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The problem considered in the present paper has its origins in the description of a laminated structure consisting of an adhesive layer and two adherents. The problem of deformation of the layered structure is described by the coupled model of linear and strain gradient elasticity. Namely, we assume that deformations of both the adhesive layer and adherents obey antiplane shear models, but, at the same time, the adhesive layer is described by the simplified strain gradient elasticity theory (or, in other words, the so-called one-parameter gradient elasticity theory), while the adherents obey the classical elasticity theory. The shear moduli of the adhesive layer and its width depend on a small parameter. We investigate the limiting passage, as this parameter tends to zero. As a result, we find out the five types of limit models with imperfect interfaces in dependence of the parameter exponent. Three of them are the same as in the classical theory, while two of them are new and essentially belong to the strain gradient elasticity. [ABSTRACT FROM AUTHOR]

We propose paradigmatic examples to show how material damage phenomena can be efficiently described as a solid-solid phase transition. Starting from the pioneering work of J.L. Ericksen (J. Elast. 5(3):191–201, 1975) and the extensions of R.L. Fosdick and other authors to three-dimensional non linear elasticity, we describe the insurgence of damage as a hard → soft transition between two material states (damage and undamaged) characterized by two different energy wells. We consider the two separate constitutive assumptions of a simple Neo-Hookean type damageable material and a more complex microstructure inspired damageable Gent type material with variable limit threshold of the first invariant. In both cases we study two different deformation shear classes, one homogeneous and the other one inhomogeneous and obtain fully analytic description of the system damage response under cyclic loading. The considered constitutive assumptions and deformation classes are aimed at attaining fully analytic descriptions. On the other hand, we remark that the proposed, Griffith type, variational approach of damage, based on two different energy density functions for the damaged and undamaged material phases, and a resulting non (rank-one) convex energy, can be extended to systems with more complex energy functions, possibly with a larger number of wells representing an increasing degree of damage. [ABSTRACT FROM AUTHOR]

The computational method of determining thin stiff shells deflections formulated on the basis of the plane-spatial problem of the FEM without using Kirchhoff’s hypothesis is developed; in virtue of the geometric properties of the finite element stiffness matrix, a tensor of flexion stiffness is introduced. A linear and a nonlinear modification of the plane-spatial problem of the FEM for calculation of small elastic deflections of thin shells are formulated. An example of calculation of fragment of sloping conical shell is given in accordance with the common principles of two-dimensional domain discretization and some elements of fractal geometry.

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. [ABSTRACT FROM AUTHOR]

In linearized 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 [V. Ehrlacher, C. Ortner, and A. V. Shapeev, Arch. Ration. Mech. Anal., 222 (2016), pp. 1217-1268] 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 antiplane 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. [ABSTRACT FROM AUTHOR]

This work presents an analytical study on the stress fields induced by cracks nucleated at the tip of radiused notches under antiplane shear and torsion loadings. Closed‐form solutions for both shallow and deep notches are derived leveraging a combination of proper curvilinear coordinate systems and the complex potential method for antiplane elasticity. Based on the exact stress field solutions, convenient expressions for the stress intensity factors are derived. The accuracy of all the analytical formulations is verified by comparison with the results from a bulk of numerical analyses. [ABSTRACT FROM AUTHOR]

Abstract In this work, the exact solution for the stress fields ahead of cracks initiated at sharp notch tips under antiplane shear and torsion loadings is derived in close form, leveraging conformal mapping and the complex potential method for antiplane elasticity. Based on the stress field distributions, relevant expressions for the mode III crack stress intensity factors are derived and their accuracy is discussed in detail taking advantage of a bulk of results from FE analyses. Highlights • General approach closed-form solution for the stress field in anti-plane shear and torsion. • Closed-form solution for V-notches weakened by a crack. • Closed-form solution for Stress Intensity Factor (SIF) and discussion on effect of notch opening, notch depth and crack. [ABSTRACT FROM AUTHOR]

The elastic field of a heterogeneous medium containing doubly periodic cylindrical inclusions is investigated in antiplane shear. The heterogeneous medium is replaced by a homogeneous medium with doubly periodic stresses not related to strains. In the regions corresponding to the inclusions of the heterogeneous medium, an equivalence condition between a problem for the heterogeneous medium (called the original problem) and the corresponding problem for the homogeneous medium (called the equivalent problem) is established. Employing displacement continuity and stress jump conditions, the equivalent problem is formulated as a doubly quasi-periodic Riemann boundary-value problem, which is solved analytically, and the elastic fields in the doubly periodic inclusions and matrix are determined. As an example of application, the present solution is used to evaluate the effective antiplane moduli of composites, which are found to be in good agreement with existing results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a variational antiplane lattice model and demonstrate that at zero temperature, there exist locally stable states containing screw dislocations, given conditions on the distance between the dislocations and on the distance between dislocations and the boundary of the crystal. In proving our results, we introduce approximate solutions which are taken from the theory of dislocations in linear elasticity and use the inverse function theorem to show that local minimizers lie near them. This gives credence to the commonly held intuition that linear elasticity is essentially correct up to a few spacings from the dislocation core. [ABSTRACT FROM AUTHOR]

This study investigates how the insertion of multimaterial circular regions embracing the tip of a finite V-notch can be used to reduce the Notch Stress Intensity Factors (NSIFs) in structures subjected to antiplane shear or torsion. Towards this goal, this work presents a novel theoretical framework to calculate stress distributions and NSIFs in closed-form. Thanks to the new solution, it is shown that by tuning multimaterial region radii and elastic properties it is possible to significantly reduce the NSIFs and stress concentrations at the material interfaces. To investigate whether the proposed multimaterial system translates into increased structural capacity even in the presence of significant nonlinear deformations, computational simulations were conducted using nonlinear hyperelastic-damage and elasto-plastic-damage models. The preliminary results show increases of structural capacity up to 46% and of nominal strain at failure of up to 86% at the expenses of only a 8% reduction in structural stiffness. It is expected that a similar approach can be extended to other loading conditions (e.g. mode I and mode II, and fatigue) and that even larger gains can be obtained by performing thorough optimization studies. [Display omitted] [ABSTRACT FROM AUTHOR]

In this paper, antiplane piezoelectricity problems with multiple inclusions are studied by the generalized finite difference method (GFDM). Developed from the Taylor series and the Moving Least Squares, the GFDM expresses the derivatives of variables as the combination of values of surrounding nodes where the sparse matrix will be obtained considering the boundary conditions and interface conditions. Stress concentration and electric field concentration are analyzed in three numerical examples, which contain circular and elliptical piezoelectric inclusions in various sizes, locations, and material parameters. The applicability and validity of the proposed method are verified through the comparison with reference results. • The GFDM is applied with DDM for antiplane piezoelectricity problems. • Inclusions in various sizes, locations and material parameters are analyzed. • The proposed method maintains high accuracy and efficiency. [ABSTRACT FROM AUTHOR]

We consider a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The adhesion of the contact surfaces, caused by the glue, is taken into account. The material is assumed to be electro-viscoelastic and the foundation is assumed to be electrically conductive. We derive a variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field, a time-dependent variational equation for the electric potential field and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators and fixed point.

The asymptotic homogenization method (AHM) yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of uni-directional circular cylindrical holes in a linear transversely isotropic piezoelectric ma-trix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The composite state is antiplane shear piezoelec-tric, that is, a coupled state of out-of-plane shear deformation and in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method. For this, the solutions are expanded in power series of Weierstrass elliptic functions, which contain coefficients that are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical for-mulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori-Tanaka approach. The results could be useful in bone mechanics. [ABSTRACT FROM AUTHOR]

A crack in a composite wedge consisting of two dissimilar anisotropic materials under concentrated antiplane loads is analyzed. The problem of a crack in an isotropic composite wedge is solved first by using the Mellin transform and the Wiener-Hopf method. Using a linear transformation of the original composite wedge into an isotropic composite wedge consisting of dissimilar isotropic materials, the antiplane displacement and stresses for the anisotropic composite wedge are obtained from the solution for the transformed wedge. The stress intensity factor for the crack in the original anisotropic composite wedge is obtained from the solution of the crack in the transformed wedge. Special attention is given to the asymptotic problem of a wedge crack in an anisotropic bimaterial. Numerical computations are carried out to obtain the energy release rate for various apex angles and anisotropic parameters as a function of crack angle. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: An oblique edge crack in an anisotropic material under antiplane shear loadings is investigated. The antiplane problems are formulated based on a linear transformation method. An anisotropic solid containing an edge crack subjected to concentrated forces is first considered. The stress intensity factor for the edge crack with concentrated forces is obtained from the solution of the transformed edge crack in an isotropic material which is solved by using conformal mapping technique and complex function theory. The solution of the edge crack under concentrated loads is used to construct the stress intensity factor for the oblique edge crack in the anisotropic material subjected to antiplane distributed loads. Some numerical computations are carried out to calculate the stress intensity factors for the edge crack in inclined orthotropic materials subjected to point forces as well as distributed tractions. [Copyright &y& Elsevier]

Abstract: An eigenfuntion expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the front of a crack/anticrack type discontinuity weakening/reinforcing an infinite pie-shaped trimaterial plate, of finite thickness, formed as a result of bimaterial (matrix/semiconductor/ARC plus reaction product/scatterer) deposit over a substrate (fiber/ARC/semiconductor). The wedge is of general (unsymmetric) geometrical configuration, and is subjected to antiplane shear (mode III) far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 or the substrate is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode III eigenvalues (or stress singularities) with various wedge aperture angles of the material 1 (reaction product/scatterer), are also presented. Hitherto generally unavailable results, pertaining to the through-thickness variations of stress intensity factors for symmetric parabolic load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, form also an important part of the present investigation. [Copyright &y& Elsevier]

Abstract: We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid. [Copyright &y& Elsevier]

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 Lamé coefficient μ satisfies infx∈Ωμ(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. [ABSTRACT FROM AUTHOR]

In the present article, based on a recently proposed model (De Tommasi et al. in J Rheol 50:495–512, 2006; Phys Rev Lett 100:085502, 2008), we analyze the influence of the microstructure properties on the damage behavior of amorphous materials. In accordance with the experimental observations, different scenarios of damage nucleation and evolution are associated to different material distributions at the microscale. In particular, we observe the possibilities of uniform or localized damage and strain geometries with a macroscopic behavior that may range from brittle to ductile or rubber-like. To describe the possibility of extending our stability analysis to three-dimensional damageable amorphous bodies we consider a simple boundary value problem of engineering interest. [ABSTRACT FROM AUTHOR]

Abstract: We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid. [Copyright &y& Elsevier]

ABSTRACT: Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: A mode III crack with a cohesive zone in a power-law hardening material is studied under small scale yielding conditions. The cohesive law follows a softening path with the peak traction at the start of separation process. The stress and strain fields in the plastic zone, and the cohesive traction and separation displacement in the cohesive zone are obtained. The results show that for a modest hardening material (with a hardening exponent N =0.3), the stress distribution in a large portion of the plastic zone is significantly altered with the introduction of the cohesive zone if the peak cohesive traction is less than two times yield stress, which implies the disparity in terms of the fracture prediction between the classical approach of elastic–plastic fracture mechanics and the cohesive zone approach. The stress distributions with and without the cohesive zone converge when the peak cohesive traction becomes infinitely large. A qualitative study on the equivalency between the cohesive zone approach and the classical linear elastic fracture mechanics indicates that smaller cracks require a higher peak cohesive traction than that for longer cracks if similar fracture initiations are to be predicted by the two approaches. [Copyright &y& Elsevier]

Abstract: We derive a general expression for an interface parameter which makes possible the design of a neutral elliptic inhomogeneity when the stress field in the surrounding matrix is a polynomial function of order and the composite is subjected to antiplane shear deformations. [Copyright &y& Elsevier]

Abstract: The effective behavior and the distribution of local mechanical fields of linearly viscous 2-D polycrystals under antiplane shear is investigated. Several microstructures are considered, and a full-field approach based on the Fast Fourier Transform technique is applied. First, the accuracy of this technique is evaluated on a strictly isotropic 2-phase microstructure. Voronoi tessellation is then used to generate artificial microstructures, and a real (fully recrystallized) polycrystalline microstructure is obtained by electron back-scattering diffraction. Ensemble averages over several configurations using eight crystalline orientations (phases) are performed. Although a slight anisotropy is obtained for the effective behavior of each individual configuration, statistically, the results are in very good agreement with the available analytical isotropic solution. At phase level, a marked asymmetry is obtained for the distribution of local stresses. The intraphase first- and second-order moments of the stress field, calculated for both microstructures are compared with corresponding self-consistent predictions. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite infinitely rigid inclusion, subjected to the far-field antiplane shear and extension/bending loadings. Four different rigid inclusion side-surface boundary conditions are considered: (i) anticrack or perfectly bonded rigid inclusion, (ii) transversely rigid inclusion (longitudinal slip permitted), (iii) rigid inclusion in part perfectly bonded, the remainder with slip, (iv) rigid inclusion located alongside a crack. The computed stress singularity for an anticrack is the same as its plane strain counterpart, while the expression for the stress intensity factor varies in the thickness direction. [ABSTRACT FROM AUTHOR]

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ2/μ1. [Copyright &y& Elsevier]

Three-dimensional asymptotic stress field in the vicinity of the front of a bimaterial pie-shaped wedge of general (unsymmetric) geometry is obtained by a recently developed eigenfunction expansion method. The bimaterial pie-shaped wedge is subjected to four combinations of wedge-side boundary conditions – free–free, clamped–clamped, free–clamped and clamped–free. Each material is isotropic and elastic, but with different material properties. Additionally, numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) with respect to the wedge opening angle of the bimaterial wedge, subjected to the aforementioned wedge-side boundary conditions, are also presented. Dependence of the same on the material properties of the component material phases is also an important part of the present investigation. [Copyright &y& Elsevier]

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a circular piezoelectric inclusion embedded in an infinite piezoelectric matrix subjected to horizontally polarized shear waves and a steady-state inplane electrical load. The problem is formulated by means of the wave function expansion method. Numerical calculations are carried out for the dynamic stress and electric field concentrations, and the results are presented in graphical form. [ABSTRACT FROM AUTHOR]

Based on the ( I) of the present work, the behavior of shear beam model at crack initiation stage andat instable propagation stage was studied. The prime results include: 1) discriminant equation which clarifies the mode of instability, snap-back or snap-through, was established; 2) analytical solution was given out for the double shear beam and the load-displacement diagram for monotonic loading was presented for a full process; and 3) the problem of the energy release induced by instability was discussed. [ABSTRACT FROM AUTHOR]

This note considers solutions to Laplace"s equation on sectors with varying vertex angles. Under Neumann conditions on the radial boundaries, there are two critical vertex angles for which classical separable solutions break down. These breakdowns have been noted in the literature and resolved. Here Dirichlet and mixed conditions are also treated. For all three boundary conditions in combination, nine further critical angles are identified and valid corresponding solutions found. [ABSTRACT FROM AUTHOR]

A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion. [ABSTRACT FROM AUTHOR]