Anti-plane Loading Research Articles

Thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load

Considering the importance of understanding the propagation of transient waves in the piezomagnetic solids, the thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load is investigated analytically in this paper. Using one-sided, two-sided Laplace transformation and Cagniard–de Hoop (CH) technique, an efficient and accurate analytical derivation for the solution of the anti-plane displacement, shear stress, magnetic potential, and induction in Laplace domain is presented. The study shows that the thermal stresses developed in x-axis and y-axis directions have significant influence on the transient response of the half-space. The magnetic induction B x increases obviously when the thermal stress is applied in x-axis direction, while it decreases when the thermal stress is applied in y-axis direction. Approaching time of magnetic induction B x and B y will become longer with higher thermal stress in x-axis direction. With the growth of the thermal stress in x-direction, contribution from the electromagnetic–elastic head (EH) wave increases, while the contribution from the shear elastic (SE) wave and the static value of shear stress decrease.

Phase-field simulation and coupled criterion link echelon cracks to internal length in antiplane shear

This paper provides a comprehensive numerical analysis of daughter crack localization in pure antiplane shear. Although antiplane shear fracture is important in various industrial applications, understanding the morphology of the resulting fragmentation remains challenging. The paper develops innovative phase-field models to induce the facets using a small spatial variation in the toughness field and examines the impact of numerical and material parameters on the newly formed daughter cracks’ shape and spacing. Through meticulous comparison to the coupled criterion, the paper reveals a compelling connection between the internal length-scale of damage regularization, Irwin’s length and the facet crack spacing. Furthermore, the effect of Poisson’s ratio on the crack form and spacing is investigated: the results reveal a significant influence and showcase comparable initiation distances between the numerical simulations and experimental measurements in pure antiplane loading.

Improved XFEM for 3D interfacial crack modeling

The conventional extended finite element method (XFEM) usually suffers from the crack tip enrichment for removing the blending elements. This may be inconvenient when treating the three dimensional (3D) and curved crack modeling since the crack tip enrichment is usually defined in the local coordinate system depending on the crack front. To avoid these shortcomings of the XFEM, an improved enrichment scheme is proposed for the 3D interfacial crack modeling. The proposed enrichment scheme removes the crack tip enrichment and two weight functions are introduced to capture the strong and weak discontinuities in the element including the crack front. The new enrichment scheme has advantages for the 3D curved interfacial cracks since all the enrichment functions are defined in the global system and it is unnecessary to introduce additional coordinate systems. Several benchmark problems including the mixed mode stress intensity factors (SIFs), anti-plane load and curved cracks are analyzed by the proposed method with the local refinement technology. Numerical results demonstrate that the proposed method can obtain accurate 3D energy release rate and SIFs. In addition, the proposed method enhances the applications of XFEM and has the potential abilities to the multiple-fields coupling and interfacial crack growth phenomena due to the fact that compatible enrichment functions are utilized.

A generalized antiplane singularity within a semi-infinite wedge of arbitrary angle

The state of stress arising within an elastic wedge generated by an antiplane singularity present within it is studied. An analytical solution is derived with a unified generic approach. The singularity may represent either an anti-plane concentrated force or a screw dislocation. To validate the general solution, two degenerate cases are presented. Further, the image force present on the screw dislocation due to the wedge free boundary is obtained. It is found that when the screw dislocations are placed in the vicinity of the wedge surface, the image force will drive dislocations to the free boundary where they will be annihilated. This implies that a dislocation-free zone may exist along the free surface of the wedge. To demonstrate the application of the fundamental solutions, a formulation for a slip band under anti-plane loading with Green’s function method is provided. The solutions developed in this study may be used as building blocks to model the damage of material near a V-notch under versatile anti-plane load conditions or torsional loading.

Investigating the effect of loading on the governing equation at the split region for a semi-infinite crack in an orthotropic material under antiplane loading

Most often there is a great disparity between experimental results and analytic results. Before now under great disparity, researchers kept suspecting the experimental procedures without investigating whether their analytic solution actually satisfy the governing equation. When a body in a plane is under loading, the loading splits the plane into regions, and the governing equation must be satisfied at these regions for the derived solution to be true. In this paper, I considered a homogeneous infinite orthotropic material containing a semi-infinite crack. A longitudinal shear load of magnitude Q is applied on the crack front. The displacement field in closed form is obtained. A verification of this solution at the split regions is carried out and shown to satisfy the governing differential equation.

Transient waves in a piezomagnetic half-space subjected to dynamic anti-plane loading

Propagation of transient waves in a piezomagnetic half-space, which is subjected to dynamic anti-plane-concentrated line force, is analyzed theoretically in this study. By employing Laplace transformation and Cagniard de-Hoop method, explicit closed forms of solution for the transverse displacement, shear stress, magnetic potential, and induction are obtained, and contribution from the mechanical and magnetic waves is thus expressed in the transient full-field solution. Propagation of the transient waves in the piezomagnetic half-space is then discussed in detail using numerical calculation. The results show that the magneto-mechanical coupling coefficient and location of the receiver have a significant influence on the transient response of this half-space. The magnitude of the transient shear stress is effectively reduced by the weakening magneto-mechanical coupling coefficient and the arrival time of the transient magnetic induction peak increases with decreasing magneto-mechanical coupling coefficient. In addition, it is found that the smaller the magneto-mechanical coupling coefficient, the higher the static value of the transient magnetic induction.

A Parabolic Notch Interacting With a Generalized Antiplane Singularity

Abstract In this study, the interaction of a parabolic notch with a generalized antiplane singularity is studied, and its analytical solution is derived. The singularity may be either an antiplane concentrated force or a screw dislocation, and separate solutions for each of these are found. The driving force present on the screw dislocation due to the notch free boundary is obtained. It is found that a dislocation-free zone may exist beneath the notch root surface when the screw dislocation is placed on the notch geometric symmetry axis, as the driving force will pull dislocations to the free boundary where they will be annihilated. The solutions developed in this study may be used as building blocks to model the damage of material near a parabolic notch under antiplane load conditions and therefore serve as a step in quantifying crack nucleation conditions, which is the novelty of the current study.

Analysis of an anti-plane crack in a one-dimensional orthorhombic quasicrystal strip

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.

Multiple rigid line inclusions (anti-cracks) in a multilayered orthotropic medium under anti-plane loading

Multiple rigid line inclusions (anti-cracks) in a multilayered orthotropic medium under anti-plane loading

Fundamental formulation for anti-plane eigenstrain problems

In this paper, the anti-plane eigenstrain problems are addressed in the framework of plane strain. A fundamental solution for an anti-plane eigenstrain nucleus located in an infinite plane is derived first. With this solution, the anti-plane eigenstrain toughening problems are formulated by Green's function method, which includes plastic strain shielding problems and transformation toughening problems. Also, with the equivalent eigenstrain principle and the Green's function method, its application to formulating inhomogeneous inclusion problems subjected to anti-plane load is demonstrated. The fundamental formulations and the related examples introduced here may pave a way to explore more complex problems in the corresponding aspects.

Failure Analysis of a Polygonal Void with an Oxide Layer in a Cracked Matrix

An oxide layer can commonly be found between a void in and the matrix of metal materials subjected to remote mechanical loading. This paper presents an evaluation of the failure behavior of a polygonal nano-void surrounded by an oxide layer in an infinite cracked matrix under a remote anti-plane load. Based on the complex variable, conformal mapping and analytical continuation with alternating methods, the stress functions are obtained in series form and the mode-III stress intensity factors (SIFs) are derived. The results indicate that a softer oxide layer (zirconia) would cause a higher SIF compared to a stiffer oxide layer (alumina). The results of this investigation provide a quantitative description of the effects of an oxide layer and inform an optimum design method for metal forming.

A novel isogeometric analysis enriched element for a V-notched one-dimensional hexagonal piezoelectric quasicrystal bi-material

A novel isogeometric analysis enriched element for accurate evaluation of singularities arising at the interface in V-notched one-dimensional hexagonal piezoelectric quasicrystals (PQCs) under anti-plane loading is developed. By taking the advantages of IGA and symplectic methodology, an enriched element centered at the notch tip is constructed by analytical symplectic eigensolutions. Explicit expressions of notch intensity factors and singular multi-physical fields within enriched element are obtained simultaneously. A comparison study between numerical predictions and analytical solutions is performed and excellent agreements are observed. Furthermore, the notch intensity factors for different V-notched PQC bi-material are presented and discussed in detail.

Analysis of the non-singular stress terms for the sharp notches under anti-plane loading

It is the singular term dominating the stress field near the vertex of a sharp notch, while the non-singular terms also play a non-ignorable role on the complete stress field. The non-singular stresses near the apex of anti-plane sharp notches are calculated and their roles on the fracture mechanics are analyzed in this paper. By coupling the singular characteristic analysis with the finite element method, the amplitude coefficients together with the singular orders and angular functions in the Williams expansion are solved. Then, different singular stress and higher-order non-singular stress terms can be determined from the asymptotic expansion according to the requirement. It is found that the solution merely considering the singular stress term can approximately reflect the stress field very close to the vertex, while it is invalid far from the notch vertex. The computational error is still greater than 5% in the solution containing the singular stress term together with the first non-singular stress term. However, the solution taking the singular stress term and the first two non-singular stress terms into account can accurately characterize the stress field far from the notch tip and its maximum error is less than 2%.

Anti-plane problem of nanocrack with surface piezoelectricity—a finite-form solution

A solution in finite form for the screw dislocation interacting with the nanocrack incorporating surface piezoelectricity under anti-plane loads is developed. By the boundary integration and the Cauchy’s integral formula to convert the problem into a first-order differential equation, the finite-form solution in physical region is established via the mapping technique. At the crack tip, the results from the present solution indicate that the electric displacements and the stresses are finite and the singularities are removed when the surface piezoelectricity is incorporated, which is against those in the classic solution and also different from other reported solutions where the singularities are weakened but non-trivial. The numerical results shows that the stresses, the electric displacements and the dislocation forces are strongly size dependent and their magnitudes reduce with the decrease in crack length, and the results approach to those by the classic solution with an increasing in the crack length.

A screw dislocation interacting with a semi-infinite crack partially penetrating a parabolic elastic inhomogeneity

We consider the interaction of a screw dislocation with a semi-infinite crack partially penetrating a parabolic elastic inhomogeneity embedded in an infinite elastic matrix subjected to anti-plane mechanical loading. Analytical solutions are derived in the case when the screw dislocation is located either in the matrix or inside the parabolic inhomogeneity. In addition, the local mode III stress intensity factor at the crack tip and the image force acting on the screw dislocation are obtained and illustrated accordingly. Moreover, we establish a dislocation emission criterion from the crack.

Impact force and moment problems on random mass density fields with fractal and Hurst effects.

This paper reports the application of cellular automata to study the dynamic responses of Lamb-type problems for a tangential point load and a concentrated moment applied on the free surface of a half-plane. The medium is homogeneous, isotropic and linear elastic while having a random mass density field with fractal and Hurst characteristics. Both Cauchy and Dagum random field models are used to capture these effects. First, the cellular automata approach is tested on progressively finer meshes to verify the code against the continuum elastodynamic solution in a homogeneous continuum. Then, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. Overall, the mean response amplitude is lowered by the mass density field's randomness, while the Hurst parameter (especially, for β < 0.2) is found to have a stronger influence than the fractal dimension on the response. The resulting Rayleigh wave is modified more than the pressure wave for the same random field parameters. Additionally, comparisons with previously studied Lamb-type problems under normal in-plane and anti-plane loadings are given. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Propagation of transient elastic waves in multilayered composite structure subjected to dynamic anti-plane loading with thermal effects

This paper investigates the transient responses of multilayered composite structure under the action of dynamical anti-plane concentrated forces with thermal effects. To obtain the transient solutions, the boundary value problem is presented by using Fourier-Laplace transform; the Cagniard’s method is adopted in inversing the solution to be expressed in time domain; and the theory of limit is employed to derive the corresponding static solution. Numerical results show that the closer the receiver's vertical position is to the surface, the slower the transient response approaches the static value in the near field of the transient waves. A sharp fluctuation appears after a short time in the intermediate and far field; and the transient response finally approaches the static value in the end. Variation of the temperature does not affect the waveform and non-dimensional arriving time of the transient shear waves. However, the higher the variation of environmental temperature, the smaller the amplitude of the transient responses.

Analysis of a mode III interface crack in a piezoelectric bimaterial based on the dielectric breakdown model

A mode III electrically conductive crack between two different piezoelectric materials under the action of anti-plane mechanical and in-plane electric loadings is analyzed. The strip dielectric breakdown (DB) model, which is free from the electric field singularity, is developed for this crack. According to this model, the electric field along a DB-zone situated in continuation of a crack is assumed to be equal to the electric breakdown strength. The DB-zone lengths are found from the condition of a finite electric field at the end point of such a zone. Using special representations of field variables via sectionally analytic functions, an inhomogeneous combined Dirichlet–Riemann boundary value problem is formulated and solved analytically. Explicit expressions for the shear stress, the electric field and the crack faces’ sliding displacement jump are derived. The stress intensity factor is determined as well. The dependencies of the mentioned values on the magnitude of the external electromechanical loading are presented.

Analysis of anti-plane interface cracks in one-dimensional hexagonal quasicrystal coating

The displacement discontinuity method is extended to study the fracture behavior of interface cracks in one-dimensional hexagonal quasicrystal coating subjected to anti-plane loading. The Fredholm integral equation of the first kind is established in terms of displacement discontinuities. The fundamental solution for anti-plane displacement discontinuity is derived by the Fourier transform method. The singularity of stress near the crack front is analyzed, and Chebyshev polynomials of the second kind are numerically adopted to solve the integral equations. The displacement discontinuities across crack faces, the stress intensity factors, and the energy release rate are calculated from the coefficients of Chebyshev polynomials. In combination with numerical simulations, a comprehensive study of influencing factors on the fracture behavior is conducted.

Interaction between a completely coated semi-infinite insulating crack and a piezoelectric screw dislocation

We derive analytic solutions to the problem of a completely coated semi-infinite insulating crack interacting with a screw dislocation under the assumption of anti-plane mechanical and in-plane electrical loading in a piezoelectric composite. The screw dislocation can be located either in the piezoelectric matrix or in the piezoelectric coating itself. We obtain explicit expressions for the resultant stress and electric displacement intensity factors, the resultant crack extension force and the image force acting on the screw dislocation. By employing orthogonality relations for two corresponding eigenvectors, we avoid tedious matrix operations in the development of the final expressions for the resultant field intensity factors, resultant crack extension force and image force.