Anti-plane Mechanical Loading Research Articles

Anti-plane fracture behavior of n nano-cracks emanating from a magnetoelectrically semi-permeable regular n-polygon nano-hole in magnetoelectroelastic materials

Based on Gurtin–Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping function, the fracture behavior on n nano-cracks emanating from a n-polygon nano-hole under far-field anti-plane mechanical load, inplane electrical load and magnetic load is studied. The analytical solutions of stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor at the crack tip are given under the boundary conditions of magnetoelectrically semi-permeable. In addition, numerical examples show that when considering the surface effect, the stress field intensity factor, electric displacement intensity factor and magnetic induction intensity factor have obvious size-dependence. The results of this paper show that when the defect size increases to a certain extent, the influence of the surface effect begins to decrease, and finally tends to the results of classical elastic theory.

Fracture analysis of circular hole edge arbitrary position crack with surface effects in magnetoelectroelastic materials

The mechanical properties of magnetoelectroelastic (MEE) materials are brittle, which is easy to produce nano-defects (hole or crack). The micro-fields of nanoscale cracks and holes are size dependent significantly. Nanoscale cracks and holes interact with each other, and the interference effect is very obvious. In this paper, the Model III fracture behavior of nanoscale circular hole edge arbitrary position radial crack is studied under anti-plane mechanical load, in-plane electric displacement load and in-plane magnetic induction load. Based on the surface elasticity theory, the conformal mapping technique and the MEE elasticity theory, the MEE fields expressions of the nano-hole and nano-crack are obtained, and the analytical solution of generalized intensity factors (stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor) at crack tip is given. The comparison between the present results and the existing research shows the correctness of the present method in this paper. The influences of the microstructure and coupling loads on the generalized intensity factors are discussed.

Fracture analysis on periodic radial cracks emanating from a nano-hole with surface effects in magnetoelectroelastic materials

In this paper, the fracture behavior of multiple radial cracks with periodic and uniform distribution around a nanoscale circular hole in a magnetoelectroelastic (MEE) solid is studied under anti-plane mechanical load, in-plane electric displacement load and magnetic induction load. Based on the surface elasticity theory, the analytical solutions of the MEE fields distribution and the MEE fields intensity factors at the crack tip are obtained by using the conformal mapping method, the boundary value theory of analytic function and the basic theory of MEE elastic solid. Considering that the defects are piezomagnetic phase or piezoelectric phase, size effects of the defects, cracks number, geometric parameters and applied MEE loads on the MEE fields intensity factors at crack tip are discussed. The size effect, interference effect and multi-fields load coupling effect of the MEE tip fields are studied in detail.

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.

A novel method for measuring AFO stiffness using motion capture

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

Surface effects of electroelastic tip fields of multiple cracks emanating from a circular hole

The problem of piezoelectric solids containing a nanoscale multi-cracks cracked hole under antiplane mechanical load and inplane electric load is investigated theoretically. The surface electroelastic constants of the defects (crack and hole) are introduced based on the Gurtin-Murdoch surface model. An exact solution of the stress fields and electric displacement fields around the defects are obtained by constructing a conformal mapping function. Closed-form solution of the stress intensity factor and the electric displacement intensity factor at crack tips are presented. Numerical examples are provided to reveal the relationship between the electroelastic fields intensity factors and the defect size, the crack-hole interaction, and the applied mechanical-electrical loads.

Poling angle effect on two mode-III semi-permeable collinear cracks in a piezoelectric strip: Strip-saturation model

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.

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.

Anti-plane crack problem of a functionally graded piezoelectric materials strip with arbitrarily distributed properties

This paper treats the anti-plane crack problem of a functionally graded piezoelectric material (FGPM) strip with arbitrarily distributed properties. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric constant of the FGPM vary continuously along the thickness of the medium, and the strip is under anti-plane mechanical and in-plane electric impact loadings. By using the Fourier transform and defining unknown discontinuous functions across the crack surfaces, the anti-plane crack problem of FGPM is reduced to a group of singular integral equations, which are solved numerically. The stress and electric displacement intensity factors are presented, and the influences of the nonhomogeneous and geometric parameters on stress and electric displacement intensity factors are also included.

Interaction of Screw Dislocation with Edge Interfacial Crack on Fine-Grained Piezoelectric Coating/Substrate

The interaction between micro- and macrocracks in a fine-grained piezoelectric coating/substrate under remote antiplane mechanical and in-plane electrical loadings was studied. The principle of superposition and a mapping function method was used to transform the fine-grained coating/substrate structure containing the screw dislocation and the edge interfacial crack into the right semi-infinite plane piezoelectric bimaterial with screw dislocation to simplify the problem. Furthermore, the electric field, displacement field, intensity factors, and image force of these two problems were established. In addition, numerical calculations were then given graphically to study the effects of the elastic modulus of the material, the size of the crack, the thickness of the coating, and the screw dislocation angle on the edge interface crack and dislocation.

Elastic interaction between a semi-infinite debonded anticrack and a screw dislocation

We study the elastic interaction between a semi-infinite debonded anticrack lying on the negative real axis and a screw dislocation under the assumption of anti-plane mechanical loading. The lower side of the anticrack (or rigid line inclusion) is perfectly bonded to the surrounding material in contrast to its upper side which is fully debonded. A simple closed-form solution to this interaction problem is developed by means of conformal mapping and the method of images. Specifically, we obtain expressions for the image force acting on the screw dislocation, the two resultant stress intensity factors respectively characterizing the r−3/4 and r−1/4 type singularities at the tip of the debonded anticrack and the resultant material force on the debonded anticrack. The dislocation emission criterion from the debonded anticrack in terms of two critical stress intensity factors is also established.

A screw dislocation near a non-elliptical piezoelectric inhomogeneity with internal uniform electroelastic field

Conformal mapping and analytic continuation are employed to prove the existence of an internal uniform electroelastic field inside a non-elliptical piezoelectric inhomogeneity interacting with a screw dislocation. We focus specifically on the case when the piezoelectric matrix surrounding the inhomogeneity is subjected to uniform remote anti-plane mechanical and in-plane electrical loading and a constraint is imposed between the remote loading and the screw dislocation. The constraint can be expressed in a relatively simple decoupled form by utilizing orthogonality relationships between two corresponding eigenvectors. The internal uniform electroelastic field is found to be independent of the presence of the screw dislocation; moreover, it can be expressed in decoupled form.

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.

A crack along a part of an interface electrode in a piezoelectric bimaterial under anti-plane mechanical and in-plane electric loadings

An electrically conducting crack along a part of an electrode in the interface of a piezoelectric bimaterial under the action of anti-plane mechanical and in-plane electric loadings is analyzed. The electrode is assumed to be much thinner than the piezoelectric material, and therefore, its mechanical properties are neglected. Using special representations of field variables via sectionally analytic functions, a 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 are derived. These quantities are also presented graphically along the corresponding parts of the material interface. The intensity factors for stress and electric field are determined as well. The dependencies of the mentioned values on the magnitude of the external electric loading and different ratios of the crack and electrode lengths are presented.

An Interface Crack with Mixed Electro-Magnetic Conditions at it Faces in a Piezoelectric / Piezomagnetic Bimaterial under Anti-Plane Mechanical and In-Plane Electric Loadings

Abstract An interface crack between two semi-infinite piezoelectric/piezomagnetic spaces under out-of-plane mechanical load and in-plane electrical and magnetic fields parallel to the crack faces is considered. Some part of the crack faces is assumed to be electrically conductive and having uniform distribution of magnetic potential whilst the remaining part of the crack faces is electrically and magnetically permeable. The mechanical, electrical, and magnetic factors are presented via functions which are analytic in the whole plane except the crack region. Due to these representations the combined Dirichlet-Riemann and Hilbert boundary value problems are formulated and solved in rather simple analytical form for any relation between conductive and permeable zone lengths. Resulting from this solution the analytical expressions for stress, electric and magnetic fields as well as for the crack faces displacement jump are presented. The singularities of the obtained solution at the crack tips and at the separation point of the mention zones are investigated and the formulas for the corresponding intensity factors are presented. The influence of external electric and magnetic fields upon the mechanic, electric and magnetic quantities at the crack region are illustrated in graph and table forms.

Asymmetric impermeable wedge crack in a piezoelectric material under anti-plane deformation

A problem of an inclined wedge crack in transverse isotropic piezoelectric material under electromechanical loading is analyzed. A single anti-plane mechanical and in-plane electrical loads are applied at a point either on wedge surface or on crack surface. By using Mellin transform, the problem based on the generalized displacement and stress vector is formulated. The derived equation is solved employing the Wiener-Hopf technique, from which the intensity factors and the energy release rates are obtained in closed forms. With material properties of PZT-5H, the intensity factors and energy release rate for wedge and crack surface loadings are calculated numerically and investigated. The closed form solutions can be used as a Green’s function for arbitrarily coupled anti-plane mechanical and inplane electrical loading cases.

Surface effects on an electrically permeable elliptical nano-hole or nano-crack in piezoelectric materials under anti-plane shear

An electrically permeable elliptical nano-hole or nano-crack embedded in an infinite piezoelectric material with surface effect is investigated based on the Gurtin–Murdoch surface/interface model, which is subjected to far-field anti-plane mechanical and in-plane electrical loads. The electric field inside the elliptical nano-hole has been taken into consideration, and exact electroelastic fields near the elliptical nano-hole are obtained by using the technique of conformal mapping. The size-dependent stress and electric displacement intensity factors at the crack tip are derived exactly for both electrically permeable and impermeable boundary conditions when the elliptical nano-hole reduces to the nano-crack. Numerical examples are illustrated to show the surface effects on the stress and electric displacement intensity factors of electrically permeable and impermeable cracks, and on the stress and electric field concentrations around the electrically permeable and impermeable holes. The results indicate that the size dependence of an electrically permeable nano-hole or nano-crack is entirely different from that of the electrically impermeable case.

Fracture characteristics of a cracked equilateral triangle hole with surface effect in piezoelectric materials

A theoretical study is conducted on the fracture behavior of a cracked equilateral triangle hole with surface effect subjected to far-field antiplane mechanical load and inplane electric load. A rigorous analytical solution to the stress and electric displacement fields are obtained based on the theory of Gurtin-Murdoch surface model and conformal mapping technique. A closed form solution to the stress intensity factor, the electric displacement intensity factor and the energy release rate at the tip of crack are presented. Numerical examples are provided to reveal the variations of the electroelastic field intensity factors and the energy release rate with the size of triangle hole, the length of crack and the applied mechanical load and electrical load. The major results of the study are as follows: (1) The electroelastic field intensity factors at crack tip are dramatically size dependent when the size of the cracked triangle hole is at nanoscale. The present solution approaches classical electroelastic theory when the length of cracked equilateral triangle hole has large characteristic dimensions. (2) With the increase of the length of crack, the electroelastic field intensity factors first increase rapidly, and then slowly decrease, and finally stabilize. (3) When considering surface effect, the stress and electric displacement intensity factors depend on the applied mechanical-electrical loads, which is different from that of the classical electroelastic fracture theory. (4) The normalized energy release rate increases with the increase of the size of the cracked triangle hole. (5) The influences of the applied mechanical load on the normalized energy release rate depend on whether the applied electrical load is positive or negative. (6) With the increase of the applied electrical load from negative to positive, the normalized energy release rate release rate first increases and then decreases. Very high positive and negative applied electrical loads shield the normalized energy release rate.

Mixed Electrical Conditions on Interface Inclusion in a Piezoelectric Bimaterial under Antiplane Mechanical and Inplane Electric Loadings

An absolutely rigid interface inclusion in a bimaterial piezoelectric space under the action of antiplane mechanical and in-plane electric loadings is analyzed. One zone of the inclusion is electrically insulated while the other part is electrically permeable. This problem is important for engineering application, but it has not been solved earlier in an analytical way. Presenting all electromechanical quantities via sectionally analytic vector functions, the combined Dirichlet-Riemann boundary value problem is formulated. An exact analytical solution of this problem is obtained. Closed form analytical expressions for electromechanical quantities at the interface are derived. Some of these values are also presented graphically along the corresponding parts of the material interface. Singular points of the shear strain and the electric displacement are found and the corresponding intensity factors are determined as well.