Strain energy distribution in ceramic-to-metal jointswhere h is the height of the ceramic, r is the...

Acta Materialia 50 (2002) 883–899www.actamat-journals.com

Strain energy distribution in ceramic-to-metal joints

J.-W. Park, P.F. Mendez, T.W. Eagar*

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 13 November 2000; received in revised form 24 September 2001; accepted 24 September 2001

Abstract

This work introduces a framework for evaluating the strength characteristics of ceramic-to-metal joints with multipleinterlayers. Strain energy in the ceramic is used as a strength metric instead of maximum tensile stress. Based on theFEM analysis and order of magnitude scaling (OMS), simple analytical formulations between the strain energy andmaterial properties are developed, which provide a guideline in designing multiple interlayers. Our analysis reveals theimportant role of multiple interlayers, which reduce the strain energy in the ceramic, increasing the strength of thejoint. Based on the proposed design rule, Si3N4 to Inconel 718 joints have been brazed with single, double and tripleinterlayers and the joint strength was evaluated using a shear test. The experimental results support the design rulesand confirm that strain energy is a good strength metric. 2002 Acta Materialia Inc. Published by Elsevier ScienceLtd. All rights reserved.

Keywords: Ceramic-to-metal; Interlayer; Brazing; Strain energy

1. Introduction

Engineering ceramics possess fascinatingproperties as structural and electronic materials dueto their excellent mechanical properties at hightemperature as well as resistance to wear, erosion,oxidation and corrosion. As novel techniques forproducing these materials have been developed,more and more interest has been focused on theiruse in advanced engineering designs [1]. There hasbeen an increasing need for complex ceramic struc-tures and ceramics have been incorporated intocritical segments of overall structures. Reliablejoining technologies are essential for full exploi-

* Corresponding author.E-mail address: [email protected] (T.W. Eagar).

1359-6454/02/$22.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S1359-6454 (01)00352-4

tation of the properties of the ceramic and for thesuccess of the overall structure.

Significant differences in chemical and physicalproperties between ceramics and metals make itextremely difficult to find an effective joining pro-cess that maintains the strength and resilience ofthe joint. Two primary factors that inhibit forma-tion of a mechanically reliable joint are the coef-ficient of thermal expansion (CTE) mismatch andthe difference in the nature of the interatomic bond.On cooling from the joining temperature, residualstresses are induced due to the CTE mismatch anddiffering mechanical responses of the ceramic andthe metal.

Generally, metals have a larger CTE and lowerelastic modulus than the ceramic. In the joints ana-lyzed in this study [Fig. 1(a)], large tensile andshear stresses are induced in the regions around the

884 J.-W. Park et al. / Acta Materialia 50 (2002) 883–899

Fig. 1. (a) A schematic of the maximum principal stress distribution under thermal loading and FEM mesh configuration of a modeljoint. (b) A schematic of stress distributions in the ceramic near the interface along the x-direction of the model joint under mechanicalor thermal loading. (c) A schematic of stress distributions in the ceramic near the interface along the y-direction of the cylindricallap joint under mechanical or thermal loading.

edge and near the interface of the ceramic, respect-ively, on cooling from the joining temperature[2,3]. These stresses enhance the propensity tofracture of the ceramic at the edges, corners, andarea adjacent to the interface [3]. Analytical andnumerical calculations have revealed that regard-less of the types of loading, the basic nature ofstresses induced in the ceramic is the same exceptits magnitude [2,4,5]. Fig. 1(b) shows the stressdistributions in the ceramic when the joint is under

tensile stress or is cooled from bonding tempera-ture [2,5]. Since both mechanical and thermal load-ing affect the same region, the lower the thermalresidual stresses are, the higher the allowable stressto the fracture of the ceramic. This is the reasonwhy, for this type of joint, the level of thermalresidual stresses can characterize the mechanicalstrength of the joint. However, this type of analysiswould not be valid, for example, in cylindrical lapjoints in which a ceramic tube is constrained by a

885J.-W. Park et al. / Acta Materialia 50 (2002) 883–899

metal shaft [6]. As can be seen in Fig. 1(c), mostparts of the ceramic are highly compressed on coo-ling from bonding temperature, which will increasethe fracture resistance of the ceramic under mech-anical loading [6].

Evans et al. [3,7–11] provided theoretical back-ground on the experimental and numerical overva-tions that (1) brittle fracture in the ceramic hasbeen found to be much more likely to happen thanfracture along the interface; and (2) the joint withlarger thermal residual stresses fractured at lowerstresses, in the same type of joints analyzed in thisstudy [Fig. 1(a)]. They calculated the fractureresistance of a small edge crack in the ceramic andat the interface when the joint is under externalloading after cooling from bonding temperature.

According to their results, large shear stressinduced at the interface suppresses the crack propa-gation along the interface [8,10]. Under certainexternal load, the fracture resistance at the cracktip in the ceramic decreases as thermal residualstresses increase [11]. Compared to the effects ofthermal residual stresses, stresses induced by exter-nal loading on the fracture resistance are negli-gible. During the mechanical test, relaxation ofthermal residual stresses or modification in originalresidual stress distribution can occur due to theadditional plastic deformation of the metal part bythe applied external load. However, it only happenswhen the external load to fracture is many timeslarger than the yield stress of the metal and suchhigh fracture strength cannot be obtained in mostof ceramic-to-metal joints [11]. Therefore, brittlefracture in the ceramic at lower stress indicates thathigher thermal residual stresses are induced in theceramic-to-metal joint [3,7].

Engineers have addressed this problem by usingductile metal interlayers to relieve the residualstresses. However, single interlayers have draw-backs; for example, copper interlayers providemaximum reduction of residual stresses, but theirapplicability in real systems is limited due to theirlow resistance to corrosion and oxidation at hightemperatures. To overcome the limitations of a sin-gle interlayer and for the further release of residualstresses, multiple interlayers [12–14] and func-tionally graded materials (FGM) have been usedfor some applications [15–17].

Currently, there is no deep understanding of thefactors determining the residual stresses in cer-amic-to-metal joints with or without interlayersbecause of the large number of variables involved.In addition, some experimental results seem toyield contradictory conclusions, although it is dif-ficult to make direct comparisons in the absence ofstandard testing methods or requirements of sizeof the specimen. For example, it is not clearlyunderstood what the most effective sequence ofdiffering layers in multiple interlayers is, eventhough this type of joint has already been used incommercial products [18]. Nevertheless, theoreti-cal study of ceramic-to-metal joints can provide auseful viewpoint for comparing the experimentalresults.

The purposes of this investigation are threefold:

� To develop a theoretical understanding andexperimental demonstration of residual stressrelief by interlayers.

� To propose improved design rules for multipleinterlayers.

� To extend strain energy as a strength metric forceramic-to-metal joints having significant plas-ticity.

There is no analytical model that can account forboth plastic deformation and the effects of variousgeometries during bonding of two dissimilarmaterials responding to a change in temperature[4]. Numerical simulations using the finite elementmethod (FEM) are performed in this work to ana-lyze residual stresses from the joining process andto provide criteria for design of the interlayer priorto joining. In this study, the recoverable elasticstrain energy in the ceramic has been used to meas-ure the residual stresses as a more reliable metricof failure, as proposed by Blackwell [4].

1.1. Metric of failure

For a purely elastic case, stress concentration atthe periphery of the ceramic [Fig. 1(a) and (b)] hasthe form of a singularity (s�Hr�l, where l is orderof singularity and H is the intensity of singularity,r is a radial distance from the center of the joint)similar to the crack tip singularity in elastic frac-


ture mechanics. Contrary to the crack tip singular-ity, l is specific for each material combination andH cannot be compared for material combinationsof different l [19], which restrict singularity con-stants (l and H) from being used as a strength met-ric. The stress intensity factor (K) and energyrelease rate (G) at the crack tip have also been usedto predict the fracture resistance of the ceramic.Since both K and G are dependent on the locationand the size of cracks, a statistical approach to theflaw distribution in the ceramic should be used [20]and mesh formulation at the crack tip is highlyrestricted [7].

As mentioned previously and can be seen in Fig.1(a) and (b), a strong stress concentration occursalong the free surface of the ceramic. Themaximum tensile residual stress has been used forthe strength metric most frequently. However, themaximum value of a certain component of residualstresses or strains alone cannot explain fracturebehavior [8]. Fracture initiation by other compo-nents of residual stresses also cannot be ignored[7,16]. Center cracks [8] are frequently observedin cylindrical joint, which are caused by radial orcircumferential stresses near the center of the joint.In addition, the presence of a singularity is prob-lematic for FEM calculations. According to Whit-comb et al. [21], the results obtained within twoelements closest to the singularity are not valid, yetthis is the region where the maximum stresses arefound. Although finer meshes would reduce thesize of the invalid region, the peak value of stressis still inaccurate and highly mesh dependent.

Blackwell [4] addressed these problems by usingstrain energy in the ceramic as a metric for failure.Using FEM, Blackwell calculated the strain energyin ceramic-to-metal bonded systems without aninterlayer having relatively large thickness/widthratios. Based on the fact that the strain energy inthe base materials is controlled by the deformationoccurring at the interface, he found that the func-tional dependency of the strain energy induced inthe ceramic could be expressed by the followingrelationship:

Ue,C�ECE2

M

(EM � EC)2·(aC�aM)2·(T (1)

�Tb)2·(r)3·f(h/r)

where h is the height of the ceramic, r is the radiusof the joint and f(h/r)�erf(h/r). EM, EC are the elas-tic modulus of the metal and ceramic, respectively,and aM, aC are CTE of the metal and the ceramic.T is the room temperature and Tb is the bondingtemperature. The advantages of this approach areless dependency on the mesh formulation and rapidconvergence of strain energy. The strain energy inthe ceramic is calculated by summating the magni-tude of sij·eij·(dV)el over all the elements in the cer-amic, where (dV)el is the volume of each element.Since it summates the magnitude of stress over thevolume of elements for which it acts, it effectivelycouples the magnitude of stress and its distri-butions [4]. Experimental results also corroboratedthat strain energy is a good strength metric [4,6].Since Eq. (1) considers only the elastic behavior ofmaterials, it cannot be applied to joints undergoinglarge temperature changes or those that have a sig-nificant thermal and mechanical discrepancybetween the ceramic and the metal. Since plasticdeformation of the metal relieves residual stressesin the joint, it is desirable to obtain generalexpressions of the type of Eq. (1), but which arevalid in the plastic range.

The strain energy calculated from numericalmodeling was summarized and generalized usingthe technique of order of magnitude scaling (OMS)[22], which provides simple and accurate closedform expressions based on the predominant para-meters. The results obtained were verified exper-imentally by shear tests of Si3N4 to Inconel 718joints with different interlayers. Experimentalresults showed that the simple formulationsdeveloped in this study capture the predominantfactors affecting the joint strength.

2. Model description and experimentalprocedures

2.1. Numerical analysis

The effects of material properties on strainenergy in the ceramic were investigated using asensitivity analysis. Fictitious model materialswere created for this study. For example, to investi-gate changes in the CTE of the interlayer, strain


energy changes were calculated for the fictitiousinterlayer materials that were designed to have thethermophysical properties of Ni [23] but differentCTE. Similar methods were used for studying theeffects of other material properties in the jointwithout an interlayer and joints with a single inter-layer. Based on the results obtained from this sen-sitivity analysis, simple analytical solutions for thejoints including plastic behavior of metals andmetal interlayers have been developed using theOMS method.

Based on the theoretical understanding, jointswith different types of multiple interlayers weremodeled. As base materials, Si3N4 and Inconel 718have been chosen. Typical types of double inter-layers are laminates of a ductile material with ahigh CTE such as Ni and hard materials such asW having a low CTE. The strain energy in the cer-amic and the stress distribution in the interlayer arecompared when two layers are placed in a differentorder between base materials. The strain energyhas also been calculated in the joint with the tripleinterlayer that has already been used industrially[18]. The effects of two kinds of double interlayersand one triple interlayer have been compared withNi single interlayers with thicknesses of 0.3 and0.9 mm. In Fig. 2, model materials and joints withsingle, double and triple interlayers used innumerical and/or experimental analyses have beenschematically described.

To simplify the calculation as a two-dimensionalproblem, the specimen was modeled as a cylindri-cal, axisymmetric, rod-shaped specimen of 12.7mm diameter and 12.0 mm height. The thicknessof each layer in the multiple interlayers is 0.3 mm.The computational mesh is shown in Fig. 1(a). Theleft boundary of the mesh is the axis of symmetry.It is composed of 60–80 rows of elements alongthe axial direction and 30 rows of elements alongthe radius. The mesh was refined closer to the rad-ial free surface and near the interface since thelargest stress and strain gradients occur in theseregions as shown in Fig. 1(a) [24].

The continuum models simulated cooling of abrazed specimen of Si3N4 to Inconel 718 from thebonding temperature to room temperature.Materials were assumed to be perfectly bonded atthe interfaces. Uniform cooling and time-inde-

pendent material properties were used. For the basemetal and metallic interlayers, elastic–plasticresponses were modeled. The temperature depen-dency of the material properties was considered[23,25]. Numerical solutions were obtained usingthe ABAQUS computer program [26].

2.2. Experimental procedure

Si3N4 was used as the ceramic material andInconel 718 was selected for the metal. Si3N4 hasbeen used frequently for structural componentswhere high performance is required [27] and Inconel718 has been used for applications in similar oper-ational environments as Si3N4. Shear test specimenswere 12.7 mm square and 10.0 mm in height forboth Si3N4 and Inconel 718. For each test, five tosix specimens were tested at room temperature witha constant crosshead speed of 0.5 mm/min. The testconfiguration is shown in Fig. 3. There is no support-ing system to prevent bending of the metal, thus, thestress imposed on the interface by this test is notpure shear [28]. The peak load in the linear displace-ment–load curve before the actual load supporteddrops sharply with increasing displacement wasdefined as the failure load.

As interlayer materials, thin foils of Ni and W(0.3 mm) were used. Ticusil1, brazing alloy wasused in the form of thin foil (�0.05 mm). Prior tojoining, the surfaces of the Si3N4 and the Inconel718 were polished using SiC emery paper and dia-mond paste to 1 µm surface finish. All thematerials were cleaned in acetone using ultrasonicvibration and dried in a warm oven. The sampleswere brazed at 880 °C for 20 min [29] at a vacuumpressure of �5×10�6 Torr with a slight load of 0.5MPa applied across the joint. The heating rate was5�8 °C/min and the samples were furnace-cooledat 2 °C/min.

3. Results

3.1. Joints without an interlayer

Strain energy in the ceramic has been calculatedfor joint combinations that have been used indus-

1 Ticusil: brazing alloy, 68.8Ag–26.7Cu–4.5wt%Ti.


Fig. 2. A schematic of joint models.

trially and are frequently referred to in the litera-ture (Table 1). The joint configuration used fornumerical calculation was described in Section 2.As seen in Table 1, the yield stress of the metalseems to be the most important factor in reducingthe strain energy in the ceramic. According toTable 1, for the same ceramic material, a metalhaving a higher yield stress by x-fold induceslarger strain energy by nearly x2. Alternatively,other parameters such as the difference in elasticmodulus and CTE seem to be not as important asthe yield stress of the metal.

As described in Section 2, a sensitivity analysiswas performed on the Si3N4–Ni joint in Table 1 toinvestigate the effect of various parameters onstrain energy development in the ceramic. Asexpected from previous calculations of strainenergy (Table 1), it has been confirmed that strainenergy changes induced by varying material

properties are negligible compared to those pro-duced due to changes in the yield stress of themetal. Increasing the radius and the h/r ratio, how-ever, is not negligible. With regard to the strainenergy changes changing the radius and h/r ratio,similar functional relationships as with the elasticcase [Eq. (1)] have been obtained; the strain energyis proportional to the radius cubed and the effectof h/r becomes negligible if the value of h/r isgreater than one, although the strain energyincreases rapidly with the ratio h/r smaller thanone.

3.1.1. Analytical expressionTo obtain an analytical expression, we used

OMS [22], which provides closed-formexpressions for systems with simultaneous drivingforces. Based on our sensitivity analysis usingFEM calculations, a scaling factor for the strain


Fig. 3. A schematic description of shear test configuration.

energy in the ceramic was obtained, and the mostrelevant dimensionless group was identified.

The strain energy in the ceramic is calculated as

Ue,C � �12sijeij(dV)el. (2)

The integral of Eq. (2) can be estimated as

Ue,C�Ue,C � scecVc (3)

where ec and sc are the characteristic strain andstress in the ceramic, and Vc is the characteristic

volume over which the stress and strain are pro-duced. Since the ceramic behaves elastically,ec = sc/EC. Numerical simulations indicate that thecharacteristic volume is of the order of r3; thus, theelastic energy stored in the ceramic can be esti-mated as:

Ue,C �s2

cr3

EC. (4)

The characteristic stress in the ceramic dependson the amount of plasticity in the metal. In theasymptotic case when all the metal near the jointis in the plastic regime, the characteristic stress inthe ceramic is given by the yield stress of themetal: sc = sYM. When some of the metal is notin the plastic regime, Eq. (4) must be corrected. Inthis case, the dimensionless group that captures theeffect of partial plasticity is the ratio of the residualstrain at the interface and the yield strain of themetal:

� �(aM�aC)�TEM

sYM

(5)

where sYM is the yield stress of the metal at roomtemperature. Based on Eqs. (4) and (5), the strainenergy in the ceramic can be calculated as:

Ue,C �s2

YMr3

EC

f(�). (6)

The correction function f(�) does not include thegeometry parameter h/r because it is intended forjoints having relatively large thickness. This func-tion can be determined by analyzing FEM calcu-lations of strain energy in the ceramic (Table 1).For values of � larger than 0.5 the followingexpression of f(�) provides values of strain energywithin 10% of the numerical calculations, as shownin Fig. 4:

f(�) � 0.035� � 0.563. (7)

3.2. Joints with a single interlayer

It is known that metal interlayers with low yieldstress reduce the strain energy in the base materials


Table 1Strain energy in the selected joint systems without an interlayer

Ceramic–metal aM�aC (×10�6/°C) EC�EM (GPa) sYM (MPa) Ue,C (×10�3, J)

Si3N4–Cu 13.7 176 75.8 4.23Si3N4–Ni 10.3 96 148 15.2Si3N4–Nb 4.2 201 240 25.9Si3N4–Inconel 600 10.3 98 250 37.8Si3N4–304SS 14.2 104 255.5 38.8Si3N4–AISI 316 14 110 289.6 49.1Al2O3–Ti 1.01 238 172 10.4Al2O3–Inconel600 5.9 152 250 30.0Al2O3–304SS 9.8 158 255.5 31.6

Fig. 4. Normalized strain energy as a function of normalizedthermal strain.

[17,24,30,31]. FEM investigations by Williamsonet al. [24] showed the important trade-off betweenresidual stress relief in the ceramic and increasingplastic deformation in the interlayer. As in Section3.1, OMS has been used for obtaining a closedform expression for the strain energy in the cer-amic. In this case, the characteristic stress in theceramic is given by the yield strength of the inter-layer sYI.

Since the interlayer adds a new degree of free-dom to the system, the correction function willdepend on two parameters instead of one as before.To determine this additional parameter, a sensi-tivity analysis was performed.

3.2.1. Effect of interlayer material properties onstress relief in the ceramic

For the selected joint systems, strain energy hasbeen calculated and the results are summarized inTable 2. The selected joint systems have a rela-tively large CTE difference between basematerials. Intensive research has investigated theeffect of interlayers with intermediate CTEbetween the ceramic and that of the metal; how-ever, materials with smaller CTE generally have ahigh yield stress. Based on the numerical analysisresults, the interlayers with a yield stress higherthan that of the metal induce even larger strainenergy than when an interlayer is not used. There-fore, joints with an interlayer of intermediate CTEbut a higher yield stress than that of the metal havebeen excluded in this work.

In order to examine the effects of interlayermaterial properties other than the yield stress, sen-sitivity analysis has been done with one of theselected joint systems (Si3N4–Ni–Inconel 718 jointin Table 2). Strain energy changes were calculatedfor fictitious interlayer materials that weredesigned to have the thermophysical properties ofNi but a different CTE. Similar methods were usedfor studying the effects of other interlayer materialproperties such as elastic modulus, the degree ofhardening and the thickness of the interlayer (t). Interms of practical applications, a thinner interlayeris generally better. In this study, the range of theinterlayer thickness of interest lay between 0.3 and2 mm, which is 1/40 and 1/6 of the height of thebase material, respectively. It was found that


Table 2Strain energy in the selected joint systems with a single interlayer

Ceramic–interlayer–metal sYM (MPa) sYI (MPa) aM�aC (×10�6/°C) Ue,C (×10�3, J)

Si3N4-Cu-In600 250 75.8 10.3 3.60Si3N4-Cu-304SS 255.5 75.8 14.2 4.15Si3N4-Cu-Inc718 1036 75.8 10.3 3.65Si3N4-Ti-In600 250 172 10.3 18.2Si3N4-Ti-304ss 255.5 172 14.2 18.5Si3N4-Ti-AISI316 289.6 172 14 18.5Si3N4–Ni–Ti 172 148 5.41 11.5Si3N4–Ni–In600 250 148 10.3 14.8Si3N4–Ni–304SS 255.5 148 14.2 16.5Si3N4–Ni–Inc718 1036 148 10.3 15.0Si3N4–Nb–In600 250 240 10.3 34.5Si3N4–Nb–304SS 255.5 240 14.2 35.5Si3N4–Nb– Inc718 1036 240 10.3 33.1Si3N4–Mo–Inc718 1036 370 10.3 74.4Al2O3–Cu–In600 250 75.8 5.9 2.56Al2O3–Cu–304SS 255.5 75.8 9.8 3.23Al2O3–Cu–In718 1036 75.8 5.9 4.18Al2O3–Ni–In600 250 148 5.9 11.5

changes in properties of the interlayer other thanCTE do not significantly affect the strain energyin the ceramic. As in joints without an interlayer,FEM calculations showed that the strain energy inthe ceramic is proportional to the radius cubed andthe effect of the h/r ratio is negligible when it isover one.

The strain energy change in the ceramic (Ue,C)and plastic strain energy change in the interlayer(Up,I) as a function of CTE are shown in Fig. 5.Since a ductile interlayer with a smaller CTE thanthe ceramic is physically unrealistic, the minimumCTE evaluated was limited to values larger thanaSi3N4 (=3.0×10�6/°C). In Fig. 5, as CTE of theinterlayer, aI, becomes larger than aM, (1) lessstrain energy is induced in the ceramic and (2)larger plastic strain energy dissipation occurs in theinterlayer. Based on these results, a new parameter� is proposed.

� � 1��aM�aI

aC�aI�m

(8)

where m = 1 when aI is larger than (aM + aC)/2and m = �1 for aI values smaller than(aM + aC)/2. Assuming that a value of � closer tozero indicates less strain energy induced in the cer-

Fig. 5. Elastic strain energy in the ceramic and plastic strainenergy in the interlayer as a function of CTE of the interlayer.

amic, the parameter shows that (1) as aI becomesmuch larger even than aM, less strain energy willbe induced in the ceramic, and (2) for otherwiseidentical interlayers, less strain energy is inducedin the ceramic, as (aM�aC) decreases.


3.2.2. Analytical expressionAs in Section 3.1, a scaling factor for the strain

energy in the ceramic and the most relevant dimen-sionless group can be identified, based on sensi-tivity analysis. As in the case of joints without aninterlayer, the dimensionless group that capturesthe effect of partial plasticity at the interface wasdefined as the ratio of residual strain at the inter-face and yield strain of the interlayer, �I. The newparameter, �, captures the effect of having threeindependent CTEs in a system with one interlayer.The relative differences in CTEs of three materials(the metal, the ceramic and the interlayer) alsoaffect the plastic deformation in the entire volumeof the interlayer and thus, the strain energy in theceramic (in Fig. 5). This effect has been capturedby the parameter, �, in the previous sensitivityanalysis. Therefore, the correction functionincludes two dimensionless groups: �I and �. Thescaling factor and two dimensionless groups aredefined as follows:

Ue,C �s2

YIr3

EC

f(�I, �) (9)

�I �(aM�aC)�TEI

sYI(10)

� � 1��aM�aI

aC�aI�m

.

The correction function can be validated by analyz-ing FEM calculations of strain energy in the cer-amic of the selected joint systems as shown inTable 2. For values of �I larger than two and �larger than 0.5, the following function of �I and� provides values of strain energy in the ceramicwithin 10% of the numerical calculations:

f(�I, �) � 0.027�I � 0.110� � 0.491. (11)

As mentioned previously, it is known that thereis a trade-off between the residual stresses in theceramic and plastic deformation of the interlayer.Since three independent CTEs in a system willchange the stress and strain distribution in theinterlayer, the accommodation of strain energy inthe ceramic with variation in � has been investi-gated in terms of the effect of these changes onplastic deformation in the interlayer.

3.2.3. Stress distribution in the interlayer and itseffect on stress relief in the ceramic

The two largest components of residual stressesin the interlayer are sxx and sxy. Since the lateraldeformation of the interlayer is highly constrainedby adhesion with the ceramic, the shear stress, sxy,is generally larger than sxy near the interface. Inaddition, as (aM�aC) increases, sxx tends toremain stable, while sxy increases significantly[31].

The residual stress distributions in the interlayerhave been analyzed and compared for the threefictitious joint systems described in Figs. 2 and 5.In joint S-1, the CTE of the Ni interlayer has beenmodified as aInconel718 + aSi3N4

/2, and in joint S-3,it is 18×10�6/°C (aInconel 718). Joint S-1 has thelargest values of � whereas joint system S-3 hasthe smallest. The distribution of sxx, sxy, syy alongthe interface in the three joints are compared inFig. 6. As seen in Fig. 6, the change in tensilestress, syy, with � is negligible compared to thechanges in the other two components. Closer to thefree edge, both sxx and syy increase rapidly, whichwill result in larger plastic deformation near thefree edge. As � becomes smaller, sxx increases fas-ter than does sxy and the distribution of sxx

becomes more uniform through the thickness of theinterlayer as seen in Fig. 7(a). Alternatively, thedistribution of sxy through the interlayer [Fig. 7(b)]becomes more symmetric as � becomes smaller,which induces less stress in base materials. Thedistributions of these two stress components in theinterlayer of joints S-1 and S-3 are shown sche-matically in Fig. 8 with the contours of plasticstrain energy density in the interlayer. Fig. 8(a)shows that stress distributions with large gradientsresult in much larger residual elastic strain remain-ing in the interlayer of joint S-1, which inducesmore strain energy in the ceramic, as in Fig. 5. Themore uniform and more symmetric distributions ofsxx and sxy in joint S-3 cause less bending in theinterlayer and cause plastic deformation to occurover a larger volume of the interlayer as seen inFig. 8(b). This induces less strain energy in theceramic as shown in Fig. 5.

As the relative CTE difference between the basematerials and the interlayer (aM�aI/aC�aI)m

approaches one [i.e. � = 1�(aM�aI/aC�aI)m is


Fig. 6. Residual stress distributions in the interlayer near theinterface along the x-direction: (a) joint S-1, (b) joint S-2, (c)joint S-3.

close to zero), (1) the stress state develops into onewhere more than one stress component dominates,and (2) the distribution becomes more uniform andsymmetric, which facilitates yielding of the inter-layer over the entire volume as well as near the inter-face.

Fig. 7. Residual stress distributions along the y-direction inthe interlayer of joints S-1, S-2, and S-3: (a) sxx, (b) sxy.

3.3. Design of multiple interlayers

Based on the results obtained in cases with sin-gle interlayers, the two types of double interlayersthat have been described in Fig. 2 can be evaluatedin two aspects: (1) the yield stress of the layer nextto the ceramic and (2) the CTE gradient from theceramic to the metal. According to the previousresults, the yield stress of the interlayer is the mostimportant parameter affecting the strain energy inthe ceramic. However, tailoring the CTE gradientfrom the ceramic to the metal represented by � isalso an important parameter to relieve the strainenergy.

In joint M-1 in Fig. 2, the CTE of the first layernext to the ceramic is closer to the CTE of theceramic, which may mitigate residual stresses dueto the abrupt increase in CTE between the ceramic


Fig. 8. A schematic description of stress distribution and thecontour of plastic strain energy in the interlayer of (a) joint S-1 and (b) joint S-3.

and the ductile second layer or the metal. However,the yield stress of the first layer is higher than thatof the ductile layer placed next to the ceramic injoint M-2 (or the joint with a single Ni interlayer),which may limit the release of residual stresses. Ifone assumes that the first and second layer in jointsM-1 and M-2 have similar mechanical propertiesbut different CTEs, the parameter � is anotherparameter to determine the most effective sequenceof layers in the double interlayer. When the CTEof a second layer [aI(2)] inserted between the metaland the first layer is used instead of aM in Eq. (8),joint M-2 in Fig. 2 has a value � = 0.15 which iscloser to zero than the value of � = 1.17 of jointM-1, in which the CTE decreases when movingfrom the metal to the ceramic. As the differencebetween aI(2) and aC decreases, � becomes closerto zero, which should improve the stress distri-bution in the interlayer, thus producing less strainenergy in the ceramic.

In Fig. 9, distributions of the major stresscomponents sxx and sxy near the interface betweenthe first layer and the ceramic of joints M-1 andM-2 have been compared with those in the jointwith a single interlayer of 0.3 mm Ni. Althoughthe CTE difference between the ceramic and thefirst interlayer is smaller in joint M-1, larger

Fig. 9. Residual stress distribution in the interlayer near theinterface with the ceramic: (a) joint with 0.3 mm Ni single inter-layer; (b) joint M-1: Si3N4/0.3 mm W/0.3 mm Ni/Inconel 718;(c) joint M-2: Si3N4/0.3 mm Ni/0.3 mm W/Inconel 718.

stresses have been induced in the interlayer due tothe high yield stress of the tungsten. In additionto the high yield stress, placing layers in order ofincreasing CTE as compared with the ceramiccauses quite a different stress state in the first layerof joint M-1 from in the joints with either a singleinterlayer or joint M-2. As seen in Figs. 9(b) and10(a), sxx is the largest stress component in thefirst layer of joint M-1 and its gradient through the


Fig. 10. Residual stress distributions through the interlayerthickness: (a) sxx, (b) sxy.

interlayer is the largest among the joints beingcompared, thus inducing a larger tensile residualstress in the ceramic. The distribution of sxy

through the layer becomes less symmetric than inthe joint with a single interlayer. In contrast, injoint M-2, sxx in the first ductile layer has beenincreased by inserting an additional hard layer hav-ing a lower CTE than the first layer; however, themagnitudes of sxx and sxy are comparable as shownin Fig. 9(c). Compared to the stress state in thesingle 0.3 mm Ni interlayer, sxx increased uni-formly through the layer as shown in Fig. 10(a)and sxy has become more symmetric as shown inFig. 10(b), which leads to less residual stress inthe ceramic.

As mentioned previously, joint M-1 with alarger � shows that sxx is the largest stress compo-nent and its distributions with a large gradientthrough the layer seem to create greater bendingin this layer. Such stress distributions resulted from

high yield stresses in the first layer and the large� of joint M-1 have induced larger strain energyin the ceramic than the single 0.3 mm Ni interlayershown in Table 3. Alternatively, lower yield stressof the first layer and the order of placement pro-ducing smaller values of � in joint M-2, makeplastic deformation occur over a larger volume ofthe first ductile layer and induce less strain energyin the ceramic than in the joint with a 0.3 mm Niinterlayer. This indicates that the additional hardlayer having a lower CTE in joint M-2 improvesthe function of the ductile interlayer by modifyingthe stress distributions in the interlayer. Therelationship between changes in stress distributionsin the first layer by addition of more interlayersand changes in the strain energy in the ceramicshow similar results as those obtained by changingthe CTE of the single interlayer in Section 3.2.

With the triple interlayer in joint M-3 (Fig. 2),plastic deformation of a third ductile interlayerbetween the second interlayer and the metal willfurther reduce the effect of the metal on the stressdistribution in the first ductile interlayer, and hencewill further reduce the strain energy in the ceramic.Calculation results in Table 3 indicate that a greatdeal of strain energy in the second interlayer[Ue,I(2), Up,I(2)] has been reduced in joint M-3 incontrast to joint M-2 due to the considerable plasticdeformation in the third ductile layer in joint M-3. Compared to the joint with a single ductile inter-layer of the same thickness (0.9 mm), more strainenergy is relieved in joint M-3 with the triple inter-layer of 0.3 mm per layer (Table 3).

3.4. Experimental validation

Shear test specimens were prepared followingthe procedures described in Section 2.2, and thetest results are shown in Table 4 with specimenconfiguration. The test specimen without an inter-layer (specimen 1) fractured under the lowest load.The degree of scatter in the measured strengthvalues was also large. The average strength of thejoints with the double interlayer of increasing CTEfrom the ceramic (specimen 4) was lower than thatof joints with a single Ni interlayer and onlyslightly higher than that of the joint without aninterlayer. The joints with an additional hard inter-


Table 3Strain energy in six model jointsa

Ue,C (×10�2, Ue,M (×10�2, Ue,I(1) Up,I(1) (J) Ue,I(2) Up,I(2) (J) Ue,I(3) Up,I(3) (J)J) J) (×10�2, J) (×10�2, J) (×10�2, J)

Joint without an 4.511 5.896 – – – – – –interlayerJoint with 0.3 mm 1.650 2.116 2.435 0.221 – – – –NiJoint M-1 2.314 1.798 3.872 0.0001 2.394 0.206 – –Joint M-2 1.304 2.514 0.243 0.103 1.685 0.015 – –Joint M-3 1.101 1.914 0.258 0.081 0.967 0.002 0.224 0.115Joint with 0.9 mm 1.570 1.480 0.480 0.222 – – – –Ni

a Ue,M, elastic strain energy in the metal; Ue,I(1), elastic strain energy in the first interlayer; Up,I(1), plastic strain energy in the firstinterlayer; Ue,I(2), elastic strain energy in the second interlayer; Up,I(2), plastic strain energy in the second interlayer; Ue,I(3), elasticstrain energy in the third interlayer; Up,I(3), plastic strain energy in the third interlayer.

Table 4Specimen configuration and test results

Specimen 1 Si3N4/Inconel 718Specimen 2 Si3N4/0.3 mm Ni/Inconel 718Specimen 3 Si3N4/0.3 mm Ni/0.3 mm W/Inconel 718Specimen 4 Si3N4/0.3 mm W/0.3 mm Ni/Inconel 718Specimen 5 Si3N4/0.3 mm Ni/0.3 mm W/0.3 mm Ni/Inconel 718

Minimum (MPa) Average (MPa) Maximum (MPa)

Specimen 1 1.5 17.0 36.2Specimen 2 10.0 32.0 48.0Specimen 3 24.8 59.0 62Specimen 4 18.24 21.0 25Specimen 5 49.6 62.0 80.6

layer next to the ductile interlayer (specimen 3)showed a nearly 30% increase in the averagestrength compared to joints with a single interlayer;however, the maximum strength of specimen 3 isnearly the same as the joints with 0.3 mm Ni inter-layer. The large difference in strength of the twodouble interlayer models (specimens 3 and 4) indi-cates the importance of interlayer sequence whenusing multiple interlayers.

As seen in Table 3, by inserting a third ductileinterlayer, the strain energy in the ceramic of speci-men 5 is reduced by 40% as compared to specimen2 with a single interlayer. The maximum strengthof these joints is increased by 35% compared tojoints with a single interlayer and the average

strength was increased almost 100%. Overall, thejoint having a 0.9 mm triple interlayer has the high-est average strength and it has the lowest calculatedstrain energy. Comparing the strength data in Table4 with the strain energy calculation in Table 3, thetest results are consistent with FEM calculations ofthe strain energy in the ceramic in that joints withless strain energy in the ceramic show higherstrength. Fig. 11 shows the relationship betweenthe strength and strain energy of the ceramic ineach joint. The plot indicates that the strength ofthe joint increases with decreasing strain energy inthe ceramic, which suggests that the strain energyin the ceramic is a reliable metric for joint strength.

Except for a few joints having no interlayer, the


Fig. 11. Shear strength vs. calculated strain energy in the cer-amic.

crack initiated in the ceramic adjacent to the inter-face, propagated and ended in the ceramic. In jointswith interlayers, no brittle interfacial fractures wereobserved. Two different types of fractured speci-mens after the test are shown in Fig. 12. In thefracture shown in Fig. 12(a), the crack initiatedvery close to the interface and propagated in theceramic with a concave shape. This is typical offractures in joints with large thermal expansionmismatch and low strength [28]. In joints withgreater strength, fracture occurs as shown in Fig.12(b). The crack path is well removed from theceramic interface, thus increasing the strength ofthe joint.

4. Discussion

It can be argued that a compressive triaxial stateof stress would make a joint stronger while thestrain energy also increases. This would make theproper use of strain energy as a joint strength met-ric dependent on the geometry of the problem. Inour system only a small portion of the volumealong the center of the axis is in this state as seenin Fig. 1(a). The smaller the compressive strain atthe interface of the ceramic and the smaller the h/r

Fig. 12. Different fracture patterns of joints with (a) relativelylow strength, (b) higher strength.

ratio, the lower the triaxial compressive state ofstress induced, and the volume where it occursdecreases. The volume over which the triaxial stateof stress occurs may affect the fracture path. How-ever, according to our FEM calculations, the mag-nitude of the strain energy calculated over the vol-ume of triaxial compressive stress is negligible(less than 15%) compared to the total strain energyin the ceramic. This small value supports the useof strain energy as an appropriate strength metric.


This conclusion is also validated by the experi-mental results summarized in Fig. 11.

Based on the FEM calculations for multipleinterlayers, it appears that properly designed mul-tiple interlayers can reduce the strain energy in theceramic more effectively than a single interlayer;however, the most desirable gradation of interlayerproperties is not a simple linear decrease from onebase material to the other. Use of rigid layers withincreasing CTE away from the ceramic interfaceand insertion of ductile layers between each rigidlayer will reduce the strain energy most effectively.As an example, in Table 3, the triple interlayeraccommodates more strain energy than a singleductile interlayer of the same thickness (0.9 mm).However, a multiple interlayer is not always desir-able in practice. A larger number of layers requiresmore joint interfaces and the probability of produc-ing defects will increase with the number of inter-faces. This can be the cause of failure even at verylow stresses.

The relative thickness ratio of each layer in themultiple interlayer affects the strain energy distri-bution in the joint. Suganuma et al. investigatedthe effect of the thickness of second interlayers[13]. For a system similar to joint M-2, they foundan optimum thickness of the second interlayer inminimizing the strain energy in the ceramic. Ourrule of design for multiple interlayers could beenhanced by considering these effects.

5. Conclusions

A multiple interlayer design rule has been pro-posed based on closed form functional relation-ships between material properties and strain energydeveloped for ceramic-to-metal joints with a singleinterlayer [Eqs. (8)–(11)]. The simple analyticalexpressions indicate that strain energy in the cer-amic scales with the square of the yield stress ofan interlayer. In addition, two factors were foundto affect the relationships: the CTE differencebetween the ceramic and the metal (captured byparameter �I), and the relative CTE differencebetween the ceramic, the metal, and the interlayer(captured by parameter �). The parameter � [Eq.(8)] quantifies the uniformity and symmetry of the

residual stress distribution through the interlayerdue to the CTE mismatch between the materialsinvolved in the joint. As � becomes closer to zero,the stress distribution in the interlayer has a smallergradient and becomes more symmetric, whichcauses a larger volume of the interlayer to deformplastically. This more homogeneous stress distri-bution causes a reduction of the strain energy inthe ceramic due to the joining process, thusstrengthening the joint.

Although the CTE of the interlayer material can-not be controlled easily, use of multiple interlayerscan create even greater reductions in strain energyby redistributing the stress and plastic strain in aductile interlayer next to the ceramic. The analyti-cal model and numerical results indicate thatinserting an additional rigid layer with lower CTEnext to the ceramic to mitigate large differences inCTE between the ceramic and the ductile interlayerinduces more strain energy than the single ductileinterlayer, which results from the larger yield stressof the rigid layer and less symmetric stress distri-bution in the interlayer. In contrast, a rigid inter-layer with a low CTE can enhance the beneficialeffect of the ductile interlayer when it is insertedbetween the metal and the ductile interlayer.Further decreases in strain energy of the ceramiccan be achieved with a third ductile interlayer byreducing the residual strain induced in the secondrigid interlayer due to mismatch with the metal.

Our experiments confirm the proposed designrules for multiple interlayers and that the strainenergy in the ceramic is a good strength metric forceramic-to-metal joints. For example, joints withan appropriately designed triple interlayer showedthe minimum strain energy in our FEM calcu-lations, and the highest strength of all joints tested.This supports the use of strain energy as a jointstrength metric.

Acknowledgement

This research was funded under the UnitedStates DOE Award No. DE-FC26-99FT400054.


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