Mechanical Strain Energy Release Rate Research Articles

Fatigue crack analysis in piezoelectric specimens by a single-domain BEM

In this paper, the fatigue crack problems in piezoelectric materials under cyclic mechanical loading or alternating electric field were analyzed by a single-domain boundary element method. To determine the direction of the crack propagation, the fracture criteria of maximum of hoop mechanical strain energy release rate was used. Meanwhile, for evaluating the remaining life of the cracked piezoelectric specimens, the Paris-type laws based on different fracture parameters were employed and compared. All the involved fracture parameters were computed by the interaction integral method. Numerical examples were considered and analyzed for cyclic mechanical and electrical loadings, respectively. The comparisons showed the efficiency of the present BEM program in analyzing fatigue cracks and the choice of the effective fracture parameter in Paris’ law for life prediction.

The Fracture Analysis of Collinear Periodic Cracks in an Infinite Piezoelectric Plate

The fracture problem of collinear periodic cracks in an infinite transversely isotropic piezoelectric plate subjected to the anti-plane shear stress and the in-plane electric load at infinity is studied. Using the complex function method, the mechanical problem is turned into the boundary value problem of partial differential equations. The solutions of the boundary value problem of partial differential equation are obtained by undetermined coefficients method. Then, considering the periodicity of cracks, the stress intensity factors and the electric displacement intensity factors for mode III near the right tip of every crack are defined, the expressions of the stress fields, electric displacement fields, displacement fields, electric potential fields and the mechanical strain energy release rate around the crack tip are obtained with the assumption that the surface of the crack is electrically impermeable. Finally, interference effect and scale effect of collinear periodic cracks and the mechanical strain energy release rate are discussed by analysis of examples. It can be seen interference effect of collinear periodic cracks is strong when 1 < b / a < 2. The scale effect of the singularity of the stress intensity factors and electric displacement intensity factors in crack tip is obvious. Stress always promotes extension of the cracks, the mechanical strain energy release rate is related to the size and direction of the electric field, the positive electric field can promote the expansion of the cracks, the negative electric field can inhibit the extension of cracks.

Numerical simulation of crack growth in piezoelectric structures by BEM

In this paper, a dual boundary element computer program is developed for numerical simulation of a crack propagating in piezoelectric plates under a combined quasi-static electric and mechanical loading. To determine the crack growth path, two fracture criteria are taken into account: the maximum of hoop stress intensity factor (HSIF) and hoop mechanical strain energy release rate (HMERR). By using the displacement extrapolation method, these fracture parameters of any small kinked crack branch are obtained and validated by comparing with the available analytical results. The critical fracture loads for some test specimens are numerically analyzed based on the maximum HMERR fracture criterion. Different electrical boundary conditions on the crack faces are considered and checked with the experimental data. Finally, one crack or a pair of cracks propagating in infinite or finite piezoelectric plates is numerically simulated. The influences of the loading conditions, the anisotropic fracture toughness and the interaction between the cracks on the crack growth paths are also studied. The comparisons with the exiting finite element results show the accuracy and efficiency of the present BEM program for numerical simulation of crack growth in piezoelectric materials.

A simplified evaluation of the mechanical energy release rate of kinked cracks in piezoelectric materials using the boundary element method

Based on the concept of the hoop field intensity factors of an initial crack prior to any kink, an apparent hoop mechanical (strain) energy release rate (MERR) is defined to approximate the MERR of a piezoelectric crack with an infinitesimal kink at any arbitrary angle. The validity and the efficiency of the simplified approximation are examined by numerical examples using the boundary element method (BEM). The generalized crack-opening-displacements or displacement jumps are computed by the traction boundary integral equations (BIEs). By using the displacement extrapolation method, the crack-tip field intensity factors of any arbitrarily kinked crack in linear piezoelectric materials are obtained and the BEM results are validated by comparing them with the available reference analytical results. Then, the differences between the conventional field intensity factors and MERR of an infinitesimally kinked crack and the hoop field intensity factors and hoop MERR of the main crack prior to any kink are numerically analyzed. Finally, the crack propagation in an infinite linear piezoelectric material is numerically simulated. The paths of the crack growth are predicted by adopting four different fracture criteria, namely, the maximum hoop stress intensity factor (SIF) and MERR fracture criteria for the main crack-tip before the next propagation, and the maximum KI and MERR fracture criteria for the kinked tip of the main crack with an infinitesimal branch at an arbitrary kinking angle evaluated by using a trial crack extension technique. The comparisons among these results show that the present simplified approximation can efficiently provide a sufficient accuracy for numerical simulation of crack growth in linear piezoelectric materials.

In-plane fracture analysis on the magneto-electro-elastic interfacial region in a multiferroic composite: Effects of volume fraction

Abstract The high temperature and pressure of the hot-pressing process of layered ferromagnetic/ferroelectric (FM/FE) composites inevitably give rise to magneto-electro-elastic (MEE) interfacial regions, where defects such as cracks often aggregate and definitely affect the performance of the composite. Fracture analysis is performed on an MEE interfacial region between an FM layer and an FE layer by the methods of Fourier integral transform and Cauchy singular integral equations. After the numerical computation is verified by the exact solution in a special case and the numerical solutions in existing literature, numerical results of MSERR (mechanical strain energy release rate) and COD (crack opening displacement) are provided to discuss the fracture behavior. The effect of FE volume fraction of the interfacial region on the fracture behavior is demonstrated from three different perspectives. Both material stiffness and constraint effect are used to explain the physical mechanisms. The obtained results may serve as guidelines for the anti-fracture design of FM/FE multiferroic composites. Because the existence of such MEE interfacial regions has been scarcely considered by existing papers, the main novelty of the present work is the incorporation of the MEE interfacial region into the mechanical model of the FM/FE composite.

Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids

By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples are presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions.

Two bonded multiferroic ceramics half-planes with coupled interfacial imperfections: In-plane fracture and its dislocation-based physical mechanisms

Abstract Essentially speaking, both crack and interfacial imperfection can be regarded as continuously distributed dislocations. Therefore, the fracture behavior of composites containing imperfect interfaces may be explained by the interactions of different distributed dislocations. The purpose of this paper is to make a try on such explanation. For this purpose, in-plane fracture analysis is performed on two bonded multiferroic half-planes with coupled interfacial imperfections by the methods of Fourier integral transform and Green's functions. Numerical results of the mechanical strain energy release rate are obtained by numerically solving the Cauchy singular integral equations of the crack problem. The effects of the three kinds of interfacial imperfections and their inter-couplings on the fracture behavior are discussed based on the variations of mechanical strain energy release rate versus the interface parameters, and the fracture behavior including shielding, anti-shielding, interference and anti-interference are revealed, respectively. Finally, the cracks and the imperfect interface are simulated as continuously distributed dislocations to construct two types of dislocation distribution models, which are then, for the first time, employed to explain the mechanisms of the fracture behavior.

Interfacial sliding in a bi-layered multiferroic ceramics: Localized sliding-prevention/promotion based mechanism of intra-layer fracture

Different types of sliding regions on the interface ( a 1 = 0 mm; b 1 = 4 mm; a 2 = 31 mm; b 2 = 33 mm; d = 2 mm; β ̂ 1 = 10; f = 3). • A prevention/promotion model is proposed to formulate interfacial sliding. • Local positive normal stress gives rise to local sliding-promotion effect. • Local negative normal stress leads to local sliding-prevention effect. • Local sliding-prevention/promotion based fracture mechanisms are revealed. Sliding is a typical failure mode of composite interfaces. According to traditional theory, interfacial sliding can be represented by the linear shear-spring model (LSSM). Although the required driving interfacial shear stress has been characterized by LSSM, the relation between interfacial normal stress and sliding is not yet reflected by it. The present paper proposes an interfacial sliding-prevention/promotion model to consider the effect of interfacial normal stress on interfacial sliding. As an example of application, the intra-layer fracture problem is analyzed on a bi-layered multiferroic ceramics. Green’s functions are derived to construct the Cauchy singular integral equations, which are further numerically solved to get mechanical strain energy release rate (MSERR) and interfacial normal stress. Parametric studies yield a new finding that non-zero normal stress may be produced in local interfacial regions by intra-layer cracks under pure in-plane shear. Local positive normal stress gives rise to local sliding-promotion effect, while local negative normal stress leads to local sliding-prevention effect. The interfacial shear imperfection and local sliding-prevention/promotion constitute the mechanisms for the variation of MSERRs of the two intra-layer cracks.

The influences of Coulomb tractions on static and dynamic fracture parameters for semi-permeable piezoelectric cracks

In this paper, the influences of the induced Coulomb tractions on the static and dynamic crack-tip fracture parameters of semi-permeable piezoelectric cracks are studied and discussed. The static crack problems are solved by a static dual boundary element method (BEM), while the corresponding crack problems under dynamic impact loading are numerically analyzed by a time-domain BEM considering the inertial effects. In the numerical implementation, a collocation method is applied for the spatial discretization together with a quadrature formula for the temporal discretization. An iterative scheme based on the quasi-Newton method is adopted to solve the corresponding non-linear boundary-value problem resulted from the semi-permeable electric boundary conditions and the induced Coulomb tractions on the crack-faces. The crack-tip facture parameters involving the field intensity factors, the energy release rate and the mechanical strain energy release rate are evaluated by a displacement extrapolation method. Some examples are presented to compare the effects of the Coulomb tractions on the static and dynamic fracture parameters.

Interfacial imperfection coupling model with application to the in-plane fracture problem of a multiferroic composite

A coupling model is proposed to characterize interfacial imperfections by generalizing the spring model and referring to the magneto-electro-elastic constitutive relations. As an application, in-plane fracture analysis is performed on a multiferroic composite, which consists of a ferromagnetic layer, a ferroelectric layer and an imperfect interfacial region. The methods of Fourier integral transform and Green’s function are employed to derive the Cauchy singular integral equations for the cracks. Numerical results of the mechanical strain energy release rate are provided to survey the effects of interfacial imperfections and their couplings. It is found that imperfection couplings affect the fracture behavior remarkably and so it is necessary to consider them in the mechanical problems of this kind of smart structures.

Crack-tip amplification and shielding by micro-cracks in piezoelectric solids – Part I: Static case

Abstract Analysis of the crack-tip amplification and shielding by micro-cracks in an unbounded two-dimensional piezoelectric solid is presented in this paper. A boundary element method (BEM) based on the hypersingular traction boundary integral equations (BIEs) is developed for this purpose. Integrals with hypersingular kernels are analytically transformed into weakly singular and regular integrals. A collocation method is applied for the spatial discretization. Quadratic quarter-point elements are implemented at all crack-tips. The amplification ratios of the field intensity factors and mechanical strain or electrical energy release rate are defined to show the crack-tip amplification and shielding. Numerical results are compared with the analytical results for isotropic materials to verify the present BEM. The influences of various loading conditions, the location and orientation angles of micro-cracks on the amplification ratios are investigated. The contours of the amplification ratios for an arbitrarily located micro-crack are also presented to show the crack-tip amplification and shielding effects.

Crack-tip amplification and shielding by micro-cracks in piezoelectric solids – Part II: Dynamic case

Abstract Analysis of the crack-tip amplification and shielding by micro-cracks in an unbounded two-dimensional piezoelectric solid is presented in this paper. A boundary element method (BEM) based on the hypersingular traction boundary integral equations (BIEs) is developed for this purpose. Integrals with hypersingular kernels are analytically transformed into weakly singular and regular integrals. A collocation method is applied for the spatial discretization. Quadratic quarter-point elements are implemented at all crack-tips. The amplification ratios of the field intensity factors and mechanical strain or electrical energy release rate are defined to show the crack-tip amplification and shielding. Numerical results are compared with the analytical results for isotropic materials to verify the present BEM. The influences of various loading conditions, the location and orientation angles of micro-cracks on the amplification ratios are investigated. The contours of the amplification ratios for an arbitrarily located micro-crack are also presented to show the crack-tip amplification and shielding effects.

Mixed mode fracture of a piezoelectric–piezomagnetic bi-layer structure with two un-coaxial cracks parallel to the interface and each in a layer

The purpose of the present work is to study the mixed mode fracture of a piezoelectric–piezomagnetic composite with two un-coaxial cracks parallel to the interface and each in a layer. Methods of generalized dislocation simulation, Green’s function, Cauchy singular integral equation and Lobatto–Chebyshev collocation are combined together to get the numerical results of mechanical strain energy release rate (MSERR). Three kinds of effects are revealed by parametric studies, i.e., the free-surface effect, the shielding effect and the interference effect, and they are used to interpret the characteristics of COD and MSERR curves. In addition, the effects of shear loading, magnetic loading and electric loading on MSERR are also disclosed, respectively, by varying the corresponding loading factor.

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique, the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface. Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable. The results can be reduced to the well-known solutions for a purely elastic material in the absence of an electric load. Moreover, when the distance between the two crack tips tends to infinity, analytic solutions of a semi-infinite crack in a piezoelectric strip can be obtained. Numerical examples are given to show the influence of the loaded crack length, the height of the strip, the distance between the two crack tips, and the applied mechanical/electric loads on the mechanical strain energy release rate. It is shown that the material is easier to fail when the distance between two crack tips becomes shorter, and the mechanical/electric loads have greater influence on the propagation of the left crack than those of the right one.

Collinear unequal crack series in magnetoelectroelastic materials: Anti‐plane case

AbstractThe purpose of the present work is to study collinear unequal crack series in magnetoelectroelastic materials, which are of more practical significance than the widely studied collinear equal cracks. Fracture analysis is performed by the combined methods of distributed generalized dislocations, Green's function and singular integral equations (SIEs). The theoretical derivation is validated by the degeneration of SIEs in special cases. Parametric studies on the numerical results of mechanical strain energy release rate (MSERR) yield the following conclusions. (a) As two unequal cracks approach each other, interference occurs in crack tip fields, resulting in an obvious increase of MSERR. The ratio between the crack space and the length of the shorter crack is a main factor affecting the interference. (b) Negative or positive electric loading may shield or enhance crack growth, respectively. In the case of positive magnetic permeability, the effect of magnetic loading is analogous to that of electric loading; however, if magnetic permeability is negative, the result is completely opposite. (c) Positive values of magnetic permeability are found to be more reasonable than the negative values.

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.

Anti-plane fracture analysis for the weak-discontinuous interface in a non-homogeneous piezoelectric bi-material structure

The concept of weak discontinuity is extended to functionally graded piezoelectric bi-material interface, and fracture analysis for the weak discontinuous interface is performed by the methods of Fourier integral transform and Cauchy singular integral equation. Numerical results of the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR) are obtained to show the effects of non-homogeneity parameters, geometrical parameters and loads. Parametric studies yield three conclusions: (1) To reduce the weak-discontinuity of the interface is beneficial to resisting interfacial fracture. The effect of the weak-discontinuity of the interface on TERR and MSERR still depends on the strip width. The wider the strip, the more sensitive the TERR and MSERR will be to the weak-discontinuity of the interface. (2) To predict the effect of electric load on crack propagation , MSERR is more appropriate than TERR to be used as a fracture parameter . To predict the effect of mechanical load on crack propagation, both of them could be used as fracture parameters, and MSERR is more conservative. (3) Mechanical load and negative electric displacement load would promote crack propagation, but positive electric displacement load would retard it. For the structure applied by combined mechanical and positive electric displacement loads, crack propagation may be impeded by appropriately selecting the strip width and the ratio of non-homogeneity parameters.

Fracture analysis and improved design for a symmetrically bonded smart structure with linearly non-homogeneous magnetoelectroelastic properties

The mechanical model was established for the anti-plane interfacial fracture problem of a symmetrically bonded smart structure with linearly non-homogeneous magnetoelectroelastic properties. The system of Cauchy singular integral equations for the interfacial crack was derived by Fourier integral transform. The numerical solutions of the Cauchy singular integral equations were obtained by the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta. The mechanical strain energy release rate and the total energy release rate were chosen as fracture parameters to discuss the effect of the non-homogeneity parameter on the extension force of the crack. A conclusion was drawn that, to reduce the weak-discontinuity of the interface in the magnetoelectroelastic structure would be beneficial to decrease the extension force of the interfacial crack. Based on this conclusion, a new improved design was suggested for the symmetrically bonded linearly non-homogeneous magnetoelectroelastic composite. The enhancement of the capability of the improved structure to resist interfacial fracture was validated by comparison between the improved and unimproved structures for their fracture responses.

Fracture Analysis of Magnetoelectroelastic Composite Materials

This paper presents a crack analysis of linear magnetoelectroelastic materials subjected to static loading conditions. To this end, an efficient boundary element method (BEM) is developed. Unlike many previous investigations published in literature, two-dimensional (2-D) linear magnetoelectroelastic materials possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered in this paper. A combination of the displacement BEM and the traction BEM is used in the present formulation. The displacement BEM is applied for the external boundary of the cracked solid, while the traction BEM is used for the crack-faces. A regularization technique is implemented to compute the strongly singular and hypersingular boundary integrals in the BEM. The electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF), the stress intensity factors (SIF), the mechanical strain energy release rate (MSERR) and the total energy release rate (TERR) are evaluated directly from the computed nodal values at discontinuous quarter point elements placed next to the crack tip. The accuracy of the BEM is verified by analytical solutions known in literature. Results are presented for a branched crack in a bending specimen subjected to combined magnetic-electric-mechanical loading conditions.