Crack propagating in a functionally graded strip under the plane loading

In the present paper, a finite crack with constant length (Yoffe type crack) propagating in the functionally graded strip under the plane loading is investigated by means of the Schmidt method. By using the Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of the material properties, the thickness of the functionally graded strip, and speed of the crack propagating upon the dynamic fracture behavior.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

Dynamic behavior of a crack in a functionally graded piezoelectric strip bonded to two dissimilar half piezoelectric material planes

Summary. The dynamic behavior of a crack in a functionally graded piezoelectric material (FGPM) strip bonded to two half dissimilar piezoelectric material planes subjected to combined harmonic anti-plane shear wave and in-plane electrical loading was studied under the limited permeable and permeable electric boundary conditions. It was assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric layer vary continuously along the thickness of the strip. By using the Fourier transform, the problem can be solved with a set of dual integral equations in which the unknown variables are the jumps of the displacements and the electric potentials across the crack surfaces. In solving the dual integral equations, the jumps of the displacements and the electric potentials across the crack surfaces were expanded in a series of Jacobi polynomials. Numerical results illustrate the effects of the gradient parameter of FGPM, electric loading, wave number, thickness of FGPM strip and electric boundary conditions on the dynamic stress intensity factors (SIFs).     

在反平面变形下功能梯度压电板条的应力强度因子分析

利用Fourier变换技术将混合边值问题化为对偶积分方程,求解对偶积分方程得到应力强度因子以及电位移强度因子的表达式。最后通过数值计算讨论了材料参数、载荷条件以及裂纹的几何参数等对功能梯度压电材料中裂纹尖端应力强度因子的影响。     

Investigation of the behavior of a mode-I crack in functionally graded materials by non-local theory

In this paper, the behavior of a mode-I crack in functionally graded materials is investigated by means of the non-local theory. The traditional concepts of the non-local theory are firstly extended to solve the mode-I crack fracture problem in functionally graded materials, in which the shear modulus varies exponentially with coordinate parallel to the crack. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are jumps of displacements across crack surfaces, not the dislocation density functions or the analysis functions. To solve the dual integral equations, the jumps of displacements across crack surfaces are directly expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The non-local elastic solutions yield a finite stress at crack tips, thus allowing us to use the maximum stress as a fracture criterion. Numerical examples are provided to show the effects of the crack length, the parameter describing functionally graded materials, the lattice parameter of materials and the material constants upon the stress fields near crack tips.     

Transient internal crack problem for a nonhomogeneous orthotropic strip (Mode I)

Supposing the material properties to be one-dimensionally dependent, this paper studied the transient internal crack problem for a functionally graded orthotropic strip. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. Numerical results show the effects of the material parameter βh, the crack configuration and the isotropic or orthotropic property on the dynamic stress intensity factors.     

A moving interface crack between two dissimilar functionally graded strips under plane deformation with integral equation methods

In this paper a finite interface crack with constant length (Yoffe-type crack) propagating along the interface between two dissimilar functionally graded strips with spatially varying elastic properties under in-plane loading is studied. By utilizing the Fourier transformation technique, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters, the graded parameter, the strip thickness and crack speed on the stress intensity factors and the probable kink angle are investigated. The numerical results show that the graded parameters, the thicknesses of the functionally graded strips and the crack speed have significant effects on the dynamic fracture behavior of functionally graded material structures.     

Non-Local Theory Solution for an Anti-Plane Shear Permeable Crack in Functionally Graded Piezoelectric Materials

In this paper, the non-local theory solution of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading is obtained for the permeable electric boundary conditions, in which the material properties vary exponentially with coordinate parallel to the crack. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variable is the jump of displacement across the crack surfaces, not the dislocation density function. To solve the dual integral equations, the jump of the displacement across the crack surfaces is directly expanded in a series of Jacobi polynomials. From the solution of the present paper, it is found that no stress and electric displacement singularities are present near the crack tips. The stress fields are finite near the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses and the electric displacements at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively. On the other hand, the angular variations of the strain energy density function are examined to associate their stationary value with locations of possible fracture initiation.     

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

The dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.     

The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Thermal fracture analysis of a functionally graded orthotropic strip with a crack

This paper is concerned with the thermal fracture problem of a functionally graded orthotropic strip, where the crack is situated parallel to the free edges. All the material properties are assumed to be dependent only on the coordinate y (perpendicular to the crack surfaces). By using Fourier transform, the thermoelastic problem is reduced to those that involve a system of singular integral equations. Numerical results are presented to show the effects of the crack position and the material distribution on the thermal stress intensity factors.     

On the dynamic propagation of an anti-plane shear crack in a functionally graded piezoelectric strip

Summary. In this paper, a finite crack propagating at constant speed in a functionally graded piezoelectric material (FGPM) is studied. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential gradients along the thickness of the strip, and that the strip is under anti-plane shear mechanical and in-plane electrical loads. The analysis is conducted on the electrically unified (natural) crack boundary condition, which is related to the ellipsoidal crack parameters. By using the Fourier transform, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, crack propagation speed, electric field, FGPM gradation, crack length, and electromechanical coupling coefficient. It reveals that there are considerable differences between traditional electric crack models and the present unified crack model.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

Two parallel penny-shaped or annular cracks in a functionally graded piezoelectric strip under electric loading

In this paper, the mixed-mode fracture problem of a functionally graded piezoelectric material strip with two penny-shaped or annular cracks is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under electric loading. The problem is formulated in terms of a system of singular integral equations, which are solved numerically. Numerical calculations are carried out, and the stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

Stresses in an orthotropic strip containing A Griffith Crack

The problem of determining the stress and displacement fields in an orthotropic elastic strip containing a Griffith crack situated symmetrically and oriented in a direction normal to the edges of the strip is considered. A general solution in terms of two potential functions is presented. The mixed boundary conditions lead to dual integral equations, which are reduced to Fredholm integral equation of second kind and are solved by the use of Gaussian quadrature formula. Numerical solutions for a fiber-reinforced composite material and some isotropic materials are carried out and the effect of orthotropy on various quantities of physical interest, in fracture mechanics, is discussed.     

A Moving Mode-III Crack in a Functionally Graded Piezoelectric Strip

In this paper, the problem of a functionally graded piezoelectric strip with a constant-velocity Yoffe-type moving crack is considered. By using the Fourier transforms, the problem is first reduced to dual integral equations and then into Fredholm integral equations of the second kind. The electroelastic field near the crack tip is obtained for electrical impermeable boundary conditions and electrical permeable boundary conditions, respectively. The results obtained show that the gradient of the material properties can increase or decrease the magnitudes of the stress intensity factors, and the velocity can disturb the stress distribution near the crack tip.