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Stress redistribution due to cracking in periodically layered composites
Engineering Fracture Mechanics 93 (2012) 225–238
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Stress redistribution due to cracking in periodically layered composites Michael Ryvkin ⇑, Jacob Aboudi School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
a r t i c l e
i n f o
Article history: Received 7 March 2012 Received in revised form 1 July 2012 Accepted 10 July 2012
Keywords: Localized damage Cracked layered composite Representative cell method High-fidelity generalized method of cells Higher-order theory
a b s t r a c t The stress redistribution caused by a brittle fracture of one or several layers in a bi-material periodically layered composite is considered. It is assumed that one of the composite constituents possess low crack resistance and is subject to cracking. In the case of a uniaxial tension parallel to the layering direction and in the presence of weak interfaces, the fracture pattern may be quite complicated including branching cracks. In particular, it can have the form of H-crack which is the case addressed in the present investigation. A direct numerical analysis of this problem by standard methods may be more time consuming due to the necessity to account for a relatively large number of degrees of freedom. This is required due to the fine composite microstructure and steep stress field gradients in the vicinity of the crack. Therefore a novel approach is employed which is based on the combined use of the high fidelity generalized method of cells model, the representative cell method and the higher-order theory. The crack existence is modeled by introducing fictitious unknown eigenstresses which are computed by an iterative procedure. This modeling is verified by a comparison with known analytical solution for a crack in a homogeneous plane. In addition, a verification by comparison with a known numerical solution for the special case of a transverse crack embedded in a periodically layered material is given. The influence of the volume fraction and elastic moduli ratio of the constituents as well as the H-crack aspect ratio on the stress field variation is examined, and the shielding effect of the interface cracks is quantified. The limiting situation of long interfacial cracks corresponding to the case of an incomplete layer is considered. The effect of a thermal loading on the cracked layered composite is demonstrated. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Multilayered materials exhibit unique resistance to transverse cracking due to their ability of crack arrest either in a soft ductile layer as in metal/ceramic laminates [19,20,9], or at the interface between two brittle constituents of ceramic matrix composites [4,12]. The exact analytical condition for the arrest of a crack approaching the interface between two dissimilar elastic materials was derived by [13]. Subsequent crack propagation scenario were examined by [7,10]. One of these scenario is the appearance of a crack branching along the weak interface causing a debonding of the layers. For some types of composites at the stage of failure initiation when damage is localized within a single matrix layer, the branching phenomenon may lead to an H-crack configuration. Therefore, the prediction of further damage propagation to the adjacent uncracked matrix layers accompanied by interface debonding and (or) transverse crack nucleation requires the analysis of stress redistribution caused by the H-crack appearance. This problem was addressed by [5] in the case of a composite with thin metallic layers modeled by interfaces with specific properties. They showed that interface debonding lowers the high stress concentration in front of the transverse crack. Consequently, for this type of cracks the prediction of fracture behavior which is ⇑ Corresponding author. Tel.: +972 3 6408130; fax: +972 3 6407617. E-mail address: [email protected] (M. Ryvkin). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.07.013
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Nomenclature a half of the crack length in the exact solution (m) dc length of the interface crack (m) df ; dm thicknesses of stiff and soft layers, respectively (m) C stiffness tensor (MPa) D damage variable (1) h; l half of the height and half of the width of the representative cell, respectively (m) H; L half of the height and half of the width of the composite domain, respectively (m) K 2 ; K 3 integers denoting the cell location (1) M2 ; M3 integers indicating the number of repetitive cells in X 2 and X 3 directions (1) S effective compliance tensor (1/MPa) tf ; tm traction vectors in stiff and soft layers, respectively (MPa) t ð2Þ ; t ð3Þ tractions at the boundaries perpendicular to the to x02 and x03 , respectively (MPa) uðx02 ; x03 ÞðK 2 ;K 3 Þ displacements vector in cell ðK 2 ; K 3 Þ uf ; um displacements vectors in stiff and soft layers, respectively (m) x02 ; x03 local coordinate system in repetitive cells (m) X2; X3 global coordinate system (m) a thermal expansion coefficient tensor (1/°C) C thermal stress tensor (MPa/ C) d2 ; d3 vectors of the displacement jumps between the opposite domain boundaries (m) strain tensor (1) q average strains tensor (1) h temperature deviation ( C) r stress tensor (MPa) r average stress tensor (MPa) r 22 remote average stress (MPa) r f22 ; r m22 remote stress in stiff and soft layers, respectively (MPa) reðK 2 ;K 3 Þ eigenstress tensor in cell ðK 2 ; K 3 Þ ((MPa) discrete Fourier transform parameters (1) /p ; /q
related to the crack reinitiation in the adjoining uncracked layer requires the analysis of the entire elastic field in the crack region. The analysis of the stress field perturbation generated by a transverse or H-crack embedded in a multilayered composite with a large number of layers represent a serious modeling problem due to the fine material microstructure and large stress gradients in the vicinity of the crack. Therefore some simplifying assumptions regarding the number of layers included in the analysis, or the specific form of the stress distribution within the layers are adopted [5,8,6,19]. In the present investigation a novel approach allowing an exact continuum modeling of arbitrarily large portion of periodically layered composites with embedded crack is employed. It should be mentioned that although the present investigation is confined to layered composites, some of the resulting issues shed light on the brittle fracture behavior of fiber reinforced composites [5]. This approach has been offered by [3] for the analysis of a distributed damage over a confined region within fiber reinforced composites. It is based on the combination of the representative cell method [15], the high-fidelity generalized method of cells micromechanical model [1] and the higher-order theory [2]. In the first one, the discrete Fourier transform is applied on the periodic composite in which the distributed damage effects have been included. As a result, a representative cell problem is obtained in the transform domain. The formulation of the specific boundary conditions that should be imposed in this problem requires the knowledge of the effective elastic moduli of the undamaged periodic composite. These elastic effective moduli have been established by the micromechanics analysis which forms the second approach. The solution of the governing equations in the transform domain is achieved by employing the higher-order theory that was originally developed for the analysis of functionally graded material. In the framework of this theory the representative cell is divided into several subcells in which a second-order expansion of the displacements in the transform domain is employed and the governing equations, interfacial and boundary conditions in this domain are imposed in the average (integral) sense. Once the solution of the representative cell problem has been established for all Fourier harmonics, an inverse transform is employed to obtain the actual elastic field in the composite. The effect of the localized damage appears in the constitutive equations in the form of eigenstresses. These field variables are not known in advance and, therefore, an iterative procedure was employed that establishes the requested elastic field in the composite to a preassigned degree of accuracy. In the present investigation this damage analysis approach is applied for the crack’s modeling. The method is verified by a comparison with an analytical solution for a crack in a homogeneous plane. Furthermore, a numerical solution (based on the Green’s functions approach) of a periodically layered composite with a transverse crack that has been established by [16] is employed to exhibit the validity of the present method of solution.
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Results are given for one and several fully debonded H-cracks in periodically layered composites. In particular, these results exhibit the stress field variation along the interface of the closest soft unbroken layer where the crack reinitiation must be predicted. A parametric study is provided which shows the effect of the H-crack aspect ratio, the volume fraction of the phases and their Young’s moduli ratio. The shielding effect of the interfacial cracks is quantified. Finally, the effect of a thermal loading of the cracked composite is shown. The paper is organized as follows: In Section 2 the problem formulation is given followed by the method of solution description in Section 3. Verifications of the suggested crack modeling method are presented in Section 4. This followed by the application Section 5 for the stress redistribution caused the H-crack. Finally, in Section 6 conclusions are given. 2. Problem statement Consider a layered periodic composite in which one of its layers is damaged by the appearance of a transverse crack perpendicular to the layering direction (taken as the 3-direction) and symmetric interfacial cracks, see Fig. 1a. The transverse crack is assumed to exist in the weak layer (denoted by m) whose width is dm , whereas the width of the stiff layer (denoted by f) is df . The length of the interfacial cracks is denoted by dc . As result of the localized damage which appears in the form of a transverse and interfacial cracks thus forming an H-crack flaw, the periodicity of the layered composite is lost and it is not possible to identify a repeating unit cell anymore.
(a)
(b)
(c)
22 at infinity. (b) A rectangular domain Fig. 1. (a) A periodic layered composite with a transverse and interfacial cracks, subjected to constant remote stress r 2H 2L of the layered composite is divided into repeating cells. These cells are labeled by ðK 2 ; K 3 Þ with M 2 6 K 2 6 M2 and M 3 6 K 3 6 M 3 , and the size of every one of which is 2h 2l (the figure is shown for M 2 ¼ M 3 ¼ 2). (c) A characteristic cell ðK 2 ; K 3 Þ in which local coordinates ðx02 ; x03 Þ are introduced whose origin is located at the center.
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The response of the considered damaged layered composite can be determined by satisfying the equilibrium equation which, in the absence of body forces, is given by
rr¼0
ð1Þ
where r is the stress tensor. In addition, the interfacial conditions that requires in the case of perfect bonding the continuity of displacements u and tractions t between the stiff and soft constituents must be imposed:
uf ¼ um ;
tf ¼ tm
ð2Þ
The constitutive equations of a thermoelastic material in the presence of damage are given, according to the principle of strain equivalence [11] by
r ¼ ð1 DÞ½C : Ch
ð3Þ
where is the strain tensor and C and C are the stiffness and thermal stress tensors of the phase, respectively, h denotes the temperature deviation from a reference temperature, and D is the current isotropic damage variable. The damage variable D ¼ 0 in the undamaged layers, whereas in transverse and interfacial crack regions it equal to 1, which represents a complete damage. Thus the cracks are modeled by regions with D ¼ 1 as will be elaborated in the next section. Finally, the far-field boundary conditions that are applied on the composite should be incorporated. 3. Method of solution Far away from the damaged layers, the periodically bi-layered composite behavior is governed by its effective moduli which can be determined by the exact expressions that have been established by [14]. To this end, let us consider a rectangular domain H 6 X 2 6 H; L 6 X 3 6 L of the composite which includes the damaged region. Although this region includes the localized damage, it is assumed that it is extensive enough such that the elastic field at its boundaries is not influenced by the damage existence and therefore, expressions of [14] are applicable. Consequently, the boundary conditions that are applied on X 2 ¼ H and X 3 ¼ L are referred to as the far-field boundary conditions. According to the representative cell method, [15], this region is divided into ð2M 2 þ 1Þ ð2M 3 þ 1Þ cells, see Fig. 1b for M 2 ¼ M3 ¼ 2. Every cell is labeled by ðK 2 ; K 3 Þ with K 2 ¼ M 2 ; . . . ; M 2 and K 3 ¼ M 3 ; . . . ; M 3 . In each cell, local coordinates ðx02 ; x03 Þ are introduced whose origins are located at its center, see Fig. 1c. The equilibrium Eq. (1) of the materials within the cell ðK 2 ; K 3 Þ takes the form
r rðK 2 ;K 3 Þ ¼ 0
ð4Þ
The present analysis is not confined to a single transverse crack and its corresponding interfacial cracks (thus forming an Hcrack), since several cracks can be considered in the region of which D ¼ 1. The constitutive equation in the cell, Eq. (3), can be written as
h
rðK 2 ;K 3 Þ ¼ C : ðK 2 ;K 3 Þ DC : ðK 2 ;K 3 Þ þ ð1 DÞChðK 2 ;K 3 Þ
i
ð5Þ
where
D¼
0 no damage
ð6Þ
1 full damage
The last term in the right-hand-side of Eq. (5) can be referred to as eigenstress which is denoted herein by
reðK 2 ;K 3 Þ ¼ DC : ðK 2 ;K 3 Þ þ ð1 DÞChðK 2 ;K 3 Þ ðK 2 ;K 3 Þ
The continuity of displacements u
and tractions t
ð7Þ ðK 2 ;K 3 Þ
between adjacent cells should be imposed. Thus,
ðK ;K Þ ðK þ1;K 3 Þ uðh; x03 Þ 2 3 uðh; x03 Þ 2 ¼0
ð8Þ
ð2Þ ðK 2 ;K 3 Þ ð2Þ ðK 2 þ1;K 3 Þ t ðh; x03 Þ t ðh; x03 Þ ¼0
ð9Þ
where K 2 ¼ M 2 ; . . . ; M 2 1; K 3 ¼ M 3 ; . . . ; M 3 ; l 6 x03 6 l, and
0 ðK 2 ;K 3 Þ 0 ðK ;K þ1Þ uðx2 ; lÞ uðx2 ; lÞ 2 3 ¼0
ð10Þ
ð3Þ 0 ðK 2 ;K 3 Þ ð3Þ 0 ðK 2 ;K 3 þ1Þ t ðx2 ; lÞ t ðx2 ; lÞ ¼0
ð11Þ
where K 2 ¼ M 2 ; . . . ; M 2 ; K 3 ¼ M 3 ; . . . ; M 3 1; h 6 x02 6 h. In the following, the appropriate form of the far-field boundary conditions that specify the tractions and displacements at the opposite sides X 2 ¼ H; X 3 ¼ L of the rectangle of Fig. 1b are presented. The tractions at the opposite sides of the rectangular domain are equal:
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ð2Þ ðM2 ;qÞ ðM2 ;qÞ t ðh; x03 Þ t ð2Þ ðh; x03 Þ ¼ 0; ð3Þ ðp;M3 Þ 0 ðp;M3 Þ 0 ðx2 ; lÞ t ð3Þ ðx2 ; lÞ ¼ 0; t
q ¼ M3 ; . . . ; M 3
229
ð12Þ
p ¼ M 2 ; . . . ; M 2
ð13Þ
The displacements at the opposite sides, on the other hand, differ by certain jumps as follows:
uðM2 ;qÞ ðh; x03 Þ uðM2 ;qÞ ðh; x03 Þ ¼ d2 ; uðp;M3 Þ ðx02 ; lÞ uðp;M3 Þ ðx02 ; lÞ ¼ d3 ;
q ¼ M3 ; . . . ; M 3
ð14Þ
p ¼ M 2 ; . . . ; M 2
ð15Þ
where d2 and d3 denote the vector of the far-field displacement differences which are given by
d2j ¼ 2H2j ;
d3j ¼ 2L3j ;
j ¼ 1; 2; 3
ð16Þ
and 2j ; 2j are the average strains of the unperturbed periodic composite which have to be determined from its effective compliance S and thermal expansion coefficient a tensors, in conjunction with the appropriate imposed traction boundary : conditions r
¼ S r þ a h
ð17Þ
In the present investigation, the effective compliance and thermal expansion coefficient tensors of the (unperturbed) layered composite has been determined by utilizing the exact expressions of the effective moduli that have been established by [14]. 22 is related to the It should be noted that in the case of a loading in the 2-direction (for example) the remote average stress r remote stresses rf22 and rm 22 in the constituents according to
r 22 ¼
df rf22 þ dm rm 22 df þ dm
ð18Þ
The double discrete Fourier of the displacement vector uðK 2 ;K 3 Þ is defined by
^ ðx02 ; x03 ; /p ; /q Þ ¼ u
M2 X
M3 X
uðK 2 ;K 3 Þ ðx02 ; x03 Þ exp iðK 2 /p þ K 3 /q Þ
ð19Þ
K 2 ¼M 2 K 3 ¼M 3
where
/p ¼
2p p ; 2M 2 þ 1
p ¼ 0; 1; 2; . . . ; M2 ;
/q ¼
2pq ; 2M3 þ 1
q ¼ 0; 1; 2; . . . ; M 3 ;
The application of this transform to the boundary problem (4)–(15) for the rectangular domain H < X 2 < H; L < X 3 < L, divided into ð2M2 þ 1Þ ð2M 3 þ 1Þ cells, converts it to the problem for the single representative cell h < x02 < h; l < x03 < l with respect to the complex valued transforms. The field equations obtained from the equilibrium and constitutive equations have the form
^¼0 rr
ð20Þ
r^ ¼ C ^ r^ e
ð21Þ
and
where the transformed eigenstress tensor is given by
r^ e ¼ DC : ^ þ ð1 DÞC^h
ð22Þ
The conditions relating the opposite boundaries of the representative cell in a similar problem were derived by [3]. Following their approach one obtains from (8)–(15)
^ ðh; x03 Þ þ d0;q ð2M 3 þ 1Þd2 expði/p M2 Þ; ^ ðh; x03 Þ ¼ expði/p Þu u ^tðh; x0 Þ ¼ expði/p Þ^tðh; x0 Þ; 3 3
l 6 x03 6 l
l 6 x03 6 l
ð23Þ ð24Þ
and
^ ðx02 ; lÞ ¼ expði/q Þu ^ ðx02 ; lÞ þ d0;q ð2M2 þ 1Þd3 expði/q M 3 Þ; u ^tðx0 ; lÞ ¼ expði/q Þ^tðx0 ; lÞ; 2 2
h 6 x02 6 h
h 6 x02 6 h
ð25Þ ð26Þ
In these equations, dp;q denotes the Kronecker delta. The representative cell boundary value problem (20)–(26) have been solved by employing the higher-order theory, [2]. According to this theory, the domain h 6 x02 6 h; l 6 x03 6 l (the representative cell) is divided into several rectangular subcells. The transformed displacement vector is expanded into a second-order polynomial, and the equilibrium equations,
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interfacial and boundary conditions are imposed in the average (integral) sense. In order to model a (line) crack, a single row of subcells filled with a fully damaged material (D ¼ 1) is introduced. Thus the value of the damage variable D is pre-determined in accordance with the crack configuration. Once the solution in the transform domain has been established, the actual elastic field can be readily determined at any point in the desired cell ðK 2 ; K 3 Þ of the considered rectangular region H 6 X 2 6 H; L 6 X 3 6 L by the inverse transform formula whose form for the displacements, for example, is:
uðK 2 ;K 3 Þ ðx02 ; x03 Þ ¼
M2 M3 X X 1 ^ ðx02 ; x03 ; /p ; /q Þ exp iðK 2 /p þ K 3 /q Þ u ð2M 2 þ 1Þð2M 3 þ 1Þ p¼M q¼M 2
ð27Þ
3
^ e to be In the application of this theory where a stationary or evolving damage is taking place, the eigenstress tensor, r used in Eq. (21) is not known. Hence an iterative solution has to be employed as follows: ^ e ¼ 0 and solve Eqs. 20,21, (23)–(26) in the transform domain. 1. Start by assuming that r 2. Apply the inverse transform formula to compute the stress field. The latter can be employed to compute the current eigenstress reðK 2 ;K 3 Þ in the actual space. 3. Compute the transform of reðK 2 ;K 3 Þ by employing Eq. (22). 4. Solve Eqs. 20,21 and (23)–(26). This procedure should be continued until a convergence to a desired degree of accuracy is achieved.
(a)
(b)
(c)
22 generated in homogeneous material with a crack. (a) Comparison between the present and the exact solution along the Fig. 2. The normal stress r22 =r 22 in the region 1:5 6 X 2 =2h 6 1:5; 1:5 6 X 3 =2l 6 1:5 computed by the present approach. (c) The crack line X 3 at X 2 ¼ 0. (b) The distribution of r22 =r 22 in the region 1:5 6 X 2 =2h 6 1:5; 1:5 6 X 3 =2l 6 1:5 computed by the closed-form expressions. distribution of r22 =r
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231
4. Verifications In all cases given in this paper, the computations of the elastic field were carried out with a square cell h ¼ l, Fig. 1c, which has been divided into 40 40 subcells. Consider a single crack embedded in an infinite homogeneous elastic isotropic plane that is subjected to a remote tensile 22 . The closed-form solution of this problem can be found in [18] and a comparison between this solution and the loading r present approach is given in Fig. 2. The stress r22 along the line of the crack ðX 2 ¼ 0Þ is given by
jX j
3 ; jX 3 j > a r22 ðX 2 ¼ 0; X 3 Þ ¼ r 22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð28Þ
X 23 a2
where 2a is the crack length. Fig. 2a shows a comparison between the normal stress r22 along the crack’s line with 2a=2l ¼ 0:5, as predicted by the present approach and the exact expression (28). Excellent agreement can be readily observed. 22 in the plane It is also possible to compare the present solution approach by comparing the stress field distribution r22 =r ðX 2 X 3 Þ with the exact solution. Fig. 2b and c present the comparison between the present and exact solutions, respectively, in the region 1:5 6 X 2 =2h 6 1:5; 1:5 6 X 3 =2l 6 1:5. Here too excellent agreement exists. Consider next the problem of a transverse crack (i.e., dc ¼ 0) located in the soft layer of a periodically layered ceramic 22 at infinity. The properties of the stiff silicon-carbide and the soft composite that is subjected to a uniform stress loading r lithium-alumino-silicate glass layers are given in Table 1 with the reinforcement volume ratio v f ¼ df =ðdf þ dm Þ ¼ 0:5. The resulting elastic field has been determined by [16] by employing the representative cell method where the Green’s function needs to be establish. A comparison between the normal stress r22 variation along the crack’s line X 3 at X 2 ¼ 0, computed by the present approach and the latter method is shown in Fig. 3. Excellent agreement between the different methods of solution exists.
Table 1 Material constants of the silicon-carbide and lithium-alumino-silicate (LAS) glass layers. Material
E (GPa)
m
a (106 °C1)
Silicon-carbide Lithium-alumino-silicate
200 85
0.19 0.25
2.2 9.5
E; m and a denote the Young’s modulus, Poisson’s ratio and the coefficient of thermal expansion, respectively.
(a)
(b)
22 , computed by the present and the Green’s function approach, along the crack line X 3 at X 2 ¼ 0, of a transverse Fig. 3. (a) The normal stress variation r22 =r 22 . The stiff crack located in the soft layer of a silicon-carbide/glass periodically layered composite that is subjected to a uniform far-field normal loading r 22 , computed along the crack line X 3 at X 2 ¼ 0, of an H-crack with dc =2h ¼ 1, located layer volume fraction is v f ¼ 0:5. (b) The normal stress variation r22 =r 22 . The stiff layer within the soft layer of a silicon-carbide/glass periodically layered composite that is subjected to a uniform far-field normal loading r volume fraction is v f ¼ 0:5.
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5. Applications The developed technique for the cracks modeling is applied to a parametric study of the stress redistribution in the bimaterial periodic composite caused by H-crack fracture patterns. The knowledge of the stress field together with the fracture characteristics of the composite constituents and the interfaces is necessary for the prediction of the failure propagation. The failure may take the form of interfacial crack growth, new transverse crack from the interface in the adjoining uncracked
22 distribution in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2 generated in the silicon-carbide/glass periodically layered Fig. 4. The normal stress r22 =r 22 . The stiff layer volume fractions are: (a) composite with H-cracks (dc =2h ¼ 1). The composite is subjected to a uniform far-field normal loading of r v f ¼ 0:1 and (b) v f ¼ 0:5.
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233
layer or crack reinitiation in the next layer of the same type as the cracked one. The established stress field allows addressing all these scenarios. The present analysis of the results however concentrates on the last one. Consider the case of a silicon-carbide/glass periodically layered composite with the thermoelastic properties specified in Table 1. One of the soft layers contains a transverse crack branching into two interfacial ones with dc =2h ¼ 1, thus forming an 22 . The dimensions of the rectangular domain for which H-crack. The composite is subjected to a far-field normal loading r the influence of the perturbation induced by the H-crack on the external boundaries is negligibly small are found to be 2H=ðdf þ dm Þ ¼ 21; 2L=ðdf þ dm Þ ¼ 11. This corresponds to a division of the rectangular domain into ð2M 2 þ 1Þ ð2M 3 þ 1Þ cells with M2 ¼ 10 and M 3 ¼ 5. For longer H-cracks and/or higher elastic moduli contrasts, the dimensions of the rectangular region have been appropriately increased. This increase is achieved by increasing the numbers M 2 and M 3 which however does not affect the size of the representative cell domain. 22 is shown in Fig. 4 in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2, in the two The resulting elastic field distribution r22 =r cases of stiff layer volume fractions of v f ¼ 0:1 and 0.5. As expected, the weak layer region which is occupied by the H-crack is unloaded and characterized by very low stresses except at the interface cracks tips. The stress concentration in the adjoining stiff layer just across the interface is lowered with respect to the single transverse crack case (dc ¼ 0). This phenomenon of a shielding effect caused by an interfacial debonding has been mentioned by Chan et al. [5] in a similar problem. The same phenomenon can be also observed by a comparison of Fig. 3a and b. It should be noted that the variation of the periodic stress state for both volume fractions in Fig. 4 is confined to the region of about three periods in X 2 as well as in X 3 directions which justifies the above choice of rectangular domain dimensions. A reinitiation of the transverse crack in the soft layer next to the cracked one is related to the tensile stress distribution. To this end, the normal stress r22 variations within the soft uncracked layer along the interface is examined. The results are presented in Fig. 5 where a family of curves for different interface crack lengths dc =2h ¼ 0; 1; 5; 7; 9; 11 are depicted. It is seen that in the vast majority of the cases the location of the maximal stress is shifted from the transverse crack line X 2 ¼ 0. It should be mentioned that this shifting phenomenon is observed even in the limiting case of a transverse crack (dc ¼ 0) which is in accordance with the known experimental [20] and theoretical [19] results. The dependence of the maximal stress upon the interface crack length dc =2h clearly illustrates the shielding effect of the interfacial debonding. The magnitude of the stress decreases monotonically until it approaches some limiting value for sufficiently large interface cracks. This limiting value represents an important fracture characteristics of the bi-material composite defining the crack resistance in the case of incomplete (damaged) layer. As expected, this value is lower for the thick stiff layers v f ¼ 0:5 than for the case of the thin ones v f ¼ 0:1. Further investigation of this dependency on different values of stiff layer volume fraction is presented in Fig. 6. Here, the maximal stress results for an incomplete layer are shown together with those for a transverse crack (dc ¼ 0). Interestingly, it is found that for the considered layered composites with volume fractions v f > 0:3 the stress concentrations resulting from the transverse crack and incomplete layer are very close. It is obvious that the variations of the stiff layer volf ð1Þ mð1Þ ume fraction v f is accompanied by corresponding changes of the remote stress r22 and r22 in the layers. The latter is given by
rmð1Þ ¼ 22
Em r 22 E2
(a)
(b)
22 generated in the silicon-carbide/glass composite along the interface X 2 within the soft layer, at Fig. 5. The variation of the normal stress r22 =r X 3 ¼ dm =2 þ df for various values of dc of H-cracks. (a) v f ¼ 0:1 and (b) v f ¼ 0:5.
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where E2 ¼ ðdf Ef þ dm Em Þ=ðdf þ dm Þ. Therefore one can obtain the values of max the values presented in Fig. 6 by utilizing the following formula ðmÞ 22 mð1Þ 22
max
r
r
ðmÞ
¼
max r22 r 22
mð1Þ rðmÞ from 22 normalized with respect to r22
r 22 rmð1Þ 22
The influence of the Young’s moduli ratio of the layers on the stress distribution in the transversely cracked and Hcracked composites with stiff layer volume fraction v f ¼ 0:1 is presented in Fig. 7a and b, respectively. As in the previous cases, the normal stresses r22 along the interface in the soft uncracked layer are addressed. The general trend exhibited in this figure is that the increase of the contrast in the elastic properties of the composite constituents leads to the stress decrease. The existence of interface cracks extends the stress concentration region, and somewhat diminishes the maximal value. However, for the considered parameter combinations with relatively short interface cracks, this effect becomes pronounced only in the case of relatively close materials properties Em =Ef > 0:3. This is separately illustrated in Fig. 8 where the
ðmÞ 22 in the soft layer, generated in the silicon-carbide/glass composite along the interface X 2 Fig. 6. The variation of the maxima of the normal stress r22 =r within the soft layer, at X 3 ¼ dm =2 þ df for various values of v f , with dc =2h ¼ 0 and 11.
(a)
(b)
22 generated in composites (v f ¼ 0:1) with various values of Em =Ef along the interface X 2 within the soft Fig. 7. The variation of the normal stress r22 =r layer, at X 3 ¼ dm =2 þ df . (a) dc ¼ 0 and (b) dc =2h ¼ 1.
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235
dependence of the maximal stresses upon the Young’s moduli ratio is given. For the materials with Em =Ef ¼ 0:6 the stress decrease is about 15%. For a more significant contrast between the materials, a longer interface crack only would provide the shielding effect. In the case of an extended composite’s damage, failure may appear in the form of several layer cracking. The present method of analysis can be readily employed to model such a situation. The results obtained for the case of three cracked soft layers in the silicon-carbide/glass composite with v f ¼ 0:1 are presented in Figs. 9, 10. In the first figure, the normal stress variation along the first intact interface within the soft layer in front of the flaw consisting of three H-cracks is presented. The cracks with dc =2h ¼ 0; 1; 11 are considered corresponding to the cases of a transverse crack, a relatively short (dc =2h ¼ 1) and long interface crack, respectively. It is readily seen that the shielding effect of the interface debonding when several layers are damaged is significantly less pronounced. On the other hand, the stress field distributions shown in Fig. 10a and b for dc ¼ 0 and dc =2h ¼ 1 exhibit very different patterns.
22 generated in composites (v f ¼ 0:1) with various values of Em =Ef along the interface X 2 Fig. 8. The variation of the maxima of the normal stress r22 =r within the soft layer, at X 3 ¼ dm =2 þ df for dc =2h ¼ 0 and 1.
22 generated in the silicon-carbide/glass composite (v f ¼ 0:1) along the interface X 2 within the soft layer, at Fig. 9. The variation of the normal stress r22 =r X 3 ¼ 3dm =2 þ 2df for various values of dc . The composite contains three H-cracks in the soft layers.
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22 distribution in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2 generated in a silicon-carbide/glass periodically layered Fig. 10. The normal stress r22 =r 22 . and stiff layer volume fractions is composite with three H-cracks in the soft layers. The composite is subjected to a uniform far-field normal loading of r v f ¼ 0:1. (a) dc ¼ 0 and (b) dc =2h ¼ 1.
In Fig. 11 the response of the considered layered composite with an H-crack pattern to the thermal loading of h ¼ þ200 C is presented. In the present case of heating, the remote periodic r22 stress field in the soft and stiff layers is compressive and tensile, respectively, which is caused by the mismatch between the coefficients of thermal expansion of the layers, see Table 1. Therefore, the transverse crack in a soft layer will close being not dangerous and the case of a stiff cracked layer
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237
Fig. 11. The normal stress r22 distribution (measured in GPA units) in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2 generated in a silicon-carbide/glass periodically layered composite with a single H-crack in the stiff layer. The composite is subjected to a thermal loading of 200 °C, with stiff layer volume fractions is v f ¼ 0:5 and dc =2h ¼ 1.
is addressed. As expected, the rectangular region occupied by the H-crack becomes unstressed and stress redistribution leads to the stress concentration in the neighboring stiff layers. 6. Conclusions A method for the simulation of cracks by damage variables has been implemented to investigate the behavior of periodically layered composites. The effect of the cracks (damage) must be localized in the sense that the far-field in the composite should not be affected by their presence. As a result specific far-field boundary conditions must be formulated and implemented. The effect of the cracks appear in the thermoelastic constitutive equations in the form of unknown eigenstresses which necessitates the application of an iterative procedure until a convergence of the solution is achieved. The method has been verified by a comparison with an the exact solution of a crack in a homogeneous medium and for a transverse crack in a periodically layered composite the solution of which can be established by different method that is based on the construction of Green’s functions. The shielding effect of the interface debonding for the soft layers cracking in a periodically layered composite is quantified. It is found to be increasing with the decreasing of the stiff layers volume fraction and with the diminishing of the contrast between the elastic properties of the phases. It appears that when the length of the interface cracks is about more than seven times the layer thickness, the stress concentration in the next soft layer becomes insensitive to the further their extension. This limiting situation corresponds to the case of an incomplete soft layer. The lowering of the stress concentration, as compared to the case of a transverse crack, is more significant for a single incomplete layer than for several ones. It should be mentioned that although the present investigation was confined to layered composites, some of the issues shed light on the brittle fracture behavior of fiber reinforced composites [5]. In these fiber reinforced systems the debond crack as it grows in the vertical direction it also grows in the circumferential direction and eventually surrounds the fiber with another matrix crack appearing in the adjacent uncracked layers. In the present investigation the cracks form a complete debonding in both normal and shear. It is possible however to generalize the present approach to obtain debonding in the shear direction whereas perfect bonding in normal direction is retained. The present analysis can be also applied to instigate periodically layered composites with any system (non periodic) of vertical and horizontal cracks. In addition, it is possible to analyze periodic layered composites with more than two different isotropic and anisotropic materials. In the present paper, layered composites with brittle constituents have been considered. Another important class of layered composites includes metallic layers where plasticity effects should be accounted for. Investigation this type of composites using the present approach seems to be a natural research topic. Finally, the present analysis can be generalized to a system of cracks in periodic fiber-reinforced composites. This three-dimensional
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generalization is straightforward and requires the utilization of the triple finite discrete Fourier transform as has been applied in the case of a single penny-shape crack [17]. References [1] Aboudi J. The generalized method of cells and high-fidelity generalized method of cells micromechanical models – a review. Mech Adv Mater Struct 2004;11:329–66. [2] Aboudi J, Pindera MJ, Arnold SM. Higher-order theory for functionally graded materials. Composites Part B: (Engng) 1999;30:777–832. [3] Aboudi J, Ryvkin M. The effect of localized damage on the behavior of composites with periodic microstructure. Int J Engng Sci 2012;52:41–55. [4] Bermejo R, Baudn C, Moreno R, Llanes L, Sanchez-Herencia AJ. Processing optimisation and fracture behaviour of layered ceramic composites with highly compressive layers. Compos Sci Tech 2007;67:1930–8. [5] Chan KS, He MY, Hutchinson JW. Cracking and stress redistribution in ceramic layered composites. Mater Sci Engng A 1993;167:57–64. [6] Hbaieb K, McMeeking RM, Lange FF. Crack bifurcation in laminar ceramics having large compressive stress. Int J Solids Struct 2007;44:3328–43. [7] He MY, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials. Int J Solids Struct 1989;25:1053–67. [8] Huang Y, Zhang HW, Wu F. Multiple cracking in metal–ceramic laminates. Int J Solids Struct 1994;31:2753–68. [9] Jha M, Charalambides PG. A finite element analysis of fracture initiation in ductile/brittle periodically layered composites. Int J Fract 1998;90:299–323. [10] Leguillon D, Lacroix C, Martin E. Interface debonding ahead of a primary crack. J Mech Phys Solids 2000;48:2137–61. [11] Lemaitre J, Desmorat R. Engineering damage mechanics. Berlin: Springer; 2005. [12] Lugovy M, Orlovskaya N, Berroth K, Kuebler J. Macrostructural engineering of ceramic-matrix layered composites. Compos Sci Tech 1999;59:1429–37. [13] Nuller B, Ryvkin M, Chudnovsky A. A closed-form solution for a crack approaching an interface. J Mech Mater Struct 2006;1:1405–24. [14] Postma GW. Wave propagation in stratified medium. Geophysics 1955;20:780–6. [15] Ryvkin M, Nuller B. Solution of quasi-periodic fracture problems by the representative cell method. Comp Mech 1997;20:145–9. [16] Ryvkin M, Aboudi J. A continuum approach to the analysis of the stress field in a fiber reinforced composite with a transverse crack. Int J Solids Struct 2007;44:6826–41. [17] Ryvkin M, Aboudi J. Three-dimensional continuum analysis for a unidirectional composite with a broken fiber. Int J Solids Struct 2008;45:4114–29. [18] Sneddon IN. Fourier transforms. New York: McGraw Hill; 1951. [19] Shaw MC. The aspect of strength probabilistic on the fracture mode of ceramic/metal multilayers. Engng Fract Mech 1998;61:49–74. [20] Shaw MC, Marshall DB, Dadkhah MS, Evans AG. Cracking and damage mechanisms in ceramic/metal multi-layers. Acta Metall Mater 1993;41:3311–22.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Stress redistribution due to cracking in periodically layered composites Michael Ryvkin ⇑, Jacob Aboudi School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
a r t i c l e
i n f o
Article history: Received 7 March 2012 Received in revised form 1 July 2012 Accepted 10 July 2012
Keywords: Localized damage Cracked layered composite Representative cell method High-fidelity generalized method of cells Higher-order theory
a b s t r a c t The stress redistribution caused by a brittle fracture of one or several layers in a bi-material periodically layered composite is considered. It is assumed that one of the composite constituents possess low crack resistance and is subject to cracking. In the case of a uniaxial tension parallel to the layering direction and in the presence of weak interfaces, the fracture pattern may be quite complicated including branching cracks. In particular, it can have the form of H-crack which is the case addressed in the present investigation. A direct numerical analysis of this problem by standard methods may be more time consuming due to the necessity to account for a relatively large number of degrees of freedom. This is required due to the fine composite microstructure and steep stress field gradients in the vicinity of the crack. Therefore a novel approach is employed which is based on the combined use of the high fidelity generalized method of cells model, the representative cell method and the higher-order theory. The crack existence is modeled by introducing fictitious unknown eigenstresses which are computed by an iterative procedure. This modeling is verified by a comparison with known analytical solution for a crack in a homogeneous plane. In addition, a verification by comparison with a known numerical solution for the special case of a transverse crack embedded in a periodically layered material is given. The influence of the volume fraction and elastic moduli ratio of the constituents as well as the H-crack aspect ratio on the stress field variation is examined, and the shielding effect of the interface cracks is quantified. The limiting situation of long interfacial cracks corresponding to the case of an incomplete layer is considered. The effect of a thermal loading on the cracked layered composite is demonstrated. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Multilayered materials exhibit unique resistance to transverse cracking due to their ability of crack arrest either in a soft ductile layer as in metal/ceramic laminates [19,20,9], or at the interface between two brittle constituents of ceramic matrix composites [4,12]. The exact analytical condition for the arrest of a crack approaching the interface between two dissimilar elastic materials was derived by [13]. Subsequent crack propagation scenario were examined by [7,10]. One of these scenario is the appearance of a crack branching along the weak interface causing a debonding of the layers. For some types of composites at the stage of failure initiation when damage is localized within a single matrix layer, the branching phenomenon may lead to an H-crack configuration. Therefore, the prediction of further damage propagation to the adjacent uncracked matrix layers accompanied by interface debonding and (or) transverse crack nucleation requires the analysis of stress redistribution caused by the H-crack appearance. This problem was addressed by [5] in the case of a composite with thin metallic layers modeled by interfaces with specific properties. They showed that interface debonding lowers the high stress concentration in front of the transverse crack. Consequently, for this type of cracks the prediction of fracture behavior which is ⇑ Corresponding author. Tel.: +972 3 6408130; fax: +972 3 6407617. E-mail address: [email protected] (M. Ryvkin). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.07.013
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Nomenclature a half of the crack length in the exact solution (m) dc length of the interface crack (m) df ; dm thicknesses of stiff and soft layers, respectively (m) C stiffness tensor (MPa) D damage variable (1) h; l half of the height and half of the width of the representative cell, respectively (m) H; L half of the height and half of the width of the composite domain, respectively (m) K 2 ; K 3 integers denoting the cell location (1) M2 ; M3 integers indicating the number of repetitive cells in X 2 and X 3 directions (1) S effective compliance tensor (1/MPa) tf ; tm traction vectors in stiff and soft layers, respectively (MPa) t ð2Þ ; t ð3Þ tractions at the boundaries perpendicular to the to x02 and x03 , respectively (MPa) uðx02 ; x03 ÞðK 2 ;K 3 Þ displacements vector in cell ðK 2 ; K 3 Þ uf ; um displacements vectors in stiff and soft layers, respectively (m) x02 ; x03 local coordinate system in repetitive cells (m) X2; X3 global coordinate system (m) a thermal expansion coefficient tensor (1/°C) C thermal stress tensor (MPa/ C) d2 ; d3 vectors of the displacement jumps between the opposite domain boundaries (m) strain tensor (1) q average strains tensor (1) h temperature deviation ( C) r stress tensor (MPa) r average stress tensor (MPa) r 22 remote average stress (MPa) r f22 ; r m22 remote stress in stiff and soft layers, respectively (MPa) reðK 2 ;K 3 Þ eigenstress tensor in cell ðK 2 ; K 3 Þ ((MPa) discrete Fourier transform parameters (1) /p ; /q
related to the crack reinitiation in the adjoining uncracked layer requires the analysis of the entire elastic field in the crack region. The analysis of the stress field perturbation generated by a transverse or H-crack embedded in a multilayered composite with a large number of layers represent a serious modeling problem due to the fine material microstructure and large stress gradients in the vicinity of the crack. Therefore some simplifying assumptions regarding the number of layers included in the analysis, or the specific form of the stress distribution within the layers are adopted [5,8,6,19]. In the present investigation a novel approach allowing an exact continuum modeling of arbitrarily large portion of periodically layered composites with embedded crack is employed. It should be mentioned that although the present investigation is confined to layered composites, some of the resulting issues shed light on the brittle fracture behavior of fiber reinforced composites [5]. This approach has been offered by [3] for the analysis of a distributed damage over a confined region within fiber reinforced composites. It is based on the combination of the representative cell method [15], the high-fidelity generalized method of cells micromechanical model [1] and the higher-order theory [2]. In the first one, the discrete Fourier transform is applied on the periodic composite in which the distributed damage effects have been included. As a result, a representative cell problem is obtained in the transform domain. The formulation of the specific boundary conditions that should be imposed in this problem requires the knowledge of the effective elastic moduli of the undamaged periodic composite. These elastic effective moduli have been established by the micromechanics analysis which forms the second approach. The solution of the governing equations in the transform domain is achieved by employing the higher-order theory that was originally developed for the analysis of functionally graded material. In the framework of this theory the representative cell is divided into several subcells in which a second-order expansion of the displacements in the transform domain is employed and the governing equations, interfacial and boundary conditions in this domain are imposed in the average (integral) sense. Once the solution of the representative cell problem has been established for all Fourier harmonics, an inverse transform is employed to obtain the actual elastic field in the composite. The effect of the localized damage appears in the constitutive equations in the form of eigenstresses. These field variables are not known in advance and, therefore, an iterative procedure was employed that establishes the requested elastic field in the composite to a preassigned degree of accuracy. In the present investigation this damage analysis approach is applied for the crack’s modeling. The method is verified by a comparison with an analytical solution for a crack in a homogeneous plane. Furthermore, a numerical solution (based on the Green’s functions approach) of a periodically layered composite with a transverse crack that has been established by [16] is employed to exhibit the validity of the present method of solution.
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Results are given for one and several fully debonded H-cracks in periodically layered composites. In particular, these results exhibit the stress field variation along the interface of the closest soft unbroken layer where the crack reinitiation must be predicted. A parametric study is provided which shows the effect of the H-crack aspect ratio, the volume fraction of the phases and their Young’s moduli ratio. The shielding effect of the interfacial cracks is quantified. Finally, the effect of a thermal loading of the cracked composite is shown. The paper is organized as follows: In Section 2 the problem formulation is given followed by the method of solution description in Section 3. Verifications of the suggested crack modeling method are presented in Section 4. This followed by the application Section 5 for the stress redistribution caused the H-crack. Finally, in Section 6 conclusions are given. 2. Problem statement Consider a layered periodic composite in which one of its layers is damaged by the appearance of a transverse crack perpendicular to the layering direction (taken as the 3-direction) and symmetric interfacial cracks, see Fig. 1a. The transverse crack is assumed to exist in the weak layer (denoted by m) whose width is dm , whereas the width of the stiff layer (denoted by f) is df . The length of the interfacial cracks is denoted by dc . As result of the localized damage which appears in the form of a transverse and interfacial cracks thus forming an H-crack flaw, the periodicity of the layered composite is lost and it is not possible to identify a repeating unit cell anymore.
(a)
(b)
(c)
22 at infinity. (b) A rectangular domain Fig. 1. (a) A periodic layered composite with a transverse and interfacial cracks, subjected to constant remote stress r 2H 2L of the layered composite is divided into repeating cells. These cells are labeled by ðK 2 ; K 3 Þ with M 2 6 K 2 6 M2 and M 3 6 K 3 6 M 3 , and the size of every one of which is 2h 2l (the figure is shown for M 2 ¼ M 3 ¼ 2). (c) A characteristic cell ðK 2 ; K 3 Þ in which local coordinates ðx02 ; x03 Þ are introduced whose origin is located at the center.
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The response of the considered damaged layered composite can be determined by satisfying the equilibrium equation which, in the absence of body forces, is given by
rr¼0
ð1Þ
where r is the stress tensor. In addition, the interfacial conditions that requires in the case of perfect bonding the continuity of displacements u and tractions t between the stiff and soft constituents must be imposed:
uf ¼ um ;
tf ¼ tm
ð2Þ
The constitutive equations of a thermoelastic material in the presence of damage are given, according to the principle of strain equivalence [11] by
r ¼ ð1 DÞ½C : Ch
ð3Þ
where is the strain tensor and C and C are the stiffness and thermal stress tensors of the phase, respectively, h denotes the temperature deviation from a reference temperature, and D is the current isotropic damage variable. The damage variable D ¼ 0 in the undamaged layers, whereas in transverse and interfacial crack regions it equal to 1, which represents a complete damage. Thus the cracks are modeled by regions with D ¼ 1 as will be elaborated in the next section. Finally, the far-field boundary conditions that are applied on the composite should be incorporated. 3. Method of solution Far away from the damaged layers, the periodically bi-layered composite behavior is governed by its effective moduli which can be determined by the exact expressions that have been established by [14]. To this end, let us consider a rectangular domain H 6 X 2 6 H; L 6 X 3 6 L of the composite which includes the damaged region. Although this region includes the localized damage, it is assumed that it is extensive enough such that the elastic field at its boundaries is not influenced by the damage existence and therefore, expressions of [14] are applicable. Consequently, the boundary conditions that are applied on X 2 ¼ H and X 3 ¼ L are referred to as the far-field boundary conditions. According to the representative cell method, [15], this region is divided into ð2M 2 þ 1Þ ð2M 3 þ 1Þ cells, see Fig. 1b for M 2 ¼ M3 ¼ 2. Every cell is labeled by ðK 2 ; K 3 Þ with K 2 ¼ M 2 ; . . . ; M 2 and K 3 ¼ M 3 ; . . . ; M 3 . In each cell, local coordinates ðx02 ; x03 Þ are introduced whose origins are located at its center, see Fig. 1c. The equilibrium Eq. (1) of the materials within the cell ðK 2 ; K 3 Þ takes the form
r rðK 2 ;K 3 Þ ¼ 0
ð4Þ
The present analysis is not confined to a single transverse crack and its corresponding interfacial cracks (thus forming an Hcrack), since several cracks can be considered in the region of which D ¼ 1. The constitutive equation in the cell, Eq. (3), can be written as
h
rðK 2 ;K 3 Þ ¼ C : ðK 2 ;K 3 Þ DC : ðK 2 ;K 3 Þ þ ð1 DÞChðK 2 ;K 3 Þ
i
ð5Þ
where
D¼
0 no damage
ð6Þ
1 full damage
The last term in the right-hand-side of Eq. (5) can be referred to as eigenstress which is denoted herein by
reðK 2 ;K 3 Þ ¼ DC : ðK 2 ;K 3 Þ þ ð1 DÞChðK 2 ;K 3 Þ ðK 2 ;K 3 Þ
The continuity of displacements u
and tractions t
ð7Þ ðK 2 ;K 3 Þ
between adjacent cells should be imposed. Thus,
ðK ;K Þ ðK þ1;K 3 Þ uðh; x03 Þ 2 3 uðh; x03 Þ 2 ¼0
ð8Þ
ð2Þ ðK 2 ;K 3 Þ ð2Þ ðK 2 þ1;K 3 Þ t ðh; x03 Þ t ðh; x03 Þ ¼0
ð9Þ
where K 2 ¼ M 2 ; . . . ; M 2 1; K 3 ¼ M 3 ; . . . ; M 3 ; l 6 x03 6 l, and
0 ðK 2 ;K 3 Þ 0 ðK ;K þ1Þ uðx2 ; lÞ uðx2 ; lÞ 2 3 ¼0
ð10Þ
ð3Þ 0 ðK 2 ;K 3 Þ ð3Þ 0 ðK 2 ;K 3 þ1Þ t ðx2 ; lÞ t ðx2 ; lÞ ¼0
ð11Þ
where K 2 ¼ M 2 ; . . . ; M 2 ; K 3 ¼ M 3 ; . . . ; M 3 1; h 6 x02 6 h. In the following, the appropriate form of the far-field boundary conditions that specify the tractions and displacements at the opposite sides X 2 ¼ H; X 3 ¼ L of the rectangle of Fig. 1b are presented. The tractions at the opposite sides of the rectangular domain are equal:
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ð2Þ ðM2 ;qÞ ðM2 ;qÞ t ðh; x03 Þ t ð2Þ ðh; x03 Þ ¼ 0; ð3Þ ðp;M3 Þ 0 ðp;M3 Þ 0 ðx2 ; lÞ t ð3Þ ðx2 ; lÞ ¼ 0; t
q ¼ M3 ; . . . ; M 3
229
ð12Þ
p ¼ M 2 ; . . . ; M 2
ð13Þ
The displacements at the opposite sides, on the other hand, differ by certain jumps as follows:
uðM2 ;qÞ ðh; x03 Þ uðM2 ;qÞ ðh; x03 Þ ¼ d2 ; uðp;M3 Þ ðx02 ; lÞ uðp;M3 Þ ðx02 ; lÞ ¼ d3 ;
q ¼ M3 ; . . . ; M 3
ð14Þ
p ¼ M 2 ; . . . ; M 2
ð15Þ
where d2 and d3 denote the vector of the far-field displacement differences which are given by
d2j ¼ 2H2j ;
d3j ¼ 2L3j ;
j ¼ 1; 2; 3
ð16Þ
and 2j ; 2j are the average strains of the unperturbed periodic composite which have to be determined from its effective compliance S and thermal expansion coefficient a tensors, in conjunction with the appropriate imposed traction boundary : conditions r
¼ S r þ a h
ð17Þ
In the present investigation, the effective compliance and thermal expansion coefficient tensors of the (unperturbed) layered composite has been determined by utilizing the exact expressions of the effective moduli that have been established by [14]. 22 is related to the It should be noted that in the case of a loading in the 2-direction (for example) the remote average stress r remote stresses rf22 and rm 22 in the constituents according to
r 22 ¼
df rf22 þ dm rm 22 df þ dm
ð18Þ
The double discrete Fourier of the displacement vector uðK 2 ;K 3 Þ is defined by
^ ðx02 ; x03 ; /p ; /q Þ ¼ u
M2 X
M3 X
uðK 2 ;K 3 Þ ðx02 ; x03 Þ exp iðK 2 /p þ K 3 /q Þ
ð19Þ
K 2 ¼M 2 K 3 ¼M 3
where
/p ¼
2p p ; 2M 2 þ 1
p ¼ 0; 1; 2; . . . ; M2 ;
/q ¼
2pq ; 2M3 þ 1
q ¼ 0; 1; 2; . . . ; M 3 ;
The application of this transform to the boundary problem (4)–(15) for the rectangular domain H < X 2 < H; L < X 3 < L, divided into ð2M2 þ 1Þ ð2M 3 þ 1Þ cells, converts it to the problem for the single representative cell h < x02 < h; l < x03 < l with respect to the complex valued transforms. The field equations obtained from the equilibrium and constitutive equations have the form
^¼0 rr
ð20Þ
r^ ¼ C ^ r^ e
ð21Þ
and
where the transformed eigenstress tensor is given by
r^ e ¼ DC : ^ þ ð1 DÞC^h
ð22Þ
The conditions relating the opposite boundaries of the representative cell in a similar problem were derived by [3]. Following their approach one obtains from (8)–(15)
^ ðh; x03 Þ þ d0;q ð2M 3 þ 1Þd2 expði/p M2 Þ; ^ ðh; x03 Þ ¼ expði/p Þu u ^tðh; x0 Þ ¼ expði/p Þ^tðh; x0 Þ; 3 3
l 6 x03 6 l
l 6 x03 6 l
ð23Þ ð24Þ
and
^ ðx02 ; lÞ ¼ expði/q Þu ^ ðx02 ; lÞ þ d0;q ð2M2 þ 1Þd3 expði/q M 3 Þ; u ^tðx0 ; lÞ ¼ expði/q Þ^tðx0 ; lÞ; 2 2
h 6 x02 6 h
h 6 x02 6 h
ð25Þ ð26Þ
In these equations, dp;q denotes the Kronecker delta. The representative cell boundary value problem (20)–(26) have been solved by employing the higher-order theory, [2]. According to this theory, the domain h 6 x02 6 h; l 6 x03 6 l (the representative cell) is divided into several rectangular subcells. The transformed displacement vector is expanded into a second-order polynomial, and the equilibrium equations,
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interfacial and boundary conditions are imposed in the average (integral) sense. In order to model a (line) crack, a single row of subcells filled with a fully damaged material (D ¼ 1) is introduced. Thus the value of the damage variable D is pre-determined in accordance with the crack configuration. Once the solution in the transform domain has been established, the actual elastic field can be readily determined at any point in the desired cell ðK 2 ; K 3 Þ of the considered rectangular region H 6 X 2 6 H; L 6 X 3 6 L by the inverse transform formula whose form for the displacements, for example, is:
uðK 2 ;K 3 Þ ðx02 ; x03 Þ ¼
M2 M3 X X 1 ^ ðx02 ; x03 ; /p ; /q Þ exp iðK 2 /p þ K 3 /q Þ u ð2M 2 þ 1Þð2M 3 þ 1Þ p¼M q¼M 2
ð27Þ
3
^ e to be In the application of this theory where a stationary or evolving damage is taking place, the eigenstress tensor, r used in Eq. (21) is not known. Hence an iterative solution has to be employed as follows: ^ e ¼ 0 and solve Eqs. 20,21, (23)–(26) in the transform domain. 1. Start by assuming that r 2. Apply the inverse transform formula to compute the stress field. The latter can be employed to compute the current eigenstress reðK 2 ;K 3 Þ in the actual space. 3. Compute the transform of reðK 2 ;K 3 Þ by employing Eq. (22). 4. Solve Eqs. 20,21 and (23)–(26). This procedure should be continued until a convergence to a desired degree of accuracy is achieved.
(a)
(b)
(c)
22 generated in homogeneous material with a crack. (a) Comparison between the present and the exact solution along the Fig. 2. The normal stress r22 =r 22 in the region 1:5 6 X 2 =2h 6 1:5; 1:5 6 X 3 =2l 6 1:5 computed by the present approach. (c) The crack line X 3 at X 2 ¼ 0. (b) The distribution of r22 =r 22 in the region 1:5 6 X 2 =2h 6 1:5; 1:5 6 X 3 =2l 6 1:5 computed by the closed-form expressions. distribution of r22 =r
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231
4. Verifications In all cases given in this paper, the computations of the elastic field were carried out with a square cell h ¼ l, Fig. 1c, which has been divided into 40 40 subcells. Consider a single crack embedded in an infinite homogeneous elastic isotropic plane that is subjected to a remote tensile 22 . The closed-form solution of this problem can be found in [18] and a comparison between this solution and the loading r present approach is given in Fig. 2. The stress r22 along the line of the crack ðX 2 ¼ 0Þ is given by
jX j
3 ; jX 3 j > a r22 ðX 2 ¼ 0; X 3 Þ ¼ r 22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð28Þ
X 23 a2
where 2a is the crack length. Fig. 2a shows a comparison between the normal stress r22 along the crack’s line with 2a=2l ¼ 0:5, as predicted by the present approach and the exact expression (28). Excellent agreement can be readily observed. 22 in the plane It is also possible to compare the present solution approach by comparing the stress field distribution r22 =r ðX 2 X 3 Þ with the exact solution. Fig. 2b and c present the comparison between the present and exact solutions, respectively, in the region 1:5 6 X 2 =2h 6 1:5; 1:5 6 X 3 =2l 6 1:5. Here too excellent agreement exists. Consider next the problem of a transverse crack (i.e., dc ¼ 0) located in the soft layer of a periodically layered ceramic 22 at infinity. The properties of the stiff silicon-carbide and the soft composite that is subjected to a uniform stress loading r lithium-alumino-silicate glass layers are given in Table 1 with the reinforcement volume ratio v f ¼ df =ðdf þ dm Þ ¼ 0:5. The resulting elastic field has been determined by [16] by employing the representative cell method where the Green’s function needs to be establish. A comparison between the normal stress r22 variation along the crack’s line X 3 at X 2 ¼ 0, computed by the present approach and the latter method is shown in Fig. 3. Excellent agreement between the different methods of solution exists.
Table 1 Material constants of the silicon-carbide and lithium-alumino-silicate (LAS) glass layers. Material
E (GPa)
m
a (106 °C1)
Silicon-carbide Lithium-alumino-silicate
200 85
0.19 0.25
2.2 9.5
E; m and a denote the Young’s modulus, Poisson’s ratio and the coefficient of thermal expansion, respectively.
(a)
(b)
22 , computed by the present and the Green’s function approach, along the crack line X 3 at X 2 ¼ 0, of a transverse Fig. 3. (a) The normal stress variation r22 =r 22 . The stiff crack located in the soft layer of a silicon-carbide/glass periodically layered composite that is subjected to a uniform far-field normal loading r 22 , computed along the crack line X 3 at X 2 ¼ 0, of an H-crack with dc =2h ¼ 1, located layer volume fraction is v f ¼ 0:5. (b) The normal stress variation r22 =r 22 . The stiff layer within the soft layer of a silicon-carbide/glass periodically layered composite that is subjected to a uniform far-field normal loading r volume fraction is v f ¼ 0:5.
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5. Applications The developed technique for the cracks modeling is applied to a parametric study of the stress redistribution in the bimaterial periodic composite caused by H-crack fracture patterns. The knowledge of the stress field together with the fracture characteristics of the composite constituents and the interfaces is necessary for the prediction of the failure propagation. The failure may take the form of interfacial crack growth, new transverse crack from the interface in the adjoining uncracked
22 distribution in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2 generated in the silicon-carbide/glass periodically layered Fig. 4. The normal stress r22 =r 22 . The stiff layer volume fractions are: (a) composite with H-cracks (dc =2h ¼ 1). The composite is subjected to a uniform far-field normal loading of r v f ¼ 0:1 and (b) v f ¼ 0:5.
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layer or crack reinitiation in the next layer of the same type as the cracked one. The established stress field allows addressing all these scenarios. The present analysis of the results however concentrates on the last one. Consider the case of a silicon-carbide/glass periodically layered composite with the thermoelastic properties specified in Table 1. One of the soft layers contains a transverse crack branching into two interfacial ones with dc =2h ¼ 1, thus forming an 22 . The dimensions of the rectangular domain for which H-crack. The composite is subjected to a far-field normal loading r the influence of the perturbation induced by the H-crack on the external boundaries is negligibly small are found to be 2H=ðdf þ dm Þ ¼ 21; 2L=ðdf þ dm Þ ¼ 11. This corresponds to a division of the rectangular domain into ð2M 2 þ 1Þ ð2M 3 þ 1Þ cells with M2 ¼ 10 and M 3 ¼ 5. For longer H-cracks and/or higher elastic moduli contrasts, the dimensions of the rectangular region have been appropriately increased. This increase is achieved by increasing the numbers M 2 and M 3 which however does not affect the size of the representative cell domain. 22 is shown in Fig. 4 in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2, in the two The resulting elastic field distribution r22 =r cases of stiff layer volume fractions of v f ¼ 0:1 and 0.5. As expected, the weak layer region which is occupied by the H-crack is unloaded and characterized by very low stresses except at the interface cracks tips. The stress concentration in the adjoining stiff layer just across the interface is lowered with respect to the single transverse crack case (dc ¼ 0). This phenomenon of a shielding effect caused by an interfacial debonding has been mentioned by Chan et al. [5] in a similar problem. The same phenomenon can be also observed by a comparison of Fig. 3a and b. It should be noted that the variation of the periodic stress state for both volume fractions in Fig. 4 is confined to the region of about three periods in X 2 as well as in X 3 directions which justifies the above choice of rectangular domain dimensions. A reinitiation of the transverse crack in the soft layer next to the cracked one is related to the tensile stress distribution. To this end, the normal stress r22 variations within the soft uncracked layer along the interface is examined. The results are presented in Fig. 5 where a family of curves for different interface crack lengths dc =2h ¼ 0; 1; 5; 7; 9; 11 are depicted. It is seen that in the vast majority of the cases the location of the maximal stress is shifted from the transverse crack line X 2 ¼ 0. It should be mentioned that this shifting phenomenon is observed even in the limiting case of a transverse crack (dc ¼ 0) which is in accordance with the known experimental [20] and theoretical [19] results. The dependence of the maximal stress upon the interface crack length dc =2h clearly illustrates the shielding effect of the interfacial debonding. The magnitude of the stress decreases monotonically until it approaches some limiting value for sufficiently large interface cracks. This limiting value represents an important fracture characteristics of the bi-material composite defining the crack resistance in the case of incomplete (damaged) layer. As expected, this value is lower for the thick stiff layers v f ¼ 0:5 than for the case of the thin ones v f ¼ 0:1. Further investigation of this dependency on different values of stiff layer volume fraction is presented in Fig. 6. Here, the maximal stress results for an incomplete layer are shown together with those for a transverse crack (dc ¼ 0). Interestingly, it is found that for the considered layered composites with volume fractions v f > 0:3 the stress concentrations resulting from the transverse crack and incomplete layer are very close. It is obvious that the variations of the stiff layer volf ð1Þ mð1Þ ume fraction v f is accompanied by corresponding changes of the remote stress r22 and r22 in the layers. The latter is given by
rmð1Þ ¼ 22
Em r 22 E2
(a)
(b)
22 generated in the silicon-carbide/glass composite along the interface X 2 within the soft layer, at Fig. 5. The variation of the normal stress r22 =r X 3 ¼ dm =2 þ df for various values of dc of H-cracks. (a) v f ¼ 0:1 and (b) v f ¼ 0:5.
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where E2 ¼ ðdf Ef þ dm Em Þ=ðdf þ dm Þ. Therefore one can obtain the values of max the values presented in Fig. 6 by utilizing the following formula ðmÞ 22 mð1Þ 22
max
r
r
ðmÞ
¼
max r22 r 22
mð1Þ rðmÞ from 22 normalized with respect to r22
r 22 rmð1Þ 22
The influence of the Young’s moduli ratio of the layers on the stress distribution in the transversely cracked and Hcracked composites with stiff layer volume fraction v f ¼ 0:1 is presented in Fig. 7a and b, respectively. As in the previous cases, the normal stresses r22 along the interface in the soft uncracked layer are addressed. The general trend exhibited in this figure is that the increase of the contrast in the elastic properties of the composite constituents leads to the stress decrease. The existence of interface cracks extends the stress concentration region, and somewhat diminishes the maximal value. However, for the considered parameter combinations with relatively short interface cracks, this effect becomes pronounced only in the case of relatively close materials properties Em =Ef > 0:3. This is separately illustrated in Fig. 8 where the
ðmÞ 22 in the soft layer, generated in the silicon-carbide/glass composite along the interface X 2 Fig. 6. The variation of the maxima of the normal stress r22 =r within the soft layer, at X 3 ¼ dm =2 þ df for various values of v f , with dc =2h ¼ 0 and 11.
(a)
(b)
22 generated in composites (v f ¼ 0:1) with various values of Em =Ef along the interface X 2 within the soft Fig. 7. The variation of the normal stress r22 =r layer, at X 3 ¼ dm =2 þ df . (a) dc ¼ 0 and (b) dc =2h ¼ 1.
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dependence of the maximal stresses upon the Young’s moduli ratio is given. For the materials with Em =Ef ¼ 0:6 the stress decrease is about 15%. For a more significant contrast between the materials, a longer interface crack only would provide the shielding effect. In the case of an extended composite’s damage, failure may appear in the form of several layer cracking. The present method of analysis can be readily employed to model such a situation. The results obtained for the case of three cracked soft layers in the silicon-carbide/glass composite with v f ¼ 0:1 are presented in Figs. 9, 10. In the first figure, the normal stress variation along the first intact interface within the soft layer in front of the flaw consisting of three H-cracks is presented. The cracks with dc =2h ¼ 0; 1; 11 are considered corresponding to the cases of a transverse crack, a relatively short (dc =2h ¼ 1) and long interface crack, respectively. It is readily seen that the shielding effect of the interface debonding when several layers are damaged is significantly less pronounced. On the other hand, the stress field distributions shown in Fig. 10a and b for dc ¼ 0 and dc =2h ¼ 1 exhibit very different patterns.
22 generated in composites (v f ¼ 0:1) with various values of Em =Ef along the interface X 2 Fig. 8. The variation of the maxima of the normal stress r22 =r within the soft layer, at X 3 ¼ dm =2 þ df for dc =2h ¼ 0 and 1.
22 generated in the silicon-carbide/glass composite (v f ¼ 0:1) along the interface X 2 within the soft layer, at Fig. 9. The variation of the normal stress r22 =r X 3 ¼ 3dm =2 þ 2df for various values of dc . The composite contains three H-cracks in the soft layers.
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22 distribution in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2 generated in a silicon-carbide/glass periodically layered Fig. 10. The normal stress r22 =r 22 . and stiff layer volume fractions is composite with three H-cracks in the soft layers. The composite is subjected to a uniform far-field normal loading of r v f ¼ 0:1. (a) dc ¼ 0 and (b) dc =2h ¼ 1.
In Fig. 11 the response of the considered layered composite with an H-crack pattern to the thermal loading of h ¼ þ200 C is presented. In the present case of heating, the remote periodic r22 stress field in the soft and stiff layers is compressive and tensile, respectively, which is caused by the mismatch between the coefficients of thermal expansion of the layers, see Table 1. Therefore, the transverse crack in a soft layer will close being not dangerous and the case of a stiff cracked layer
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Fig. 11. The normal stress r22 distribution (measured in GPA units) in the region 2 6 X 2 =2h 6 2; 2 6 X 3 =2l 6 2 generated in a silicon-carbide/glass periodically layered composite with a single H-crack in the stiff layer. The composite is subjected to a thermal loading of 200 °C, with stiff layer volume fractions is v f ¼ 0:5 and dc =2h ¼ 1.
is addressed. As expected, the rectangular region occupied by the H-crack becomes unstressed and stress redistribution leads to the stress concentration in the neighboring stiff layers. 6. Conclusions A method for the simulation of cracks by damage variables has been implemented to investigate the behavior of periodically layered composites. The effect of the cracks (damage) must be localized in the sense that the far-field in the composite should not be affected by their presence. As a result specific far-field boundary conditions must be formulated and implemented. The effect of the cracks appear in the thermoelastic constitutive equations in the form of unknown eigenstresses which necessitates the application of an iterative procedure until a convergence of the solution is achieved. The method has been verified by a comparison with an the exact solution of a crack in a homogeneous medium and for a transverse crack in a periodically layered composite the solution of which can be established by different method that is based on the construction of Green’s functions. The shielding effect of the interface debonding for the soft layers cracking in a periodically layered composite is quantified. It is found to be increasing with the decreasing of the stiff layers volume fraction and with the diminishing of the contrast between the elastic properties of the phases. It appears that when the length of the interface cracks is about more than seven times the layer thickness, the stress concentration in the next soft layer becomes insensitive to the further their extension. This limiting situation corresponds to the case of an incomplete soft layer. The lowering of the stress concentration, as compared to the case of a transverse crack, is more significant for a single incomplete layer than for several ones. It should be mentioned that although the present investigation was confined to layered composites, some of the issues shed light on the brittle fracture behavior of fiber reinforced composites [5]. In these fiber reinforced systems the debond crack as it grows in the vertical direction it also grows in the circumferential direction and eventually surrounds the fiber with another matrix crack appearing in the adjacent uncracked layers. In the present investigation the cracks form a complete debonding in both normal and shear. It is possible however to generalize the present approach to obtain debonding in the shear direction whereas perfect bonding in normal direction is retained. The present analysis can be also applied to instigate periodically layered composites with any system (non periodic) of vertical and horizontal cracks. In addition, it is possible to analyze periodic layered composites with more than two different isotropic and anisotropic materials. In the present paper, layered composites with brittle constituents have been considered. Another important class of layered composites includes metallic layers where plasticity effects should be accounted for. Investigation this type of composites using the present approach seems to be a natural research topic. Finally, the present analysis can be generalized to a system of cracks in periodic fiber-reinforced composites. This three-dimensional
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