an optimal a priori error estimate is obtained. To solve the discrete nonlinear problem an inexact primal-dual active set strategy is introduced. Finally, numerical examples confirming the theoretical result for the a priori error estimate and illustrating the performance of the algorithm are presented. The results can easily be generalized to the case of Coulomb friction. For this case a numerical example is given.