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This study focuses on analyzing a slanted lip-shaped crack within a thermoelectric plate that experiences both a temperature gradient and an electric potential gradient. Based on the electrically insulated and thermally permeable crack model, we derive analytical solutions for the thermoelectric field and the associated thermal stress field in the cracked thermoelectric material by employing the complex variable method. Subsequently, we evaluate the stress intensity factors at the tips of the lip-shaped crack. Under the impact of thermoelectric loading, both mode-I and mode-II stress intensity factors occur at the tips of a slanted lip-shaped crack. Numerical examples are provided to demonstrate the influence of crack width, crack angle and thermal conductivity of the filling medium on the stress intensity factors at the crack tip. [ABSTRACT FROM AUTHOR]

A mode-I crack in a one-dimensional (1D) orthorhombic quasicrystal (QC) strip under in-plane phonon and phason stress loading is considered. Fourier transforms are applied to reduce the mixed boundary value problem of the mode-I crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and phason stresses and displacement fields near the crack tips have been obtained in an explicit form. The crack-tip singularities of the mode-I crack have been investigated and the intensity factors of the stresses in the phonon and phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack kinking phenomenon. The effect of the thickness ratio of the cracked strip on the SIFs and energy release rates has been investigated. When the thickness of the cracked strip becomes infinite large, the results obtained for the crack problem can be reduced to the analytic solution for a mode-I crack in an infinite 1D orthorhombic QC media. [ABSTRACT FROM AUTHOR]

In this paper, a central crack in a one-dimensional (1D) orthorhombic quasicrystal strip under anti-plane loadings has been studied. Fourier transform is applied to reduce the mixed boundary value problem of the crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and the phason fields near the crack tips have been obtained. The crack-tip singularities of the anti-plane crack have been investigated, and the intensity factors of the stresses in the phonon and the phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack skewing phenomenon. This crack skewing phenomenon is different from the mode-III crack problems in isotropic elastic materials and transversely isotropic elastic materials where the crack propagates along the original crack plane. [ABSTRACT FROM AUTHOR]

In this work, the bending response of Euler–Bernoulli beam is analyzed with three nonlocal strain gradient model with high-order boundary conditions and modified nonlocal strain and stress gradient models, in which the integral constitutive equation is transformed to an equivalent differential equation with two extra constraint equations. The corresponding differential equations are solved uniquely by the Laplace transformation and its inverse version. Under different boundary and loading conditions, the inconsistency of a nonlocal strain gradient model is examined extendedly, and consistent softening and toughening responses can be observed while using modified nonlocal strain and stress gradient model. [ABSTRACT FROM AUTHOR]

Based on the complex variable techniques combined with the boundary collocation method, a semi-analytical procedure is proposed to explore the anti-plane shear behavior of non-uniformly periodic fibrous composites under uniform remote shear loadings. Specific series with unknown coefficients are introduced to describe the complex potentials of the representative unit cell (RUC) of the composite. The unknown coefficients are determined from the continuity conditions on the interface, the periodic boundary conditions imposed on the edge of the unit cell and the remote loading conditions. Once the complex potentials are determined, the effective moduli of the composite are obtained according to the average-field theory. Extensive numerical examples are provided to investigate the influence of the periodic distribution of the inclusions, the moduli of each component, and the volume fraction of the inclusions on the local stress fields and effective shear moduli of the composite. Numerical results show that the inhomogeneous distribution of the inclusions may induce a continuous area with high equivalent stress in the matrix under certain loadings and material properties of each component. However, we could overcome this disadvantage by changing the components or the periodic arrangement of the inclusions. For the case of a RUC with double inclusions, the two inclusions with the same shear moduli arranged symmetrically in the direction of 45 degrees may induce the transverse isotropy of the composite. Nevertheless, when the two inclusions are horizontally arranged and the volume fraction of the inclusions is large enough (more than 10%), the composite generally shows orthotropic characteristics. [ABSTRACT FROM AUTHOR]

In the mechanical analysis of a structure/composite with periodic holes/inhomogeneities based on analytic techniques, the holes/inhomogeneities are usually assumed to be circular. In this paper, we develop an efficient method (based on complex variable techniques) to calculate the surface tension-induced stress field in a porous material containing a periodic array of unidirectional holes of arbitrary shape. In this method, we use conformal mapping and Faber series techniques to address a finite representative unit cell (RUC) consisting of a single arbitrarily-shaped hole with a constant surface tension imposed on the hole’s boundary and periodic deformations imposed on the edge of the RUC. Several numerical examples are presented to verify the accuracy of our method and to study the influence of the shape and volume fraction of the periodic holes on the stress concentration in the structure. We show that the maximum hoop stress around periodic holes of some shapes (such as triangle, pentagon or hexagon) may appear exactly at the point(s) of maximum curvature when the hole volume fraction exceeds a certain value. Moreover, when the hole volume fraction falls below about 7%, it is found that the surface tension-induced stress concentration around periodic holes can be treated approximately as that around a single hole with the same hole shape and size in an infinite plane. [ABSTRACT FROM AUTHOR]

This paper studies wave propagation in a poroelastic solid bar with polygonal cross-section under plane-strain conditions. The boundary conditions on the surface of the cylinder whose base curve is polygon are satisfied by means of the Fourier expansion collocation method. The frequency equations are discussed for both symmetric and antisymmetric modes in the framework of Biot's theory of poroelastic solids. For illustration purposes, sandstone saturated materials and bony material are considered. The numerical results were computed as the basis of relevant material data. Phase velocity is computed against the wavenumber for various cross-sections and results are presented graphically. [ABSTRACT FROM AUTHOR]

We consider the anti-plane shear of a composite containing a periodic array of circular inclusions which incorporate separate interface effects in the presence of uniform remote loading. Using complex variable methods, the corresponding stress distributions and effective shear modulus of the composite are obtained by analyzing a representative unit cell subjected to periodic boundary conditions imposed on its edge. We present several examples to illustrate the interfacial stress field and effective shear modulus relative to the interface parameter and volume fraction of the inclusions. We show that when the volume fraction of the inclusions falls below approximately 9 %, the interfacial stress recovers effectively to that of a single inclusion with the same interface parameter in an infinite plane. We find also that when the shear modulus of the inclusions exceeds twice the shear modulus of the matrix, we can essentially treat each inclusion-matrix interface as being perfectly bonded without inducing significant errors in the effective shear modulus of the composite. Finally, we show that the use of effective medium theories may induce significant errors in the determination of the effective shear modulus of the composite when the inclusions are much softer than the matrix and simultaneously the volume fraction of the inclusions exceeds 50 %. [ABSTRACT FROM AUTHOR]

We develop a new method to construct periodic inclusions with uniform internal hydrostatic stress in an elastic plane subjected to uniform remote in-plane loading. The method is based on two particular conformal mappings which lead to a system of nonlinear equations from which the inclusion shapes are determined. We illustrate our results with several examples. In particular, we show that the ratio of the inclusion size to the period of the inclusion-matrix system changes with the inclusion aspect ratio and, in the specific case of uniform remote shear loading, the orientation of the inclusions is also altered. Finally, we show that if the period of the inclusion-matrix system exceeds roughly six times the inclusion size, such periodic inclusions can be treated approximately as periodic elliptical inclusions with specific aspect ratio and orientation determined by the corresponding elastic constants and uniform remote loading. [ABSTRACT FROM AUTHOR]

We incorporate the mechanics of the interface to construct optimal shapes of periodic inclusions which achieve uniform internal strain fields in an elastic plane subjected to uniform remote anti-plane shear loading. These shapes are determined by solving a problem of the existence of a holomorphic function which is defined outside the unit circle in an infinite imaginary plane with specific boundary value on the unit circle. We illustrate such shapes using several examples. We show that the incorporation of interface mechanics has a significant effect on the design of such shapes and hence on the existence of these inclusions at the nanoscale. In addition, we show that if the period of the inclusion-matrix system exceeds roughly seven times the inclusion size, such shapes can be treated essentially as being equivalent to those of a single inclusion enclosing the same uniform internal strain in the presence of identical bulk and interface material constants, inclusion size and remote loading. [ABSTRACT FROM AUTHOR]

In existing literature, it remains an unexplored question whether any inclusion shape can achieve a uniform internal strain field in an elastic half-plane under either given uniform remote loadings or given uniform eigenstrains imposed on the inclusion. This paper examines the existence and construction of such single or multiple non-elliptical inclusions that achieve prescribed uniform internal strain fields in an elastic half-plane under given uniform anti-plane shear eigenstrains imposed on the inclusions. Such non-elliptical inclusion shapes in a half-plane can be determined by solving the original problem of an unknown holomorphic function in a multiply connected half-plane, which is transferred to an equivalent problem of an unknown holomorphic function in a multiply connected whole plane based on analytic continuation techniques. Extensive numerical examples are shown for single inclusion, multiple inclusions and two geometrically symmetrical inclusions, respectively. It is found that the inclusion shapes which achieve uniform internal strain fields depend on the given uniform eigenstrains, and the inclusion shapes that achieve uniform internal strain fields for arbitrarily given uniform eigenstrains do not exist. Moreover, specific conditions are derived on the given uniform eigenstrains and prescribed uniform internal strain fields for the existence of two geometrically symmetrical inclusions that achieve uniform internal strain fields. [ABSTRACT FROM AUTHOR]

The two-dimensional problem of a crack in thermoelectric materials is studied in this research. The general solution is derived based on the complex variable method. For the case of a crack subjected to remote electric current and heat flow, the solutions are obtained in closed-form. The results show that the fields of heat flow, electric current, and stress exhibit traditional square-root singularity at the crack tip. The remote electric current produces both type I and II stress intensity factor. Furthermore, the stress intensity factor has a linear relationship with the heat flux, but a non-linear relationship with the electric current. [ABSTRACT FROM PUBLISHER]

Abstract: Based on the assumption that the elastic strain of electrostrictive materials is a higher-order small quantity, this paper studies the 3D problem of an infinite electrostrictive solid with a flat elliptical crack which is electrically permeable. According to existing solutions of similar problems in pure elastic materials, with the displacement function method, we first derived explicit expression for displacement potential function and obtained stress field near the crack and open displacement of crack surface. Then, the general solution for the stress intensity factor was derived, and the corresponding solutions were also presented for a penny-shaped crack and a permeable line-crack as two special cases of the present problem. Finally, numerical results were given to discuss the effect of environment at infinity and electric field inside the crack on the stress-intensity factors. [Copyright &y& Elsevier]

The 2D problems of a soft ferromagnetic solid with collinear cracks are studied based on the complex variable method. First, the general solutions for potential functions of magnetic fields and stress fields are derived by using magnetically permeable, impermeable and conducting crack models, respectively. Then, for special cases of one and two cracks, more explicit solutions are given for potential functions and intensity factors of total stresses. Finally, numerical results are presented to discuss the effects of applied magnetic loading on fracture behavior of the ferromagnetic solid. It is found that the total stresses always have the classical singularity of the r^{ - 1/2} type at the crack tips for the considered three crack models, and the applied magnetic loading may either enhance or retard crack growth depending on the adopted crack models. [ABSTRACT FROM AUTHOR]

Abstract: Based on the strip dielectric breakdown (DB) model, the two-dimensional problem for a conductive crack in an infinite electrostrictive material is studied by means of the complex variable method. Explicit solutions are obtained for the electric fields and electroelastic fields, respectively. Then, the intensity factors of total stresses are presented in closed-form. Finally, all the solutions are extended to the case of a soft electrode in an infinite electrostrictive material. It is found that for a conducting crack, the applied electric field parallel to the crack retards its growth before DB, but it may enhance its growth once DB takes place ahead of the conductive crack. For the soft electrode, it can be seen that DB can produce the singularity of electric fields ahead the tip of electrode and induce the cracking of the electrostrictive material since the tensile stress has a traditional singularity of r −1/2 at the tip of the electrode. [Copyright &y& Elsevier]

Abstract: This paper presents a rigorous treatment on a two-dimensional problem of an elliptic hole or a crack in an infinite electrostrictive solid subjected to remote electric fields based on the complex variable method. Firstly, the general solutions are obtained for the electric fields inside the elliptical hole and the complex potentials in the solid, respectively. Secondly, numerical results of stresses around the hole are given in order to discuss the effect of the electric fields inside the hole on the fracture behaviors of the solid. Finally, explicit and closed-form solutions are obtained for an electrically permeable/impermeable crack when the hole degenerates to a crack. It is found that: (1) in general, for a permeable crack, the total stress may have a traditional type singularity at the crack tip under pure electric loads, and the applied electric loads may enhance or retard crack propagation, which is dependent on the Maxwell stresses on the crack faces and at infinity; (2) when the interior of the permeable crack is filled with the same medium as that at infinity, the magnitude of the Maxwell stress on the crack faces is equal but opposite to that on the solid surface at infinity, and as a result the applied electric field has no effects on crack growth; and (3) for an impermeable crack, the total stresses still have a type singularity at the crack tips under pure electric loads, and the applied electric field perpendicular to the crack surface may enhance crack propagation. [Copyright &y& Elsevier]

This work revisits the generalized 2D problem of electrically permeable collinear interface cracks in piezoelectric materials with two motivations: one is to present a more explicit approach to the considered problem; the other is to derive some new results for periodical interface crack problem in piezoelectric materials with the use of the new approach. Based on the Stroh formalism, the mechanical–electric coupling boundary equations are decoupled into two equations: the first one is related only to the applied mechanical loads, and the second one only to the applied electric field. According to the traditional method or available results, the solution for the first equation can be given, and then the solution for the second equation can be directly written out by using the results of the first equation. Furthermore, the solutions for infinite number of periodical collinear interface cracks are at the first time presented in closed form. The solutions include the field intensity factors and the electric fields both inside and outside the cracks. It is shown that under the electric loading only, the electric fields are uniform not only in the materials but also inside the cracks, while the stress is zero wherever. However, when the combined mechanical–electric loadings are applied at infinity, the electric fields inside the cracks may be singular and oscillatory, and such is the case for the stresses near the crack tips, but the intensities of all singularities depend on the material properties and the applied mechanical loads, not on the applied electric loads. Finally, a numerical example is given for the case of a single interface crack, and the electric fields inside the crack is shown analytically and graphically. [Copyright &y& Elsevier]

A theoretical model is developed to analyze the stress transfer between fiber and matrix through the interphase with finite thickness. The Young's modulus of interphase is assumed to be homogeneous uniform or power-graded along radial direction while other material parameters are constants. The bonds between fiber and interphase as well as between interphase and matrix are perfect. The geometrical equations are strictly satisfied except that the radial displacement gradient with respect to the axial direction is neglected, as its magnitude is much smaller than that of the axial displacement gradient with respect to the radial direction. The equilibrium equations along radial direction are strictly satisfied, while the equilibrium equations along axial direction are satisfied in the integral forms. In addition, both the interfacial displacement and stress continuity conditions as well as stress boundary conditions are enforced exactly. Two coupled 2nd-order ordinary differential equations can be obtained in terms of average axial stresses in fiber and matrix. Finite element analysis (FEA) with refined mesh for single-fiber composite containing uniform interphase with finite thickness is developed to validate the present model. Series of parameter studies are performed to investigate the influence of interphase properties and thickness as well as the fiber volume content and model length on the stress distribution in composites. [ABSTRACT FROM AUTHOR]

We present a rigorous treatment of the problem of collinear periodic cracks in an infinite electrostrictive plane subjected to plane strain deformation in the presence of a uniform remote in-plane electric field. Using complex variable methods, exact electric and stress fields in the electrostrictive plane are obtained for both permeable and impermeable cracks. We show that the ratio of the mode-I stress intensity factors for periodic cracks to those for a single crack is determined by only the ratio of the period of the periodic crack-system to the crack length. Additionally, we find that the stress intensity factors for periodic cracks and the stress field in the proximity of periodic cracks can be treated essentially as those for a single crack when the period exceeds four times the crack length. [ABSTRACT FROM AUTHOR]

A formula is derived to calculate the phase angle of interfacial mixed mode fracture in the indentation delamination test, in which the delaminated film buckles. Interfacial fracture toughness depends on the phase angle and calculating the phase angle is necessary for the determination of mode I interfacial fracture toughness. [Copyright &y& Elsevier]

Abstract: In terms of crack opening, the present work studies the self-consistent, energetically consistent, and electrostatic traction approaches. In the self-consistent approach, crack will not open if no mechanical load is applied. The energetically consistent approach under a given electric field yields a threshold stress for crack opening and a bifurcation stress, which is higher than the threshold stress. Between the bifurcation and threshold points there are two solutions for crack opening. The electrostatic traction approach considers electrostatic tractions along crack faces and infinite boundaries. The electrostatic traction along infinite boundaries is equivalent to an additional mechanical load for a given electric field, which is tensile and promotes crack opening, if the dielectric constant of the infinite medium is smaller than that of the material. Energy release rate is also comprehensively analyzed in each of the three approaches. [Copyright &y& Elsevier]