Investigation of Antiplane Shear Behavior of Two Collinear Permeable Cracks in a Piezoelectric Material by Using the Nonlocal Theory

Abstract

Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, May 17, 2001; final revision, Oct. 19, 2001. Associate Editor: A. K. Mal. Discussion on the paper should be addressed to the Editor, Prof. Lewis T. Wheeler, Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. In the theoretical studies of crack problems, several different electric boundary conditions at the crack surfaces in piezoelectric materials have been proposed by numerous researchers (1234). However, these solutions contain stress and electric displacement singularity. This is not reasonable according to the physical nature. To overcome the stress singularity in the classical elastic theory, Eringen 5 used the nonlocal theory to study the state of stress near the tip of a sharp line crack in an elastic plate subjected to antiplane shear. The solution did not contain any stress singularity. Recently, the same problems have been resolved in Zhou’s papers (6) by using the Schmidt method. In this paper, the behavior of two collinear symmetric cracks subjected to the antiplane shear loading in the piezoelectric materials is investigated by using the Schmidt method and the nonlocal theory for permeable crack surface conditions. The traditional concept of linear elastic fracture mechanics and the nonlocal theory are extended to include the piezoelectric effects. As expected, the solution in this paper does not contain the stress and electric displacement singularity at the crack tip. As discussed in 7, for the antiplane shear problem, the basic equations of linear, nonlocal piezoelectric materials can be written as follows: (1)∂τxz∂x+∂τyz∂y=0(2)∂Dx∂x+∂Dy∂y=0(3)τkzX=∫Vα|X′−X|[c44w,kX′+e15ϕ,kX′]dVX′,k=x,y(4)DkX=∫Vα|X′−X|[e15w,kX′−ε11ϕ,kX′]dVX′,k=x,ywhere the only difference from classical elastic theory is that in the stress and the electric displacement constitutive Eqs. (34), the stress τzkX and the electric displacement DkX at a point X depends on w,kX and ϕ,kX, at all points of the body. w and ϕ are the mechanical displacement and electric potential. c44,e15,ε11 are the shear modulus, piezoelectric coefficient, and dielectric parameter, respectively. α|X′−X| is the influence function. As discussed in the papers (56), α|X′−X| can be assumed as follows: (5)α|X′−X|=1π β/a2 exp[−β/a2X′−XX′−X]where β is a constant and a is the lattice parameter. Consider an infinite piezoelectric plane containing two collinear symmetric permeable cracks of length 1-b along the x-axis. 2b is the distance between two cracks. The boundary conditions of the present problem are (6)τyz1x,0+=τyz2x,0−=−τ0,b⩽|x|⩽1(7)Dy1x,0+=Dy2x,0−,ϕ1x,0+=ϕ2x,0−,|x|⩽∞(8)w1x,0+=w2x,0−=0,0<|x|<b,1<|x|(9)wkx,y=ϕkx,y=0,forx2+y21/2→∞,k=1,2.Note that all quantities with superscript kk=1,2 refer to the upper half-plane and the lower half-plane. As discussed in 7, the general solutions of Eqs. (12) satisfying (9) are, respectively, w1x,y=2π ∫0∞A1se−sy cosxsds,(10)ϕ1x,y−e15ε11 w1x,y=2π ∫0∞B1se−sy cosxsdsw2x,y=2π ∫0∞A2sesy cosxsds,(11)ϕ2x,y−e15ε11 w2x,y=2π ∫0∞B2sesy cosxsdswhere A1s,B1s,A2s,B2s are to be determined from the boundary conditions. For solving the problem, the gap functions of the crack surface displacements and the electric potentials are defined as follows: (12)fwx=w1x,0+−w2x,0−(13)fϕx=ϕ1x,0+−ϕ2x,0−.Substituting Eqs. (1011) into Eqs. (34), (1213), applying the Fourier transform and the boundary conditions (678), it can be obtained as (14)1π ∫0∞sf¯wserfcεscossxds=τ0c44,b⩽|x|⩽1(15)1π ∫0∞f¯wscossxds=0,0<|x|<b,1<|x|<∞and f¯ϕs=0,fϕx=0, for all s and x. ε=a/2β,erfcz=1−Φz,Φz=2/π∫0z exp−t2dt.As discussed in 6, the Schmidt method (8) can be used to solve the triple-integral Eqs. (1415). The gap functions of the crack surface displacement can be represented by the following series: (16)fwx=∑n=0∞anPn1/2,1/2x−1+b21−b21−x−1+b221−b221/2,forb⩽x⩽1,y=0(17)fwx=0,for0<x<b,1<x,y=0where an is unknown coefficients to be determined and Pn1/2,1/2x is a Jacobi polynomial. The Fourier transformation of Eq. (16) is (18)f¯ws=∑n=0∞anQnGns 1s Jn+1s 1−b2Qn=2π Γn+1+12n!,(19)Gns=−1n/2 coss 1+b2,n=0,2,4,6,…−1n+1/2 sins 1+b2,n=1,3,5,7,…where Γx and Jnx are the Gamma and Bessel functions, respectively. By substituting Eq. (18) into Eqs. (1415), respectively, Eq. (15) can be automatically satisfied. Then the remaining Eq. (14) reduces to the form (20)∑n=0∞anQn∫0∞ erfcεsGnsJn+1s 1−b2cossxds=πc44 τ0.Equations (20) can now be solved for the coefficients an by the Schmidt method (8). τyz and Dy along the crack line can be expressed as (21)τyz=τyz1x,0=−c44π ∑n=0∞anQn∫0∞ erfcεsGnsJn+1s 1−b2cosxsds(22)Dy=Dy1x,0=−e15π ∑n=0∞anQn∫0∞ erfcεsGnsJn+1s 1−b2cosxsds=e15c44 τyz1x,0.So long as ε≠0, the semi-infinite integration and the series in Eqs. (20) is convergent for any variable x. Equations (21) and (22) give finite stress and electric displacement all along y=0, so there are no stress and electric displacement singularity at the crack tips. The results are plotted in Figs. 1 and 2. From the results, the dimensionless stress field is found to be independent of the material parameters. They just depend on the length of the crack and the lattice parameter. However, the electric displacement field is found to depend on the stress loads, the shear modulus, the length of the crack, the lattice parameter and piezoelectric coefficient except the dielectric parameter ε11. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions.