Fracture analysis of a functionally graded interfacial zone between two dissimilar homogeneous materials

Transcription

1 540 Siene in China Series G: Physis, Mehanis & Astronomy 006 Vol.49 No DOI: 0.007/s Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials CHENG Zhanqi, & ZHONG Zheng. Shool of Aerospae Engineering and Applied mehanis, Tongji University, Shanghai 0009, China;. College of Civil Engineering, Zhengzhou University, Zhengzhou 45000, China Correspondene should be addressed to Zhong Zheng (zhongk@mail.tongji.edu.n) Reeived Deember 3, 005; aepted May 3, 006 Abstrat In this paper the plane elastiity problem for a funtionally graded interfaial zone ontaining a rak between two dissimilar homogeneous materials has been onsidered. It is assumed that in the interfaial zone the reiproal of the shear modulus is a linear funtion of the oordinate, while Possion s ratio keeps onstant. By utilizing the Fourier transformation tehnique and the transfer matrix method, the mixed boundary problem is redued to a system of singular integral equations that are solved numerially. The influenes of the geometri parameters and the graded parameter on the stress intensity fators are investigated. The numerial results show that the graded parameters, the thikness of interfaial zone, the rak size and loation have signifiant effets on the stress intensity fators. Keywords: funtionally graded material, interfaial zone, frature, stress intensity fator. In designing omponents involving funtionally graded materials (FGMs) the frature behavior of FGMs is an important aspet of onsideration. The frature of FGM may exhibit omplex behavior indued from the nonhomogeneous properties of FGMs. Due to the lak of symmetry in the material properties the frature mode of an embedded rak in FGMs is inherently mixed in many ases. Perhaps the most important task in this area is to obtain solutions for stress intensity fators of FGMs similar to those available for homogeneous materials. Signifiant efforts have been made in the study of the frature behavior of FGMs over the past two deades. For example, Delale and Erdogan [] used the exponential funtion to model the elasti modulus variation and found that the asymptoti rak tip stress field possessed the same inverse-square-root-singularity as that for homogeneous materials. Eishen [] and Jin and Noda [3] respetively affirmed that the leading term in the rak tip stress field is of the inverse-square-root-singularity for any form of elasti modulus variation. Previous studies (Delale and Erdogan [4], Jin and

2 Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials 54 Batra [5], Erdogan et al. [6] ) were onerned with different rak problems of FGMs. Their studies indiated that the elasti modulus has a signifiant effet on the rak-tip stress field, while the effet of Possion s ratio an be ignored. Some authors used other funtions, suh as power funtion, to model the elasti modulus variation (Craster and Atkinson [7], Gerasoulis and Srivastav [8] ). There are also some approximate methods to study the frature behavior of FGMs with arbitrarily varying properties, for example, a pieewise multi-layered model with onstant shear modulus in eah sub-layer (Wang et al. [9], Itou [0] ) and a multi-layered linear model with the elasti modulus varying linearly in eah sub-layer (Huang et al. [], Wang et al. [] ). The two multi-layered models then were used to study a series of frature problems. In reent years, more and more studies are onerned about the rak problems for layered strutures. Some important ontributions inlude those of Delale and Erdogan [3], Erdogan and Ozturk [4], Chen and Erdogan [5], Shbeeb and Binienda [6], Sheeb et al. [7], Choi [8,9], Choi et al. [0] and Ueda [], among others. These literatures onfirmed that the FGMs an be used as interfaial layer or oating to improve the interfaial bonding strength and toughness. In this paper, a funtionally graded interfaial zone between two dissimilar homogeneous materials is studied by assuming that the reiproal of the shear modulus is a linear funtion of the oordinate, while Possion s ratio keeps onstant. By means of the Fourier transformation tehnique and the transform matrix method, the mixed boundary value problem is redued to a system of singular integral equations. The stress intensity fators (SIFs) are alulated and the effets of the FGMs graded properties, the size and loation of the rak are studied. Formulation of the problem Consider a funtionally graded interfaial zone of thikness h sandwihed between two bonded dissimilar elasti half-planes whose shear modulus and Possion s ratio are () () (3) (3) ( μ, ν ) and ( μ, ν ), respetively. The funtionally graded interfaial zone ontains a through rak of length that is parallel to the interfaes, as shown in Fig.. Quantities defined in the two half planes will subsequently be designed by supersripts () and (3), while quantities for the interlayer are denoted without supersripts. With referene to a set of Cartesian oordinates ( x, y ), as shown in Fig., the equilibrium equations an be expressed as σ σ xx xy σxy σ + = 0, + = 0, () y y where ( σ xx, σ xy, σ ) are stress omponents, and the strain-displaement relationships are u u x y u u x y ε xx =, ε =, γ xy = +, () x y y where ( ε xx, ε, ε xy ) and ( u x, u y ) are strain omponents and displaement omponents,

3 54 Siene in China Series G: Physis, Mehanis & Astronomy respetively. The ompatibility equation is Fig.. A rak in a funtionally graded interfaial zone. ε ε xx γ xy + =. (3) y x x y Under the plane strain state, the onstitutive equations an be written as ( ν) ν εxx = ( σxx σ ) μ ν ( ν) ν ε = ( σ σxx ), (4) μ ν γ xy = σxy μ where μ and ν are the shear modulus and Possion s ratio, respetively. We suppose that the FGMs in the interfaial zone possess the following nonhomogeneous properties: the shear modulus μ and Possion s ratio ν are both the funtions of the oordinate y. To satisfy the equilibrium equation (), the Airy stress funtion is introdued as F F F σxx =, σ,. = σ xy = (5) y y Substituting eqs. (4) and (5) into eq. (3) results in the following governing equation of the problem: 3 3 ( ν) 4 μ ( ν) ν F μ ( ν) ν F F + 3 μ μ μ yx μ μ y (6) [ μμ ( μ ) ]( ν ) ν μ ν F ν [ μμ ( μ ) ] μ ν F + 0, = μ μ μ y μ μ where μ = d μ/d y, μ = d μ/d y, ν = dν d y, ν = d ν dy. The problem of a raked struture subjeted to remote loadings an be regarded as the superposition of the following two problems: () the struture free of rak is subjeted to remote loadings, induing shear and normal trations σ = σ ( x ) and xy

4 Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials 543 σ = σ ( x ) at y = h ; () the struture without remote loadings is loaded at the rak fae by σ = σ ( x ) and σ = σ ( x ). Sine the problem () ontributes nothing to xy the singular fields at the rak tip, we will only onsider the problem () and treat σ ( x) and σ ( x) as known funtions. Then, on the rak fae, we have σ ( x, h ) = σ ( x ), σ xy ( x, h ) = σ ( x ) ( x ). (7) And other boundary and ontinuity onditions of the rak problem are σ (3) ( x,0) = σ ( x,0), σ (3) ( x,0) = σ ( x,0), (8) xy u (3) ( x,0) = u ( x,0), u (3) ( x,0) = u ( x,0), (9) x x () σ ( x, h) = σ ( x, h ), () x y xy y xy () xy σ ( x, h) = σ ( x, h ), (0) u ( x,0) = u ( x,0), u ( x,0) = u ( x,0), () + x x () y σ ( x, h ) = σ (, h ), σxy ( x, h ) = σxy ( x, h ) ( x > ), () + x y + + ux( x, h ) = u ( x, h ), uy( x, h ) = uy( x, h ) ( x > ). (3) Solution The governing equation (6) is a partial differential equation with spatial varied oeffiients, and it an be redued to a partial differential equation with onstant oeffiients only if the shear modulus μ is an exponential funtion of the oordinate y. In most previous investigations the shear modulus was assumed to be an exponential funtion, while in the present paper we suppose that the FGMs possess the following graded manner: (3) μ μ ( y) =, (4) + py (3) () where p= ( μ / μ )/ h is a graded parameter and Possion s ratio ν keeps onstant. Substituting eq. (4) into eq. (6) yields p F + ( F) = 0, + py y (5) where = + is the two dimensional Laplae operator. y By applying the Fourier transformation with respet to x, eq. (5) an be redued to 4 3 d F ( y) p d F ( y) d F ( y) p d F ( y) 4 + s s + s F ( y) = 0, (6) 4 3 dy + py dy dy + py dy where + isx Fsy (, ) = Fxye (, ) dx and ~ indiates the Fourier transform. Introduing s( + py) ξ =, eq. (6) an be fur- p ther redued to is the Fourier transform of the Airy stress funtion,

5 544 Siene in China Series G: Physis, Mehanis & Astronomy 4 3 d F ( ξ) d F( ξ) d F( ξ) d F( ξ) + + F ( ξ ) = 0. (7) 4 3 dξ ξ dξ dξ ξ dξ The general solution of eq. (7) is obtained as F = Bφ + B φ + Bφ + B φ, (8) where ξ ξ ξ ξ ξ ξ e e 3 e Ei e 4 e Ei e φ =, φ =, φ = ( ξ) ln ξ, φ = ( ξ) lnξ. (9) e x e Ei( x ) is the exponential integral funtion, Ei( x) = dt = dt x t ; B,, t and are the unknown oeffiients to be determined. B 4 t t B B3 From eqs. (), (4) and (5), the Fourier transforms of the displaements and stress omponents an be obtained as the following matrix form: where S = ux uy σ σxy, with its elements being given as { S} = [ T]{ B}, (0) { },,, { B} = [ B, B, B, B ] and [ ] ν φl l T T T 3 4 d ( y) νs = + φl( y), isμ( y) dy iμ( y) 3 ( ) l ( ) p l ( ) l l 3 T is a 4 4 matrix ν d φ ( y) ν d φ ( y) ν d φ ( y) ν p = φ l ( y), s μ( y) dy s dy μ( y) dy 3l = s φl, T4l dφl = is ( l =,,3, 4 ). dy As to the homogeneous materials, both the elasti modulus and Possion s ratio are onstant. Aordingly, from eq. (6), the governing equations for the elasti homogeneous half-planes, y < 0 and y > 0, an be obtained as k 4 ( F ) ( x, y) = 0 ( k =,3). () The orresponding Fourier transforms of Airy stress funtions an be obtained as () F (, s y) = ( A + A y) e s y ( h< y < ), () (3) (, ) = ( + ) s y ( y 0) F s y C C y e < <, (3) where A, A, C and C are the unknown oeffiients to be determined from boundary onditions (8) (). Consequently, the Fourier transforms of the displaement and stress omponents in the elasti homogeneous materials an also be obtained and written as a matrix form: () () S H = [ A ] ( < y < ), (4) (3) (3) S H = [ C ] ( < y < 0), (5) () () () () () (3) (3) (3) (3) (3) where S = u x u y σ σ xy, S = u x u y σ σ xy, and

6 Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials 545 s s is s s () iμ μ H = e ( ν ) s ) ( ys + ) ( ν y s ) is( + y s ) s y iμ s μ s s is s s (3) iμ3 μ3 H = e ( ν ) s ) ( ys ) ( ν + y s ) is( y s ) s y iμ3 s μ3 A C [ A] = A, [ C] = C. From eq. (0), the Fourier transforms of the displaement and stress omponents of the two strips, 0 < y< h and h < y < h, an be obtained as the following matrix form: { S} = [ T( y) ] { B} = [ T( y), T( y), T3( y), T4( y) ]{ B} { S} [ T y ] { B} [ T y T y T y T y ] B 3 4 s y s y ( 0 < y< h ), (6) = ( ) = ( ), ( ), ( ), ( ) { } h < y < h), (7) where { } and { } orrespond to Strip ( 0 y h ( < < ) and Strip ( h < y <h), respetively. From the ontinuity onditions, (8) (), the following relations an be obtained as (3) () [ T(0) ]{ B} = H (0) [ C], [ T( h) ]{ B} H ( h) [ A] =. (8) Using eqs. () and (3), in the transformed domain, we have S S = Δ S, (9) where Δ S = Δu x, Δu y,0,0, Δu x and { } { } { } Δu y are the Fourier transform of the jumps of displaements aross the rak fae. Using eqs. (0), (8) and (9), the equations of unknown oeffiients { B } with respet to { Δ S} an be obtained as { B} [ E] { S} = Δ, (30) { B} [ E] { S} = Δ, (3) where [ E] = [ V(0) ][ D ][ Q] [ V( h) ][ T( h )], [ E] = [ V(0) ][ D ][ Q] [ V( h) ][ T( h )] () (3) () (3) +I, [ V ] [ T ] { H } { H }} [ V h ] { H h } { H h }} [ T h ] (0) (0) (0) (0), () () () (), = = { } [ D] [ V(0) ][ V( h) ][ D], [ D] [ D3] = [ D3], [ D] = [ D4] [ D4 ], [ D3 ] and [ D4 ] = ,, Q = = 0 0 0,

7 546 Siene in China Series G: Physis, Mehanis & Astronomy By substituting eq. (30) into eq. (6) and onduting the inverse Fourier transform, the stress and displaements in Strip ( 0 < y< h ) an be obtained as { ux uy σ σxy } = [ T( y) ][ E] { S} exp( isx)ds π Δ. (3) From eq. (7), we have ϖ(, s h) { ux, uy} exp( isx)d s { σ(), x σ() x } π Δ Δ = ( x ), (33) 4 3 where ϖ (, sh) [ D][ Th ( )][ E] [ D] =. Considering the ontinuity ondition of displaements, eq. (), we have { Δ x Δ y} u, u exp( isx)ds = 0, ( x > ). (34) Eqs. (33) and (34) are the dual integral equations of the mixed boundary value problem, whih an be redued to a system of singular integral equations by introduing the following disloation density funtions: ( ) η x = ( Δu ) x, ( ) η x = ( Δu ) y ( x ). (35) Applying Fourier transform with respet to x on both sides of eq. (35), we have Δ u = ( is) η ( α)exp( isα)dα, x Substituting eq. (36) into eqs. (33) and (34), we obtain πi Δ u = ( is) η ( α)exp( isα)dα. (36) + s ϖ(, sh) { η( α), η( α) } exp [ isx ( α)d ] αd s= { σ(), x σ() x}, (37) y { η( α), η( α) } du = 0. (38) Through detailed asymptoti analysis of the exponential integral funtion, it is easy to prove where β = μ( h ) ( ν ) [ ] lim s ϖ( sh, ) = β I, (39) s and [ ] possesses the following property: I is a j + l ϖ s identity matrix. The matrix of ϖ (, s h ) ϖ( s, h ) = ( ) (, h ) ( jl=,,). (40) Using the property of analyti funtion i sgn( s)exp( is( α x))ds =. (4) α x Eq. (37) an be transformed into a set of singular integral equations, π ( α x) 0 Z( α, x) Z ( α, x) η ( α) dα 0 ( α x) + Z( α, x) Z( α, x) η ( α) σ ( x) = β ( x σ ( x) ), (4)

8 Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials 547 where Z jl ( α, x) = ( is) ϖ ( )os( ( ) d 0 jl s s α x ) β s (j l ), s ϖ jl () s Z jl ( α, x) = ( + )sin ( s( α x) ) ds (j = l ). 0 β The methods of Erdogan [] an be used to solve eqs. (4) and (38). Noting that the disloation density funtions, η ( x) and η ( x), have the square-root singularity at the rak tips, we an express them as f( α) η ( α) = α ( ), η( α) = f ( α) ( α ). (43) By using Chebyshev polynomials to expand the disloation density funtions, η ( x) and η ( x) an be expressed as j η j( α) = GT i i( α ) (j =,), (44) ( α ) i= 0 j where T i are Chebyshev polynomials of the first kind and G i are the unknown oeffiients to be determined. Substituting eq. (43) into eqs. (4) and (38), the following algebrai equations are obtained as M f j( αn) + fl( αn) Zl( αn, xr) = σ j( xr) M n= αn xr l= β ( j =, ), (45) M f j( αn) = 0 ( j =,), (46) M n= π πr where α n = os (n ), xr = os, r =,, M, and M is the total M M number of the disrete point of the unknown funtions f j ( α ) in the interval of (, ). Through solving the algebrai equations, f ( α ) an be derived, and the value of f j ( α ) in other points an be obtained by interpolation. The stress intensity fators (SIFs) at the rak tip are defined as ± ± Ι = lim ± ( x, h ), KΙΙ x σ xy x ± x ± K x σ Using the following property of Chebyshev polynomials, j n = lim ± ( x, h ). (47) ( )d π Tn u u x x x = x ( u x) u x x x ( n= 0,,, x > ), (48) we obtain ± Ι ± ΙΙ n K = β f ( ± ), K = β f ( ± ). (49)

9 548 Siene in China Series G: Physis, Mehanis & Astronomy 3 Examples and disussion As an example, the material properties are given as (3) 0 () (3) μ(0) = μ = 0 N / m, ν = ν = ν = 0.3, (50) where N is the fore in Newton and m is the length in meter. Firstly, we onsider a FGMs interfaial layer with a rak subjeted to a uniform normal load σ ( ) = σ or uniform shear load σ ( ) = τ. The geometry of the problem x 0 x 0 being onsidered is shown in Fig., where the rak is loated at the midline of the interfaial zone, i.e. h = h/. To ensure enough auray of the omputational results and to avoid too muh CPU time, proper value of M should be arefully hosen for a partiular problem. In the present ase we hoose M = 50. Fig. displays the variations of the normalized stress intensity fators KΙ / σ 0 and KΙΙ / σ 0 with the normalized graded parameter ph for different rak sizes / h Fig.. Influene of the graded parameter ph on the normalized SIFs of a midline rak in a FGM interfaial zone for different rak sizes under normal loading. (a) Mode I; (b) mode II.

10 Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials 549 in ase of h = h/ under uniform normal loading σ ( x) = σ 0. The figure shows essentially the effet of the graded parameter of the funtionally graded material on the SIFs. It is observed that both KΙ / σ 0 and KΙΙ / σ 0 inrease with the inrease of the graded parameter ph and the material graded parameter has an important influene on the stress intensity fators. Fig. also shows that both KΙ / σ 0 and KΙΙ / σ 0 inrease with the inrease of the ratio / h. This means that for a definite rak size an inrease of the thikness of interfaial zone will redue the stress intensity fators. Fig. 3 shows the effet of normalized graded parameter ph on the normalized stress intensity fators KΙ / τ 0 and KΙΙ / τ 0 for the ase of uniform shear loading σ ( x) = τ 0 for the same geometri and material onditions as those for Fig.. Similarly as that for normal loading, the normalized stress intensity fators, both KΙ / τ 0 and KΙΙ / τ 0, inrease with the inrease of the graded parameter ph and the normalized rak size / h. Fig. 3. Influene of the graded parameter ph on the normalized SIFs of a midline rak in a FGM interfaial zone for different rak sizes under shear loading. (a) Mode I; (b) mode II.

11 550 Siene in China Series G: Physis, Mehanis & Astronomy Seondly, in order to investigate the influene of the loation of the rak on the stress intensity fators, the normalized stress intensity fators of the funtionally graded material interfaial layer for some seleted values h / h under uniform normal loading or uniform shear loading with / h= are plotted in Figs. 4 and 5, respetively. As an be seen from Figs. 4 and 5, if the rak loation approahes to the less stiff side, the normalized stress intensity fators inrease, while they tend to derease as the rak loation approahes to stiffer side. 4 Conlusion remarks The problem of a finite rak in a funtionally graded interfaial zone between two homogeneous elasti half-planes is studied. It is assumed that the reiproal of the shear modulus is a linear funtion of the oordinate, while Possion s ratio keeps onstant. The obtained results show that the graded parameter, the thikness of the FGMs interfaial layer, the length and the loation of the rak have important effets on the frature behavior of FGMs. Fig. 4. Influene of the graded parameter ph on the normalized SIFs of a rak in a FGM interfaial zone for different rak loations under normal loading. (a) Mode I; (b) mode II.

12 Frature analysis of a funtionally graded interfaial zone between two dissimilar homogeneous materials 55 Fig. 5. Influene of the graded parameter ph on the normalized SIFs of a rak in a FGM interfaial zone for different rak loations under shear loading. (a) Mode I; (b) mode II. Aknowledgements This work was supported by the National Natural Siene Foundation of China (Grant Nos and 0509). Referenes Delale F, Erdogan F. The rak problem for a nonhomogeneous plane. J Appl Meh-Trans ASME, 983, 50: Eishen J W. Frature of nonhomogeneous material. Int J Frat, 987, 34: 3 3 Jin Z H, Noda N. Crak tip singular fields in nonhomogeneous materials. J Appl Meh-Trans ASME, 994, 6: Delale F, Erdogan F. On the mehanial modeling of the interfaial region in bonded half-planes. J Appl Meh-Trans ASME, 988, 55: Jin Z H, Batra R C. Some basi frature mehanis onepts in funtionally graded materials. J Meh Phys Solids, 996, 44: 35[DOI] 6 Erdogan F, Kaya A C, Josehp P F. The rak problem in bonded nonhomogeneous materials. J Appl Meh-Trans ASME, 99, 58: Craster R V, Atkinson C. Mixed boundary value problems in non-homogeneous elasti materials. Q J Math,

13 55 Siene in China Series G: Physis, Mehanis & Astronomy 994, 47: Gerasoulis A, Srivastav R P. A Griffith rak problem for a nonhomogeneous medium. Int J Eng Si, 980, 8: 39 47[DOI] 9 Wang B L, Han J C, Du S Y. Craks problem for nonhomogeneous omposite material subjeted to dynami loading. Int J Solids Strut, 000, 37: 5 74[DOI] 0 Itou S. Stress intensity fators around a rak in a nonhomogeneous interfaial layers between two dissimilar elasti half-plane. Int J Frat, 00, 0: 3 35[DOI] Huang G Y, Wang Y S, Gross D. Frature analysis of a funtionally graded oatings: Plane deformation. Eur J Meh A-Solids, 003, : Wang Y S, Huang G Y, Gross D. On the mehanial modeling of funtionally graded interfaial zone with a Griffith rak: Plane deformation. Int J Frat, 004, 5: 89 05[DOI] 3 Delale F, Erdogan F. Interfae rak in a nonhomogeneous elasti medium. Int J Eng Si, 988, 6(6): [DOI] 4 Erdogan F, Ozturk M. Diffusion problems in bonded nonhomogeneous materials with an interfae ut. Int J Eng Si, 99, 30(0): [DOI] 5 Chen Y F, Erdogan F. The interfae rak problem for a nonhomogeneous oating to a homogeneous subtrate. J Meh Phys Solids, 996, 44: [DOI] 6 Shbeeb N I, Binienda W K. Analysis of an interfae rak for a funtionally graded strip sandwihed between two homogeneous layers of finite thikness. Eng Frat Meh, 999, 64: [DOI] 7 Shbeeb N I, Binienda W K, Kreider K. Analysis of the driving fore for a generally oriented rak in a fun- tionally graded strip sandwihed between two homogeneous half planes. Int J Frat, 000, 04: 3 50[DOI] 8 Choi H J. An analysis of raking in a layered medium with a funtionally graded nonhomogeneous interfae. J Appl Meh-Trans ASME, 996, 63: Choi H J. Bonded dissimilar strips with a rak perpendiular to the funtionally graded interfae. Int J Solids Strut, 996, 33: 40 47[DOI] 0 Choi H J, Lee K Y, Jin T E. Collinear raks in a layered half-plane with a graded nonhomogeneous interfaial zone-part (I): Mehanial response. Int J Frat, 998, 94: 03 [DOI] Ueda S. The surfae rak problem for a layered plate with a funtionally graded nonhomogeneous interfae. Int J Frat, 00, 0: 89 04[DOI] Erdogan F. Complex funtion tehnique. In: Continuum Physis. New York: Aademi Press, 975