Anti-plane Shear Loading Research Articles

Vzaimodeystviya treshchin v uprugom bimateriale pod deystviem antiploskogo sdviga

Vzaimodeystviya treshchin v uprugom bimateriale pod deystviem antiploskogo sdviga

Nano-inclusion with uniform internal strain induced by a screw dislocation

This paper addresses the question of whether it is possible to design a nanoinclusion (characterized here by the incorporation of interface effects along the material interface) to achieve a screw dislocation-induced uniform internal strain field when a composite is subjected to anti-plane shear deformation. We demonstrate the existence of such an inclusion by identifying its shape via a conformal mapping with unknown coefficients obtained through a system of nonlinear equations. Our numerical examples verify that the inclusion shape is dependent on its size and the specific uniform internal strain field. We show also that the inclusion shape is available even with increasing distance between the inclusion and dislocation. This latter fact leads to the additional conclusion that non-circular nano-inclusions which achieve uniform internal strain fields do indeed exist in a composite subjected to uniform remote anti-plane shear loading.

Uniform strain fields inside periodic inclusions incorporating interface effects in anti-plane shear

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.

Non-circular nano-inclusions with interface effects that achieve uniform internal strain fields in an elastic plane under anti-plane shear

This paper verifies the existence of a single non-circular nano-inclusion with interface effect that achieves a uniform internal strain field in an elastic plane under uniform remote anti-plane shear loadings. The uniform strain field inside such a non-circular inclusion is prescribed via perturbations of the uniform strain field inside the analogous circular inclusion, and the unknown (non-circular) inclusion shape is characterized by a conformal mapping whose unknown coefficients are determined by a system of nonlinear equations. Numerical examples show various shapes of non-circular nano-inclusions with interface effects that achieve uniform internal strain fields. It is found that such a non-circular inclusion shape depends on the inclusion size and the specific uniform internal strain field. In particular, for given interface shear modulus and residual tension, it is shown that the minimum inclusion size required to guarantee the existence of such a non-circular inclusion usually increases as the shear modulus of the inclusion approaches that of the matrix. Moreover, the relationship between the shear traction jump across the interface and the curvature of the interface is discussed in detail for such non-circular inclusions.

Fracture surface evolution and apparent delamination toughness in split composite beam specimens subjected to mixed mode I–III loading

The influence of specimen twisting during global anti-plane shear loading in composite split beam specimens is studied. Tests were conducted on specimens with different thicknesses and delamination lengths to produce different amounts of specimen twisting prior to fracture. It is shown that specimen twisting causes mode I stresses to develop, thereby producing mixed mode I–III conditions along the delamination front. This causes near-tip transverse cracks to initiate, prior to delamination advance, at an orientation related to the mode mix. Unlike in homogeneous materials, transverse crack extension is accompanied by planar delamination advance, and transverse crack rotation during extension is restricted by the laminate’s fibers. The overall fracture surface evolution is therefore strongly controlled by specimen geometry. The influence of these findings on the apparent delamination toughness as obtained from composite split beam and other types of mode III tests is discussed.

Peridynamics for antiplane shear and torsional deformations

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

An analytical method for predicting the anti-plane effective magnetoelectroelastic coefficients of composites containing doubly periodic multicoated fibers

The present paper deals with the magnetoelectroelastic composites containing a doubly periodic array of multicoated fibers under anti-plane shear loads and in-plane electromagnetic loads. By introducing the generalized eigenstrain, the heterogeneous magnetoelectroelastic medium is equivalent to a homogeneous magnetoelectroelastic medium with the periodically distributed generalized eigenstrains. Then the homogeneous magnetoelectroelastic medium with the generalized eigenstrain is solved analytically under the applied load conditions, the generalized stresses and strains in the fibers, coatings and matrix are derived. Based on the average-field theory, the solutions of the generalized stresses and strains are applied to determine the anti-plane effective magnetoelectroelastic properties of the composites. Two-phase (fiber/matrix) and three-phase (fiber/coating/matrix) magnetoelectroelastic composites are examined, and the comparison between the obtained results and the existing results shows the accuracy of the proposed method. Several four-phase magnetoelectroelastic composites with epoxy matrix are studied, and the influences of the composites microstructures on the effective magnetoelectric coefficient are discussed.

Interaction between the doubly periodic interfacial cracks in a layered periodic composite: Simulation by the method of singular integral equation

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.

Stress field of a radially functionally graded panel with a circular elastic inclusion under static anti-plane shear loading

The elastic stress field around a circular elastic inclusion in an infinite functionally graded material (FGM) panel that is subjected to a uniform static anti-plane shear loading at infinity is considered. Assuming the rigidity modulus of the FGM panel to be exponential, the analytical solutions of stress and strain distribution around the circular elastic inclusion are obtained by using the variable separation method. Due the generality of the elastic inclusion, other geometrical discontinuities, such as hole and rigid inclusion, can be seen as special cases of circular elastic inclusion when the inclusion rigidity modulus takes a suitable value. Therefore, the present solutions ana- lytically reduce to some classical solution in some special cases. The present analytical solutions are compared with the existing exact solutions to verify them. Finally, the effects of the relative inclusion rigidity modulus ratio and the inhomogeneous rigidity modulus ratio on the stress distribution and the concentration factor are systematically investigated.

Surface effects in anti-plane deformations of a micropolar elastic solid: integral equation methods

The theory of linear micropolar elasticity is used in conjunction with a new representation of micropolar surface mechanics to develop a comprehensive model for the deformations of a linearly micropolar elastic solid subjected to anti-plane shear loading. The proposed model represents the surface effect as a thin micropolar film of separate elasticity, perfectly bonded to the bulk. This model captures not only the micro-mechanical behavior of the bulk which is known to be considerable in many real materials but also the contribution of the surface effect which has been experimentally well observed for bodies with significant size-dependency and large surface area to volume ratios. The contribution of the surface mechanics to the ensuing boundary-value problem gives rise to a highly nonstandard boundary condition not accommodated by classical studies in this area. Nevertheless, the corresponding interior and exterior mixed boundary-value problems are formulated and reduced to systems of singular integro-differential equations using a representation of solutions in the form of modified single-layer potentials. Analysis of these systems demonstrates that the classical Noether theorems reduce to Fredholms theorems leading to results on well-posedness of the corresponding mathematical model.

Use of 2D Graphene Nanoplatelets (GNP) in cement composites for structural health evaluation

This study investigates the use of Graphene Nanoplatelets (GNP) in cement composite to quantify the material damage extent. The damage sensing capability of this new cement composite is demonstrated by experimentally measuring the electric potential across prisms with a known notch depth and comparing it with the finite element simulations. Meanwhile, the damage extent is directly related to the change in electric potential based on a mathematical analogy between the electrostatic field and the elastostatic field under anti-plane shear loading. It is shown that the fractional change in electric potential arising from damage is equivalent to the fractional change in elastic compliance, which can be exploited for structural health evaluation.

The Anti-Plane Shear Problem of Two Symmetric Cracks Originating from an Elliptical Hole in 1D Hexagonal Piezoelectric QCs

Using the complex variable function method and the technique of conformal mapping, the fracture mechanics of two symmetric collinear cracks originating from an elliptical hole in a one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) are investigated under anti-plane shear loading and electric loading. The crack is assumed to be either electrical impermeable or permeable. The exact solutions in closed-form of the stress intensity factors (SIFs) of the phonon field and the phason field, and the electric displacement intensity factors (EDIFs) are obtained. In the limiting cases, the new results such as Griffith crack, a circular hole with equal two edge cracks and cross crack can be obtained from the present solutions. In the absence of the phason field, the obtainable results in this paper match with the classical results.

Analysis of bonded anisotropic wedges with interface crack under anti-plane shear loading

The antiplane stress analysis of two anisotropic finite wedges with arbitrary radii and apex angles that are bonded together along a common edge is investigated. The wedge radial boundaries can be subjected to displacement-displacement boundary conditions, and the circular boundary of the wedge is free from any traction. The new finite complex transforms are employed to solve the problem. These finite complex transforms have complex analogies to both kinds of standard finite Mellin transforms. The traction free condition on the crack faces is expressed as a singular integral equation by using the exact analytical method. The explicit terms for the strength of singularity are extracted, showing the dependence of the order of the stress singularity on the wedge angle, material constants, and boundary conditions. A numerical method is used for solving the resultant singular integral equations. The displacement boundary condition may be a general term of the Taylor series expansion for the displacement prescribed on the radial edge of the wedge. Thus, the analysis of every kind of displacement boundary conditions can be obtained by the achieved results from the foregoing general displacement boundary condition. The obtained stress intensity factors (SIFs) at the crack tips are plotted and compared with those obtained by the finite element analysis (FEA).

Intrinsic coupling of near-tip matrix crack formation to mode III delamination advance in laminated polymeric matrix composites

Laminated carbon/epoxy specimens are loaded in anti-plane shear to investigate the relationship between near-tip matrix crack formation and the apparent mode III delamination toughness. Specimens are tested with different insert lengths to various load levels and examined fractographically. Near-tip matrix cracks are found to initiate and propagate intralaminarly before the onset of planar growth. These cracks are inclined at approximately 45° to the delamination plane and are perpendicular to the direction of maximum near-tip tensile stress. It is found that this represents an intrinsically coupled sequence of events for anti-plane shear loading of continuous fiber laminated polymeric composites when a preexisting delamination is bounded by plies that have their fiber direction aligned with the direction of macroscopic advance. This sequence of events violates the assumptions used in the reduction of data from common mode III tests. It therefore invalidates the associated toughness measurements, and may account for or strongly contribute to the common observation that laminated polymeric composites exhibit an apparent mode III delamination toughness that is dependent on test geometry.

Crack front échelon instability in mixed mode fracture of a strongly nonlinear elastic solid

In order to assess the role of elastic nonlinearity in gel fracture, we study the échelon instability in gelatin under mixed mode tensile and antiplane shear loading —i.e. the emergence of segmented crack front structures connected by steps. We evidence the existence of an energy-release-rate–dependent mode mixity threshold. We show that échelons appear via nucleation of localized helical front distortions, and that their emergence is the continuation of the cross-hatching instability of gels and rubbers under pure tensile loading, shifted by the biasing effect of the antiplane shear. This result, at odds with the direct bifurcation predicted by linear elastic fracture mechanics, can be assigned to the controlling role of elastic nonlinearity.

A Line Inhomogeneity in an Elastic Half Plane Under Anti-Plane Shear Loading

An elastic homogeneous isotropic material with a right line inhomogeneity embedded in the material under Anti-shear is analyzed; the mathematical model of the problem is a boundary value problem formulated using the mellin transform and solved by the Wiener-Hoph Techniques. A closed form solution for displacement is obtained from which the stress intensity factor is calculated. The stress field were found to have square-root singularity at the inner tip. As a result of this, micro-cracking can initiate at the inner tip of the line inhomogeneity in the matrix depending on the applied loads. The outer tip showed no singularly.

Anti – plane crack emanating from the interface in a bounded smart PEMO- elastic structure

The magnetoelectroelastic analysis of two bonded dissimilar piezo-electro-magneto-elastic ceramics with a crack perpendicular to and terminating at the interface is made. By using the Fourier integral transform (in perpendicular directions in each materials), the mixed boundary conditions and continuity conditions are transformed to a singular integral equation with generalized Cauchy kernel, the solution of which has been well studied, and classical methods are directly applicable here to obtain the closed form solution. The results are presented for a permeable crack under anti-plane shear loading and in-plane electric and magnetic loadings, as prescribed electric displacement and magnetic inductions or electric and magnetic fields. The results indicate that the magnetoelectroelastic field near the crack tip in the homogeneous PEMO- elastic ceramic is dominated by a traditional inverse square-root singularity, while the coupled field near the crack tip at the interface exhibits the singularity of the power law r<sup>--α</sup> , r being the distance from the interface crack tip and α depending on the material constants of a bimaterial. In particular, electric and magnetic fields have no singularity at the crack tip in a homogeneous solid, whereas they are singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order, field intensity factors and energy release rates. The results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures.

Some remarks on the Neuber rule applied to a control volume surrounding sharp and blunt notch tips

ABSTRACTThe 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.

The cyclically symmetric anti-plane fracture analysis for the arc-shaped interface in a non-homogeneous bimaterial cylindrical structure

A hollow cylinder, which consists of an inner functionally graded elastic substrate and an outer functionally graded elastic layer with cyclically symmetric cracks (a special case of multiple cracks), is considered under anti-plane shear load. The method of variable separation is employed to reduce the mixed boundary value problems to a Cauchy singular integral equation, which is solved numerically by the Lobatto–Chebyshev quadrature technique. Numerical results are presented to show the effects of geometrical and physical quantities on the stress intensity factors (SIFs). Parametric studies are conducted on the SIFs, and practical guidelines are given for the optimization of the fracture performance: (a) the SIFs depend on the ratio between the outer and inner radii, and the ratio should be at least 1.1; (b) the outer elastic layer should be stiffer than the inner elastic substrate; (c) large non-homogeneity parameter of the outer graded layer and small non-homogeneity parameter of the inner graded substrate are beneficial to SIFs reduction; (d) there is a strong interference between the stress fields around multiple cracks when the cyclically symmetric parameter increases.

Stress concentration factors and stress-gradients due to circular holes in radially functionally graded panels subjected to anti-plane shear loading

Closed-form expressions for the elastic fields (stresses, stress-gradients, strains, and displacements) in radially graded continua without and with circular holes are derived. The conventional definition of the stress concentration factor, which is the ratio of the maximum tangential stress on the circumference of the hole to the far-field applied traction, leads to physically unrealistic interpretations in functionally graded materials. A novel definition is derived for the stress concentration factor in graded panels. The stress concentration factor increases when the rigidity modulus progressively decreases away from the center of the hole (softening materials). The stress concentration desirably reduces in hardening materials in which the rigidity modulus progressively increases away from the hole. The gradient of the tangential stress ahead of the circular hole introduces an affected zone that is size-dependent (depends on the size of the hole, even in homogeneous media). The affected zone size decreases, while the stress-gradients increase with increase in the inhomogeneity length-scale and results in the increasing propensity of damage in softening materials. In hardening materials, the propensity of damage reduces with increase in the inhomogeneity length-scale due to the stress dilution.