Search

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.

When the dimension of a structure falls to the micro-/nanoscale, surface effect is significant and plays a key role in affecting the mechanical behavior. This article studies the influence of surface elasticity on the stress intensity factor of an antiplane shear crack embedded in an elastic strip made of functionally graded materials. Surface elasticity is applied on the strip surfaces and crack faces, and classic elasticity is invoked for the strip interior. An antiplane shear crack problem is solved for a symmetric FGM with a crack parallel to the strip surfaces. The associated problem is converted to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces through the Fourier transform and then to a singular integro-differential equation with Cauchy kernel. The Galerkin method is applied to expand the crack face displacement as a Chebyshev series, and the singular integro-differential equation reduces to a system of algebraic linear equations. Stress intensity factors at the crack tips and the out-of-plane displacement on the crack faces are calculated numerically. It is found that surface elasticity and gradient index strongly alter the bulk stress and its intensity factors near the crack tips. Positive surface shear modulus decreases the mode III stress intensity factors and negative surface shear modulus has an opposite behavior. The influence of the variation of material gradient on the mode III stress intensity factors is expounded in graph.

Considering the piezoelectric effect, the electro-elastic field of an infinite one-dimensional quasicrystal medium with two circular cylindrical inclusions is derived under antiplane shear and inplane electric loading. The boundary value problem of the composite material with circular cylindrical inclusions is analytically solved by the use of the conformal mapping technique and analytical continuation theory. The stresses in the phonon and phason fields and the electric displacements are obtained explicitly in the form of a power series both for the matrix and the inclusions. Some typical examples are analyzed to show the effect of the geometric parameters, material properties, and electro-mechanical loading on the electro-elastic fields in the matrix, inclusions, and interfaces. The limiting cases of circular cavities and rigid circular inclusions have also been investigated and discussed.

Scale-dependent stress intensity factors in an anti-plane cracked orthotropic material layer are evaluated using strain gradient theory. Both volumetric and surface strain gradient material characteristic lengths represented as l and $$l^{'}$$ , respectively, are employed to obtain semi-analytical solutions. The surface strain gradient effect is considered for both positive and negative $$l^{'}$$ values. The layer edges are assumed stress-free and oriented parallel to the crack plane. The presence of orthotropy can either increase or decrease the stress intensity factors depending on if it is greater or smaller than unity. The volumetric strain gradient effect reduces the stress intensity factor and it is more pronounced for smaller layer thickness. It was found that the negative surface gradient leads to a more complaint crack, while the positive surface gradient increases crack stiffness. Overall, the surface gradient effect is less significant in comparison with the volumetric gradient effect.

A mode-III crack embedded in a homogeneous isotropic elastic layer of nanoscale finite thickness is studied in this article. The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem, where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation. Using the Fourier transform, we convert the problem to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces. By expanding the out-of-plane displacement as series of Chebyshev polynomials, the Galerkin method is invoked to reduce the singular integro-differential equation with Cauchy kernel to a set of algebraic linear equations for the unknown coefficients. An approximate solution is determined, and the influences of surface elasticity on the elastic field and stress intensity factor are examined and displayed graphically. It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus.

Detailed expressions are derived in closed-form for the field components at a general point in the deformation of magnetoelectroelastic media by a stack of parallel, antiplane shear cracks to which...

An analytical method is proposed to solve the problem of an infinite elastic matrix containing a doubly periodic array of coated cylindrical inclusions under antiplane shear. The elastic fields in the inclusions, the coatings/interphases and the matrix are derived, which are used to investigate the stresses and the effective stiffness coefficients of the nanofiber composites. Numerical examples demonstrate the size dependence of the stress and the effective stiffness coefficient, and the effects of the interphase thickness and stiffness and array configurations of the inclusions on the effective stiffness coefficient. A finite element analysis is used to benchmark the effective stiffness coefficient predicted by the proposed model, in which excellent agreement is observed. When letting the interphase be thin enough, the proposed coated inclusions model can be used to simulate the zero-thickness interface model, which is validated by the results comparisons of the two models. Instabilities of the stress fields are observed under certain conditions in simulating the zero-thickness interface model.

The dynamic behavior of an arc-shaped interfacial crack in an orthotropic functionally graded annular bi-material structure is investigated. In order for the analysis to be executable, the material...

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.

A theoretical study is performed on the fracture characteristics for the problem of two symmetrical edge cracks emanating from a circular hole with surface effect under far-field antiplane shear loading. Based on the theory of Gurtin–Murdoch surface model, a rigorous analytical solution of the stress intensity factor and the strain energy release rate at a crack tip is presented by using the complex potential function and a conformal mapping technique. When the cracked hole becomes nano-sized, the stress intensity factor and the strain energy release rate show significant size dependence. The interaction effects between cracks and hole on the stress intensity factor and the strain energy release rate are discussed in providing numerical examples.

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.

A virtual element method approach is presented for solving the unit cell problem, in application of the asymptotic homogenization method, and computing the antiplane shear homogenized material moduli of a composite material reinforced by cylindrical inclusions of arbitrary cross section. Validation of the proposed numerical method is proved by comparison with analytical and numerical reference solutions, for a number of micro-structural arrays and for different grading properties of the material constituents. A point on numerical efficiency is also made with respect to the possibility of local refinement granted by the innovative numerical procedure which relies on a mesh conformity concept ampler than the one of classical finite element method. The flexibility of the method allows for a large variety of microstructure shapes.

Due to the increasing ratio of surface to bulk volume, surface/interface effects play an important part in the mechanical performance of devices and materials when the characteristic size approaches nanoscale. In this work, a size-dependent three-phase cylinder model (fiber/matrix/composite) is proposed for magnetoelectroelastic (MEE) nanocomposites based on the theory of Gurtin–Murdoch surface/interface model. Exact closed-form solutions of elastic, electric, and magnetic fields with interface effect are derived under far-field antiplane mechanical and in-plane electric and magnetic loadings via the Laurent expansion technique. The effective MEE moduli of nanocomposites with interface effects are predicted by the generalized self-consistent method. As special cases, several size-dependent models such as two-phase MEE cylinder model and hollow cylinder model can be deduced from the present model. Numerical examples are illustrated to show the effects of two kinds of interface material properties and fiber volume fraction on the effective MEE moduli of the piezoelectric, piezomagnetic, and MEE nanocomposites.

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

As outlined in the previous paper, an eigenstrain state due to an interstitial atom, acting as misfitting inclusion in the crystal lattice, may interact with a load stress state and/or a stress state due to a defect. Since the interstitial atoms can be assembled in a cylinder, as in the established case of a so-called Cottrell cloud, the according stress state can be investigated by a cylindrical inclusion with a general eigenstrain state. Both eigenstrain components belonging to the antiplane shear, however, were not dealt with in the past. According stress and deformation equations are now offered in analytical form in this addendum.

© 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 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.

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.

This contribution investigates the stress fields in orthotropic plates featuring lateral notches under anti-plane shear loading. Four different notch geometries are considered and the relevant analytical expressions for the stress distribution are derived in closed form. For each geometry, the main features of the stress fields and the accuracy of the analytical expressions developed are discussed comparing theoretical results and numerical data from FE analyses carried out on finite plates under longitudinal shear.

We investigate diffraction of reduced traction shear waves applied at the faces of a stationary crack in an elastic solid with microstructure, under antiplane deformation. The material behaviour is described by the indeterminate theory of couple stress elasticity and the crack is rectilinear and semi-infinite. The full-field solution of the crack problem is obtained through integral transforms and the Wiener–Hopf technique. A remarkable wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves moving along the crack, localized waves irradiating from the crack-tip with, possibly, super-Rayleigh speed and body waves scattered around the crack-tip. Interestingly, the localized wave solution may be greatly advantageous for defect detection through acoustic emission. Dynamic stress intensity factors are presented, which generalize to Elastodynamics the corresponding results already obtained in the static framework. The correction brings out the important role of wave diffraction on stress concentration.

Analytical displacement and stress fields with stress concentration factors (SCFs) are derived for linearly elastic annular regions subject to inhomogeneous boundary conditions: an infinite class of the mth order polynomial antiplane tractions or displacements. The solution of the Laplace equation governing the out-of-plane problem covers both rigid and void circular inclusions forming the core of the annulus. The results show first that the SCF and the loading order are inversely proportional. In particular, the SCF approaches value 2 when either the outer boundary of the annulus tends to infinity or the order of the polynomial loading increases. Second, the number of peculiar points on the inner contour having null stress increases with the increasing loading order. The analytical solution is confirmed and extended to noncircular enclosures via finite element analysis by exploiting the heat-stress analogy. The results show that the closed-form solution for a circular annulus can be used as an accurate approximation for noncircular enclosures. Altogether, the results shown can be exploited for analyzing complex loading conditions and/or multiple rigid or void inclusions for enhancing the design of hollow and reinforced composites materials.

Amicromechanicsmodel composed of a three-phase confocal elliptical cylinder of the nanocompositeswith interface effect is proposed.Ageneralized self-consistent method for the piezoelectric nanocomposites accounting for fiber section shape under far-field mechanical–electrical loads is presented based on the model. By using the theory of Gurtin–Murdoch surface/interface and the conformal mapping technique, a closedform solution of the effective electroelastic constants is obtained. The present solution can be degenerated into the existing solution. The results show that the effective electroelastic constants are dramatically size dependent when the size of the fiber is on the order of nanometers. The effective electroelastic constants decrease monotonically with the fiber section aspect ratio $${\gamma}$$ increasing from 0 to 1. The elastic constant and dielectric constant obtained by the present solution are very similar to the results obtained by the classical electroelastic theory, whereas the piezoelectric coupling constant obtained by the present solution is very different from the results obtained by the classical electroelastic theory.

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.

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.

The medium is composed of an elliptic inclusion and many confocal elliptic layers. The crack is embedded in the elliptic inclusion. The remote loading is applied at the remote place of the matrix. Complex variable method and conformal mapping are used to study the mentioned problem. This paper provides a numerical solution for the mentioned crack problem. The continuity condition for the traction and displacement along the interface is reduced to a relation of two sets of Laurent series coefficients for the complex potentials defined in the interior or exterior to the interface. This formulation is called the matrix transfer method in this paper. From the following three conditions, the traction-free condition along creak face, the continuity condition for the traction and displacement along the interfaces and the remote loading condition, the problem is finally solved. Servable numerical examples are provided. For the exterior finite matrix case, the relevant solution is also provided.

Effective anti-plane quasistatic moduli for 2D piezoelectric phononic crystals of arbitrary anisotropy are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in one dimension with direct numerical integration along the other direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases.

In this paper, an analytical study is carried out on the work-hardening, elastic-plastic stress distributions in a cracked body under antiplane shear deformation. A modified Ramberg-Osgood law is introduced to describe the material behaviour, and stress and strain fields are derived in closed form. Compared with the conventional Ramberg-Osgood formulation, the new law includes the effect of a new parameter, κ, which allows the transition from the ideally elastic behaviour (low stress regime) to the power law behaviour (large stress regime) to be controlled, thus providing 1 more degree of freedom to better fit the actual behaviour of engineering materials. A discussion is carried out on the features of stresses and strains close to and far away from the crack tip.

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.

An iterative method has been applied to analyze the free edge interlaminar stresses of piezo-bonded composite laminates. Electric fields are applied to two symmetrically bonded piezo actuators that can generate induced strain, resulting in pure extension of the whole structure. The stresses, which satisfy the traction-free and free edge boundary conditions, are obtained by taking the principle of complementary virtual work, and conducting the extended Kantorovich method. In order to obtain accurate interlaminar stress distributions, static and kinematic continuity conditions are applied at the interfaces between plies through iterations. The stress components were obtained under the plane strain assumption as well as antiplane shear assumption. The results were compared with those obtained by the finite element method, to demonstrate the validity of the proposed method. Through the iteration processes, the interlaminar stresses converged well, and predicted maximum peak stress at the interface of piezo and composite layers. The present method provides accurate stresses distribution near the free edges, of piezo-bonded composite laminates.

Representations for the stress fields created around an infinite row of collinear, antiplane shear cracks moving within one-dimensional hexagonal quasicrystals, and the resulting stress intensity factors and the J -integral, are determined in closed-form and discussed, using an extended method of dislocation layers. The solutions for a finite quasicrystalline plate containing a single moving crack and a plate with a moving edge crack are also provided by this analysis.

The circumferential shear stress along the boundary of a traction-free circular hole in an infinite isotropic solid under antiplane shear is twice the circumferential shear stress along the corresponding circle in an infinite solid without a hole, under the same loading conditions. This result, originally derived by Lin et al. (Yielding, damage, and failure of anisotropic solids. EGF Publication, Mechanical Engineering Publications, London, 1990) by invoking the extended circle theorem of Milne-Thomson, is rederived in this paper using the Fourier series analysis, in the spirit of the original Kienzler and Zhuping (J Appl Mech 54:110–114, 1987) derivation of the plane strain result for the hoop stress around the hole. The formula is also deduced by the complex potential method, without explicitly invoking the extended circle theorem. The ratio of the circumferential stresses along the boundary of a noncircular hole and the congruent curve in a solid without a hole depends on the loading. For an elliptical hole in an infinite solid under remote loading $${{\sigma_{xz}^{0}}}$$ , this ratio is constant and equal to 1 + b/a; under $${{\sigma_{yz}^{0}}}$$ it is equal to 1 + a/b, where a and b are the semi-axes of the ellipse. These constant values coincide with the values of the stress concentration factor in the two cases. If $${{\sigma_{xz}^{0}}}$$ and $${{\sigma_{yz}^{0}}}$$ are both applied, the stress ratio varies along the boundary of the ellipse; a closed form expression for this variation is determined. The portions of the elliptical boundary along which the strain energy density is increased or decreased by the creation of the hole are determined for different types of loading and different aspect ratios of the ellipse. An analytical expression for the stress concentration factor is then obtained for any orientation of the ellipse relative to the applied load.

The stress and displacement fields are determined inside and outside a circular inclusion located in the vicinity of a circular void in an infinite elastic solid, within a circular cylinder, or near the free surface of a half-space, in the case when the inclusion is characterized by a uniform eigenstrain of the antiplane shear type. The fields are obtained as the sum of their infinite-medium stress fields and the calculated auxiliary fields. It is shown that the fields outside the inclusion follow directly from the extended Milne-Thomson circle theorem, but not the fields inside the inclusion. The overall fields are interpreted as the superposition of the infinite-medium fields from the actual and the image inclusion of the appropriate location, radius, and eigenstrains. The stress amplification is evaluated for the inclusion approaching the boundary of the void, cylinder, or half-space. The configurational forces are then evaluated, associated with a relative translation of the void and inclusion, or the expansion of the void or inclusion. The J and M integrals along the boundary of the void are evaluated without using the solution of the entire boundary-value problem, but only the stress field for an inclusion in an unvoided infinite medium. This is accomplished by incorporating the result that the circumferential shear stress along the boundary of a traction-free circular void in an infinite isotropic solid under antiplane shear is twice the circumferential shear stress along the corresponding circle in an infinite solid without a void, under the same loading conditions. The energy release rate associated with a self-similar expansion of the inclusion is calculated from the determined elastic field around the inclusion and from the evaluated total strain energy of the system. The configurational forces on the inclusion in a circular cylinder and near the free surface of a half-space are also determined and discussed.

The integral equations of antiplane shear deformation of anisotropic solids with periodic sets of thin ribbonlike inclusions are constructed. A modified boundary-element method is used to obtain the numerical solutions of specific problems. We compute the stress intensity factors for anisotropic bodies with one, two, or three columns of parallel defects. The comparison with the analytic solutions confirms the validity of the obtained results for cracks and absolutely rigid inclusions. We also study the effect of changes in the degree of anisotropy (orthotropy) of the material on the stress-strain state of the body and on the stress intensity factors in the vicinity of the tips of thin inhomogeneities.

First, an eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite crack, subjected to the far-field extension/bending, in-plane shear/twisting and antiplane shear loadings, distributed through the plate thickness. A complex variable approach in conjunction with this eigenfunction expansion technique is employed here to derive the heretofore-unavailable through-thickness distribution of stress intensity factors and energy release rates for a center-crack weakening an infinite plate. Interesting numerical results for three-dimensional stress intensity factors and energy release rates for five different far-field loading scenarios are also presented.

An overview of the results for Mode III interface crack problems available in the literature, as well as those obtained by the authors, is presented in the paper. The analytical and semi-analytical solutions for the main fracture characteristics (i.e. stress intensity factors and energy release rates) are discussed with respect to their dependency on bimaterial constants. Detailed formulation for the antiplane shear problem for an interface crack interacting with internal cracks is presented using the methods of complex potentials and singular integral equations. Some particular solutions (e.g. for the problem of a singularity interacting with an interface and for the Mode III interface crack) are obtained, and these can serve as benchmark solutions or as parts of other similar problems. The approximate closed-form solution is derived for a special case of the interaction between the interface crack and internal micro-cracks. The resulting form of the solution for the stress intensity factors is rather simple and its structure is easily analyzed. A theoretical analysis of the explicit form of expressions for the stress intensity factors shows the conditions under which the solution for the interface crack depends on the bimaterial mismatch, and those under which it does not depend. This formula also allows to define the condition, when a small internal crack does not influence the stress intensity factor of the interface crack (i.e. invisible for the interface crack).

The paper deals with the small scale yielding estimation of nonlinear stresses and strains at the root of sharp and blunt notches through the mechanical model of antiplane shear loadings. The frame stems from the relation existing between the elastic and plastic averaged strain energy densities evaluated over the control volume drawn by the energy contour lines ahead of the notch tip. The analysis proves that there exist different relationships in terms of point-wise elastic and plastic stresses and strains at the notch tip depending whether the notch is sharp (small notch tip radius) or blunt. For sharp notches, the analysis confirms previous results obtained by the present authors, according to which . This equation accounts for the influence of the material law through the hardening exponent n. Differently, when the notch can be regarded as blunt, calculations over the control volume give , in agreement with the Neuber rule.

A theoretical study on piezoelectric coated nano-inclusion composites with interface effect under antiplane shear is reported. Based on the theory of Gurtin–Murdoch surface/interface theory and the generalized self-consistent method, a closed-form solution of the effective electroelastic moduli are obtained. The numerical results reveal the size dependence of the effective electroelastic moduli when the size of the coating and inclusion are on the order of nanometer. The effects of the coating thickness and inclusion radius on the effective electroelastic moduli of the composites are discussed.

Despite extensive studies of inclusions with simple shape, little effort has been devoted to inclusions of irregular shape. In this study, we consider an inclusion of irregular shape embedded within an infinite isotropic elastic matrix subject to antiplane shear deformations. The inclusion–matrix interface is assumed to be imperfect characterized by a single, non-negative, and constant interface parameter. Using complex variable techniques, the analytic function that is defined within the irregular-shaped inclusion is expanded into a Faber series, and in conjunction with the Fourier series, a set of linear algebraic equations for a finite number of unknown coefficients is determined. With this approach and without imposing any constraints on the stress distribution, a semi-analytical solution is derived for the elastic fields within the irregular-shaped inclusion and the surrounding matrix. The method is illustrated using three examples and verified, when possible, with existing solutions. The results from the calculations reveal that the stress distribution within the inclusion is highly non-uniform and depends on the inclusion shape and the weak mechanical contact at the inclusion/matrix boundary. In fact, the results illustrate that the imperfect interface parameter significantly influences the stress distribution.

The elastostatic antiplane problem of a Dugdale crack at the interface of two different materials is considered. Using integral transform, the problem is reduced to a single integral equation. The integral equation is solved numerically. The evolution of the crack for different values of the physical and geometrical parameters of the problem is studied. A comparison between the results obtained with the Griffith and Dugdale models is presented.

A recently developed eigenfunction expansion technique, based in part on separation of the thickness-variable and partly utilizing a modified Frobenius type series expansion in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of crack/anticrack/junction front of an infinite tricrystal plate, made of monoclinic/hexagonal/orthorhombic/cubic phases, of finite thickness and subjected to far-field antiplane shear loading. Such tricrystals are used in solar cells and superconductivity application among others. Crack/anticrack-face boundary and/or interface contact conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the tri-crystal plate are exactly satisfied. Five different through-thickness boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) tri-crystal junction, and (iv and v) rigid inclusion located alongside a crack. Numerical results pertaining to the variation of the mode III order of stress singularity with various wedge aperture angles of the material 1 (e.g., scatterer in a solar cell), are also presented. Finally, results, pertaining to the through-thickness variations of mode III stress intensity factors or stress singularity coefficients for symmetric exponentially growing distributed load and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the tricrystal plate under investigation, also form an important part of the present investigation.

The state of stress in a three-quarter-plane undergoing antiplane shear deformation is studied, due to the presence of a screw dislocation along one of the projection lines extended from the free surfaces. A simple, accurate formula for the state of stress along the line is found, providing a useful kernel for the solution of crack and contact edge slip problems.

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.

In elasticity theory, a neutral inhomogeneity is defined as a foreign body which can be introduced into a host solid without disturbing the stress field in the solid. The existence of circular neutral elastic nano-inhomogeneities has been established for both antiplane shear and plane deformations when the interface effect is described by constant interface parameters, and the surrounding matrix is subjected to uniform external loading. It is of interest to determine whether noncircular neutral nano-inhomogeneities can be constructed under the same conditions. In fact, we prove that only the circular elastic nano-inhomogeneity can achieve neutrality under these conditions with the radius of the inhomogeneity determined by the corresponding (constant) interface parameters and bulk elastic constants. In particular, in the case of plane deformations, the (uniform) external loading imposed on the matrix must be hydrostatic in order for the corresponding circular nano-inhomogeneity to achieve neutrality. Moreover, we find that, even when we relax the interface condition to allow for a nonuniform interface effect (described by variable interface parameters), in the case of plane deformations, only the elliptical nano-inhomogeneity can achieve neutrality.

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

The elastic, electric and magnetic fields created within transversely isotropic magnetoelectroelastic media around an infinite row of uniformly-moving, collinear, antiplane shear cracks are studied, using an extension of the powerful method of dislocation layers. This analysis additionally provides the solutions for a finite magnetoelectroelastic plate containing a single crack and a plate with an edge crack.

An interfacial crack in an anisotropic bimaterial wedge under antiplane shear is investigated. An interfacial wedge crack in an isotropic bimaterial with concentrated forces is first solved by using the Mellin transform and the Wiener–Hopf technique. The solution of an interfacial crack in an anisotropic composite wedge is then derived from the isotropic solution based on a linear transformation method. An expression for the stress intensity factor for the interfacial crack is obtained. The asymptotic problem of an interfacial wedge crack is also analyzed. The stress intensity factor is obtained in terms of the stress intensity factor of a wedge without cracks. Numerical calculations of the stress intensity factor are carried out for various wedge types such as symmetric and asymmetric wedges as well as wedges with edge cracks and kinked cracks.

A recently developed eigenfunction expansion technique, based in part on separation of the thickness-variable, is first developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/reinforcing an infinite monoclinic plate, of finite thickness and subjected to far-field antiplane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the anisotropic (monoclinic) plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. The three-dimensional stress intensity factor for a center-crack, and stress singularity coefficients for/on a center-anticrack are then derived by incorporating an extension of the Stroh type approach in the present analysis. Through-thickness distribution of stress intensity factor and stress singularity coefficient for a crack and an anticrack, respectively, is also presented. Additionally, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed. Hitherto unavailable expressions for three-dimensional energy release and absorption rates for center-cracks and anticracks are derived by using Irwin’s crack closure and Eshelby’s eigenstrain approach, respectively. A heretofore unavailable expression for the energy release rate on the super-rigid inclusion is derived, using an approach which is analogous, in a reverse sense, to Irwin’s crack closure method. Finally, a new mode III crack deflection/bifurcation criterion is also derived. The crack deviation under antiplane shear loading is strongly correlated with the elastic stiffness constant, c45, of the monoclinic single crystal or off-axis composite lamina concerned.