Optimization of 316 stainless steel alumina functionally...

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Materials Science and Engineering A252 (1998) 117 – 132 Optimization of 316 stainless steel/alumina functionally graded material for reduction of damage induced by thermal residual stresses M. Grujicic *, H. Zhao Program in Materials Science and Engineering, Department of Mechanical Engineering, 241 Flour Daniel Building, Clemson Uni6ersity, Clemson, SC 2963 -0921, USA Received 19 November 1997; received in revised form 23 February 1998 Abstract Development of material damage due to the thermal residual stresses in a 316 stainless steel/Al 2 O 3 functionally graded material [FGM] model system during cooling from the processing temperature (900°C) has been analyzed using the commercial finite element package ABAQUS. Specifically, the effect of the material concentration profile of a 316 stainless steel/Al 2 O 3 graded layer between the pure 316 stainless steel and pure Al 2 O 3 regions on redistribution and reduction of thermal residual stresses and material damage has been investigated. For each condition of the interlayer material concentration profile analyzed, the stress and damage reductions have been quantified by comparing the magnitudes of specific stress components and damage parameters (interface decohesion, porosity, loss of materials stiffness, etc.) with their counterparts in the nongraded (sharp interface) 316 stainless steel/Al 2 O 3 case. An optimization analysis of the concentration profile showed that the maximum stress and damage reductions are achieved for nonlinear material concentration profiles represented by the material concentration exponent p =4. In this concentration profile the largest gradient in the material properties is located in the metallic portions of the graded region characterized by the lower values of the Young’s modulus and higher plasticity. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Material damage; Thermal residual stress; Stainless steel/Al 2 O 3 ; Functionally graded material; Model system; Cooling 1. I. Introduction Due to differences of thermal and mechanical prop- erties in ceramics and metals, residual stresses develop in regions near the ceramic/metal interfaces during fabrication and under thermal and mechanical loading in service. These stresses affect performance and the lifetime of the ceramic/metal bonded systems and can cause cracking within ceramic, plastic deformation ac- companied by formation and growth of the voids in metal and/or ceramic/metal decohesion. Over the last decade, numerous experimental and theoretical investi- gations [1 – 6] have established the relationship between the magnitude and the distribution of residual stresses, the failure mechanisms, materials thermo-mechanical properties and specimen/structure geometry. It is well-established [7,8] that residual stresses due to ceramic/matrix property mismatch can be reduced in magnitude and redistributed in a desired manner if an intermediate layer is placed between the two materials. The microstructure and properties within this layer are graded, i.e. varied in a more gradual fashion from those of one material to the ones of the other. The concept of grading was initially applied, with limited success, to thermal barrier coatings for turbine and engine compo- nents [7,8]. Over the last several years, particularly in Japan [9,10] the same grading approach has been uti- lized to develop the so-called ‘functionally gradient materials’ (FGM) for variety of applications. While extensive experimental and theoretical studies conducted in the past have helped identify factors which control the magnitude and the distribution of residual stresses in ceramic/metal materials with sharp interfaces, the behavior of graded microstructures is considerably less investigated and not well understood [11,12]. In the present paper, the finite element method * Corresponding author. 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S09 21- 5093(98)006 1 8 -2

Transcript of Optimization of 316 stainless steel alumina functionally...

Materials Science and Engineering A252 (1998) 117–132

Optimization of 316 stainless steel/alumina functionally gradedmaterial for reduction of damage induced by thermal

residual stresses

M. Grujicic *, H. ZhaoProgram in Materials Science and Engineering, Department of Mechanical Engineering, 241 Flour Daniel Building, Clemson Uni6ersity, Clemson,

SC 2963-0921, USA

Received 19 November 1997; received in revised form 23 February 1998

Abstract

Development of material damage due to the thermal residual stresses in a 316 stainless steel/Al2O3 functionally graded material[FGM] model system during cooling from the processing temperature (900°C) has been analyzed using the commercial finiteelement package ABAQUS. Specifically, the effect of the material concentration profile of a 316 stainless steel/Al2O3 graded layerbetween the pure 316 stainless steel and pure Al2O3 regions on redistribution and reduction of thermal residual stresses andmaterial damage has been investigated. For each condition of the interlayer material concentration profile analyzed, the stress anddamage reductions have been quantified by comparing the magnitudes of specific stress components and damage parameters(interface decohesion, porosity, loss of materials stiffness, etc.) with their counterparts in the nongraded (sharp interface) 316stainless steel/Al2O3 case. An optimization analysis of the concentration profile showed that the maximum stress and damagereductions are achieved for nonlinear material concentration profiles represented by the material concentration exponent p=4. Inthis concentration profile the largest gradient in the material properties is located in the metallic portions of the graded regioncharacterized by the lower values of the Young’s modulus and higher plasticity. © 1998 Elsevier Science S.A. All rights reserved.

Keywords: Material damage; Thermal residual stress; Stainless steel/Al2O3; Functionally graded material; Model system; Cooling

1. I. Introduction

Due to differences of thermal and mechanical prop-erties in ceramics and metals, residual stresses developin regions near the ceramic/metal interfaces duringfabrication and under thermal and mechanical loadingin service. These stresses affect performance and thelifetime of the ceramic/metal bonded systems and cancause cracking within ceramic, plastic deformation ac-companied by formation and growth of the voids inmetal and/or ceramic/metal decohesion. Over the lastdecade, numerous experimental and theoretical investi-gations [1–6] have established the relationship betweenthe magnitude and the distribution of residual stresses,the failure mechanisms, materials thermo-mechanicalproperties and specimen/structure geometry.

It is well-established [7,8] that residual stresses due toceramic/matrix property mismatch can be reduced inmagnitude and redistributed in a desired manner if anintermediate layer is placed between the two materials.The microstructure and properties within this layer aregraded, i.e. varied in a more gradual fashion from thoseof one material to the ones of the other. The concept ofgrading was initially applied, with limited success, tothermal barrier coatings for turbine and engine compo-nents [7,8]. Over the last several years, particularly inJapan [9,10] the same grading approach has been uti-lized to develop the so-called ‘functionally gradientmaterials’ (FGM) for variety of applications.

While extensive experimental and theoretical studiesconducted in the past have helped identify factorswhich control the magnitude and the distribution ofresidual stresses in ceramic/metal materials with sharpinterfaces, the behavior of graded microstructures isconsiderably less investigated and not well understood[11,12]. In the present paper, the finite element method* Corresponding author.

0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0921-5093(98)00618-2

M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132118

is used to investigate the development of material dam-age due to the residual stresses in 316 stainless steel/Al2O3 model system and determine the materialconcentration profile within the graded region, whichresults in smallest magnitudes and the optimum distri-bution of the thermal residual stresses and minimumdamage. Rather than assuming a continuous variationof the materials volume fractions throughout thegraded region, which is practically unattainable viapowder metallurgy processing, the graded region istaken to consist of a finite number of layers, each withfixed volume fractions of the two materials. Differentgrading profiles are obtained by varying the thicknessesof these layers.

The organization of the present paper is as following:in Section 2.1, details of the finite element procedureused in the present work are discussed. The methodsused for determination of the effective thermo-elasticproperties of the FGM and their dependence on thematerials microstructure are given in Section 2.2. Themodels used to represent the plastic response of differ-ent layers in FGM are presented in Section 2.3. Adescription of the constitutive models for the interfacesbetween different layers of FGM is presented in Section2.4. In Section 3, the results obtained are presented anddiscussed. The main conclusions resulting from thepresent work are listed in Section 4.

2. Computational procedure

2.1. Finite element analysis

2.1.1. Problem descriptionIn the present work, the commercial finite element

package ABAQUS [13] is used to study the develop-ment of thermal residual stresses and material damagein an axisymmetric disk-shape specimen containing 316stainless steel and Al2O3. It is assumed that the speci-men was fabricated at a temperature of 900°C andcooled to room temperature. Cooling is assumed to beuniform throughout the specimen, and the effect oftime-dependent materials evolution processes such ascreep is ignored. These assumptions are generally con-sistent with the conditions encountered during relativelyfast furnace cooling following hot pressing of powdermetallurgy processed 316 stainless steel–Al2O3 com-posites and FGMs.

2.1.2. Specimen geometry and finite element meshesA disk-shape specimen with a diameter of 20 mm and

thickness of 10 mm shown in Fig. 1 is analyzed in thepresent work. For the analysis of the sharp interfacebetween the metal and the ceramic, the specimen isconsidered to consist of two 5-mm thick layers, onecontaining pure 316 stainless steel (top layer, Fig. 1(a)),

and the other pure Al2O3 (bottom layer, Fig. 1(a)). Thespecimen used in the analysis of the effect of the gradedinterlayers is taken to consist of seven layers while eachlayer contains fixed volume fractions of 316 stainlesssteel and Al2O3. The top and the bottom layers, each2.5 mm thick, are taken to be pure 316 stainless steeland pure Al2O3, respectively. The intermediate region istaken to consist of five layers with fixed volume frac-tions of 316 stainless steel (0.9, 0.7, 0.5, 0.3 and 0.1,listed from top to bottom). The thickness of each layeris determined using the following procedure. First, it isassumed that the ‘smoothed-out’ variation in the vol-ume fraction of 316 stainless steel throughout thegraded region can be represented using a power-lawtype (continuous) function:

f316=�y

t�p

(1)

where y is the distance from the bottom (pure Al2O3)layer and t is the thickness of the graded region (5 mmin the present case). Variations of the volume fractionof 316 stainless steel throughout the graded region for pvalues of 0.125, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0 are shown inFig. 2(a). For a given value of p, the values of y/tcorresponding to the volume fraction of 316 stainlesssteel of 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0 are determined.These six y/t values are next used to determine thethickness of the five layers as shown in Fig. 2(b).

To simplify the calculation, the computational modelis considered to be axisymmetric. The finite elementmeshes for both the sharp-interface case and the casesinvolving a graded region are constructed in such a waythat the element size is minimized in the regions wherethe stress and strain gradients are the largest. Specifi-cally, for the sharp-interface case, the element size isrefined in the region near the radial free edge (the rightedge of the mesh in Fig. 3(a)), near the z-axis (the leftedge in Fig. 3(a)) and around the 316 stainless steel/Al2O3 interface. To overcome the potential numerical

Fig. 1. The geometry of the circular disk-shape 316 stainless steel/Al203 specimens containing: (a) a sharp ceramic/metal interface and(b) a graded ceramic/ metal region.

M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132 119

Fig. 2. (a) Variation of the volume fraction of 316 stainless steelthroughout the graded region for several values of the exponent p inEq. (1). (b) The procedure used to determine the thickness of the fivelayers (marked as 1, 2, 3, 4, 5) within the graded region.

and strain gradient region is reduced by 50%. The meshshown in Fig. 3(b) is used for the volume fractionexponent value p=2.

It should be noted that the polycrystalline materialsproperties and constitutive relations used in the presentwork impose a lower limit in the size of the finiteelements. That is the smallest dimension of any elementshould not be smaller than the edge length of thecorresponding representative material element (RME),i.e. the element which captures all the essential featuresof the polycrystalline material microstructure and prop-erties. Determination of the RME edge size was donefollowing the procedure outlined in Ref. [14] for thetypical 316 stainless steel particle size of 5 mm andAl2O3 particle size of 0.5 mm. The results of this proce-dure are shown in Fig. 4.

2.1.3. Computational procedureAs stated earlier all the calculations were carried out

using the finite element package ABAQUS [13]. Withinthis package the Cauchy’s equilibrium equations aresolved in Lagrangian form. Except for the left edge ofthe meshes shown in Fig. 3, which corresponds to theaxis of symmetry, all other edges are set free so thatbending is permitted to take place during cooling. Thecooling was done in fine increments (10°C) and, due tothe nonlinear (plastic) material behavior, solution hadto be iterated at each temperature until the equilibriumis obtained.

To examine the sensitivity of the results to the size ofthe mesh used, few calculations were carried out usingthe meshes obtained by halving thickness and the width

Fig. 3. Finite element meshes and materials used in the analysis of thesharp ceramic/metal interface and graded ceramic/metal intermediateregion (p=2).

difficulties resulting from the elements with an aspectratio significantly different than one, the width of theelements in ten rows adjacent to the interface is reducedby 50%. This doubled the number of elements in theserows. For the cases of a graded region separating the316 stainless steel and Al2O3 layers, the elements arealso refined near the radial free edge and near the z-axisas well as in portions of the graded region where thegradient in the volume fraction of the two materialschanges is the greatest. As in the case of the sharpinterface, the width of the elements in the high stress

M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132120

Fig. 4. Variation of the representative material element (RME) edgelength with 316 stainless steel volume fraction.

Table 2Effect of temperature on Young’s modulus and Poisson’s ratio of 316stainless steel [15]

Young’s modulus (GPa)Temperature (°C) Poisson’s ratio

25 0.263198.6193.793 0.278189.6149 0.297

0.306185.52040.314181.3260

316 0.319176.5171.7371 0.325

427 166.9 0.331162.0482 0.334157.2538 0.341

593 153.1 0.338148.2 0.328649

0.316143.4704760 137.9 0.298816 131.7 0.274

123.0 0.243900

ume fractions of the two materials but also on thematerials microstructure. As suggested by the schematicshown in Fig. 5, the FGM microstructure in the regionsof large and small volume fractions of the two materialscan be described as discrete inclusions of one materialembedded in the matrix of the other material. Con-trary, in the region where the volume fractions of thetwo materials are comparable, the microstructure canbe described in terms of intertwined clusters of the twomaterials. In the intermediate regions, the FGM mi-crostructure can be represented as a mixture of the twobasic microstructures described above.

Most of the averaging schemes used to determine theeffective thermo-mechanical properties of the ceramic/metal composite for the two basic microstructures assummarized in Table 5 are well-established and widelyused and hence will not be discussed in the presentwork. However, the use of the Voronoi cell finiteelement method (VCFEM) to determine the effectiveelastic and plastic properties for the microstructure

of the elements shown in Fig. 2. Maximum change instress and strain due to mesh refinement was found tobe within 5%. These differences are deemed as minorand hence all the subsequent calculations were doneusing the original meshes shown in Fig. 2.

2.2. Determination of the effecti6e thermo-elasticproperties

The linear coefficient of thermal expansion and elas-tic properties (the Young’s modulus and the Poisson’sratio) and their dependence on temperature for 316stainless steel [15] and Al2O3 [16] are summarized inTables 1–4.

The effective thermo-elastic properties of the ce-ramic/metal composite material in the graded regionhave been determined using different analytical andnumerical averaging schemes as summarized in Table 5.The first step in the procedure for determination of theeffective materials properties is to recognize that theseproperties are dependent not only on the relative vol-

Table 3Effect of temperature on mean linear thermal expansion coefficient ofalumina [16]

Expansion coefficient (10−6 mm mm−1 °C−1)Temperature (°C)

27 5.60127 6.03

6.55227327 6.93427 7.24527 7.50627 7.69727 7.83

7.97827927 8.08

Table 1Effect of temperature on mean linear thermal expansion coefficient of316 stainless steel [15]

Expansion coefficient (10−6 mm mm−1 °C−1)Temperature (°C)

14.725200 15.3

16.2400600 16.9800 17.6

18.31000

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M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132 121

Table 4Effect of temperature on Young’s modulus and Poisson’s ratio ofalumina [16]

Poisson’s ratioYoung’s modulus (GPa)Temperature (°C)

385.4 0.2625500 369.1 0.25

352.4 0.248001000 330.3 0.23

Fig. 5. (a) Schematic representation of the microstructure of a gradedceramic/metal region. The microstructure consisting of dispersedparticles embedded in a continuous matrix is observed for large andsmall volume fractions of the metal and ceramic, as in (b) and (d). Inthe intermediate region where the volume fractions of the two materi-als are comparable to each other, the microstructure consists ofintertwined clusters of the two phases.

consisting of intertwined clusters of the two materialshas been recently proposed by Grujicic and Zhang [14]and will be briefly review below.

The first step in the application of VCFEM is toidentify the representative material element (RME)which contains all the essential features of the materialsmicrostructure. An example of the RMEs used in thepresent work is shown in Fig. 6. The microstructurewithin the RME is next discretized using Voronoi cells,and each cell is assigned a given set of materials proper-ties depending on the nature of the material constitut-ing the cell. By treating each cell as a finite element, ahybrid stress/displacement finite element formulation[14] is next used to determine the stiffness matrix ofeach element and the elements are assembled to formthe entire computational domain (RME). Lastly, theRME is analyzed using different boundary value prob-lems (e.g. plane-strain tension, simple shear etc.) andthe effective RME (i.e. FGM) properties determinedfrom the FEM solution. To include the statistical ef-fects of microstructure, for each layer of FGM, tenRMEs are generated and analyzed and the average oftheir properties taken as the effective FGM properties.This procedure was found by Zhang [29] to yield asignificantly better agreement with the experiment. For

example, the Young’s modulus for the 316 stainlesssteel+50% Al2O3 two phase system predicted by the‘discrete inclusion’ method are typically 25–30% lowerthan its experimental counterpart. On the other hand,the experimental Young’s modulus data and the onepredicted by the VCFEM method are in agreement towithin 5–10%.

2.3. Plastic materials constituti6e relations

The nonreversible (plastic) response of various por-tions of the graded and nongraded regions of FGM is

Fig. 6. Typical representative materials element (RME) consisting ofVoronoi cells used for determination of effective elastic and plasticmaterials properties of the regions of FGM whose microstructureconsists of intertwined clusters of the two materials as shown in Fig.4(c).

Table 5Averaging schemes used for determination of effective thermome-chanical properties of two basic microstructures in FGMs

Material property Type of microstructure/averaging scheme

Intertwined clustersDiscrete inclusions

Thermal conduc- Kerner’s equation Kerner’s equation [17][17]tivityKerner’s Equation Turner’s equation [18]Thermal expansion

coefficient [17]Elastic properties Self consistent Voronoi cell FEM [14]

method [19]Effective plastic Self consistent Voronoi cell FEM [14]

method [19]propertiesFailure stress/ Self consistent Voronoi cell FEM [14]

strain method [19]

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Table 6Effect of temperature on uniaxial tensile properties of 316 stainless steel

n (no unit)e* (%)Temperature (°C) Yield stress (MPa) Uniform tensile strength (MPa) Uniform elongation (%) k (MPa)

572 0.66525 232 544 0.18075.00.330535 0.17593 52.0197 478

509 0.110204 162 440 42.0 0.168502 0.0375316 142 435 41.0 0.160

0.0211502 0.156371 40.0134 43540.0 473 0.0111 0.148482 123 413

408 0.00853593 109 359 40.0 0.1410.0100294 0.133704 41.090 272

0.1260.0516816 72 169 45.0 187114 0.689900 63 105 48.0 0.120

Coefficients k, e* and n appear in the flow stress (s) vs. plastic strain (epl) relation: s=k(epl+e*)n.

analyzed using the materials constitutive models givenbelow.

2.3.1. Stainless steelPlastic response of 316 stainless steel is modelled

using the revised version of the Gurson’s porous plas-ticity model [20–22]. This model takes into accountthe effect of initial porosity as well as the porositywhich may develop in the high strain/high hydrostaticstress regions. Following Tvergaard [22], the yield po-tential parameters are set as following: q1=1.5, q2=1.0, and q3=q1

2. The void nucleation parameters aredefined as: initial porosity fN=0.05, mean void nucle-ation strain eN=0.1, the nucleation strain standarddeviation sN=0.05. The plastic properties (the yieldstress, the ultimate tensile strength and the percentageuniform elongation and their dependence on tempera-ture) for 316 stainless steel [15] are given in Table 6.The coefficients k, e* and n appearing in the flowstress (s) versus plastic strain (epl) relation: s=k(epl+e*)n are also shown in the Table 6.

2.3.2. Al2O3

The behavior of Al2O3 was analyzed using theHilleborg’s model for brittle solids [23–25]. Accordingto this model, the plastic response of Al2O3 is asfollows: in compression, after the initial linear elasticresponse, a small nonrecoverable straining occurs be-fore the ultimate strength is reached. Further strain-ing in compression leads to materials softening untilthe material can no longer support the load. Whenloaded in tension, Al2O3 initially responds as a linearelastic material until a level of stress (taken as 10% ofthe ultimate compressive strength) at which crackingtakes place. The damage caused by cracking manifestsitself in the loss of materials stiffness, and not as anonreversible strain. In other words, the cracks areassumed to close completely during unloading. Itshould be noted that in the present model the physi-

cal cracks are smeared into the continuum materialand their existence is evidenced only by the effectthey have on the stiffness at a given material point.

Temperature dependence of the compressive plasticproperties of Al2O3 [16] is given in Table 7. The com-pressive stress–plastic strain relationship has beenfitted to a function s c=A(e c

pl−C) e−B(ecpl−C) and the

temperature dependence of the parameters A, B, andC is also given in Table 7. Temperature dependenceof the tensile properties of Al2O3 is given in Table 8.The post cracking behavior of Al2O3 is representedusing a function s t=s t

US e−D(e−etcrack), where sUS

t ande crack

t are the stress and total strain levels at whichcracking occurs. Temperature dependence of parame-ter D is also given in Table 8.

2.3.3. Stainless steel matrix/Al2O3 inclusionsThe plastic response of the FGM region consisting

of the 316 stainless steel matrix and discrete inclu-sions of Al2O3 is also analyzed using the revised ver-sion of Gurson’s model. The porosity parameters, fN

and eN, are altered to vary, in linear fashion, with thevolume fraction of Al2O3 in order to account for thefact that voids nucleation is facilitated by the metal/ceramic interfaces and that void growth by plasticdilation can occur only in the metallic matrix. Theyield stress and its dependence of plastic strain aredetermined using the Self Consistent averagingscheme [19].

2.3.4. Intertwined cluster of 316 stainless steel andAl2O3

The effective plastic responses of the graded regionconsisting of intertwined clusters of the two materialsare determined using the procedure analogous to theone applied to the case of the 316 stainless matrix/Al2O3 inclusions except that the yield stress versusplastic strain relationship is determined using theVCFEM procedure as described in Section 2.2.

M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132 123

Table 7Temperature dependence of uniaxial compression yield stress (sy

c), ultimate strength (sUSc ), plastic strain corresponding to ultimate strength, and

the stress at the plastic strain level of 0.006 of alumina

sc (eplc =0.0006) (MPa) A (GPa) BTemperature (°C) Csy

c (MPa) sUSc (MPa) epl,US

c

−0.00021824.6569424 50.81016 2540 0.00146.2 5178 824.6300 −0.00021924 2310 0.001

−0.00021824.66164500 55.01100 2750 0.00147.6 5335 824.6800 952 −0.000212380 0.00148.2 54021000 964 824.62410 −0.000210.001

Parameters appearing in the stress–strain relationship sc=A(ecpl−C) e−B(ecpl−C) are also shown.

2.3.5. Al2O3 matrix/316 stainless steel inclusionsIn this case, the overall plastic behavior of the mate-

rial is analyzed using Hilleborg’s model. The effect of316 stainless steel inclusions, is included by reducingthe rate of the loss of load carrying capacity past thepoint of cracking in tension to account for the crackbridging effect of the 316 stainless steel inclusions. Thiswas done by increasing (in a linear fashion) the strainat which the yield stress after cracking is 1% of theultimate tensile strength so that this strain is 5% forAl2O3 containing 30% 316 stainless steel. This valuewas obtained in our ongoing experimental work [26].Similarly, the rate of the loss of load carrying capacitypast the peak stress in compression is also reduced toaccount for the effect of plasticity in the metallic inclu-sions. This was done by increasing (in a linear manner)the strain at which the yield softening has reduced theyield stress to 1% of the ultimate compressive stress, sothat this strain is twice as large as that in pure Al203.

2.4. Interface constituti6e beha6ior

The interfaces between different layers have beenmodelled using the cohesive zone framework intro-duced by Needleman [27]. The cohesive zone is as-sumed to have negligible thickness when compared toother characteristic lengths of the problem, such as theRME edge length, typical lengths associated with thegradient of the fields, etc. The mechanical behavior ofthe cohesive zone is characterized by a traction–dis-placement relation which is introduced through thedefinition of an interface potential, C.

Stable equilibrium for the interface corresponds to aperfectly bonded configuration, where the potential hasa minimum and all tractions vanish. For any otherconfiguration, the value of the potential is taken todepend only on the displacement jump across the inter-face. For axisymmetric problems, the interface dis-placement jump is expressed in terms of its normalcomponent, Un, and a tangential component, Ut, wherethe latter lies in the r–z plane of the polar coordinatesystem.

Differentiating the function C=C. (Un, Ut) with re-spect to Un and Ut yields, respectively, the normal and

tangential components of F, the traction per unit inter-face area in the deformed configuration:

Fn(Un, Ut)=−(C. (Un, Ut)

(Un

(2)

Ft(Un, Ut)=−(C. (Un, Ut)

(Ut

(3)

The interface constitutive relations are thus fullydefined by specifying the form for the potential func-tion C. (Un, Ut).

The interface potential of the following form initiallyproposed by Socrate [28], is used in the present study:

C. (Un, Ut)

=�

−esmaxdn+12

tmaxdt log�

cosh�

2Ut

dt

�n��

e−Un

dn�

1+Un

dn

n�(4)

where the parameters smax and tmax are respectively thenormal and tangential interfacial strengths, and dn anddt are the corresponding characteristic interfacelengths. Differentiation of Eq. (4) with respect to Un

and Ut yields the expressions for the interfacial trac-tions:

Fn(Un, Ut)

=�

esmax−12

tmax

dt

dn

log�

cosh�

2Ut

dt

�n��Un

dn

e−Un

dn�

(5)

Ft(Un, Ut)=�

tmaxtanh�

2Ut

dt

���e−

Un

dn�

1+Un

dn

n�(6)

Graphical representation of the two functions definedby Eqs. (5) and (6) is given in Fig. 7.

If Fn given by Eq. (5) is expressed for the case ofpurely normal interface decohesion, and the Ft for thecase of pure sliding, one obtains:

Fn(Un, Ut=0)=F0n(Un)=esmax

�Un

dn

e−Un

dn�

(7)

Ft(Un=0, Ut)=F0t (Ut)=tmax tanh

�2

Ut

dt

�(8)

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Table 8Temperature dependence of ultimate tensile strength of alumina

Ultimate tensile strength, scrackt (MPa) Cracking strain, ecrack

t (%) DTemperature (°C)

254 0.066 46051.7244605.1.70.062300 231

0.0745 46051.7500 27546051.70.0675800 23846051.7241 0.0731000

Temperature dependence of parameters D appearing in the post-cracking stress–strain relation s t=s tUS e−D(e−etcrack).

Inspection of Eqs. (7) and (8) shows that: smax is thepeak normal traction for purely normal interface deco-hesion; dn as the normal interface separation whichcorresponds to this peak traction; tmax as an asymptoticshear traction for interface sliding; and dt as a charac-teristic length in pure sliding which corresponds to ashear traction within 1% tmax (F t

0(dt)#0.99tmax).Integration of Eq. (7) from the fully bonded configu-

ration (Un=0) to complete decohesion (Un��), yieldsthe work for normal separation:

F=esmaxdn (9)

The normal behavior of the interface is thus character-ized in terms of its strength smax, and the work ofseparation F. The normal interfacial strength is gener-ally taken to be 0.01 to 0.02 times the material Young’smodulus, while F is in the range of 1–10 J m−2

(unpublished research). In the present work each inter-face is taken that smax=2 GPa and F=5 J m−2.

The tangential response of the interface is assumed tobe of a frictional type so that purely tangential interfacedecohesion is not allowed. Rather, the effect of shearinterfacial displacements is reflected in a reduced capa-bility of the interface to withstand normal decohesion.In other words, the peak normal strength decreaseswith increasing tangential displacements. The interfaceshear strength is generally considered to be a fraction ofthe normal strength: i.e. tmax=atsmax, with at51, andthe characteristic length in shear is expressed as afraction of the characteristic length for normal decohe-sion: dt=btdn. This allows the effects of different tan-gential behaviors of the interface to be assessed byvarying the magnitudes of at and bt parameters. In ourwork, it was observed that the tangential behavior ofthe interface plays a minor role, as compared to theinterface behavior for normal decohesion. All the re-sults presented in the next section pertain to the condi-tion at=bt=0.1.

The cohesive model discussed above is used to definethe constitutive behavior for the interfacial finite ele-ments separating different layers in the FGM. Theprocedure for numerical implementation of the cohe-sion model into the user element (UEL) subroutine ofABAQUS is recently presented by Grujicic and Lai(unpublished research).

3. Results and discussion

It is well-established that ceramic-coated disk-shapemetal components can fail by several different mecha-nisms, the main one being: (a) ceramic cracking in thedirection normal to the metal/ceramic interface; (b)metal/ceramic interface decohesion which results in de-lamination and spelling of the coating; (c) excessiveplastic deformation and void formation and growth inthe metal; etc. These failure mechanisms are mainlycontrolled by the magnitude and the distribution of theaxial stresses at or near the radial free edge of thespecimen and by the magnitude and distribution of theradial or circumferential stresses near the specimen’saxis of symmetry (z-axis). To quantify the extent ofmaterial damage and relate it to the magnitude and thedistribution of the thermal residual stresses, for each ofthe cases analyzed in the present work, the followingquantities were monitored: (a) The magnitude of thenormal displacements across the interface between 316stainless steel, alumina and five 316 stainless steel/Al2O3-based composite layers; (b) The existence of thecracks in Al2O3 and the Al2O3-based layers in the axialand/or radial directions: (c) the magnitude of the equiv-alent plastic strain and the volume fraction of voids in316 stainless steel and 316 stainless steel based layers,and (d) the magnitude of the axial and radial normalstresses throughout the entire specimen.

3.1. Case I. Sharp interface

The results pertaining to the extent and distributionof damage and the magnitude of thermal residualstresses for the case of sharp 316 stainless steel/Al203

interface are shown in Fig. 8(a)–(f).The magnitude of the normal interfacial displace-

ment (Fig. 8(a)) is normalized relative to the character-istic normal interface separation distance, dn (Eq. (7)).The normal interface displacement Un is maximum atthe free edge of the interface where it approaches thevalue of dn. However, since along the entire 316 stain-less steel/Al2O3 interface UnBdn, no interfacial decohe-sion occurs.

The distribution of damage in Al2O3, shown in Fig.8(b), reveals that it is primarily concentrated along the

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Fig. 7. Normal and tangential components of the traction per unit interface area, as a function of the normal and tangential components of theinterface displacements.

bottom face of the specimen and in the interface regionnear the free radial edge of the specimen. The numbersused in the contour plot in Fig. 8(b) designate thenumber of principal directions in which cracks haveformed. It should be noted that since formation of thecracks causes a degradation in materials stiffness ratherthan imparting a complete loss of load carrying capac-ity, the extent of damage can be further assessed byanalyzing the magnitude of the stresses (and the extentof their reduction) in the ‘damaged’ region. A compari-son of the results given in Fig. 8(b) with those given inFig. 8(e) and (f) shows that formation of the cracksalong the bottom edge of the specimen and along thespecimen’s free radial edge gives rise to a significantreduction of the stresses, especially of the radial stress.This suggest that the extent of damage (reduction inmaterials stiffness) caused by cracking in Al2O3 is sig-nificant. A comparison of the results in Fig. 8(e) and (f)with those published by Williams et al. [11,12] for asimilar specimen geometry and similar materials consti-

tutive relations shows, after accounting for the differ-ences in the elastic properties of the two metal–ceramicsystems, that the stresses, while displaying a similardistribution throughout the specimen, are lower in thepresent case by 20–40%.

The magnitude and the distribution of the equivalentplastic strain and porosity in 316 stainless steel areshown in Fig. 8(c) and (d), respectively. It should benoted that in the initial condition, both 316 stainlesssteel and Al2O3 were deformation free and contained5% of porosity, uniformly distributed throughout thespecimen. As expected the maximum level of plasticstrain in 316 stainless steel occurs in the interfacialregion near the free edge of the specimen (Fig. 8(c)).However, only a minor increase (up to 0.6%) in thelevel porosity has resulted from this deformation. Sinceextensive void coalescence occurs typically at porositylevels between 10 and 20%, no extensive porosity-in-duced damage in 316 stainless steel is expected to occurin the present case.

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Fig. 8. The finite element simulation results of thermally induced: (a) interface decohesion, (b) crack in Al203, (c) plastic strain, and (d) voids in316 stainless steel, (e) radial residual stress, sr, and (f) axial residual stress, sz, in the case of the sharp 316 steel/Al2O3 interface. The numbersin (b) refer to the number of cracks in the elements in question.

3.2. Case II. Graded interface cases

Previous work [11,12] has established that beneficialeffects of grading are obtained only when the gradientin materials properties is the largest in the regions ofthe specimen where the material has low rigidity and

can undergo plastic deformation. In other words, thelargest gradient in materials properties should be lo-cated in the metallic regions. According to Fig. 2(a),this is accomplished through the use of the materialsconcentration exponent p which are greater thanunity.

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Fig. 9. The finite element simulation results of thermally induced: (a) interface decohesion, (b) cracks in Al2O3 (layer 1), 90% Al203/10% 316stainless steel (layer 2) and 70% Al203/30% 316 stainless steel (layer 3), (c) plastic strain, and (d) voids in 316 stainless steel, (e) radial residual stresssr, and (f) axial residual stress, sz, in the case of the graded 316 stainless steel/Al203 interface characterized by the exponent p=1.0. The 1/2, 2/3,etc. in (a) refer to the interfaces separating layers 1 and 2, 2 and 3, etc.

The results for the four graded interface cases associ-ated with the values of the material concentration expo-nent p= l, p=2, p=4, and p=8 are shown, respectively,in Figs. 9–12. For each figure, part (a) shows the normalinterfacial displacements for six interfaces in the seven-layer graded specimen; part (b) shows the distribution ofdamage in pure Al2O3, 90% Al2O3/10% 316 stainless steel

and 70% Al2O3/30% 316 stainless steel layers; parts (c)and (d) show, respectively, the distribution of equivalentplastic strain and porosity in pure 316 stainless steel, 90%316 stainless steel/10% Al2O3, 70% 316 stainless steel/30% Al2O3 and 50% 316 stainless steel/50% Al2O3 layers;parts (e) and (f) show respectively the distribution of theradial, sr, and the axial, sz, stresses.

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Fig. 10. The finite element simulation results of thermally induced: (a) interface decohesion, (b) cracks in Al203 (layer 1), 90% Al203/10% 316stainless steel (layer 2) and 70% Al2O3/30% 316 stainless steel (layer 3), (c) plastic strain, and (d) voids in 316 stainless steel, (e) radial residualstress sr, and (f) axial residual stress sz, in the case of the graded 316 stainless steel/Al2O3 interface characterized by the exponent p=2.0. The1/2, 2/3, etc. in (a) refer to the interfaces separating layers 1 and 2, 2 and 3, etc.

The results displayed in Fig. 9(a), Fig. 10(a), Fig.11(a) and Fig. 12(a) show that, while the normal dis-placements peak at the free radial edge of the specimen,their magnitude is only a small fraction (0.2–0.3) of thecharacteristic normal separation distance dn. Hence, the

interface decohesion is not expected to make a signifi-cant contribution to the materials damage in either ofthe four graded-interface cases. It should be also notedthat due to the introduction of the five graded layersbetween Al2O3 and 316 stainless steel, the maximum

M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132 129

Fig. 11. The finite element simulation results of thermally induced: (a) interface decohesion, (b) cracks in Al203 (layer 1), 90% Al203/10% stainlesssteel (layer 2) and 70% Al2O3/30% stainless steel (layer 3), (c) plastic strain, and (d) voids in 316 stainless steel, (e) radial residual stress sr, and(f) axial residual stress sz, in the case of the graded 316 stainless steel/Al203 interface characterized by the exponent p=4.0. The 1/2, 2/3, etc. in(a) refer to the interfaces separating layers 1 and 2, 2 and 3, etc.

normal interfacial displacement at the free radial edgeof the specimen has been reduced by a factor of 3–4relative to its counterpart in the sharp interface case.This can be established by comparing the results shownin Fig. 8(a) with those shown in Fig. 9(a), Fig. 10(a),Fig. 11(a) and Fig. 12(a). It should be noted that due to

a lack of experimental data for the interface constitu-tive relations in the 316 stainless steel/Al2O3 system, allfive interfaces in the present work are assigned the sameadhesion characteristic. Therefore one can anticipatethat the outcome of the optimization procedure couldbe different than the one reported in the present work,

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Fig. 12. The finite element simulation results of thermally induced: (a) interface decohesion, (b) cracks in Al2O3 (layer 1), 90% Al2O3/10% 316stainless steel (layer 2) and 70% Al2O3/30% 316 stainless steel (layer 3), (c) plastic strain, and (d) voids in 316 stainless steel, (e) radial residualstress sr, and (f) axial residual stress sz, in the case of the graded 316 stainless steel/Al203 interface characterized by the exponent p=8.0. The1/2, 2/3, etc. in (a) refer to the interfaces separating layers 1 and 2, 2 and 3, etc.

should one or more interfaces be particularly strongeror weaker than the remaining ones.

The distribution of the damage due to formation ofthe cracks in Al2O3 and the two Al2O3-based layersshown in Fig. 9(b), Fig. 10(b), Fig. 11(b) and Fig. 12(b)indicates that for each of the four graded-interface

cases the damage occurs at the bottom side of each ofthe three layers. Furthermore, the extent of damage(represented by the number of principal directions con-taining cracks) is higher either near the symmetry axisof the specimen or near the free radial edge of thespecimen. As mentioned earlier further insight into the

M. Grujicic, H. Zhao / Materials Science and Engineering A252 (1998) 117–132 131

extent of damage can be obtained by examining thedistribution of the stresses which are being relaxed asdamage (a reduction in materials stiffness) takes place.For each of the Figs. 9–12, a comparison of the resultsshown in part (b) with those shown in parts (e) and (f)indicates that cracking-induced reduction in materialsstiffness gives rise to the reduction in stresses. However,based on the results given in Figs. 9–12, parts (b), (e)and (f) alone, it is difficult to determine for which of thefour graded-interface cases the extent of damage is theleast. Neither of the four graded-interface cases has theextent of damage lower in each of the three Al203-basedlayers in relative to the extent of damage in the otherthree cases. In addition, it should be noted that thepresence of metallic particles in 90% Al2O3/10% 316stainless steel and 70% Al2O3/30% 316 stainless steelgives rise to crack bridging and hence, as stated inSection 2.3, delays fracture. To take this effect intoaccount, the extent of damage is re-expressed in thefollowing way. For each of the three Al2O3-based lay-ers, the largest equivalent tensile strain is determined,expressed as a fraction of the fracture strain in thatlayer and named the maximum relative damage strain.The fracture strain is expressed as the strain at whichthe tensile strength of the ‘damaged’ material is 1% ofthe materials ultimate tensile strength. The variation ofthe maximum relative damage strain in the three Al2O3-based layers for the four graded-interface cases isshown in Fig. 13. The results show that the p=4 caseis characterized by a minimum amount of damage ineach of the three Al2O3-based layers in comparison tothe other three cases. This case appears to achieve a

trade off between concentrating the largest materialproperty gradients in the metallic (316 stainless steel-based) regions characterized by lower stiffness andhigher plasticity and minimizing the overall magnitudeof largest property gradient.

The distribution of the equivalent plastic strain andporosity depicted, respectively, in Fig. 9(c), Fig. 10(c),Fig. 11(c), Fig. 12(c) and Fig. 9(d), Fig. 10, (d)Fig.11(d) and Fig. 12(d), show that the excessive growthand coalescence of the voids is in 316 stainless steel andthe three 316 stainless steel based layers is not likely tooccur in either of the four graded-interface cases.

In summary, the extent of formation of the cracksand the resulting material damage in Al2O3 and twoAl2O3-based layers is the least for the graded-interfacecase characterized by the material concentration expo-nent p=4. It should be noted that further reductions inthe extent of material damage can be obtained byincreasing the number of layers in the graded region.Our preliminary work has shown that increasing thenumber of layers by a factor of 2, decreases the maxi-mum relative damage strain by about 15–20% in thethree Al2O3-based layers.

One must recall again that the optimization proce-dure carried out in the present work was done underthe assumption that all five interfaces have the sameinterfacial properties. We found that an increase in thenormal and shear interfacial strengths by a factor of 5did not significantly alter the results shown in Figs.9–12 and did not change the outcome of the optimiza-tion procedure. Likewise, a decrease in the normal andshear interfacial strengths by a factor of 5 did notsignificantly alter the results shown in Figs. 9–12 anddid not change the outcome of the optimization proce-dure. However, if one of the interfaces is made weakerthan the rest by reducing the normal and shear interfa-cial strengths by a factor of 5, the optimization proce-dure showed that the case p=2 becomes optimal. Inour ongoing PhD study, Zhang [29] is carrying out anexperimental characteristic of the interfacial strengthsin the 316 stainless steel/Al203 system and the results ofthis work will be included in future communication.

4. Conclusions

Based on the results presented in the present work,the following conclusions can be drawn:

Formation of cracks in Al203 and the two-Al2O3

based layers appears to be the main damage mechanismin both the sharp-interface and graded-interface cases.Interface decohesion and void-induced damage in 316stainless steel and the three 316 stainless steel layersappear to be less of a concern.

The extent of damage is dependent on the way thegrading in materials properties is done. In the present

Fig. 13. Variation of the maximum relative damage strain with thematerial concentration exponent p for pure Al2O3, 90% Al203/10%316 stainless steel and 70% Al20d3/30% 316 stainless steel.

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case, when grading us done by varying the thickness offive layers of fixed composition, the minimum extent ofdamage is obtained for the grading profile characterizedby the material concentration exponent p=4.

Acknowledgements

This material is based upon work supported by theUS Army Research Office under Grant numberDAAH04-96-1-0197. Additional support was providedby the National Science Foundation under GrantsDMR-93 17804 and CMS-953 1930. The authors areindebted to Dr Wilbur C. Simmons of ARO and DrsBruce A. MacDonald and William A. Spitzig of NSFfor the continuing interest in the present work. Con-structive discussions with Professors Judd Diefendorf,Paul Joseph and Lonny Thomson are greatlyappreciated.

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