International Journal of Solids and Structures 42 (2005) 6335–6355

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A numerical study on interface crack growthunder heat flux loading

Ashwin Hattiangadi, Thomas Siegmund *

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907, United States

Received 9 July 2004; received in revised form 25 May 2005Available online 10 August 2005

Abstract

A numerical framework for the coupled thermo-mechanical analysis of crack growth in structures loaded by anapplied heat flux is outlined. Using a thermo-mechanical cohesive zone model (TM-CZM), load transfer behavior iscoupled to heat conduction across an interface and the corresponding interface crack. Nonlinear effects occur due tothe coupling between the mechanical and thermal problem introduced by the conductance–separation response betweencrack faces as well as through the temperature dependence of material constants of the CZM. The description of theload transfer behavior uses a traction–separation law with an internal residual property variable that characterizesthe extent of damage caused by mechanical loading. The description of thermal transport includes a representationof the breakdown of interface conductance with increase in material separation. The current state of interface failure,the presence of gas entrapped in the crack, the radiative heat transfer across the crack and a contact conductancebetween crack faces determine the cohesive zone conductance. The TM-CZM is implemented into a finite element codeand applied in the study of interface crack growth between an oxidation protection coating on a thermal protectionmaterial.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Crack growth; Heat transfer; Interface; Cohesive zone

1. Introduction

In the design of engineering components for high temperature applications high heat flux loading is ofpotential concern. Examples of relevant engineering structures are thermal protection systems for spacevehicles (Dimitrienko, 1999; Columbia Accident Investigation Report, 2003), actively cooled first wall

0020-7683/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijsolstr.2005.05.050

* Corresponding author. Tel.: +1 765 494 9766; fax: +1 765 496 7536.E-mail address: siegmund@ecn.pudue.edu (T. Siegmund).

6336 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

fusion reactor liners (Tivey et al., 1999), shell structures in propulsion systems (Brockmeyer, 1993;John et al., 1999), and high performance brakes (Krenkel and Henke, 1999). Heat fluxes of up to 150MW/m2 are expected to occur in certain applications (Tivey et al., 1999). Under thermal loading cracksand delaminations are discontinuities in the temperature field, impede heat flow and subsequently redistrib-ute temperature. As a consequence thermal stresses are induced that can lead to failure of the structureunder consideration.

Early investigations on the failure behavior of cracked structures loaded by temperature fields were con-ducted by Florence and Goodier (1959, 1963) and Sih (1962). These studies were concerned with conditionsof steady state and remote uniform heat flux, and assumed fully isolated cracks in infinite homogeneousisotropic bodies. The energy release rate for the mode II crack tip field present in such a case is proportionalto the square of the temperature gradient present. These result, and subsequent others by Sih and Bolton(1986), Barber (1979) and Chen and Huang (1992), demonstrated that the energy release rate associatedwith thermal effects could indeed be substantial. Later, Hasebe et al. (1986), Lee and Shul (1991), Kokiniand Reynolds (1991), Itou (1993), Yan and Ting (1993), Chao and Chang (1994), and Lee and Park (1995),extended the linear-elastic fracture mechanics analyses to cracks at bi-material interfaces, as well as to arbi-trary cracks geometries. More importantly in context of the present analyses, crack heat flux was includedinto the analyses by Sih and Chen (1986) and by Barber and Comninou (1983), Martin-Moran et al. (1983),Kuo (1990) and Hutchinson and Lu (1995). While these past results showed a significant dependence of theenergy release rate on the crack conductance, only rather simple models for the crack conductance wereused. The crack conductance was commonly assumed as constant, not only along the entire crack but alsoindependent of crack opening, and rarely based on physically motivated input. Furthermore, only station-ary cracks were considered such that sequential solutions of heat transfer and stress analysis were sufficient.If, however, integrated multi-physics life cycle predictions of critical components are the goal realistic fail-ure criteria allowing for coupled thermo-mechanical simulations including crack growth are needed. Theaim of the present study is to propose a coupled thermo-mechanical model for material failure that simul-taneously considers changes in the load transfer process together with changes in heat transfer as a result ofdamage, crack initiation, crack growth, crack opening, and crack face contact.

For studies of crack growth the cohesive zone model approach offers many advantages (Elliot, 1947;Barenblatt, 1962; Needleman, 1987). In this method, a broad range of physical processes of material sep-aration can be incorporated within a single numerical method. Two constitutive equations, a volumetric(hardening) constitutive equation and a cohesive (softening) constitutive equation, describe the solid underconsideration. The parameters in the traction–separation law, cohesive strength and cohesive energy,are characteristic of the underlying material separation processes. A length-scale, i.e., the cohesive energydivided by cohesive strength, is part of the constitutive equation.

Most past applications of cohesive zone models were directed towards the analysis of mechanical loadingonly. Only few studies are available where cohesive zone models were used in coupled problems. That is, thethermo-mechanical contact formulations of Zavarise et al. (1992a) and Pantuso et al. (2000), and the hydro-gen diffusion—interface debonding model of Liang and Sofronis (2003). Furthermore, the present model isrelated to previous work by Hattiangadi and Siegmund (2004) on a thermo-mechanical crack bridgingmodel.

For the coupled thermo-mechanical analyses, including the creation of new free surfaces due to crackgrowth and the concurrent changing heat transfer conditions, a description of the cohesive zone conduc-tance is needed. For the present study the focus is on cohesive zone conductance laws capturing the energytransport across the material in front of the current crack tip, across the fracture process zone, as well asacross the already formed crack. Energy dissipation due to the fracture process itself (Costanzo andWalton, 2002; Gurtin, 1979) is neglected. Under the heat flux loading considered the amount of energy dis-sipated by the material separation process is small relative to the energy input from external sources. It isreasonable to assume a temperature dependence of the cohesive zone properties (e.g., Jin and Batra, 1998;

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6337

Costanzo and Walton, 2002). On the other hand, material deterioration taking place in the cohesive zoneleads to changes in the stress carrying capability and the cohesive zone conductance. Both effects introducecoupling between the thermal and the stress analysis.

In the present study interface crack growth is studied in a thermal protection system based on a carbon–carbon (C–C) composite substrate and a protective coating. Several approaches have been proposed to pro-tect C–C composites from oxidation (Tsou and Kowbel, 1995; Wu et al., 1996; Isola et al., 1998; Cairoet al., 2001). Here, a silicon carbide (SiC) coating is considered. Interface delaminations emerging from sur-face cracks need to be avoided as they increase the oxidation and burn-out of the C–C composite substrate(Jacobson et al., 1999).

2. The thermo-mechanical cohesive zone model

2.1. General considerations

Linear-elastic fracture mechanics describes the stresses, r, at the tip of a crack in an elastic solid sub-jected to mechanical loads by a square-root singularity, r � K=

ffiffir

p, with r the distance from the crack

tip. The stress intensity factor, K � r1ffiffiffia

p, characterizes the load with r1 the remote stress and a the crack

length. In the cohesive zone model approach to fracture the stress singularity is removed by introducing aprocess zone in place of the ideal crack tip. There, a constitutive law motivated by the actual physical pro-cess of material separation correlates the material separation D to cohesive surface tractions, Tcz. Materialseparation is related to the displacement field through D = (Dnn + Dtt) = u+ � u�. Here, (+,�) representthe two cohesive surfaces, u+ and u� are displacements of corresponding points on the opposing cohesivesurfaces, and n and t are the normal and tangential direction characterizing the geometry of the cohesivesurface. The traction vector Tcz possesses components Tn and Tt.

Similarly, in the presence of an applied thermal load, the heat flux per unit area, q, at an ideal crack tip ischaracterized by a square-root singularity, q � H 0=

ffiffir

p(Sih, 1965, and also Tzou, 1990). The heat flux inten-

sity factor, H 0 � �mffiffiffia

p ðoh=oyÞ1, depends on the remote applied temperature gradient in the direction per-pendicular to the crack, (oh/oy)1, and the crack length. Furthermore, the magnitude of the thermaldisturbance induced by the crack, i.e., the temperature jump across the crack relative to the applied tem-perature gradient, is characterized as the parameter m. In the thermo-mechanical cohesive zone model ap-proach proposed here the singularity in the heat flux is removed again by considering a process zone insteadof an ideal crack tip. Cohesive zone constitutive laws describing heat transfer across the interface, the pro-cess zone, as well as across the crack are needed. Regardless of the physical process, heat transfer is as-sumed to occur only in the normal direction of the local cohesive coordinate system. The heat flux perunit area across the cohesive zone thus is qcz = qczn. Furthermore, h+ and h� are the absolute temperaturesat two corresponding locations on opposing cohesive surfaces, respectively. The temperature jump acrossthe cohesive zone is defined as Dh = h+ � h�, and the temperature of the cohesive zone as hcz = (h+ + h�)/2.With k+ and k� the thermal conductivities of the material below and above the crack the heat flux densitiesgoing out of the cohesive surfaces are q+ and q�, respectively:

qþ ¼ kþohon

����þ¼ qcz; q� ¼ k�

ohon

�����¼ �qcz. ð1Þ

Heat transfer across the cohesive zone then is described as

qcz ¼ hczðhþ � h�Þ ¼ hczDh. ð2Þ

The cohesive zone conductance hcz is a positive number and is defined in its dependence on temperature,displacement jump and cohesive zone tractions.

6338 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

In the present study the thermo-mechanical constitutive relationships for the cohesive zone are devel-oped for the case of crack propagation along a bi-material interface.

2.2. Load transfer

The load transfer between the cohesive surfaces is defined through the dependence of the cohesive sur-face traction, Tcz, on the displacement jump across the cohesive surface, D, and the average temperature ofthe cohesive zone, hcz.

The traction–separation model used to characterize the interface uses a bi-linear relationship between thetraction vector Tcz and the displacement jump vector D. This model is based on the use of an internal resid-ual property variable, s (Geubelle and Baylor, 1998). The internal residual property variable records themonotonic increase in damage and enforces irreversibility in the material separation process. The normaland tangential components of Tcz, Tn and Tt, are given in dependence of the normal and tangential dis-placement jumps across the crack by

T n ¼s

1� sDn

drmaxðhczÞ and T t ¼

s1� s

Dt

drmaxðhczÞ if s > 0. ð3Þ

With this formulation, the cohesive tractions Tcz are parallel to the displacement jump vector D. This con-dition was shown in Costanzo (1998) as a necessary requirement for this type of model. The parameters ofthe cohesive zone constitutive equation are the cohesive strength, rmax, and cohesive length, d. The internalresidual property variable is a measure of the normalized displacement jumps across the cohesive zone:

s ¼ min½sini;maxð0; 1� kÞ� with k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDn

d

� �2

þ Dt

d

� �2s

. ð4Þ

The quantity k is the norm of the non-dimensional displacement jump vector. The residual property var-iable is assigned an initial value of sini close to unity and reaches zero when complete loss of load carryingcapacity has occurred at the location of consideration within the cohesive zone. The condition s = 0 definesthe current location of the crack tip. For isothermal conditions, the cohesive strength and cohesive lengthallow one to define the cohesive energy C, i.e., the fracture toughness as

Cðhcz ¼ constantÞ ¼ 12sinirmaxðhczÞd. ð5Þ

Thermal softening of the cohesive zone is accounted for through a temperature dependence of the cohe-sive strength, rmax. A linear decay of rmax with temperature and a residual value of rmax at high temper-ature, rmax,res, is assumed:

rmax ¼ max ðrmax;0 � krhczÞ; rmax;res½ �; ð6Þ

with rmax,0 the interface strength at zero temperature, kr the corresponding temperature coefficient, andrmax,res the residual strength at high temperature.

In the traction–separation law of Eq. (2) a value of sini close to unity provides traction–separation func-tions with high initial stiffness. Since the material degradation is coupled to the heat transfer problem, it isimportant to predict a realistic shape of the crack opening profile. A high initial stiffness of the traction–separation law results in a crack opening profile with significant opening in the wake of the crack tip only.From a physical perspective, an initial value of sini < 1 is interpreted as the result of the presence of a certainamount of processing related porosity at the coating–substrate interface (Friedrich et al., 2002).

Already formed crack surfaces start to interact with each other once the normal separation is lessthan the combined surface roughness of the crack faces, 2x. A frictionless soft contact model is definedto describe this interaction:

Fig. 1.condit

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6339

pc ¼ 0 if 0 < Dn < 2x;

pc ¼ kcDn if Dn < 0.ð7Þ

The contact penalty factor, kc, is taken to be equal to the initial stiffness of the traction–separation law inpure normal separation:

kc ¼rmax;0

dð1� siniÞ: ð8Þ

The isothermal traction–separation response is shown in Fig. 1(a) and (b) for normal and tangential trac-tions for a range of mode mixities, w = tan�1(Dt/Dn). During mixed mode loading, the resulting peak trac-tion values are considerably lower than the values under pure normal or tangential separation. Theisothermal normal traction–separation response at a low temperature, hcz = h0, and at a high temperature,hcz = h1 is compared to the non-isothermal case in Fig. 1(b). A linear temperature increase from h0 to h1 isimposed concurrently with the increase in Dn. The thermal softening leads to an additional loss of load car-rying capacity of the interface.

The traction–separation law: (a) normal and (b) tangential response for a range of values of mode mixity and isothermalions, (c) normal traction–separation response including thermal softening.

6340 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

2.3. Heat transfer

A model for the heat transfer across the cohesive zone is introduced. As the present study is concernedwith the situation of an applied thermal gradient in the direction perpendicular to the crack plane, heattransfer in the direction normal to the cohesive zone dominates over heat flow along the cohesive zone.Consequently, heat flow along the cohesive zone is neglected in the development of the cohesive zonemodel. A thermal circuit representing this model contains two branches, one for the interface and onefor the interface crack (Fig. 2). The formulation of the cohesive zone conductance is connected to themechanical part of the cohesive zone model by the quantities s, sini, Dn, and pc. Through the residual prop-erty variable the cohesive zone conductance model is irreversible with respect to the mechanical loads.

For part of the cohesive zone representing an intact or partially intact segment of the interface, s > 0, thecohesive zone conductance is given by a linear combination of contributions representing conductanceacross solid–solid bonds between the materials adjacent to the interface, hint, and through—potentiallygas filled—porosities at the interface:

hcz ¼ hintsþ kg;0 þ Cghcz� � ð1� sÞ

max½n;Dn�. ð9Þ

The introduction a lower bound for the normal material separation, min(Dn) = n, regularizes the formula-tion of the interface conductance and ensures that the contribution of the gas to hint is small compared tothat of the solid–solid bonds for as long as s � sini. From a physical perspective the parameter n representsthe value of the average initial diameter of the porosities found at interfaces between composite substratesand their oxidation protection coatings (Friedrich et al., 2002). As material separation progresses damage isincurred across the interface the interface conductance decreases from its initial value, hint while simulta-neously the contribution of gas is introduced. The cohesive zone conductance is temperature dependentthrough the temperature dependent thermal conductivity of the gas, kg = kg,0 + Cghcz, as well as throughthe effect of temperature on the material separation.

At s = 0, the interface has failed and solid–solid bonds no longer contribute to the conductance acrossthe cohesive zone. The cohesive zone conductance for an open crack with normal separation larger than theaverage surface roughness of the crack faces, 2x, is given by conduction through the gas present in thecrack and radiative heat transfer between the cohesive surfaces:

hcz ¼ hgðD; hCZÞ þ hrðhþ; h�Þ ¼½kg;0 þ Cghcz�

Dnþ Sb

2=em � 1ðhþ þ h�Þ½ðhþÞ2 þ ðh�Þ2� if Dn > 2x. ð10Þ

hghint hr

θ -

hg+hc

s>0 s=0

hg

θ+

Fig. 2. Thermal resistor model for the cohesive zone conductance.

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6341

The contribution to the cohesive zone conductance due to radiation, hr, is dependent on the crack surfacetemperatures, the Stephan Boltzman constant, Sb = 5.67 · 10�8 W/m2 K4, an assumed emissivity of thecrack surfaces of, em = 1.0. While hr was considered in all computations, the contribution to the specimenresponse was found to be small.

Eq. (9) holds as long as the normal displacement jump, Dn, is larger than the average surface roughnessof crack faces, 2x. If the normal displacement jump is less than the combined surface roughness of thecrack faces, Dn 6 2x, a thermal contact conductance model is employed in analogy to the mechanical con-tact model in Eq. (7). When two crack surfaces are in contact the real contact is always concentrated at thetop zones of the surfaces asperities. Contact conductance then is a combination of a localized pressuredependent spot conductance between contacting surface asperities and the conductance through the gasin the cavities formed by contacting asperities. The contact conductance between solid contacts dependson the contact pressure, pc, the effective height of the gas filled asperities, the asperity slope, #, and the prop-erties of the solids adjacent to the cohesive zone (Mikic, 1974; Zavarise et al., 1992b). In the present studythe effective height of the gas filled cavities is taken to be constant and equal to the combined surface rough-ness, 2x. This assumption is made as the expected contact pressures, Eq. (7), between crack surfaces aresmall since the external loading does not include an externally applied pressure. In summary, the contactconductance model is written as

0.0

0.1

0.2

0.3

0.4

0.5

-2.0 0.0 2.0 4.0 6.0 8.0 10.0

∆n /ω

(1) (2) (3)

hC

Z/h

int

nt

YX

ψψψψ θ+

θ−

0.00.0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.0

1.0∆∆∆n/ δδ

ψψ =75o

ψ =15o

ψ =0o

ψ =25oψ =60o

ψ =45o

nt

YX

ψ θ+

θ−

hC

Z/h

int

b

a

Fig. 3. (a) Interface conductance breakdown for loading under different values of mode mixity sini = 0.9 and, (b) crack conductance independence of material separation in the presence of gas.

6342 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

hCZ ¼ 1.55keq tan#

x

� � ffiffiffi2

ppc

Eeq tan#

" #1.06

þ ½kg;0 þ Cghcz�2x

if Dn < 2x; ð11aÞ

1=Eeq ¼ ð1� m2cÞ=Ec þ ð1� m2s Þ=Es; 1=keq ¼ 1=kc þ 1=ks; ð11bÞ

with Ec, mc, kc and Es, ms, ks Young�s moduli, Poisson�s ratios and thermal conductivities of coating and sub-strate, respectively. The surface roughness of the crack faces, x, and the corresponding asperity slope value,#, are connected by tan# = x0.4.

Fig. 3(a) depicts the cohesive zone conductance for a segment representing an initially bonded interfaceas a function of the applied normal displacement jump for several different values of mode mixity andkg = 0 (vacuum). Initially, the conductance is hcz = hintsini. Subsequently, hcz decreases as s increases duringmechanical loading. Finally, the interface becomes insulating as s = 0. The break-down in conductance isslowest for w = 0�, and but occurs already under small values of Dn in the cases of high mode mixity.Fig. 3(b) shows the relationship between the cohesive zone conductance and the normal displacement jumpfor a failed element with s = 0. For an open crack, Dn > 2x, the cohesive zone conductance is due to thepresence of a gas only, (1). For a crack with crack faces in contact, Dn < 2x, the conductance due to gasis limited by the restriction of the minimum in displacement jump as imposed by the crack surface rough-ness. Following the mechanical contact interaction defined in Eq. (7), for 0 < Dn < 2x the contact conduc-tance is pressure independent and dependent only on the gas conductance in the cavities formed by thecontacting asperities, (2), while for the overclosure case, Dn < 0, a pressure dependent contact conductanceis given, (3).

3. Formulation and numerical procedure

Interfaces and interface cracks are considered as internal cohesive surfaces with partial load and heattransfer capabilities. In a finite element formulation, the mechanical equilibrium equations include the con-tribution of the cohesive zone as integral over the internal surface, Sint. The principle of virtual work is writ-ten as

Z

Vs : dFdV �

ZSint

TCZ � dDdS ¼ZSext

Te � dudS; ð12Þ

with the nominal stress tensor, s = F�1det(F)r, the Cauchy stress, r, the deformation gradient, F, and thedisplacement vector, u. Traction vectors are related to r by T = nr, with n being the surface normal. Te isthe traction vector on the external surface of the body.

The variational statement of thermal equilibrium

ZSint

ðdhþqþ þ dh�q�ÞdS ¼ZSint

ðdhþqcz � dh�qczÞdS ¼ZSint

dðDhÞqcz dS; ð13Þ

contains the cohesive zone contribution such that the cohesive zone contributions are described by the inte-gral over the internal surface, Sint, as the product of the cohesive zone heat flux, qcz, and the variation of thetemperature jump across the cohesive zone. Then, the variational form of the energy balance under use ofFourier�s law and including the contribution of the cohesive zone is expressed by

Z

VqcpðohÞdhdV �

ZSint

dðDhÞqcz dS þZV

odhox

kohox

dV ¼ZSext

dhqe dS; ð14Þ

with the absolute temperature, h, heat capacity, cp, density, q, the conductivity matrix, k, and the heat fluxper external unit area into the body, qe.

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6343

The implementation of the thermo-mechanical cohesive zone elements includes a non-symmetric elementstiffness matrix, Kcz, representing the coupled constitutive formulation:

KCZ ¼Kmf2�2g Kmtf2�1g

Ktmf1�2g Kthf1�1g

� �f3�3g

. ð15Þ

Within the element stiffness matrix, the submatrix Km{2·2} represents the derivatives of the traction–sepa-ration response with respect to the displacement jumps; Kmt{2·1} represents the thermal softening of thecohesive zone through the dependence of the cohesive tractions on the average cohesive zone temperature;Ktm{1·2} the dependence of the cohesive zone conductance on damage, material separation and contactpressure; and Kth{1·1} the temperature dependence of the cohesive zone conductance on temperature.

Calculations were conducted with cohesive zone elements possessing 4-nodes and with linear interpola-tion functions for the displacement variables, and nodal values for temperatures. The thermo-mechanicalcohesive zone element was implemented into the finite element program ABAQUS v6.3.1 through the UELcapability. In this code transient solutions are computed using a using a backward-difference scheme todetermine the temperatures, and the coupled system is solved using Newton�s method (ABAQUS, 2003).A convergence study was performed to evaluate the effect of the mesh size on the solution. The lengthof the cohesive zone element considered ranged from 4 to 50n. Predictions of the onset of crack growth weremore sensitive to element length than those of the subsequent crack growth rates. Similar observations weremade in studies of dynamic fracture (Siegmund et al., 1997). Based on the current convergence study andthe results in Siegmund et al. (1997), the length of the cohesive elements was selected as 10d.

4. Crack growth simulation

4.1. Model definition

The sample analyzed consists of a carbon–carbon composite substrate of thickness ts = 5 mm protectedby a SiC coating of thickness tc = 1 mm on both sides of the substrate. This choice of geometry is motivatedby data described in the Columbia Accident Investigation Report (2003). The length of the laminate is2L = 50 mm. Fig. 4 depicts the half-model geometry of the structure with a symmetry plane at [x = 0].Cohesive zone elements are placed between top coating and the substrate. A pre-existing edge type crackof initial length a/L = 1/3 is assumed at the center of the structure. Such crack geometries were reportedin, e.g., Jacobson et al. (1999) and Friedrich et al. (2002).

The specimen is initially stress free. No external tractions are applied, Eq. (16a), rigid body translationsand rotations are constraint through a built-in type boundary condition, Eq. (16b), and symmetry condi-tions are imposed at the centreline of the model, Eq. (16c):

Te ¼ 0; ð16aÞux ¼ uy ¼ 0 at x ¼ 0; y ¼ 0; ð16bÞux ¼ 0 on x ¼ 0. ð16cÞ

The initial condition assume the structure at an initial temperature of h0 = 300 K in a stress free state. Thethermal boundary conditions assume radiative cooling at the top surface, [y = tc], with a surface emissivityof e = 0.7 and a surrounding (sink) temperature equal to h0, Eq. (17a), and an absolute zero temperature hz,of convective cooling at the bottom, y = �(tc + ts), with a film coefficient of h = 4 · 104 W/m2 and a sinktemperature of 300 K, Eq. (17b), and insulated sides, [x = ±L], Eq. (17c). Symmetry boundary conditionsare imposed at [x = 0], Eq. (17d). Through the convective cooling on the bottom surface of the model thetemperature at that location remains close to the initial temperature, hbot � h0 throughout the loading. Such

Fig. 4. The model geometry.

6344 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

a situation can be envisioned in an actively cooled thermal protection system. Loading of the structure isthrough an applied distributed heat flux at y = tc, Eq. (17e). The applied heat flux magnitude increaseslinearly from qe = 0 at t = 0 at a rate of _qe. Such a loading profile represents a high energy atmosphericre-entry event, with most computations were conducted for _q ¼ 0.2 MW/m2 s (Roumeas et al., 1998). Insummary the thermal boundary conditions are

TableMater

h [K]

300773127315001800

q ¼ eðh� h0Þ4 on y ¼ tc; ð17aÞq ¼ hðh� h0Þ on y ¼ �ðts þ tcÞ; ð17bÞq ¼ 0 on x ¼ �L; ð17cÞq ¼ 0 on x ¼ 0; ð17dÞqe ¼ _qet on x ¼ 0. ð17eÞ

The simulations considers the C/C composite laminate as a transversely isotropic solid and the SiC coatingsas isotropic solid. The longitudinal direction of the C/C composite is assumed to be parallel to the interfacebetween the C/C substrate and the SiC coating. The elastic and thermal properties characterizing thebehavior of the two materials are summarized in Tables 1 and 2, respectively.

As no actual data for the failure behavior of the SiC–C/C interface was available in the literature,simulations were conducted with cohesive zone parameters estimated within a potentially reasonable prop-erty data space. We choose the relationship rmax = max[(90 � 9.25 · 10�2hcz), 2.5] MPa to describe the

1ial properties of the SiC (coating), density q = 3100 kg/m3

Ec [GPa] mc kc [W/m K] ac [10�6 K�1] cp,c [J/kg K]

415 0.16 114 1.1 715404 0.159 55.1 4.4 1086392 0.157 35.7 5.0 1240387 0.157 31.3 5.2 1282380 0.156 26.3 5.5 1336

Table 2Material properties of the C–C composite (substrate), density q = 1700 kg/m3

h[K]

ET

[GPa]EL

[GPa]GT

[GPa]GL

[GPa]mT mL kT

[W/m K]kL[W/m K]

aT[10�6 K�1]

aL[10�6 K�1]

cp[J/kg K]

300 6 17 2.7 2.7 0.3 0.13 3.9 42 0.5 2.0 7201300 5.0 43 0.5 2.5 20001800 5.8 45 0.6 2.9 2000

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6345

temperature dependence of the cohesive strength based on recent data from push out tests in a SiC/C/SiCcomposite by Yang et al. (2002). In computations with a temperature independent cohesive strength thevalue rmax = 30 MPa was used. The cohesive length, d, was assumed to be 5 lm (Andrews and Kim,1998) for all calculations presented and independent of temperature. An initial value for the internal resid-ual property variable of sini = 0.85 is chosen here. With these parameter values the cohesive energy isC = 63.75 J/m2.

Thermal analyses of the model were carried out for a mechanically perfectly bonded interface and inter-face conductance hint = 0.0025, 0.025 and 0.25 MW/m2 K. These analyses were performed for steady-stateconditions with a temperature gradient of 1200 K applied across the model and a fully insulating initialcrack. Fig. 5(a) shows the distribution of the heat flux per unit area at locations in the C/C laminate adja-cent to the interface. For large values of hint the 1=

ffiffir

psingularity dependence of q (Sih, 1962) is approached.

For lower values of the interface conductance, hint < 0.25 MW/m2 K, the heat flux concentration at thecrack tip diminishes and disappears as the interface becomes insulating. The temperature jump acrossthe interface is shown in Fig. 5(b) for several values of hint. For large values of the interface conductance,hint > 0.25 MW/m2 K, the temperature jump is less than 5% of the applied temperature gradient, but in-creases significantly as hint decreases. The value of hint = 0.25 MW/m2 K was used for most numerical sim-ulations except for a parametric study. This value of hint is such that the interface introduces little thermaldisturbance.

Furthermore, a crack surface roughness of x = 1 lm, and an asperity slope with tan# = 1.63 rad wasassumed. The temperature dependent conductivity for air is kg = 0.013 + 6.5 · 10�5hcz [W/m K]. In all sim-ulations a value of n = d/3 = 1.6 lm was assumed based on estimates from micrographs in Friedrich et al.(2002) and Smeacetto et al. (2003).

012

34567

89

10

0 0.2 0.4 0.6 0.8 1x-ai [mm]

hint [MW/m2K]

~r-1/2

0.25

0.025

0.0025

q [

MW

/m2]

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8 9 10

x-ai [mm]

0.25

hint [MW/m2K]

0.025 0.0025

∆θ/(

top-

bo

t)θ

θ

a b

Fig. 5. (a) Distribution of total heat flux and (b) temperature drop ahead of the crack tip across an intact interface for different valuesof interface conductance (kg = 0).

6346 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

4.2. Results

The predicted specimen behavior for the parameter set _qe ¼ 0.2 MW/m2 s, kg = 0, hint = 0.25 MW/m2 Kand rmax = 30 MPa is presented first. Fig. 6 depicts the contour plots of the heat flux component, qy, at fourdifferent times. Fig. 7 depicts the corresponding contour plots of the temperature distribution. As the tran-sient thermal loading is applied, the temperature on the top surface of the structure increases relative to thetemperature at the bottom, creating a thermal gradient. The presence of the initial crack affects the heattransfer in the center of the specimen over a distance 2ai, and introduces a temperature jump across thecrack. The intact interface, however, introduces no notable disturbance. The temperature gradient acrossthe structure induces mechanical loading through bending as well as the differential thermal expansions be-tween the coating and substrate. As the loading increases, growth of the crack along the interface initiatesat s = 4.6 s. The contour plots of Fig. 6(a) and Fig. 7(a) at s = 4.75 s depict early stages of crack growth.Crack growth then changes the heat flux and temperature distribution, Fig. 6(b)–(d) and Fig. 7(b)–(d). Theincreased disturbance in the thermal field together with the increasing amount of applied load then drivescrack growth further. As the crack length increases, a more and more significant part of the specimen be-comes insulating. Finally, the applied heat is conducted over only a small remaining ligament, and a sig-nificant concentration of heat flux at the crack tip is present at that stage, Fig. 6(d). Most of the cracksurface is insulating, except for locations closest to the symmetry line where contact between crack facesoccurs. In the temperature fields, crack growth is manifested by an increase in the temperature jump acrossthe crack, Fig. 7(c) and (d). The insulating action of the crack leads to a significant higher temperature ofthe coating located above the crack compared to the coating located along the still intact interface.

Details of the cohesive zone heat flux are given in Fig. 8(a) and (b) for a crack in vacuum and gas-filled,respectively. For both cases, it is observed that the magnitude of predicted peak value of the cohesive zoneheat flux is neither constant nor monotonically increasing. In fact, the peak value of the heat flux concen-tration at the still stationary crack (s < 4.6 s for vacuum and s < 4.75 for the gas filled crack) is larger thanthat found for the propagating crack in its early stages. Only at later times does the magnitude in heat fluxat the current crack tip increase again, and surpass the peak found at the initial crack tip. This behavior is

Fig. 6. Contour plots of the through thickness heat flux at times: (a) 4.75 s (b) 5 s, (c) 6 s and (d) 7 s; vacuum conditions.

Fig. 7. Contour plot of temperatures: (a) 4.75 s, (b) 5 s, (c) 6 s and (d) 7 s; vacuum conditions.

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6347

due to the fact that the crack growth rate is higher than the thermal reference velocity in the coating. Thelag between the thermal front evolving due to the change in the heat transfer conditions caused by crackgrowth can be estimated as

l ¼ aðsÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikc=ðqcpÞcðs� siniÞ

q; ð18Þ

with sini the time at crack initiation. Fig. 9 shows the predicted crack extension together with the time evo-lution of the lag between thermal front and crack tip for the computations with vacuum and gas in thecrack, respectively. For the present model geometry, loading and material parameters considered, the lagbetween crack tip and thermal front increases linearly from crack initiation on, and reaches a maximumvalue of l = 7.5 mm at Da = 12.5 mm or (s � sini) � 1.1 s. This point corresponds to a ratio of crack lengthto specimen length a/L = 4/5. After that, the crack growth rate slows as the free edge is approached and thethermal front can approach the crack tip again, thus raising the magnitude of the peak value of the cohesivezone conductance. This estimate is in reasonable agreement with the predicted evolution of the peak heatflux values of Fig. 8. Comparing the peak flux values depicted in Fig. 8 to the prediction of Eq. (18), thenumerical data suggest that the turning point in the lag occurs at slightly shorter times. The cohesive zoneheat flux is significantly affected by the presence of air in the crack. Air reduces the peak values of flux at thecrack tip and allows for heat transfer in the wake of the growing crack.

Details of the heat flux predictions in the vicinity of the crack tip are depicted in Fig. 10. When compar-ing the flux field at the still stationary crack tip, Fig. 10(a), to that for a rapidly growing crack, Fig. 10(b),the reduction in the magnitude of the heat flux at the crack tip is clearly visible. Also, in Fig. 10(b) the flux ismore closely concentrated at the current crack tip. The heat flux field depicted in Fig. 10(c) finally belongsto a situation where the lag between thermal and crack tip front has been reduced. Then the crack tip fluxfield is more similar to that of the stationary crack.

The predicted traction distributions for _qe ¼ 0.2 MW/m2 s, hint = 0.25 MW/m2 K, gas present in thecrack for temperature independent and dependent cohesive strength are given in Figs. 11 and 12, respec-tively. The traction distribution is crack like, with the peak value of stress at the location of the current

0.0

1.0

2.0

3.0

4.0

5.0 10.0 15.0 20.0 25.0

x [mm]

4.5

4.0

4.654.75

5.0

5.5

6.0

6.5

7.0τ [s]

qC

Z[M

W/m

2]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

5.0 10.0 15.0 20.0 25.0

x [mm]

τ [s]

4.5

4.0

4.75 4.8 4.855.0

5.5

6.0

6.5 7.0

qC

Z[M

W/m

2]

b

a

Fig. 8. Spatial distribution of the cohesive zone heat flux for (a) vacuum and (b) gas in the crack.

6348 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

crack tip, as well as at the free edge. Crack initiation at the edge does, however, not occur. The failure pro-cess occurs under a high value of mode mixity with the tangential tractions significantly larger than the nor-mal tractions. Loading of the interface is mainly a result of the mismatch in thermal expansion betweencoating and substrate. The maximum value of Tt remains below rmax due to the mixed mode loading con-ditions, but peak values change little during crack growth. The level of normal tractions is low for most ofthe interface and reaches significant values close to the free edge, only. At times, s < 5.0 s negative values ofTn are predicted along the initial crack indicating contact between crack faces. At the location of the currentcrack tip alternating positive and negative normal tractions are observed. Positive values of Tn indicate thatthe crack is open at the location of the current crack tip. Similar traction distributions were obtained also ina study of dynamic shear crack growth by Coker et al. (2003). At later stages beyond 5 s, contact betweencrack faces is no longer present except for locations at the symmetry line of the specimen, and the peakvalues of Tn slightly increased. If the temperature-dependence of the cohesive strength is accounted for,the traction distribution reflects the coupling between the mechanical and thermal fields. While the cracktip fields remain dominated by the tangential tractions, the predicted peak values now decrease during

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0.0 1.0 2.0 3.0 4.0

τ-ττ ini [s]

∆∆ ∆∆a

[mm

]

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

Lag

[mm

]

gasvacuum

Fig. 9. Crack extension and lag between the crack tip and the thermal front, respectively, in dependence of the time after crack growthinitiation.

Fig. 10. Details of heat flux at (a) stationary crack tip, s = 4 s, (b) a rapidly advancing crack tip, s = 4.75 s, and (c) a slowly movingcrack tip s = 6.0 s. Flux vectors are scaled to the current value of applied flux density indicated.

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6349

the crack growth stage. The increase in temperature caused by the applied heat load and the increased insu-lating action of the crack leads to a significant additional degradation of the load carrying capacity of theinterface. At, e.g., s = 7.0 s, thermal softening reduces the maximum level of tangential tractions from20 MPa to 4 MPa. A similar reduction in normal tractions is present. When comparing the time historyof the locations of the stress peaks between Figs. 11 and 12, it becomes obvious that the thermal softeningalso affects the crack growth history.

-1.0

-0.5

0.0

0.5

1.0

5.0 10.0 15.0 20.0 25.0

x [mm]

4.5 4.75 5.0 6.0 7.09.0

τ [s]

T t/σ

max

-0.2

-0.1

0.0

0.1

0.2

0.3

5.0 10.0 15.0 20.0 25.0

x [mm]

4.54.75

5.0

6.0 7.09.0

τ [s]

T n/σ

max

a

b

Fig. 11. Predicted distribution of (a) tangential and (b) normal tractions, temperature independent cohesive strength; gas present incrack.

6350 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

The crack growth history for four different parameter combinations is given in Fig. 13. Crack growthrates are a maximum at crack initiation, and diminish as the crack approaches the free edge. Under vacuumcondition crack growth starts earlier than for the case of a gas filled crack. The gas contributes to a reduc-tion in temperature difference between substrate and coating thereby lowering the thermal stresses evolving.The predictions suggest that the Biot number Bi = (hcztc)/kc can be used to rank different parameter sets intheir influence on crack initiation. The situation is reversed for the crack growth rate. For an air filledcrack a higher crack growth rate is found than for the crack in vacuum. A thermal reference velocityvref = hcz/(qcp)c can be used to rank different parameter sets. The specific form of thermal softening onlylittle affects the time to crack initiation, but increases the subsequent crack growth rate.

To determine the dependence of crack initiation on the cohesive zone parameters a parametric study wasconducted with variations in the cohesive energy and the interface conductance. The dependence of sini onthe cohesive energy, C, is given in Fig. 14(a) assuming a gas filled crack and temperature independent cohe-sive zone properties. The value of C was varied by either changing the cohesive strength or the cohesive

-1.0

-0.5

0.0

0.5

1.0

5.0 10.0 15.0 20.0 25.0

x [mm]

4.5 4.755.0

6.0

7.0

τ [s]

T t/σσ

max

-0.2

-0.1

0.0

0.1

0.2

0.3

5.0 10.0 15.0 20.0 25.0

x [mm]

4.5

4.755.0

6.0

7.0

τ [s]

T n/σ

max

a

b

Fig. 12. Predicted distribution of (a) tangential and (b) normal tractions, temperature dependent cohesive strength; gas present incrack.

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6351

strength, with the parameter values given in Fig. 14(b). The computations predict the existence of a thresh-old in the initiation time, sini ¼ 2.5 s at _qe ¼ 0.2 MW/m2 s. Even for a very weak interface crack growth willnot occur at times less than sini. For larger values of C the predicted dependence of sini on C can be de-scribed by a power law sini = CCn. Computations conducted with different values of (rmax,d) but the samevalue of C lead to identical values in the prediction of sini. Fig. 14(b) depicts the dependence of the time tocrack initiation, sini, on the interface conductance for a gas filled crack and for vacuum, respectively. Anincrease in hint relative to the value of this parameter used in calculations of the results presented in theprevious figures, does not lead to any change in the predicted values of sini. However, as the value ofhint—and thus the Biot number—is reduced, the time to crack initiation is reduced. This dependence of sinion hint was found to be more pronounced for the crack without gas than for cracks filled with gas.

Results from simulations considering the effects of different magnitudes of applied heat flux rates areshown in Fig. 15 for gas filled cracks. As the applied heat flux rate increases, the rate of temperature

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

4.5 4.6 4.7 4.8 4.9 5 5.1 5.2ττ [s]

∆a [m

m]

vacuum

gas

max =f(θ)

max =const.σ

σ

Fig. 13. Predicted crack growth history for vacuum and gas-filled cracks for both the temperature independent and dependent cohesivezone strength.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 50 100 150Γ [J/m2]

(0.5/5.0)

(1.0/5.0)

(5.0/5.0)

(10.0/5.0)

(20.0/5.0)

(30.0/5.0)(40.0/5.0)

(10.0/10.0)(15.0/10.0)

(20.0/10.0)(30.0/10.0)

max [MPa]/δδσ [µm]

τ ini[

s]

3.0

3.5

4.0

4.5

5.0

1.E-04 1.E-02 1.E+00 1.E+02

hint [MW/m2K]

vaccum

gas

τ ini[s

]

ba

Fig. 14. Time to crack initiation in dependence of (a) the cohesive energy (gas present in crack), and (b) the interface conductance.

6352 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

increase in the structure and thus the rate in build-up of thermal stresses increases. The time to crack ini-tiation decreases while the subsequent crack growth rates increase with an increase in _qe. Fig. 15(a) depictsthe predicted crack extension for three values of applied heat flux rate. The results on sini of these and sev-eral other computations are summarized in Fig. 15(b). As the value of _qe is reduced a significant increase insini is present, while for high values of _qe the predictions of sini become less sensitive to the value of _qe. Thetime to crack initiation scales essentially as sini / ðconstant= _qeÞ. Furthermore Fig. 15(b) shows an averagecrack growth rate in dependence of _qe. The average crack growth rate is obtained from the time it takes the

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

2.0 4.0 6.0 8.0 10.0 12.0τ [s]

∆a

[mm

]

0.1

0.2

0.4

2eq [MW/m s]

0.00.0 0.5 1.0 1.5 2.0

2.0

4.0

6.0

8.0

10.0

12.0

0

50

100

150

200

250

300

avg

(da/

dt)

[mm

/s]

τ ini

[s]

2eq [MW/m s]a b

Fig. 15. (a) Predicted crack growth history for loading rates and (b) summary of crack initiation time and crack growth rate; gaspresent in crack.

A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355 6353

crack to extend 2 mm. The simulations predict a nearly linear dependence of the crack growth rates on theapplied heat flux rate, _qe.

5. Conclusion

A thermo-mechanical cohesive zone model is introduced and used to simulate interface crack growth be-tween a SiC coating and a C–C composite substrate under transient thermal loading. Coupling betweenmechanical and heat transfer analysis is due to changes in load transfer and conductance across the interfacedue to material separation. A crack conductance formulation complements the model. Additional couplingbetween the mechanical and thermal analysis is incorporated via thermal softening of the cohesive strength.

For the geometry and the loading considered material separation is primarily shear dominated by thedifferences in thermal expansions of coating and substrate. The traction distribution during crack growthexhibits peak values at the location of the current crack tip with the magnitude of the peak given by thecohesive strength. Thermal softening can significantly affect the traction distribution. The distributionsof the flux across the cohesive zone also exhibit peak values of heat flux at the crack tip. However, no upperlimit value for the magnitude of the peak value of heat flux exists, and the thermal conditions of the crack(vacuum or gas) significantly affect the temperature field.

It was found that characteristic numbers combining thermal parameters are useful in order to scale theobserved crack growth and its relation to the evolution of the thermal field. Scaling relations have beendeveloped to estimate the lag between thermal front and crack tip, the time to crack initiation and an aver-age crack growth rate. The actual values of the scaling parameters, however, depend on the choice of themechanical cohesive zone parameters.

Acknowledgement

The authors would like to acknowledge the support from Sandia National Laboratories through thejoint NSF-Sandia Life Cycle Engineering Program.

6354 A. Hattiangadi, T. Siegmund / International Journal of Solids and Structures 42 (2005) 6335–6355

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Further reading

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