Semi-analytical analysis of thermally induced damage in thin ceramic coatings

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  • V.A. Shevchuk , V.V. Silberschmidta

    generalized boundary conditions of thermomechanical conjugation of the substrate with environment via the coating.

    gether with a signicant mismatch in coecients of thermal expansion of coatings and substrates could

    * Corresponding author. Tel.: +380 322 646818; fax: +380 322 636270.E-mail address: (V.A. Shevchuk).

    International Journal of Solids and Structures 42 (2005) 47384757$ - see front matter 2005 Elsevier Ltd. All rights reserved.Eciency of the suggested approach is illustrated by an example of damage evolution in the alumina coating on thetitanium-alloy and tungsten substrates under uniform heating. 2005 Elsevier Ltd. All rights reserved.

    Keywords: Brittle coatings; Thin ceramic coatings; Damage; Analytico-numerical approach; Generalized boundary conditions

    1. Introduction

    The process of deposition of ceramic coatings onto (usually metallic) substrates results in formation ofspecic microstructures in such brittle coatings. Characteristic features of these coatings include manufac-ture-induced porosity and microcracks, and anisotropy in thermomechanical properties. These factors to-Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine,

    Lviv 79060, Ukraineb Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough,

    Leicestershire LE11 3TU, UK

    Received 4 June 2004; received in revised form 25 January 2005Available online 19 March 2005


    An ecient procedure to analyze damage evolution in brittle coatings under inuence of thermal loads is suggested.The approach is based on a general computational scheme to determine damage evolution parameters, which incorpo-rates an analytical solution of the appropriate interim boundary-value thermoelasticity problem. For thin inhomoge-neous coatings, the simplication in the analysis is achieved by application of the mathematical model withSemi-analytical analysis of thermally induced damagein thin ceramic coatings

    a,* bdoi:10.1016/j.ijsolstr.2005.02.002

  • result in initiation and evolution of damage and crack generation even under purely thermal loading, in theabsence of mechanical loads.

    An adequate analysis of thermally induced damage evolution thus should include an additional variable,characterizing the damage state. This can be implemented in terms of Continuum Damage Mechanics(CDM). A CDM model for a damage evolution in brittle media was suggested by Najar (1987) basedon thermodynamic considerations. This approach was modied in (Silberschmidt and Najar, 1998; Sil-berschmidt, 2003) where a computational scheme for estimation of damage evolution in bulk ceramics withdierent types of random microstructure is suggested. Application of this approach to a general case ofthermally loaded brittle coatings presupposes numerical modeling, and respective algorithms are discussedin (Silberschmidt, 2002; Zhao and Silberschmidt, 2005). Such a cumbersome procedure to estimate param-eters of the damage evolution process as well as stress, strain and temperature elds can be simplied (andenhanced) by incorporation of an analytical solution of the appropriate interim boundary-value problem ofthermoelasticity into the scheme, thus resulting in an analytico-numerical approach (Shevchuk and Sil-berschmidt, 2003).

    To obtain such type of the analytical solution for thermoelasticity problems in a case of thin inhomo-geneous coatings, various approaches can be used that employ the consideration of smallness for coatingthickness (Lyubimov et al., 1992; Tretyachenko and Barilo, 1993; Suhir, 1988; Elperin and Rudin, 1998).One of the most eective schemes is based on representing the inuence of thin-walled elements of struc-tures by means of special boundary conditions (Podstrigach and Shevchuk, 1967; Pelekh and Fleishman,1988; Shevchuk, 1997, 2000, 2002a). This approach allows reduction of the solution of a boundary-valueproblem for a non-homogeneous body to the problem of a homogeneous body, but with a generalizedboundary, on which parameters of the homogeneous body must satisfy some complicated boundary con-ditions. These conditions provide an approximate relation between components of a stress tensor and a dis-placement vector at the bodycoating interface with prescribed surface loading at the coatingenvironmentboundary, also accounting for thermal strains in coating. Such boundary conditions describe the eect ofthin coatings, simulated as shells with respective geometrical and physical properties, on the thermome-chanical state of the bodycoating system.

    These constraints for mechanical parameters together with generalized boundary conditions (GBCs) forheat transfer (Shevchuk, 1996, 2002b) allow us to formulate and solve non-classical boundary-value prob-lems of thermoelasticity, and to determine the stressstrain state of bodies with thin coatings under tran-sient thermal loads.

    The suggested approach, based on the use of the GBCs, was validated by comparing the approximate andexact solutions of several test problems including (a) heat conduction in a plate with a three-layer coating(Shevchuk, 2002b), (b) the test Lame problem for mechanical loading of a solid cylinder with a three-layercoating (Shevchuk, 2000), and (c) a thermal stress state of a solid cylinder with a three-layer coating underuniform heating (Shevchuk, 2002a). This comparison has shown a fair agreement for all the analyzed cases.

    The GBCs can be derived by means of dierent techniques. Depending on the type of boundary condi-tions, the following methods can be used:

    (i) the operator method, avoiding any preliminary hypotheses for the transverse distribution of thesought functions in coatings (Podstrigach and Shvets, 1978);

    (ii) the approach, using a priori assumptions for the transverse distribution of the sought functions incoatings (Podstrigach et al., 1975);

    (iii) the discrete approach based on appropriate approximations for normal derivatives by expansions(Pelekh and Fleishman, 1988).

    In this study, the original model is modied to take into consideration additional features: anisotropy of

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4739ceramic coatings, inhomogeneity of the initial porosity, temperature dependence of thermomechanical

  • paramdama




    ure is characterized by non-vanishing residual stiness. Also, we take into account the temperature depen-

    dence of thermomechanical and physical parameters.

    A fully analytical way to estimate the critical level of temperature changes linked to initiation of the localfailure in the coating is not possible due to coupling of deformational and damage processes, and the problemis solved as an iterative sequence of steps, implemented until attainment of the damage threshold Dm. Thus,under uniformheating the following iterative procedure to determine the change of damage parameter is used:

    DT dT D0 expEi ei T ;BjlDT 2

    2W T

    !; 2corresponds to the local failure event, i.e. a transfer from the disperse accumulation of damage to failurelocalization and initiation of microscopic cracking. For purely thermal loading of the substrate-coating sys-tem, linked to the increase in temperature T, the main cause of damage is the mismatch in coecients ofthermal expansions of its components.

    The inuence of damage on the materials behavior is introduced into the model in terms of the lineardecrease of Youngs moduli with the growth of damage Ei! Ei(1 D) that is based on one of the mainprinciples of CDM. With Dm < 1, a transition from the disperse damage accumulation to macroscopic fail-wher) for details). In this paper we deal with monotonous loading due uniform loading hence excluding-history eects from consideration.attainment of the threshold value of damage Dm due to the increase in the external load/deformationversely isotropic coating is based on the application of the theory of anisotropic shells (Ambartsumyan,1974) with an account for a normal transversal strain component (Vasilenko, 1999).

    Eciency of the suggested approach (briey outlined by Shevchuk and Silberschmidt, 2003) is illustratedin this paper by analysis of damage evolution in alumina coating on a titanium-alloy and tungsten sub-strates under uniform heating.

    2. General computational scheme

    The damage parameter, introduced into the continuum-mechanical model, characterizes the macro-scopic development of failure in ceramic coatings linked to the evolution of defects at the microscopic level.The current value of the damage parameter D is determined by the local strain level and the initial damageD0 according to the following relation for the anisotropic case:

    D D0 expEi ei 22W


    with the summation over all the non-negative principal elastic strains ei . Here angle brackets are Macaulay

    brackets, i.e. hxi x if x > 00 if x 6 0

    ;W is the specic energy linked with the damage evolution; Ei are Youngs

    moduli of the undamaged transversely isotropic material. This equation generalizes damage evolution law,initially introduced for the isotropic case (Najar and Silberschmidt, 1998; Silberschmidt, 2002). The generalvariant of this CDM model fully describes deformational behavior and damage accumulation in brittlematerials for loading and unloading/reloading conditions (see Najar (1987) and Najar and Silberschmidteters of the coating and substrate as well as the dependence of the coatings elastic moduli on thege level.e derivation of GBCs for mechanical conjugation of the body with its environment via a thin trans-4740 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757e Bjl are elastic coecients for transversely isotropic body.

  • The general ow diagram of the computational scheme used to study damage evolution in the coating isshown in Fig. 1. It starts with the input of initial state. At each step of heating linked to the temperatureincrement dT, calculations of the stress and the strain elds and corresponding damage distribution(according to the Eq. (2)) are performed. This scheme includes a sequence of interim thermoelasticityboundary-value problems for a medium with damage. It can be implemented using the modied nite-ele-ment procedures with an account for damage evolution. Such implementation is rather cumbersome andshould be based on the models accounting for the damage parameter either in a parametric form or withuse of special nite elements with an additional degree of freedom (see models and respective discussions in(Silberschmidt, 2002; Zhao and Silberschmidt, 2005)). To overcome this complicacy and to obtain an eec-tive solution algorithm, this paper utilizes a new, alternative approach based on the substitution of thenite-element solution with an analytical solution of the interim thermoelasticity problem for the bodycoating system at each iteration step of a general computational iterative procedure. It should be noted thatthe analytical part of this semi-analytical computational procedure will consist of two consecutive stages:(1) solution of the non-classical interim thermoelasticity boundary-value problem for a body with GBCs;(2) determination of stress and strain elds in a coating by means of relations that are referred to as the

    INPUT Properties of coating & substrate

    GeometryInitial temperature T0Initial distribution D0


    Temperature increment T

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4741OUTPUT Critical temperature Tcr

    Stress and strain fieldsDamage distribution according to Eq. (2)

    Local failure criterion(attainment mD )Fig. 1. Flow diagram of the general computational scheme.

  • restoration relations below (Sections 3 and 4). The details of this analytical part will be discussed in respec-tive sections below.

    3. Generalized boundary conditions and restoration relations

    The object under study is a body with a thin ceramic coating of thickness h. Here the coating is consid-ered as a thin shell referred to a mixed coordinate system (a1,a2,c), the axes of which coincide with the linesof principal curvatures of the bodycoating interface and its normal (Fig. 2).

    The ceramic coating is transversely isotropic with respect to its deposition direction, which coincideswith the axis c in Fig. 2. Deposition processes are usually linked with the non-vanishing porosity levelsin briand t


    and t


    4742 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757U c2a1; a2; c 1 k2cvca1; a2 cA2

    owca1; a2oa2


    U c3a1; a2; c wca1; a2 e3a1; a2c;


    U 1a1; a2; c 1 k1cuca1; a2 A1 oa1 ;c c owca1; a2gs moduli of a transversal isotropic coating, we shall take into account the normal strain e3 as an addi-l degree of freedom (Vasilenko, 1999; Burak et al., 1978). Then geometrical relations have the formUc Ub; rc3 rb3 at c 0: 4Here indices c, b, and e refer to the coating, the body, and the environment, respectively; r3 is the stressvector, which acts on the surface c = const, rc3 rc13e1 rc23e2 rc33e3; e1, e2, e3 are unit vectors of the coor-dinate trihedron linked to the base surface S0 of the shell; Uc U c1e1 U c2e2 U c3e3 is the displacement vec-tor of points of the coating; Ub = ube1 + vbe2 + wbe3 is the displacement vector of points of the body on theboundary of contact with the coating.

    We assume that the distribution of temperature T(a1,a2,c) in the coating is prescribed.The derivation of GBCs of the mechanical conjugation of the body with its environment via a thin coat-

    ing is implemented on the basis of the theory of anisotropic shells (Ambartsumyan, 1974; Grigorenko andVasilenko, 1981; Vasilenko, 1999).

    To derive these conditions, we use the KirchhoLove hypothesis; however, due to the dierence betweenttle coatings. Hence, the Young moduli are considered as linearly dependent on the level of porosity;he properties of the coating are also temperature-dependent.e assume that the vector of tractions at the coatingenvironment boundary is prescribed:

    rc3 re3 at c h; 3he following interfacial conditions of ideal mechanical bonding between the coating and body holdFig. 2. Scheme of the studied object.

  • and appropriate relations for strains in case of a thin shell are





    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4743T T Ttion of an operand in respective expressions; symbol T is a sign of transposition.Constitutive equations suggested by Vasilenko (1999) are modied here, and they take the form with an

    account for relations (6)

    h Ke K3e3 hT ; 8nding coordinate al, e.g., Aj,l = oAj/oal; oj = o/oaj, j, l = 1,2; the empty parentheses ( ) denote the loca-

    a comma in a subscript followed by a subscript index denotes a partial derivative with respect to the cor-Ni;Nij;QiMi;Mij;Mi3

    Z h0

    rcii; rcij; r


    n o1 kjc



    dc; i 1; 2; j 3 i;

    N 3 Z h0

    rc331 k1c1 k2cdc;A1A2 o1A2 o2A1 n N 1;N 2;N 12;N 21;Q1;Q2;M1;M2;M12;M21T; n3 N 3;M13;M23T;

    B q1; q2; q3;m1;m2;m3T;

    qj rej31 k1h1 k2h rbj3; mj hrej31 k1h1 k2h; j 1; 2; 3:C and C3 are the matrices of dierential operators:

    C 1A1A2

    o1A2 A2;1 A1;2 o2A1 k1A1A2 0 0 0 0 0A1;2 o2A1 o1A2 A2;1 0 k2A1A2 0 0 0 0

    k1A1A2 k2A1A2 0 0 o1A2 o2A1 0 0 0 00 0 0 0 A1A2 0 o1A2 A2;1 A1;2 o2A1 0 0 0 0 0 A1A2 A1;2 o2A1 o1A2 A2;10 0 0 0 0 0 k1A1A2 k2A1A2 0 0



    C3 1A1A2

    0 0 0

    0 0 0

    0 0 0

    0 0 0

    0 0 0


    3777777775;components of strains e1, e2, e12, j1, j2 and j12 are related to the base surface S0.Equilibrium equations for such shells (in the absence of body forces), given in Vasilenko (1999), can be

    written in the following form:

    Cn C3n3 B; 7where n and n3 are the column vectors of forces and moments, which arise in the coating:A1 and A2 are the Lame parameters. Principal curvatures k1 and k2, displacements uc,vc and wc, andec11 e1 j1c k1ce3 e1; ec22 e2 j2c k2ce3 e2; ec33 e3; ec12 e12 j12c; 6h N 1;N 2;N 3; S;M1;M2;H ; hT N 1T ;N 1T ;N 3T ; 0;MT ;MT ; 0 ; e e1; e2; e12; j1; j2; j12 ;

  • 0 0 1 1

    2 36 7


    tively; T is the initial strain-free temperature distribution.W

    6ch c 2c 3c 4c 3c

    Then 1 2



    4744 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757Mj3 0

    rj3c 1 k3jc dc Qj 2 rj3 rj3 12 ; j 1; 2: 11

    ituting (11) into (7), excluding transverse forces Q1 and Q2 from Eq. (7) and neglecting terms of ther orders of smallness, we present these equations in the transformed form:

    rbj3 F jN 1;N 2; S;M1;M2;H rej3; j 1; 2;rb33 F 3N 1;N 2;M1;M2;H re33 hKre13; re23;h2rb33 h


    12Krb13; rb23 k1 h2N 1 M1

    k2 h2N 2 M2 N 3 h2 re33 h212Kre13; re23;8>: 12e

    F ju1;u2;u3;u4;u5;u6 A11 A12 ojAluj Al;jul olAju3 Aj;lu3 kjojAlu3j Al;ju3l 2olAju6 2klAj;lu6;Z hc h e b h2rcj3 Qj h3 rej3 h h2 r

    bj3 1 h h2 ; j 1; 2: 10

    , neglecting the terms of the higher orders of smallness (taking into account k h,k h 1)ilenko, 1981; Ambartsumyan, 1974):


    2 0

    e assume a quadratic distribution for shear stresses rcj3 along the shell thickness (Grigorenko and Vas-where

    S N 12 k2M21 N 21 k1M12; H M12 M21=2:

    In Eq. (8) K is the matrix of elastic constants; e is the column vector of strain components of the basesurface; h is the column vector of forces and moments in the coating; hT is the column vector of parameterslinked with temperature strains; elastic coecients Bij for transversely isotropic body are given in AppendixA; b1 and b3 are coecients of thermal expansion in the plane of isotropy and along its normal, respec-Gmij 0

    Bijcm dc; m 0; 1; 2; ij 11; 12; 13; 33; 66;

    Z h

    0 0 G166 0 0 2G


    0G11 G12 0 G11 G12 0

    G112 G111 0 G

    212 G

    211 0

    6664 7775G13 k1G11 k2G12G113 k1G212 k2G211

    6664 7775


    G12 G11 0 G12 G11 0

    G013 G013 0 G

    113 G

    113 0

    0 0 G066 0 0 2G166

    1 1 2 2


    777777777; K3 G013 k1G112 k2G111G033 k1 k2G113

    01 2 2

    6666666677777777:G011 G012 0 G

    111 G

    112 06 7 G013 k1G111 k2G1122 3N 1T ;MTN 3T

    Z h0

    B11 B12U1 B13U32B13U1 B33U3

    1; c


    dc; UjT

    Z TT 0

    bjT 0dT 0; j 1; 3; 9j 1; 2; l 3 j; 13

  • Subst

    1 2 6 7

    betweinto transformed equilibrium Eq. (12) and taking into account the continuity condition for displacements

    b e e e>>>>: 15

    expressions for dierential operators Ljl(j, l = 1,2,3) are given in Appendix B,

    L4j A1j ~g1j G013 oj n3j11 ~g13j G013 A1j ~g2j G113 kj;j; j 1; 2;

    L43 k21 k22eG111 2k1k2 eG112 k1 k2G013 X2m1

    ~g2m G113 Pm;

    pje A1j g1j G013 oj n3j11 G111 G112 kj k3j A1j G111 kj;j G112 k3j;j;on the bodycoating interface (4), we obtain the relations

    rbj3 Lj1ub Lj2vb Lj3wb pjee3 rej3 A1j ojN 1T kjojMT ; j 1; 2;8>>>k1A2;1A1A2

    o2k2 A2

    1A2 o2o2 A2







    1A1A2 o1o2


    o1 A2;1A2 o2

    6664 7775en components of strains of the reference bodycoating interface and displacements of this surfaceP o1k1 A1


    1A1 o1o1 A1


    A A2o2

    666 777F 3u1;u2;u3;u4;u5 k1u1 k2u2 A11 A12X2j1

    oj A1j ojAlu2j Al;ju2l olAju5 Aj;lu5 h i


    Ku1;u2 A11 A12 o1A2u1 o2A1u2:

    ituting expressions (8) and geometrical relations (Grigorenko and Vasilenko, 1981)

    e Puc; vc;wcT at c 0; 14o1 A1



    o2 A2








    37777777p4e G33 k21 k22eG11 2k1k2 eG12 k1 k2eG13 G13 ;

  • Exclu


    its enaccou



    Substthe c


    4746 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757e3 1p4eh2re33 rb33


    Kre13 rb13; re23 rb23 L41ub L42vb L43wb

    k1 k2 h2N 1T MT

    N 3T

    : 17

    itution of the rst continuity condition given in Eq. (4) into Eq. (14) gives other strain components ofoatingbody interface:

    e Pub; vb;wbT: 18itution of relations (17) and (18) into Eq. (6) makes it possible to determine the total strains, and,y, to obtain appropriate relations for the principal elastic strains in the coating: 2vironment via a thin coating. These conditions enhance the ones given by Shevchuk (2002a) due to annt for the eect of the normal strain in the, the solution of the interim boundary-value thermoelasticity problem is reduced to a solution of theem for the body with the GBCs (16). After solving this problem, it is possible to use restoration rela-for stressstrain factors in the coating in terms of the boundary values for components of the stressr and displacement vector of the body at the bodycoating interface and known parameters linkedthe prescribed temperature change.e restoration relations can be obtained in the following way. It follows from Eq. (15)3 thatthermal strains in the coating, they can be interpreted as the generalized boundary conditions for mechan-ical variables of the body. In other words, they represent GBCs of mechanical conjugation of a body withrelations (16) establish the connection between components of the stress tensor and the vector of dis-ments at the boundary of the body with the prescribed components of surface loading that account forrbj3 h2 djrb33 h2

    12djKrb13; rb23 eLj1ub eLj2vb eLj3wb rej3 h2 djre33 h212 djKre13; re23

    h2k1 k2dj A1j oj

    N 1T djk1 k2MT N 3T A1j kjojMT ; j 1; 2;

    1 h2d3

    rb33 h


    12d3Krb13; rb23 eL31ub eL32vb eL33wb

    1 h2d3

    re33 h h



    Kre13; re23 k1 k2d3 DMT

    k1 k2 1 h2 d3

    N 1T d3N 3T ;



    dj pjep4e

    ; eLji Lji djL4i; j; i 1; 2; 3:gj kjG11 k3jG12 ; ~gj kj eG11 k3j eG12 ;Pj A1j ojA1j oj nj12o3j; njn1n2

    Aj;3jAn1j A


    ; j 1; 2; n1; n2 0; 1; 2; . . . :

    ding e3 from these equations, we obtaineG1j G1j 2G1j ; j 1; 2; 3; m 1; 2;m m m m m mm m h m1D 1A1A2


    o1 o2 A1A2 o2 ; Dj 1



    oj ; j 1; 2;

    ei ei jic kice3 ei U1T ; i 1; 2; e3 e3 U3T : 19

  • The u




    4. Ca



    rej3 A1j ojN 1T kjojMT ; j 1; 2;>>>>>>

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 47471 p33rb33 p31rb11 p32rb22 p34rb12 p3TUbT b p3ee3 re33 hKre13; re23 k1 k2N 1T DMT ;

    p41rb11 p42rb22 p43rb33 h


    12Krb13; rb23 p4TUbT b p4ee3

    h2re33 h


    12Kre13; re23 k1 k2 h2N 1T MT

    N 3T ;


    where expressions for dierential operators pjl(j = 1,2,3; l = 1,2,3,4) are given in Appendix C, Tb is theboundary value of temperature at the bodycoating interface,

    1 0 0>UbT T 0

    bbT dT ; 23

    Eb, mb and bb are the Young modulus, Poissons ratio and the coecient of linear thermal expansion of thebody, respectively; djl is the Kronecker symbol.

    Substituting Eq. (21) into equilibrium Eq. (12) and taking into account relations (23), we get

    rbj3 pj1rb11 pj2rb22 pj3rb33 pj4rb12 pjTUbT b pjee38>>>Z T0 0ejl Eb rjl Eb rlldjl UbT djl; 22

    where1 mb mb

    The DuhamelNeumann relations for the material of the body have the formts for the body

    h Keb11; eb22; 2eb12; 0; 0; 0T K3e3 hT : 21the forces and moments in the coating can be expressed only in terms of the boundary values of strain com-this case, due to continuity of tangential strains across the coatingbody interface

    e1 eb11; e2 eb22; e12 2eb12; 20For the particular case of the stressstrain state, characterized by the absence of bending strains andtwisting of the bodycoating interface (j1 = j2 = j12 = 0), we can obtain a simplied variant of the GBCsthat contain only components of the stress tensor.determination of the stress and strain elds in a body as a solution of the interim boundary-value ther-moelasticity problem with GBCs (16);determination of the principal elastic strains in a coating using restoration relations (19) with anaccount for Eqs. (17) and (18);calculation of the damage distribution according to Eq. (2).

    se of absence of bending and twisting strainscoating. For the suggested analytical procedure, the bloc Calculations of the general computationalscheme (see Fig. 1) comprises the following steps:se of relations (17) and (18) in constitutive Eq. (8) allows calculation of forces and moments in thepjT Aj G11 G12 oj; j 1; 2;

  • In th



    4748 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757e1 1Eb rb11

    mbEbrb22 rb33 UbT ;

    e2 1Eb rb22

    mbEbrb11 rb33 UbT ;

    e12 21 mbEb rb12:


    ly, the principal elastic strains in the coating are determined by relationsSubstitution of Eq. (22) into Eq. (20) gives other components of strains:om Eq. (24)3 follows

    e3 1p4eh2re33


    12Kre13 rb13; re23 rb23 p41rb11 p42rb22 p43rb33 p4TUbT

    k1 k2 h

    2N 1T MT

    N 3T

    : 26the boundary values only of the stress tensor components for the body at the bodycoating interface andknown parameters linked with the prescribed temperature change.pm5 12 dm; pm6 2 dm; pm8 dm; m 1; 2; 3;

    pm7 h2k1 k2dm A1m om; m 1; 2; p37 k1 k2 1



    pm9 k1 k2dm A1m kmom; m 1; 2; p39 k1 k2d3 D;

    ~pmn pmn p4ndm; m 1; 2; 3; n 1; 2; 3; T :is case it is possible to formulate restoration relations for stressstrain factors in the coating in terms of 1 p36re33 h p35Kre13; re23 p37N 1T p38N 3T p39MT ;


    h2 hrbj3 ~pj1rb11 ~pj2rb22 ~pj3rb33 pj4rb12 ~pjTUbT b pj5Krb13; rb23 rej3 pj6re33 pj5Kre13; re23 pj7N 1T pj8N 3T pj9MT ; j 1; 2;

    1 ~p33rb33 ~p31rb11 ~p32rb22 p34rb12 ~p3TUbT b p35Krb13; rb23

    8>>>>>>>>>:25Excluding e3 from these equations, we obtain nallyp4j E1b G013 1 mb ~g1j mb~g13j j 1; 2;

    p43 h=2 mbE1b 2G013 k1 k2eG111 eG112 ;p4T 2G013 k1 k2eG111 eG112 :p3T k1 k2G011 G012 G111 G112 D;ei ei kice3 ei U1T ; i 1; 2; e3 e3 U3T : 28

  • and tTh



    In thir = R


    33 r0 11xed ends) and DuhamelNeumann relations (22), we obtain nally stresses in the cylinder in the form:rb33 rb22 1 2mb~p31 ~p32 ~p33 1p37N 1T p38N 3T p39MT ~p3T ~p31EbUbT ;rb11 2mbrb33 EbUbT :


    Substituting Eqs. (32) and (29) into restoration relations (26)(28), we nd the principal non-negative elas-We write the solution of equilibrium equations for the cylinder in this case in the form (Timoshenko andGoodier, 1970)

    rb33r abr2 EbUbT 21 mb ; r

    b22r a

    br2 EbUbT 21 mb ; 31

    where a and b are unknown constants. We found these constants by substituting Eq. (31) into the GBC (30)and by using condition rb

    $$ 6 1. Then using equality eb 0 (which follows from the condition for thetic str37 R 2 38 39 RR R

    p 1 1 h d3

    ; p d3; p d3 ; T b T :p3T G011 G012 ; p4T 2G013

    eG11 eG12 ;1 1rb12 rb13 rb23 0: 29s case, the rst two GBCs (25) are satised identically, and the third one takes the following form for:

    1 ~p33rb33 ~p31rb11 ~p32rb22 ~p3TUbT b p37N 1T p38N 3T p39MT ; 30for this case

    p3e R1G013 R2G111 ; p4e G033 R1eG113 G113 R2 eG211 ;p33

    mbG011 G012 REb

    ; p43 h2 mbE1b 2G013

    eG111 eG112R


    p3j G01l mbG01j

    REb; p4j


    EbG013 1 mb

    eG11l mb eG11jR

    0@ 1A j 1; 2; l 3 j;all the tractions rej3j 1; 2; 3 on the coatingenvironment boundary vanish.Due to axial symmetry of the problem, shear stresses vanish:an example, we consider the problem for a solid cylinder of radius R with a perfectly bonded ceramicg under conditions of uniform heating. The ends of the cylinder are xed in the axial directions, and5. Problem of a coated cylinderaking into account j1 = j2 = j12 = 0.e general form of the analytical procedure, discussed in Section 3, remains, with substitution of Eqs.(19) with Eqs. (25)(28), respectively.Respectively, the stress factors in the coating can be determined from Eq. (8) using relations (26) and (27)

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4749ains in the coating

  • e3 e3 U3T ;e2 e2 R1ce3 e2 U1T ;



    e3 1p4e2mbp41 p42 p43rb33 p4T p41EbUbT 0:5hR1N 1T R1MT N 3T ;

    e 1 m 1 2mb rb U T





    k ? kand E are elastic moduli along the axial and radial directions, respectively. This ratio is varied in the inter-



    T (C)


    4750 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 4738475751.2 5.02 4.4 8.6450.1 5.54 4.4 9.1448.8 6.06 4.4 9.47Alumina Tungsten Ti6Al4V, according to Sevostianov and Kachanov (2001) we have m? = mkE?/Ek.

    1rature-dependent material properties of coating and substrates used in numerical calculations

    W (kJ/m3) Coecients of thermal expansion b1 = b3, 106 (C1)?val 14 in calculations for Figs. 6 and 7 with the xed value Ek = 150 GPa. The inuence of damage on thematerials behavior is implemented by transitions (Silberschmidt, 2002; Sevostianov and Kachanov, 2001)


    ! E?k


    1 D:vide the necessary data for the right-hand part of Eq. (2) to calculate the damage distribution at each step ofuniform heating.

    6. Numerical results

    Analysis of damage evolution in ceramic coating on the cylindrical metallic substrate based on thesolution obtained in the previous section is implemented according to the suggested semi-analytical schemegiven in Section 2. A case of a long cylinder with radius 5 cm, xed at its ends and coated with a 500 lmlayer of alumina, is considered. Three types of the through-thickness distribution of the initialdamage linked to the manufacture-induced porosity are analyzed: (1) uniform with the magnitudeD0 = (Dmin + Dmax)/2, (2) linearly increasing from Dmin at the coating-substrate interface to Dmax at theexternal free surface (referred to as Type 1 below), and (3) linearly decreasing from Dmax at the interfaceto Dmin at the surface (Type 2). In numerical calculations Dmin = 0.02 and Dmax = 0.10 are used, resultingin D0 = 0.06. These three types of coatings are studied for two cases of the substrates material: tungstenand Ti6Al4V alloy. We consider the system under purely uniform thermal loading and assume a stress-and strain-free initial state.

    The properties for the coating and substrates are given in Tables 1 and 2. In the numerical part of ouranalysis we use cubic spline approximation for the temperature-dependent parameters from Table 1. Trans-versal isotropy of the alumina coating is accounted in terms of the ratio of Youngs moduli E /E , where Ethermal strains Uj(T)(j = 1,3,b) and quantities N1T, N3T, MT, dened by relations (9) and (23), aren since temperature T is a prescribed parameter at uniform, Eq. (33) of the closed-form solution of the interim boundary-value thermoelasticity problem pro-2 b Eb 33b47.5 6.58 4.4 9.74

  • The type of the substrate considerably aects the character of damage accumulation in ceramics (Fig. 3),though the main stages of the process are present in both cases: a relatively slow damage development at theinitial stage is succeeded by a sharp increase in the damage level at the nal stage. The specic positionatthe interfaceis chosen since calculations have shown that it is the area with the highest rate of the tem-perature-induced damage growth. The analysis of results demonstrates that the type of the through-thick-ness distribution of damage remains the same (Figs. 4 and 5) for thin coatings, treated here, in contrast tothe case of thick ones (Silberschmidt, 2002). This can be explained by low variations of strains across thick-ness for such thin coatings. But there is a common feature of the process: non-uniform distributions of ini-tial porosity are more dangerous with regard to the crack initiation than the uniform one.

    An important parameter of purely thermal loading of coated components is the critical temperature Tcrof the macroscopic fracture initiation. It could be related to attainment of the critical damage level Dm insome part of the coating; overcoming this threshold means a transition from the disperse damage accumu-

    Table 2Elastic properties of coating and substrates used in numerical calculations

    Alumina Tungsten Ti6Al4V

    Ek (GPa) 150 400 114E? (GPa) 100 400 114mk 0.24 0.28 0.33

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4751lation to failure localization in the form of a macroscopic crack. It is follows from the simulations that thedecrease in Dm, corresponding to the local failure initiation, results in the decline of Tcr for both cases ofsubstrates and all types of coatings (Fig. 5).

    It is obvious that the transition to macroscopic fracture is controlled mainly by the highest local level ofthe initial porosity, hence the considerable dierences between the uniform distribution and two non-uni-form ones. The dierence between the latter two cases is rather small for thin coatings with Type 2 stillbeing the most dangerous one as was also the case in (Silberschmidt, 2002). The suggested semi-analyticalscheme reproduces the main features of thermally induced damage accumulation obtained by means ofFig. 3. Inuence of temperature on damage evolution near interface for various types of initial distribution of damage for case oftitanium-alloy (solid curves) and tungsten (dashed curves) substrates.

  • 4752 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757niteinfor(Zhao

    Thing otrast,is negto va1). Frtions

    Fig. 4dT = 5elements for three studied types of alumina coatings (Silberschmidt, 2002) and provides quantitativemation on both damage distribution and critical temperature that could be experimentally measuredand Silberschmidt, 2005).e next stage of the study is to analyze the eect of manufacturing-induced anisotropy of alumina coat-n damage evolution, which is essential for the case of the tungsten substrate (Figs. 6 and 7). In con-the inuence of anisotropy for all studied cases of an alumina coating on the titanium-alloy substrateligibly small (deviations do not exceed 0.1%). The dierence between results for two substrates is duerious types of the mismatch in coecients of thermal expansion of the substrate and coating (see Tableom Eqs. (33), (34) and (32) we can obtain the following asymptotic expressions (with account for rela-from Appendix A) for elastic parts of the principal strains for a very thin coating (h/R! 0):

    e3 Ekm?

    E?1 mk 2U1T 1 mbUbT ;

    e2 1 mbUbT U1T :

    . Damage distributions induced by thermal changes dT = 150 C for case of titanium-alloy substrate (solid curves) and00 C for case of tungsten substrate (dashed curves) for various types of initial distribution of damage.

    Fig. 5. Critical temperatures for varying damage threshold: (a) tungsten substrate and (b) titanium-alloy substrate.

  • V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4753Obviously, for the titanium-alloy substrate only the circumferential strain will be positive for a given type ofthermal excursions, with the positive radial strain being the main factor for the tungsten substrate.

    The decrease in the anisotropy accelerates damage accumulation for all types of coatings (i.e., distribu-tions of the initial porosity) with the uniform coating being the most resistant to thermal damage from threeanalyzed types. Fig. 7 demonstrates damage development under conditions of the increasing temperature inthe coating with a uniform distribution of the initial porosity. Here, a two-fold increase in the level ofanisotropy causes halving of damage accumulation rate. As calculations show, the similar eect of ratioEk=E? is observed for other two types of coatings.

    Fig. 6. Inuence of ratio Ek/E? on level of damage near interface induced by thermal loading of dT=500 C for various types of initialporosity (tungsten substrate).

    Fig. 7. Damage evolution near interface for case of uniform initial porosity for various values of Youngs moduli Ek/E? (tungstensubstrate).

  • witheral cincor

    and substrate as well as the dependence of the coatings elastic moduli on the damage level.

    4754 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757The suggested analytico-numerical approach has the following advantages as compared to the use of thenite-element analysis and other direct numerical methods: (i) it notably simplies calculations of structureswith thin coatings thus ensuring essential reduction of computational eorts; (ii) it provides a possibility ofimportant a priori qualitative and quantitative evaluation of the eects of various geometrical and thermo-mechanical parameters of the bodycoating system on the character of the damage evolution process, whileparametric studies which use universal numerical methods are exceedingly cumbersome; (iii) the eciencyof the approach based on the GBCs increases with the decrease of the coating thickness in contrast to directmethods without preliminary transformation of initial problems; (iv) it allows us to verify other computa-tional techniques.

    Based on the suggested approach, the performed numerical calculations allowed us to investigate essen-tial features of damage evolution processes under uniform heating of the component with the alumina coat-ing for two dierent substrates. It has been established that

    the type of the substrate considerably aects the character and rate of damage accumulation in ceramiccoatings;

    the character of the through-thickness non-uniformity of the initial damage distribution changes insig-nicantly for thin coatings in contrast to the case of thick ones;

    the non-uniform distribution of initial porosity is more dangerous that uniform; the eect of manufacturing-induced anisotropy of the alumina coating on damage evolution is absent inthe case of a titanium-alloy substrate, although it is essential for the tungsten substrate due to a dierenttype of the mismatch in coecients of thermal expansion between the substrates and coating.

    Appendix A

    Expressions for elastic coecients Bij for transversely isotropic body follow from the anisotropic onesgiven by Vasilenko (1999):

    B11 X1a11a33 a213; B12 X1a213 a12a33; B13 X1a12 a11a13;

    B33 X1a211 a212; B66 a166 ; X a11 a12a11 a12a33 2a213;where elastic constants aij are expressed by means of engineering constants (Youngs moduli, shear modulusand Poissons ratio) as follows:each iteration step. For inhomogeneous thin ceramic coatings, the simplication in the solution procedureis achieved by means of application of the mathematical model with generalized boundary conditions forthermomechanical conjugation of the substrate with environment via the coating. These conditions forceramic coatings take into account such features as their transversal anisotropy, spatial (through-thickness)inhomogeneity of the initial porosity, temperature-dependent thermomechanical properties of the coatingceramic coatings under inuence of thermal loads has been developed. The approach employs a gen-omputational scheme as an iterative procedure for determining damage evolution parameters, whichporates an analytical solution of the appropriate interim boundary-value thermoelasticity problem at7. Conclusions

    In this paper, the ecient semi-analytical approach to analyze damage evolution in structural elementsa11 1=E1; a33 1=E3; a12 m21=E1; a13 m31=E1 m13=E3; a66 1=G12:

  • Hererespetensioof isofor th



    G G0 G0 A G0 A G

    1 1

    L G11 o3 G12 2G66 o o2 G11 o A3A2

    o2 G1nj G1 2G1


    o o


    o3j n31 n00 G11 G12 2G66 n23n00 o3j g3jn11

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4755 A11 A12X2m1


    (om n

    3m12 A3j;j

    o3m A13mo3m A3j;jA3m

    G166 om A2m o3mAmAj;3jA3m

    G21i om A1m omA3jAm


    om A3j


    ! o3m A13mn3j01 kj;j

    " #);3jA3j

    j AjA23j

    j 3j A23jj j 3j j 11 22 12 66 13 1 2

    G111 n3j31 o23j fj G111 n3j14 g3j00 A11 A12 o3j nj11 G112 A2j A13joj n3j01 oj

    G111 3j 3j 1 1 1 j 3j ! 0 3j G11 G12 2G66 n31 o3j fj 12 11A2j A3joj n01 oj





    ! G

    111 G112 2G166

    A2j A3jo3j Ajoj A13j


    166 A3j;12AjA



    G011 G012

    kj k3j

    n3j11 A1j G011 kj;j G012 k3j;j


    1 1 1 11 1 1 3j 2 G G 3j AjAl A3jA3l A1A2 Aj A1A2 A3l AjAl

    G011 G066A1A2


    oj Al;jAl ol

    G01q n3j11 n3l11 G066 nj11nl11 G01sA1A2











    11s A3j;jA2l A3l


    Lj3 G111


    G112 2G166AjA





    !o2j G111 nj22 G112 2G166


    o1o2 !Ljl 1q ojol 66 o3jo3l 11 oj 3l oj 66 o3j j o3j 1q kl;lojE1 = Ek and E3 = E? are Youngs moduli in the plane of isotropy and in a direction normal to it,ctively; m21 = mk is Poissons ratio, characterizing transverse contraction in the plain of isotropy whenn is applied in the plane; m13 = m? is Poissons ratio, characterizing transverse contraction in the plaintropy when tension is applied in a direction normal to the plane of isotropy; G12 is the shear moduluse plane of isotropy.

    ndix B

    pressions for dierential operators Ljl (j,l = 1,2,3) given by Shevchuk (2000) for an isotropic case areformed for a transversely isotropic shell as follows:

    0 1

  • Here

    1 2 Aj A3j

    0; j 6 l

    Expressions for dierential operators pjl(j = 1,2,3; l = 1,2,3,4) given by Shevchuk (2000) for an isotro-

    p3j 1 mbG12 G11 ojn01 o3jn01

    1 1 1 1 1 1

    4756 V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 G11 mbG12 ojAj A3joj G12 mbG11 o3jAjA3jo3j ;

    j; l 1; 2; q 2 djl; s 1 djl;

    p33 mbG111 G112

    EbD mbG

    011 G012 k1 k2


    p 4 1 mb G1o o o n1 o n2 :Eb Eb opjl A1jmbG

    01s G01qEb

    oj n3j111 mbEb

    1jlG011 G012 ;

    pj3 A1jmbG011 G012

    Eboj; pj4

    21 mbEb

    G066 A13jo3j 2nj11


    g0j mbg03j A11 A12 1 1 3j jnpic case are transformed for a transversely isotropic shell as follows:Appendix Cj; l 1; 2; q 2 djl; s 1 djl; i 2 djm; n1; n2 0; 1; 2; . . . ;

    djl 1; j l;

    is the Kronecker symbol:X o1o2 n110o1 n210o2;L33 X2m1

    (G211 Dm G212 D3m g1m

    Pm G111 Dm G112 D3m

    km G111 G112A1A2

    omn3m01 km k3m

    G211 G212A1A2

    omn3m01 P1 n3m10 P2 2G266A1A2

    om 2nm21X o3m


    ) G011 k21 k22

    2G012 k1k2:


    gjn n A3j;3jn1 n2

    ; fj A1j g0j G111 ojA11 A12 ojA1j A3j 2G166 A11 A12 o3jnj01;34 A1A2 Eb66 1 2 1 10 2 10

  • References

    V.A. Shevchuk, V.V. Silberschmidt / International Journal of Solids and Structures 42 (2005) 47384757 4757Ambartsumyan, S.A., 1974. General Theory of Anisotropic Shells. Nauka, Moscow (in Russian).Burak, Ya.I., Galapats, B.P., Gnidets, B.M., 1978. Physicomechanical Processes in Electroconductive Bodies. Naukova Dumka, Kiev

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    Analysis of Shells, vol. 4. Naukova Dumka, Kiev (in Russian).Lyubimov, V.V., Voevodin, A.A., Spassky, S.E., Yerokhin, A.L., 1992. Stress analysis and failure possibility assessment of multilayer

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    and Applications. Springer, WienNew York, pp. 233294.Najar, J., Silberschmidt, V.V., 1998. Continuum damage and failure evolution in inhomogeneous ceramic rods. Archives of Mechanics

    50 (1), 2140.Pelekh, B.L., Fleishman, F.N., 1988. Approximate method for the solution of problems in the theory of elasticity for bodies with thin

    curvilinear coatings. Izv. Akad Nauk SSSR Mekh. Tverd. Tela 5, 3641.Podstrigach, Ya.S., Shevchuk, P.R., 1967. Temperature elds and stresses in bodies with thin coverings. Teplovye Napryazhenia

    v Elementakh Konstruktsij (Thermal Stresses in Structural Elements) 7, 227233.Podstrigach, Ya.S., Shevchuk, P.R., Onufrik, T.M., Povstenko, Yu.Z., 1975. Surface eects in solid bodies taking into account

    coupling physico-mechanical processes. Physicochemical Mechanics of Materials 11, 3643 (in Russian).Podstrigach, Ya.S., Shvets, R.N., 1978. Thermoelasticity of Thin Shells. Naukova Dumka, Kiev (in Russian).Sevostianov, I., Kachanov, M., 2001. Plasma-sprayed ceramic coatings: anisotropic elastic and conductive properties in relation to the

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    Semi-analytical analysis of thermally induced damage in thin ceramic coatingsIntroductionGeneral computational schemeGeneralized boundary conditions and restoration relationsCase of absence of bending and twisting strainsProblem of a coated cylinderNumerical resultsConclusionsAppendix AAppendix BAppendix CReferences


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