The temperature field induced by the dynamic stresses of a transverse crack in periodically layered composites

International Journal of Engineering Science 49 (2011) 732–754

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The temperature field induced by the dynamic stresses of a transverse crack in periodically layered composites 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 14 March 2011 Accepted 30 March 2011 Available online 22 April 2011 Keywords: Periodic layered composites Dynamic stresses Temperature field Wave propagation in composites Coupled heat equation Transverse cracks Thermomechanical coupling

a b s t r a c t The temperature field induced by the dynamic application of a far-field mechanical loading on a periodically layered material with an embedded transverse crack is investigated. To this end, the thermoelastically coupled elastodynamic and energy (heat) equations are solved by combining two approaches. In the first one, the dynamic representative cell method is employed for the construction of the time-dependent Green’s functions generated by the displacement jumps along the crack line. This is performed in conjunction with the application of the double finite discrete Fourier transform on the thermomechanically coupled equations. Thus the original problem for the cracked periodic composite is reduced to the problem of a domain with a single period in the transform space. The second approach is based on wave propagation analysis in composites where full thermomechanical coupling in the constituents exists. This analysis is based on the coupled elastodynamic-energy continuum equations where the transformed time-dependent displacement vector and temperature are expressed by second-order expansions, and the elastodynamic and energy equations and the various interfacial and boundary conditions are imposed in the average (integral) sense. The time-dependent thermomechanically coupled field at any observation point in the plane can be obtained by the application of the inverse transform. Results along the crack line as well as the full temperature field are given for cracks of various lengths for Mode I and Mode II deformations. In particular the temperature drops (cooling) at the vicinity of the crack’s tip and the heating zones at its surroundings are generated and discussed. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The generation of heat near the tip of a propagating crack has been discussed by Broberg (1999) where extensive list of references has been presented. In most of these investigations, the effect of thermoplasticity on the heat generation by running cracks in metals is considered. The heat generated by moving cracks in a glassy polymer has been experimentally studied by Fuller, Fox, and Field (1975). Weichert and Schonert (1978) modeled the crack tip of a propagating crack as a moving heat source of rectangular shape thus predicting temperatures of the order of 1000 K in brittle material (glass). Rittel (1998) emphasized the thermoelastic cooling effect near the tip of the crack. By assuming adiabatic conditions, he obtained a temperature drop of about 32 °C for both PMMA and steel. In addition, Rittel (1998) carried out experimental investigations for the measurement of the cooling effect at the tip of the crack that is embedded in a brittle material, and observed that this effect is followed by a temperature rise. A combined numerical-experimental treatment of the transient thermoelastic effect near the tip of a cracked PMMA subjected to a dynamic loading, has been presented by Bougaut and Rittel (2001), where in ⇑ Tel.: +972 3 6408131; fax: +972 3 6407617. E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.03.020

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the employed finite element numerical procedure, finite conductivity has been assumed. In the framework of this procedure, the dilatation is estimated from the solution of the mechanical equations, after which the coupled heat equation with the already calculated dilatational term is solved. They concluded that the cooling effect prevails during crack initiation and that adiabatic conditions can be reasonably assumed. Asymptotic near crack tip temperatures for propagating cracks with full thermomechanical coupling expressed in terms of the corresponding stress intensity factors, have been presented by Atkinson and Craster (1992). In a previous article (Aboudi & Ryvkin, 2011), the stresses created by the dynamic loading of periodically layered material with an embedded crack have been predicted. This problem is a generalization to the dynamic case of the many investigations of fiber reinforced materials with several broken fibers that are subjected to a static loading. These problems are quite complicated due to the loss of periodicity caused by the crack’s existence, but have been received considerable attention due to their engineering importance. In the static case, their analyses are usually based on the shear lag approximation (see Beyerlein & Landis, 1999 for a review), or a finite element procedure (see Nedele & Wisnom, 1994, for example). It should be mentioned that in some of these investigations, the authors employed two-dimensional models of layered composites to get insight about the behavior of cracked fiber reinforced composites. More recently, two and three-dimensional continuum approach has been offered by Ryvkin and Aboudi (2007, 2008) for the analysis of a layered material with several broken layers, and a fiber reinforced material with several broken fibers, respectively, which are subjected static loadings. The time-dependent analysis of Aboudi and Ryvkin (in press) is based on the combination of two approaches. In the first one the representative cell method (which was originally formulated in the static case by Ryvkin & Nuller (1997)) has been generalized to the dynamic case. In the framework of the dynamic representative cell method, the crack problem is modeled by a superposition of weighted unit displacement jump time-dependent solutions, everyone of which forms a Green’s function. The method is based on the double finite discrete Fourier transform which allows the reduction of the problem of the periodic domain to a finite one in the time-dependent transform space. The second approach upon which the transformed time-dependent field variables is solved, is based on transient wave propagation analysis that was originally offered by Aboudi (1987, 1988) but was restricted to wave motions that possess certain types of symmetry. In the special case of multi-layered composites, the accuracy of its predictions was verified in Aboudi (1987, 1988) by comparisons with ray theory solutions. Furthermore, in this latter situation of multi-layered composites, extensive applications and verifications of the accuracy of the model were presented by Clements, Johnson, and Hixon (1996), Clements, Johnson, and Addessio (1997), Clements, Johnson, Addessio, and Hixon (1997) Clements, Johnson, and Addessio (1998), including comparisons with measured data. The analysis of Aboudi (1987) has been generalized by Aboudi and Ryvkin (in press) to accommodate any type of wave motion irrespective whether symmetry exists or not. In the framework of this analysis, the transformed time-dependent displacement vector is expressed by a second order expansion in terms of local coordinates. The elastodynamic equations and the relevant interfacial and the so called Born–von Karman type boundary conditions result in a system of second-order ordinary differential equations which are solved at any time by an incremental procedure. The actual time-dependent elastic field is obtained by the inverse transform formula. The accuracy of the solution has been verified by comparisons with the dynamic stresses of a crack embedded in homogeneous material in Mode I and Mode III deformations for which analytical solutions are available. In the present investigation, the fully coupled elastodynamic and the energy (heat) equations are solved in the presence of a crack embedded in a layered composite. To this end, the isothermal wave propagation analysis that was presented in Aboudi and Ryvkin (in press), is presently generalized to include the full thermomechanically coupled equations. The materials are assumed to be brittle such that any irreversible (plasticity) effects are neglected. The composite is subjected to the sudden application of far-field loadings which generate either Mode I or Mode II deformations. As a result of the present fully coupled thermomechanical analysis, the temperature drops (cooling) created at the vicinity of the crack’s tip and the heating zones induced at its surroundings are generated. The presented applications illustrate the induced temperature field caused by the applied far-field mechanical loading of the composite with various crack lengths. The corresponding analysis that is based on the adiabatic assumption is assessed. The present paper is organized as follows. In Section 2, the dynamic solution of a transverse crack in a homogeneous material with two-way thermoelastic coupling is derived. To this end, time-dependent Green’s functions that correspond to unit jumps applied at the crack’s line are generated. For a layered composite with a transverse crack, these functions are appropriately superposed in Section 3 to obtain the sought thermoelastically coupled solution at any time step. In Section 4, the generalized theory for two-dimensional wave propagation in composites with full thermoelastic coupling, which is applicable to general types of loadings, is presented. This is followed by the Applications section where results for transverse cracks in a periodically layered material which is subjected to remote tensile and axial shear loadings are given, which provide the induced temperature field caused by Mode I and Mode II deformations, respectively. In the conclusion section, several generalizations to the present approach are offered.

2. The analysis of the problem of sudden appearance of a crack in a homogeneous plane In order to illustrate our approach for the analysis of the time-dependent elastic field and the induced temperature created by the dynamic stresses of a transverse crack in a periodically layered material, let us consider first the problem of the sudden appearance of a crack in an infinite thermoelastic plane that is subjected to a constant remote loading that generates

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Mode I and Mode II deformation. These dynamic crack problems are solved by combining the representative cell method and a wave propagation in composites theory. The solutions are achieved by constructing for every mode type a set of Green’s functions as described in the following. 2.1. Modeling of a crack in an infinite plane using unit jump dynamic Green’s functions  2m , Consider a homogeneous thermoelastic isotropic plane that is subjected to a remote tensile or in-plane shear loading r m = 2, 3, see Fig. 1(a), where (in this section) identical material properties should be attributed to both the stiff and soft constituents. A traction-free crack suddenly appears at time t = 0 and the generated transient stress and temperature fields are sought. The crack region a 6 x3 6 a is divided into N segments whose length is D = 2a/N, on everyone of which a unit displacement jump at time t = 0 is applied. The location of the middle point of each segment is denoted by xi3 , i = 1, . . . , N. As a result, N   elastic field distributions are generated. Let G xi3 ; xj3 ; t denote the time-dependent stress components r2k at x3 ¼ xi3 and x2 = 0 caused by the application of the unit displacement jump at segment j at time t = 0. In order to model a traction-free surface of a crack in the region a 6 x3 6 a these Green functions are superposed in the following form N X

   2m ; ci ðtÞG xi3 ; xj3 ; t ¼ r

j ¼ 1; . . . N

ð1Þ

i¼1

where m = 2, and 3 correspond to loading in Mode I and Mode II, respectively. This forms a linear system of N algebraic equations for the determination of the time-dependent coefficients ci(t), i = 1, . . . , N. Once these coefficients have been determined, the elastic field U(x2, x3, t) at any point of the cracked plane can be determined at time t as follows

Uðx2 ; x3 ; tÞ ¼

N X

ci ðtÞU i ðx2 ; x3 ; tÞ þ U 0 ðx2 ; x3 Þ

ð2Þ

i¼1

where Ui(x2, x3, t) is the elastic field that generated at point (x2, x3) and time t by the application of the displacement jump at segment i and U0(x2, x3) is the corresponding constant elastic field at this point generated by the applied loading at infinity. 2.2. The determination of the dynamic Green’s functions The determination of these Green’s functions is obtained by the combined use of the representative cell method and wave propagation in thermoelastic composites theory. In accordance with the representative cell method the homogeneous plane is viewed as an assemblage of bonded identical cells labeled by the two indices (K2, K3) where K2, K3 = 0, ±1, ±2, . . . , see Fig. 1(b) in which, as mentioned above, the existence of the stiff layers should be ignored. The infinite plane is described with   respect to global coordinates (x2, x3). In addition, in each cell local coordinates x02 ; x03 are introduced whose origin is located in its center, see Fig. 1(c). The thermoelastic field generated by a unit displacement jump, applied along the segment D=2 6 x03 6 D=2 at x02 ¼ 0, at time t = 0 in cell (0, 0) is sought. The formulation of this plane problem is presented as follows. In the absence of body forces, the elastodynamic equations in any cell (K2, K3) are given by



rjk;k

ðK 2 ;K 3 Þ

ðK 2 ;K 3 Þ jk

 ðK 2 ;K 3 Þ €j ¼q u ;

j; k ¼ 1; 2; 3;

K 2 ; K 3 ¼ 0; 1; 2; . . .

ð3Þ

ðK ;K Þ uj 2 3

where r and are the stress and displacement components, q is the mass density of the material and a dot denotes a time differentiation. The stress components of the thermoelastic material are given by

½rjk ðK 2 ;K 3 Þ ¼ C jklm ½lm ðK 2 ;K 3 Þ  ChðK 2 ;K 3 Þ dj;k

ð4Þ

where Cjklm are the elements of the stiffness tensor, C = 3Ka is the thermal stress (where K and a are the bulk modulus and ðK ;K Þ thermal expansion coefficient of the material) and dj,k is the Kronecker delta, In Eq. (3), jk 2 3 and hðK 2 ;K 3 Þ are the strain tensor components and temperature deviation from a reference temperature T0, respectively. The energy (heat) equation for the temperature, with the coupling term for the heating caused by volumetric deformation rate is given by

qcv h_ ðK 2 ;K 3 Þ ¼ ½qk;k ðK 2 ;K 3 Þ  T 0 C½_ kk ðK 2 ;K 3 Þ

ð5Þ

where cv and qk are the specific heat and heat flux k-component, respectively. The latter is given according to Fourier’s law by

 ðK 2 ;K 3 Þ @h ½qk ðK 2 ;K 3 Þ ¼ j @xk

ð6Þ

with j being the thermal conductivity of the material. The strains are related to the displacement gradients in the standard form

½jk ðK 2 ;K 3 Þ ¼

 ðK ;K Þ 1 @uj @uk 2 3 þ 2 @xk @xj

ð7Þ

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

735

 2j , j = 2, 3, at infinity. (b) Fig. 1. (a) A periodically layered composite with an embedded crack normal to the layering subjected to constant remote stress r The infinite plane is divided by repeating cells, labeled by (K2, K3) the size of every one of which is H  L. (c) A characteristic cell (K2, K3) in which local    ðbÞ ðc Þ coordinates x02 ; x03 are introduced. This cell is discretized into Nb  Nc subcells. (d) A typical subcell (bc) in which local system of coordinates  x2 ;  x3 is introduced the origin of which is located at the center. The size of the subcell is hb  lc.

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In addition, continuity of displacements, tractions, normal heat fluxes and temperature between adjacent cells should be imposed. Thus,



 ðK 2 ;K 3 Þ 

 ðK 2 þ1;K 3 Þ L L uj x02 ; x03 ¼ ; t  uj x02 ; x03 ¼  ; t ¼ 0; 2 2





 ðK 2 ;K 3 Þ 

 ðK 2 ;K 3 þ1Þ H H  uj x02 ¼  ; x03 ; t ¼ 0; uj x02 ¼ ; x03 ; t 2 2 



L 2

r2j x02 ; x03 ¼ ; t





H 2

 ðK 2 ;K 3 Þ

 

r3j x02 ¼ ; x03 ; t

 ðK 2 ;K 3 Þ

 



L 2

r2j x02 ; x03 ¼  ; t

H 2

 ðK 2 þ1;K 3 Þ

r3j x02 ¼  ; x03 ; t

H H 6 x02 6 2 2

ð8Þ

L L 6 x03 6 2 2

ð9Þ



¼ 0;



H H 6 x02 6 2 2

ð10Þ

L L 6 x03 6 2 2

ð11Þ

 ðK 2 ;K 3 þ1Þ ¼ 0;



 ðK 2 ;K 3 Þ 

 ðK 2 þ1;K 3 Þ L L  q2 x02 ; x03 ¼  ; t ¼ 0; q2 x02 ; x03 ¼ ; t 2 2







 ðK 2 ;K 3 Þ 

 ðK 2 ;K 3 þ1Þ H H q3 x02 ¼ ; x03 ; t  q3 x02 ¼  ; x03 ; t ¼ 0; 2 2

H H 6 x02 6 2 2

ð12Þ

L L 6 x03 6 2 2

ð13Þ





 ðK 2 ;K 3 Þ 

 ðK 2 þ1;K 3 Þ L L h x02 ; x03 ¼ ; t  h x02 ; x03 ¼  ; t ¼ 0; 2 2



H H 6 x02 6 2 2

ð14Þ



 ðK 2 ;K 3 Þ 

 ðK 2 ;K 3 þ1Þ H H h x02 ¼ ; x03 ; t  h x02 ¼  ; x03 ; t ¼0 2 2



L L 6 x03 6 2 2

ð15Þ

where j = 1, 2, 3. The applied unit displacement jump in the x02 -direction at the segment D=2 6 x03 6 D=2 at x02 ¼ 0 can be represented by the relations

  0þ 0   ðK 2 ;K 3 Þ 0 uj x2 ; x3 ; t  uj x0 ¼ dK 2 ;0 dK 3 ;0 Hv ðtÞ 2 ; x3 ; t



D D 6 x03 6 2 2

ð16Þ

where Hv(t) is the Heaviside step function and j = 2 and j = 3 correspond to loadings in Mode I and Mode II, respectively. The infinite plane can be replaced by a sufficiently large finite rectangular domain with an embedded central crack such that the reflected waves from its boundaries do not affect the stress field at the observation region of interest. The finite domain consists of M2  M3 cells such that Kp = 0, ±1, ±2, . . . , ±Mp with p = 2, 3. The formulated problem, Eqs. (3)–(16) is reduced to a single cell problem by the application of the shifted finite Fourier transform: M2   X ^ j x02 ; x03 ; /2ðm2 Þ ; /3ðm3 Þ ; t ¼ u

M3 X

 ðK 2 ;K 3 Þ  0 x2 ; x03 ; t

uj

h i ðM Þ ðM Þ  exp i K 2 /2 2 þ K 3 /3 3

ð17Þ

K 2 ¼M2 K 3 ¼M3

where pÞ /ðM ¼p p

2pmp ; 2M p þ 1

mp ¼ 0; 1; 2; . . . ; M p ;

p ¼ 2; 3

As a result, we obtain a representative cell problem for the transforms of the field variables in the finite region H=2 6 x02 6 H=2; L=2 6 x03 6 L=2. The governing equations in this region that correspond to Eqs. (3)–(16) are

r^ jk;k ¼ q

d

2

dt

2

^j ; u

j; k ¼ 1; 2; 3

r^ jk ¼ C jklm ^lm  C^hdj;k ; j; k; l; m ¼ 1; 2; 3 qcv

d^ d ^k;k  T 0 C ^kk ; h ¼ q dt dt

^k ¼ j q

^jk ¼

@ ^h ; @xk

 ^j @ u ^k 1 @u ; þ 2 @xk @xj

ð18Þ ð19Þ ð20Þ

ð21Þ

ð22Þ

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754



 L L ^j x02 ; x03 ¼  ; /2 ; /3 ; t ¼ 0; ^ j x02 ; x03 ¼ ; /2 ; /3 ; t  expði/3 Þu u 2 2



 H H ^ j ðx02 ¼  ; x03 ; /2 ; /3 ; tÞ ¼ 0; ^ j x02 ¼ ; x03 ; /2 ; /3 ; t  expði/2 Þu u 2 2



L 2



L 2



H H 6 x02 6 2 2



H 2





H 2

ð24Þ

H 2

H 2

ð25Þ

L 2

L 2

ð26Þ

r^ 2j x02 ; x03 ¼ ; /2 ; /3 ; t  expði/3 Þr^ 2j x02 ; x03 ¼  ; /2 ; /3 ; t ¼ 0;  6 x02 6

ð23Þ

L L 6 x03 6 2 2



r^ 3j x02 ¼ ; x03 ; /2 ; /3 ; t  expði/2 Þr^ 3j x02 ¼  ; x03 ; /2 ; /3 ; t ¼ 0;  6 x03 6



 L L ^2 x02 ; x03 ¼  ; /2 ; /3 ; t ¼ 0; ^2 x02 ; x03 ¼ ; /2 ; /3 ; t  expði/3 Þq q 2 2





 H H ^3 x02 ¼ ; x03 ; /2 ; /3 ; t  expði/2 Þq ^3 x02 ¼  ; x03 ; /2 ; /3 ; t ¼ 0; q 2 2



 ^h x0 ; x0 ¼ L ; / ; / ; t  expði/ Þ^h x0 ; x0 ¼  L ; / ; / ; t ¼ 0; 2 3 3 2 3 2 3 2 3 2 2



 ^h x0 ¼ H ; x0 ; / ; / ; t  expði/ Þ^h x0 ¼  H ; x0 ; / ; / ; t ¼ 0; 2 3 2 2 3 2 3 2 3 2 2

H H 6 x02 6 2 2

ð27Þ

L L 6 x03 6 2 2

ð28Þ





737

H H 6 x02 6 2 2

ð29Þ

L L 6 x03 6 2 2

ð30Þ



The displacements jump conditions given by Eq. (16) take the form

  0 ^ 0 0 ^ j x0þ u 2 ; x3 ; /2 ; /3 ; t  uj ðx2 ; x3 ; /2 ; /3 ; tÞ ¼ H v ðtÞ;



D D 6 x03 6 2 2

ð31Þ

It should be noted that all field variables in the transform domain (hat-quantities) form complex quantities. Eqs. (23)–(30) refer to as the Born-von Karman type boundary conditions. It is readily seen that one needs to solve Eqs. (18)–(31) for the representative cell where the identity of the cell (K2, K3) disappeared. The solution of this problem in the transform domain at any time t is carried by employing the analysis of wave propagation in thermoelastic composite materials which will be described in the next section. In the framework of this theory, the representative cell domain H=2 6 x02 6 H=2; L=2 6 x03 6 L=2 is divided into rectangular subcells, Fig. 1(c), in everyone of which the transformed displacements and temperature are represented by second-order polynomials. The transformed elastodynamic equations, displacements, tractions, heat fluxes and temperature continuity conditions between the subcells are imposed in the average (integral) sense. Once this solution has been achieved, the actual elastodynamic and temperature fields at the relevant time can be readily determined at any point of the considered rectangular region (M2 + 1/2)H 6 x2 6 (M2 + 1/2)H, (M3 + 1/2)  L 6 x3 6 (M3 + 1/2)L by the inverse transform formula whose form for the displacements, for example, is:

 ðK 2 ;K 3 Þ  0 x2 ; x03 ; t

uj

¼

1 ð2M 2 þ 1Þð2M3 þ 1Þ

M2 X

M3 X

h  i ^ j ðx02 ; x03 ; /2ðm2 Þ ; /3ðm3 Þ ; tÞ exp i K 2 /2ðm2 Þ þ K 3 /3ðm3 Þ u

ð32Þ

m2 ¼M2 m3 ¼M 3

3. Crack in a periodically layered composite with full thermoelastic coupling The analysis of layered composites with a transverse crack as shown in Fig. 1(a), in which the widths of the stiff and soft layers are denoted by df and dm, respectively, is carried out in the same manner as for a crack in an infinite homogeneous plane. The thermoelastic stiff and soft constituents are assumed to be isotropic with Young’s moduli Ef, Em and Poisson’s ratios mf, mm. This problem represents a two-dimensional model of a layered composite with broken layers. In the case of a homogeneous plane, both dimensions of the representative cell (L and H) defined by the translational symmetry of the plane were arbitrary. In the present case, however, the translational symmetry in the x3-direction is determined by the periodic layers arrangement. It is convenient to locate the vertical boundaries of the cells (K2, K3) at the middle line between the stiff layers as shown in Fig. 1(b). In addition, since each cell (K2, K3) will include two types of materials (stiff and soft) the stiffness matrix Cjklm, which is independent on (K2, K3), is not uniform within the cells. Accordingly, the representative cell in the transform domain is divided into subcells. The stiff and soft layers are incorporated by a suitable filling of the subcells by the their material properties. It should be noted that in the case of a crack in a homogeneous plane all Green’s functions calculated from applied displacement jumps at different segments are identical except for a shift in the x3-direction. In the presence of the stiff and soft layers this property does not exist. Consequently, one has to calculate all these Green’s functions that correspond to displacement jumps applied over all segments. Furthermore, in the present case of layered

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

 2m is redistributed in the stiff and soft constituents. The exact expressions in the stiff composites, the average far-field stress r and soft far-field stresses can be determined in terms of the material parameters, geometrical properties of the stiff and soft phases and the type of the applied remote loading.  22 (Mode I), these expressions can be well approximated in conjunction with the For an applied remote tensile loading r mixture’s law by

rf22ð1Þ ¼

Ef r 22 ; E

rmð1Þ ¼ 22

Em r 22 E

ð33Þ

where

E2 ¼ ðdf Ef þ dm Em Þ=ðdf þ dm Þ being the effective Young’s modulus of the (uncracked) doubly periodic layered composite in the x2-direction. For an applied  23 (Mode II), the continuity of r32 in the stiff and soft phases yields that remote axial shear loading r

 23 ; rf23ð1Þ ¼ r 23 ; rmð1Þ ¼r 23

ð34Þ

Consequently, Eq. (1) takes the form N X

  ci ðtÞG xi3 ; xj3 ; t ¼

(

r

i¼1

f ð1Þ

in the stiff layers

mð1Þ 2m

in the soft layers

r2m

ð35Þ

with j = 1, . . . , N and m = 2 and 3 for crack deformation in Mode I and Mode II, respectively. Several distinct situations must be addressed. Consider first the case in which the crack is located within the cell (K2 = 0, K3 = 0) (i.e., 2a < L). Here the number of Green’s functions NG at and time t is equal to the number of segments, namely NG = N = 2a/D. In the extreme case of a crack extending over the entire cell (i.e., 2a = L), NG = N = L/D. On the other hand, when the crack extends beyond this cell (i.e., 2a > L) the number of Green’s functions is still equal to NG = L/D and N = 2a/D. This is because due to the translational symmetry of the composite the following equality holds

    G xi3 ; xj3 ; t ¼ G xi3 þ nL; xj3 þ nL; t

ð36Þ

where n = 0, ±1, ±2, . . . Thus any value of the dynamic Green’s function generated by jumps applied outside the cell (K2 = 0, K3 = 0) can be expressed in terms of the values obtained from the jumps applied within this latter cell. It should be emphasized that the present investigation, based on a continuum approach, enables the determination of the thermoelastic field anywhere in the stiff and soft layers without the utilization of any simplifying assumption regarding the constitutive relations of the materials and the governing elastodynamic and energy equations. 4. The analysis of two dimensional wave propagation in composite materials with full thermoelastic coupling In a previous investigation, (Aboudi & Ryvkin, 2011), a method of solution for the two-dimensional elastodynamic equations has been presented for the analysis and prediction of the response of multiphase elastic materials that are subjected to arbitrary impulsive loading. This method of solution resulted into a system of second-order ordinary differential equations in the transform domain, which can be solved by an explicit incremental procedure in time. It is not necessary to present here the details of this analysis because the addition of the thermal stress terms (cf., Eq. (4)) do not add further difficulties. It is necessary however to analyze and solve the energy Eq. (5) which is considered in the following. As in the previous investigation, the case of two-dimensional wave propagation is considered such that there is no dependence of any field variable on the direction x1. In this two-dimensional theory all variables depend only on the in-plane coordinates in addition to the time t. As discussed in Section 2, the elastodynamic problem formulated in the x2  x3-plane can be reduced, in conjunction with the finite discrete Fourier transform, to the representative cell problem in the x02  x03 domain. The representative cell region H=2 6 x02 6 H=2; L=2 6 x03 6 L=2 is divided into Nb  Nc subcells with b = 1, . . . , Nb,   ðcÞ are local coordinates whose origin is located at the center of the subcell (bc), see c = 1, . . . , Nc. In addition, xðbÞ 2 ; x3 Fig. 1(d). Like the displacement field in subcell (bc), the temperature deviation is approximated by a second-order expansion   ðbÞ ðcÞ in the local coordinates  x2 ;  x3 as follows (hereafter the ‘hat’ above the variables has been omitted): ðbcÞ

h

¼

ðbcÞ hð00Þ

þ

ðbcÞ xðbÞ 2 hð10Þ

þ

cÞ x3ðcÞ hðb ð01Þ

ðbcÞ

! ! 2 2 lc ðbcÞ hb ðbcÞ 1 1 ðbÞ2 ðcÞ2 h h 3x2  3x3  þ þ 2 4 ð20Þ 2 4 ð02Þ

ð37Þ

where the time-dependent hð00Þ is the area average temperature in the subcell which together with the higher-order timeðbcÞ dependent terms hðmnÞ ; (m + n > 0); must be determined.

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The resulting heat flux components in subcell (bc) are given by ðbcÞ

q2

ðbcÞ

q3

  ðbcÞ ðbÞ ðbcÞ ¼ jðbcÞ hð10Þ þ 3x2 hð20Þ   ðbcÞ ðcÞ ðbcÞ ¼ jðbcÞ hð01Þ þ 3x3 hð02Þ

ð38Þ

By averaging Eq. (5) over the area of subcell (bc), the following relation is obtained

    ðbcÞ ðbcÞ _ ðbcÞ þ W _ ðbcÞ ðqcv ÞðbcÞ h_ ðbcÞ ¼ 3jðbcÞ hð20Þ þ hð02Þ  T 0 CðbcÞ W 2ð10Þ 3ð01Þ ðbcÞ

ð39Þ

ðbcÞ

where W 2ð10Þ and W 3ð01Þ are displacement microvariables, see Aboudi and Ryvkin (in press). In Aboudi and Ryvkin (in press) the surface-average tractions and displacements have been defined. Here the surfaceaverage of the temperature and heat fluxes are similarly defined: ð2Þ ðbcÞ

h

1 lc

¼

1 hb

¼

1 lc

ð3Þ ðbcÞ

h

ð2Þ ðbcÞ

qk

ð3Þ ðbcÞ

qk

¼

 hb ðbÞ ðcÞ hðbcÞ x2 ¼  dx3 2 lc =2

ð40Þ

 lc ðcÞ ðbÞ hðbcÞ x3 ¼  dx2 2 hb =2

ð41Þ

 hb ðcÞ xðbÞ dx3 ¼  2 2

ð42Þ

 l x3ðcÞ ¼  c dxðbÞ 2 2

ð43Þ

Z

¼

lc =2

Z

Z

hb =2

lc =2

ðbcÞ

qk

lc =2

1 hb

Z

hb =2

hb =2

ðbcÞ

qk

Substitution of the temperature expansion (37) in Eq. (40) and (41) reveals that these surface-average displacements are reðbcÞ lated to the microvariables hðmnÞ as follows 2

ð2Þ ðbcÞ

ðbcÞ

¼ hð00Þ 

h

ð44Þ

2

ð3Þ ðbcÞ

h

hb ðbcÞ hb ðbcÞ þ h h 2 ð10Þ 4 ð20Þ

ðbcÞ

¼ hð00Þ 

lc ðbcÞ lc ðbcÞ þ h h 2 ð01Þ 4 ð02Þ

ð45Þ

Manipulation of every pair in these equations results in the following ðbcÞ hð10Þ

1 ¼ hb

ðbcÞ hð01Þ

1 ¼ lc

"

"

ð2Þ þðbcÞ

h

ð2Þ ðbcÞ



ð3Þ þðbcÞ

ð46Þ

h

ð3Þ ðbcÞ



h

#

# ð47Þ

h

Table 1 Elastic and thermal parameters of the isotropic glass layer. E (GPa)

m

a (106/K)

k (W/(mK))

q (kg/m3)

cv (J/(kg K))

72

0.2

12

0.89

2800

800

E, m, a, k and q and cv denote the Young’s modulus, Poisson’s ratio, coefficient of thermal expansion, thermal conductivity, mass density and the specific heat at constant volume, respectively. The compressional and shear wave speed are: cGp ¼ 5300 m=s and cGs ¼ 3200 m=s, respectively, and the thermomechanical coupling coefficient is d = 0.0035.

Table 2 Elastic and thermal parameters of the isotropic epoxy layer. E (GPa)

m

a (106/K)

k (W/(mK))

q (kg/m3)

cv (J/(kg K))

3.45

0.33

54

0.18

1218

1050

E, m, a, k and q and cv denote the Young’s modulus, Poisson’s ratio, coefficient of thermal expansion, thermal conductivity, mass density and the specific heat at constant volume, respectively. The compressional and shear wave speed are: cEp ¼ 2060 m=s and cEs ¼ 1030 m=s, respectively and thermomechanical coupling coefficient is d = 0.014.

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

ðbcÞ

hð02Þ ¼

"

ð2Þ þðbcÞ

2

ðbcÞ

hð20Þ ¼

h

2

hb 2

ð2Þ ðbcÞ

þ

"

ð3Þ þðbcÞ

2

lc

h



ð3Þ ðbcÞ

þ

#

h #



h

4 2

hb

4 2

lc

ðbcÞ

hð00Þ

ð48Þ

ðbcÞ

ð49Þ

hð00Þ

Substitution of Eqs. (48) and (49) in (39) yields

ðq

ðbcÞ cv ÞðbcÞ h_ ð00Þ

ðbcÞ

þ 12j

1 2

hb

þ

1 2

lc

!

( ðbcÞ hð00Þ

where the surface-average displacements

ðbcÞ

¼ 6j

1

"

ð2Þ þðbcÞ

ð2Þ ðbcÞ

#

1

"

ð3Þ þðbcÞ

ð3Þ ðbcÞ

#)

h þ h h þ h þ 2 2 hb lc   1 d ð2Þ þðbcÞ ð2Þ ðbcÞ 1 d ð3Þ þðbcÞ ð3Þ ðbcÞ þ u2  u2 u2  u2  CðbcÞ T 0 hb dt lc dt

ð2Þ ðbcÞ

u2

and

ð3Þ ðbcÞ

u3

ð50Þ

have been defined in Aboudi and Ryvkin (in press).

(a)

Fig. 2. (a) The variations of the induced normalized (with respect to the epoxy properties) temperature deviation ~ h along the crack line x2 = 0, caused by the  22 far-field of the cracked monolithic epoxy material at time: cEp t=2a ¼ 0:5; 1; 1:5 and 2. (b) The corresponding distribution of ~ application of r h in the cracked epoxy in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cEp t=2a ¼ 1.

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

Fig. 2 (continued)

This differential equation can be solved by employing an implicit difference in time, (Mitchell & Griffiths, 1980), yielding

" ðbcÞ

ð2Þ þðbcÞ

ðbcÞ

hð00Þ ðt þ DtÞ ¼ X 1 hð00Þ ðtÞ þ X 2

ð2Þ ðbcÞ

hðtÞ þ hðtÞ

#

"

ð3Þ þðbcÞ

þ X3

ð3Þ ðbcÞ

# þ X4

hðtÞ þ hðtÞ

" # " # d ð2Þ þðbcÞ ð2Þ ðbcÞ d ð3Þ þðbcÞ ð23Þ þðbcÞ u2 ðtÞ  u2 ðtÞ þ X 5 u3 ðtÞ  u3 ðtÞ dt dt ð51Þ

where

1 X1 ¼ X0 X2 ¼

( ðqcv Þ

6jðbcÞ Dt 2

hb X 0

ðbcÞ

 6 Dt j

X3 ¼

;

ðbcÞ

1 2

hb

6jðbcÞ Dt 2

lc X 0

þ

;

1 2

lc

!) ;

X4 ¼ 

ðbcÞ

X 0 ¼ ðqcv Þ

CðbcÞ T 0 Dt hb X 0

;

ðbcÞ

þ 6 Dt j

X5 ¼ 

1 2

hb

þ

1 2

lc

! ;

CðbcÞ T 0 Dt

ð52Þ

lc X 0

and Dt is a time increment. The surface-average definitions of the heat fluxes in Eqs. (42) and (43) yield, in conjunction with Eq. (38) that ð2Þ ðbcÞ

q2

ð3Þ ðbcÞ

q3

 3hb ðbcÞ ðbcÞ ¼ jðbcÞ hð10Þ  hð20Þ ; 2

 3l c ðbcÞ ðbcÞ ¼ jðbcÞ hð01Þ  hð02Þ 2

ð53Þ

By employing Eq. (46)–(49), the following expressions for the heat fluxes at time t + Dt can be established ð2Þ ðbcÞ

q2

ð3Þ ðbcÞ

q3

ðbcÞ

¼

¼

jðbcÞ hb

jðbcÞ lc

" ð1  3Þ " ð1  3Þ

ð2Þ þðbcÞ

h

ð2Þ ðbcÞ

þð1  3Þ

ð3Þ þðbcÞ

h

h

ð3Þ ðbcÞ

þð1  3Þ

where hð00Þ ðt þ DtÞ is given by Eq. (51).

#

h



6jðbcÞ ðbcÞ h hb ð00Þ

ð54Þ



6jðbcÞ ðbcÞ hð00Þ lc

ð55Þ

#

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

(a)

(b)

Fig. 3. (a) The variations of the induced normalized (with respect to the glass properties) temperature deviation ~ h along the crack line x2 = 0, caused by the  22 far-field of the cracked monolithic glass material at time: cGp t=2a ¼ 0:5; 1; 1:5 and 2. (b) The corresponding distribution of ~ application of r h in the cracked G glass in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cp t=2a ¼ 1.

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

743

These two equations can summarized in the compact form

8 9ðbcÞ ð2Þ þ > > > q > 2 > > > > > > > > > < ð2Þq  > = 2

ð3Þ þ > > > > > q3 > > > > > > > > ð3Þ : þ> ; q3

¼ jðbcÞ

8 4  > > > hb > > < h2 b

> 0 > > > > : 0

 h2b

0

4 hb

0

0

 l4c

0

2 lc

8 9 8 6 9 ð2Þ þ >ðbcÞ > > 0 >> hb > > > h > > > > > > > > > > > > ð2Þ  > > > > > < 6 0 =< h = hb þ jðbcÞ ð3Þ þ > 6 > >  l2c > > > > > > h > > > >> lc > > > > > > > > > ; 4 > : 6 : ð3Þ  > ; lc lc h

ðbcÞ 9 hð00Þ > > > > ðbcÞ > = h > ð00Þ

ðbcÞ > hð00Þ > > > > ; ðbcÞ > hð00Þ

ð56Þ

ðbcÞ

where b = 1, . . . , Nb, c = 1, . . . , Nc and values of hð00Þ are given by Eq. (51). The continuity of temperature and heat flux between the subcells are given by ð2Þ þðbcÞ

ð2Þ ðbþ1;cÞ

¼

h ð3Þ þðbcÞ

ð2Þ þðbcÞ

q2

ð3Þ þðbcÞ

q3

b ¼ 1; . . . ; Nb  1;

c ¼ 1; . . . ; Nc

ð57Þ

b ¼ 1; . . . ; Nb ;

c ¼ 1; . . . ; Nc  1

ð58Þ

;

b ¼ 1; . . . ; Nb  1;

c ¼ 1; . . . ; Nc

ð59Þ

;

b ¼ 1; . . . ; Nb ;

c ¼ 1; . . . ; Nc  1

ð60Þ

;

ð3Þ ðb;cþ1Þ

¼

h

h

¼ ¼

h ð2Þ ðbþ1;cÞ

q2

ð3Þ ðb;cþ1Þ

q3

h;

(a)

Fig. 4. (a) The variations of the induced normalized (with respect to the glass properties) temperature deviation ~ h along the crack line x2 = 0, caused by the  22 far-field of the cracked glass/epoxy layered material with a broken glass layer at time: cGp t=L ¼ 0:5; 1; 1:5 and 2. (b) and (c) The application of r corresponding temperature distributions ~ h in the cracked glass/epoxy composite in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cGp t=L ¼ 0:5 and 1, respectively.

744

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

Fig. 4 (continued)

ð2Þ ðbcÞ ð3Þ ðbcÞ

In the present implementation of this theory in the complex transform domain there are 8NbNc unknowns: h , h . On the other hand, the interfacial temperature and heat flux conditions (57)–(60) provide 4(NbNc  Nc) and 4(NbNc  Nb) relations. Finally, the Born-von Karman type boundary conditions (27)–(30) provide 4Nb + 4Nc relations. Thus the total number of equations is 8NbNc. In conclusion, the method of solution of the mechanical dynamic equations that has been presented in Aboudi and Ryvkin (in press) (to which the thermal effects need to be incorporated) together with method of solution of the coupled energy equation as presented in this section, enable the solution of the entire time-dependent thermomechanical system of Eqs. (18)–(30), in conjunction with (31), in the transform domain. The actual solution is obtained by employing the inverse transform (cf. Eq. (32)).

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

745

5. Applications The developed theory is applied herein to investigate the induced temperature by the thermomechanical coupling caused by the dynamic mechanical loading of a glass/epoxy layered composite with a transverse crack. The thermoelastic properties of the isotropic glass and epoxy layers are given in Tables 1 and 2, respectively. The compressional and shear wave speeds in the glass are denoted by cGp and cGs , whereas in the epoxy they are denoted by cEp and cEs . The standard (non-dimensional) thermomechanical coupling coefficient of the material is given by

d¼

Eð1 þ mÞa2 T 0 ð1  2mÞð1  mÞqcv

ð61Þ

As expected, the values of d given in Tables 1 and 2, indicate that the thermomechanical coupling of the epoxy is higher, which implies that the induced temperature caused by the mechanical loading should be more significant than that in the glass. All results given in the following were computed for a glass volume ratio of df/(df + dm) = 0.52. The results presented in the following show the induced non-dimensional temperature deviation ~ h which is given by:

~h ¼ qcv h aT 0 r 22

ð62Þ

(a)

Fig. 5. (a) The variations of the induced normalized (with respect to the glass properties) temperature deviation ~ h along the crack line x2 = 0, caused by the  22 far-field of the cracked glass/epoxy layered material with a half broken epoxy layer at time: cGp t=L ¼ 0:5; 1; 1:5 and 2. (b) and (c) The application of r corresponding temperature distributions ~ h in the cracked glass/epoxy composite in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cGp t=L ¼ 0:5 and 1, respectively.

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

Fig. 5 (continued)

where q, cv and a corresponds to the material properties with respect to which the normalization has been performed, and T0 = 300 K. It should be noted that the same level of discretization has been applied to obtain the results in the following. Therefore it is possible to make comparisons between the magnitudes of the temperature in the various considered cases. 5.1. Temperature induced by Mode I deformation Fig. 2(a) shows the induced temperature deviations in the monolithic cracked epoxy material caused by the gradual appli 22 that commences at time t = 0 and reaches its maximum value at time cEp t=2a ¼ 0:5. These temperacation of a far-field r tures (normalized with respect to the epoxy material constants) are shown along the crack’s line (i.e. along x3-axis) at: cEp t=2a ¼ 0:5; 1; 1:5 and 2. As expected, the maximum values are obtained at the closest point to the crack tip, i.e., at x3/

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

747

L = 0.02 (at the crack’s tip the temperature is theoretically singular). The maximum temperature drop ~ h ¼ 70 is obtained at  22 ¼ 100 MPa can be esticEp t=2a ¼ 0:5. With a fracture toughness KIc = 0.6 MN/m3/2 and a crack size a = 10 lm, a far-field of r mated (e.g., Kundu, 2008). The induced temperature deviation by this far-field is: h =  88.7K which is quite cold. The temperature deviation however decreases rapidly away from the crack’s tip as well with increasing time. The distribution of the induced temperature ~ h is shown in Fig. 2(b) at time cEp t=2a ¼ 1 in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1. The cold spots in the vicinity of the crack’s tip and the hot regions above and below the crack’s plane can be well observed. The maximum values of the temperature exhibited in these hot regions at time cEp t=2a ¼ 0:5 and 1 are: ~ h ¼ 270 and 24, respectively, and continue to decay with increasing time approaching the value of ~ h ¼ 8 at cEp t=2a ¼ 2. As expected, the induced temperature in the monolithic cracked glass is far less appreciable. This is shown in Fig. 3(a) where the variation of ~ h (normalized with respect to the glass thermoelastic parameters) caused by the application of a  22 are shown at: cGp t=2a ¼ 0:5; 1; 1:5 and 2. Fig. 3(b) shows the corresponding ~ far-field r h distribution in the plane of the crack. The maximum value of temperature drop ~ h ¼ 25 obtained at cGp t=2a ¼ 0:5 at the closest point to the tip of the crack corresponds to an induced temperature deviation of h = 4 K which results in a relatively quite moderate cooling effect as compared to the epoxy polymer. Similarly, the maximum values of the hot spots which are detected above and below the crack’s line at time cGp t=2a ¼ 0:5 and 1 are: ~ h ¼ 40 and 12, respectively.

(a)

Fig. 6. (a) The variations of the induced normalized (with respect to the glass properties) temperature deviation ~ h along the crack line x2 = 0, caused by the  22 far-field of the cracked glass/epoxy layered material with a broken epoxy layer at time: cGp t=L ¼ 0:5; 1; 1:5 and 2. (b) and (c) The application of r corresponding temperature distributions ~ h in the cracked glass/epoxy composite in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cGp t=L ¼ 0:5 and 1, respectively.

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

Fig. 6 (continued)

Thus far, the effect of mechanical loading on the temperature field has been investigated in cracked homogeneous materials. In the next three figures, Figs. 4–6, the induced temperature field is investigated in the cracked layered glass/epoxy composite. In Fig. 4(a), the temperature deviation variation ~ h (normalized with respect to the glass thermoelastic properties) along the crack line and its distribution in the plane are shown for the case referred to as broken glass layer. Here the crack extends along df 6 x3 6 df/2 rendering a broken glass layer. It is already observed that at time cGp t=L ¼ 0:5 at which the ap 22 reaches its maximum value, a significant value ~ plied far-field r h ¼ 161 is detected at the closest point to the crack tip.  22 ¼ 100 MPa this value corresponds to a temperature drop of 26 K. In comparison to For an applied far-field loading of r the induced temperature in the plain glass, this high value reflects the effect of the neighboring epoxy layer. As in the previous cases, the induced temperature decreases with time increase. Fig. 4(b) and (c) show the entire temperature distribution ~ h in the plane at time cGp t=L ¼ 0:5 and 1. The cooling effect at the vicinity of the crack’s tip and the heating in the surrounding

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

area is well exhibited. The maximum values of the temperature rise are ~ h ¼ 88 and 29 at cGp t=L ¼ 0:5 and 1, respectively, whereas along the crack’s line itself the temperature is negligible. The next considered case is when the crack extends along (df + dm)/2 6 x3 6 (df + dm)/2 which is referred to as half broken epoxy layer with a crack’s length of 2a/L = 1. The resulting temperature deviations are shown in Fig. 5(a) along the crack line at several times. It can be observed that at time cGp t=L ¼ 0:5 the cooling effect takes place at the glass–epoxy interface behind the crack tip. Ahead of the crack tip, on the other hand, there is relatively slight temperature drop. This cooling effect moves further toward the crack’s tip as the time increases and is most significant at time cGp t=L ¼ 1:5, and at time cGp t=L ¼ 2 the temperature drop passes the crack’s tip. The temperature distributions at time cGp t=L ¼ 0:5 and 1 are shown in Fig. 5(b) and (c). Fig. 5(b) shows that the coldest regions ~ h ¼ 657 are detected not ahead and behind the crack’s tips (i.e., at a/ L = ±0.5), but at the vicinity of glass–epoxy interface above and below the crack’s line at x2/L = ±0.1, x3/L = ±0.3 minor hot regions can be observed. Fig. 5(c) on the other hand exhibits completely different picture. Here hot spots of ~ h ¼ 226 can be observed which are quite severe, but with less cold regions. The previously discussed case forms an intermediate situation between a broken glass layer and that of a crack which is arrested by the stiff glass layer. In this latter final considered case, the crack extends along (df/2 + dm) 6 x3 6 (df/2 + dm) and is referred to as broken epoxy layer with a crack’s length 2a = df + 2dm. The temperature variations along the crack’s line at various times in Fig. 6(a) and its distribution in the plane exhibit a quite complex pattern. As in the previous case, the cooling region appears behind the crack’s tip at time cGp t=L ¼ 0:5 and 1, but it advanced toward the tip as the time increases. The corresponding temperature distributions at cGp t=L ¼ 0:5 and 1 are shown in Fig. 6(b) and (c). Here both cold and hot spots can be observed above and below the crack’s line. At the vicinity of the crack’s tip x3/L = ±0.75, neither cold nor hot spots can be observed. In Table 3, a summary of the maximum values of the temperature drops and elevations that are induced at various times in the glass/epoxy composite with different transverse crack lengths is presented. 5.2. Temperature induced by Mode I deformation – adiabatic conditions In order to estimate the induced temperature in the vicinity of the crack, it is often assumed that the thermoelastic process is adiabatic namely, that the thermal conductivity of the material is zero. This approach has been followed by Rittel (1998) for the estimation of the amount of cooling in the vicinity of a crack embedded in PMMA polymer. The adiabatic assumption, stems from the difficulty of analyzing the full thermomechanically coupled elastodynamic and energy equations. It should be interesting to investigate the accuracy of the adiabatic assumption by comparing the induced temperature deviations predicted by the full thermomechanical coupling and by the adiabatic assumption. With the adiabatic assumption, Eq. (5) provides the following expression for the temperature deviation

had ¼ 

aT 0 ðr þ r22 þ r33 Þ qcv 11

ð63Þ

where r11 + r22 + r33 should be computed directly from the solution of the isothermal (uncoupled) elastodynamic equations (Aboudi & Ryvkin, 2011). Consequently, the corresponding normalized temperature ~ had (to be compared with ~ h that is predicted by the fully coupled equations) is given according to Eq. (62) by

~had ¼  ðr11 þ r22 þ r33 Þ r 22

ð64Þ

For the plain epoxy for which the induced temperature deviation has been discussed in Fig. 2(a), a cooling temperature deviation h = 88.7 K at the closest point to the crack’s tip, induced by the application of a far-field normal stress

Table 3 Maximum temperature drops and elevations induced at various times in the glass/epoxy composite with different transverse crack lengths. cGp t=L

~ hmin

Broken glass layer 0.5 1.0 2.0

161 65 64

88 29 41

Half broken epoxy layer 0.5 1.0 2.0

657 103 54

51 226 116

Broken epoxy layer 0.5 1.0 2.0

472 147 85

788 419 164

~ hmax

750

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

Fig. 7. Comparisons between the variations of the induced normalized (with respect to the glass properties) temperature deviation ~ h and ~ had along the crack line x2 = 0 at time cGp t=L ¼ 0:5. (a) and (b) Broken glass layer, (c) and (d) half broken epoxy layer, (e) and (f) broken epoxy layer.

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

751

Fig. 8. Comparisons between the variations of the induced normalized (with respect to the glass properties) temperature deviation ~ h and ~ had along the crack line x2 = 0 at time cGp t=L ¼ 2. (a) and (b) Broken glass layer, (c) and (d) half broken epoxy layer, (e) and (f) broken epoxy layer.

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

r 22 ¼ 100 MPa at time cEp t=2a ¼ 0:5, has been obtained. By assuming an adiabatic process, the corresponding induced temperature is had = 100 K which forms an error of 13%. The error of assuming an adiabatic process in the glass/epoxy layered composite can be investigated by examining Figs. 7 and 8. These figures show comparisons between the induced temperature deviations ~ h and ~ had along the crack’s line for broken glass layer, half broken epoxy layer and broken epoxy layer at cGp t=L ¼ 0:5 and 2. It can be readily observed that the adiabatic assumption overestimate the induced temperature but, as expected, the error decreases as the time increases. Let us examine the difference between ~ h ¼ 161 and ~ had ¼ 245 for broken glass layer at cGp t=L ¼ 0:5 at the closest point to the crack’s tip, see Fig. 7(a) and (b). The corresponding temperature deviations generated by a far field of 100 MPa are: h = 26 K and had = 40 K which forms a significant error of 54%. The maximum value of the elevated temperatures generated in the regions above and below the crack is ~ h ¼ 88 which is almost identical to ~ had as predicted on the basis of adiabatic assumption. As to the broken epoxy layer, an error of about 100 percent between ~ h and ~ had can be observed, see Fig. 7(e) and (f).

(a)

(b)

~ along the crack line x2 = 0, Fig. 9. (a) Mode II. The variation of the induced normalized (with respect to the epoxy properties) temperature deviation h  23 far-field of the cracked monolithic epoxy material at time: cEs t=2a ¼ 1. (b) The corresponding distribution of ~ caused by the application of r h in the cracked epoxy in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cEs t=2a ¼ 1.

J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

753

5.3. Temperature induced by Mode II deformation It should be interesting to examine the effect of full thermomechanical coupling on the induced cracked materials when they are subjected to Mode II deformation. Here, the non-dimensional temperature deviation ~ h is obtained from h according to

~h ¼ qcv h aT 0 r 23

ð65Þ

which replaces Eq. (62). Fig. 9(a) shows the variation of the temperature deviation ~ h for a cracked monolithic epoxy along the crack line x2 = 0 at time cEs t=2a ¼ 1, whereas Fig. 9(b) exhibits the entire distribution in the plane at this same time. The antisymmetric patterns can be observed in both figures. In the vicinity of the two tips hot and cold equal temperatures exist. For a far-field loading of

(a)

(b)

Fig. 10. (a) Mode II. The variation of the induced normalized (with respect to the glass properties) temperature deviation ~ h along the crack line x2 = 0,  23 far-field of the cracked glass/epoxy layered material with a broken glass layer at time: cGs t=L ¼ 1. (b) The corresponding caused by the application of r distribution of ~ h in the cracked glass/epoxy composite in the region 1 6 x2/L 6 1, 1 6 x3/L 6 1 at time cGs t=L ¼ 1.

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J. Aboudi / International Journal of Engineering Science 49 (2011) 732–754

r 23 ¼ 100 MPa, the maximum absolute value of these temperatures is h = 12.6 K. This temperature deviation is quite low as compared to the 88.7 K cooling that has been previously observed in Mode 1 deformation. Finally, Fig. 10 shows the induced temperature deviation ~ h along the crack’s line at cGs t=2a ¼ 1 and its distribution at this time in a glass/epoxy layered composite with a broken glass layer. Here too, the antisymmetric patterns of the temperature can be immediately observed. In the vicinity of the crack’s tip a temperature deviation of h = ±1.6 K caused by a far-field load 23 ¼ 100 MPa is detected. This is compared to h = 26 K that was previously obtained when the composite was subing of r jected to Mode I deformation. It is interesting to note that the effect of finite thermal conductivities in the present case of Mode II deformations does not seem to be of significance. This implies that an adiabatic assumption may be sufficient in this type of deformations. 6. Conclusions Full (two way) thermomechanicaly coupled analysis has been established from the solution of the elastodynamic and energy equations. As a result, the temperature field in the plane of a transverse crack embedded in a periodically layered composite can be generated. This field is induced by the dynamic application of far-field mechanical loadings that correspond to Mode I and Mode II deformations. The locations and values of the temperature drops in the vicinity of the crack and the elevated temperatures in its surrounding were generated. The temperature fields in the plane of the crack embedded in homogeneous materials and in layered glass/epoxy composites with a broken glass layer, half broken and broken epoxy layers were determined. It turned out that as compared to the other cases, the values of the generated temperature by the thermomechanical coupling at the cold and hot spots in the composite with a broken epoxy layer (the crack is arrested by the stiff glass layer) are significant. It has been shown that unlike the induced temperature results in cracked monolithic materials, the cold spots are not necessarily located at the immediate vicinity of the tip of the crack. The adiabatic assumption has been shown not to be valid for Mode I deformations at short times after the application of the far-field loading. In the present investigation plasticity effects have been ignored so that the derived coupled thermomechanical analysis is applicable to brittle materials. Plasticity effects would significantly reduce the induced temperatures at the vicinity of the crack’s tips. A generalization to accommodate these effects is a subject for a future research. 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