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Composite Parameters and MechanicalCompatibility of Material Joints

T. SUGA, G. ELSSNER AND S. SCHMAUDER

Max-Planck-Institut fü."rMetallforschungStuttgan, Ui'st Germany

(Received March Tl. 1984)(Revised February 7. 1988)

ABSTRACT

The elastic behaviour of abimaterial interface with interfacial cracks, misfit dislocationsand interfacial thermal stresses can be described in a simple manner by using the composite parameters C( and ß, and the effective modulus of elasticity E *, assuming a planedeformation of ideaIly bonded isotropie materials. A coefficient Kr for the thermally induced stress intensity at the interface serves as a measure of the mechanical compatibilityof two bonded materials. An examination of these parameters for many compositematerials shows that the values of the composite parameters C( and ß are limited to a narrow range and that the material transition can be classified into six groups with regard totheir mechanical compatibility.

INTRODUCTION

THE MECHANICAL BEHAVIOUR of the interfaces between different materials isof great importance both for the technical application of composite materialsand in basic studies of material transitions. The bonding properties of dissimilarmaterials depend not only on their chemical but also on their mechanical compatibility. Differences in the thermal expansion of the components affect themacroscopic strength of composite materials or materials joints which aremanufactured at high temperatures, such as ceramic-to-metal joints [1]. Thermarstresses in the interfacial region may lead to the formation of microcracks andpartial debonding. High stress concentration at the bonding edges or aroundinterfacial flaws of a joint may arise from differences in the elastic properties ofits components. Usually composite materials or material joints fail by the initiation and propagation of flaws in such highly stressed interfacial regions.

Although the bonding region of real materials is characterized by a diffusionzone of finite thickness or by thin layers of reaction products its approximation bya sharp interface between two elastic and ideally bonded materials can provide auseful tool for the description of bond strength and mechanical compatibility.Dundurs [2,3] derived two composite parameters, C( and ß, from the elastic constants of the materials comeonents and ~how~d that the stress field ()f a c~mpo_site

Reprinted from Journal of COMPOSITE MATERIALS, Vol. 22 - October 1988

918 T. SUGA, G. ELSSNER AND S. SCHMAUDER

in astate of plane deformation depends only on these two parameters. The reduction in the number of elastic constants simplifies considerably the description ofa stress state in an interfacial region. It seems, however, that the compositeparameters are not widely used to analyze practical problems related to the bondstrength and mechanical compatibility.

In the present paper an etfective modulus of elasticity E* is introduced in addition to a and ß, and the utility of these composite parameters is demonstrated.As an example of the application of the composite parameters a coefficient KT ofthe thermally induced stress intensity at the interface will be derived, whichserves as a parameter of the mechanical compatibility of bonded materials. Correlations between the parameters a, ß and Kr ca1culated for various materialcombinations are examined to classify the material combinations according totheir mechanical compatibility.

COMPOSITE PARAMETERS

The Dundurs' composite parameters [2,3] are defined for a combination oftwoisotropie elastic materials 1 and 2 by

k(Xl + I) - (X2 + 1)a= k(Xl + I) + (X2 + 1)(1)k(Xl - 1) - (X2 - I) ß= k(Xl + 1) + (X2 + 1)

and k = JL2/ JLl

where JLj is the shear modulus of material j (j = 1,2), and Xj = (3 - Vj)/

(1 + vJ for plane stress, Xj = 3 - 4vj for generalized plane strain, Vj being thePoisson ratio. For convenience two subsidiary parameters rand 'Y are introduced:

l+ar=~

Under the physical restrictions

1 + ß

'Y=I-ß(2)

o < Vj < 0.5 and JLj > 0 (3)

all values of the composite parameters a and ß are contained in a parallelogramin the a-ß-plane (Figure 1) [4,5]. The four elastic constants Vj and JLj (j = 1,2)for a pair of materials determine a unique point in the a-ß-diagram, but one pointin the a-ß-diagram may correspond to an infinite number of material combinations. The origin a = ß = 0 represents combinations of identical materials, and

Composite Parameters anti Mechanical Compatibility 01 Material Joints

ß

919

k > 1

I_a-1.0

0.01~.O

k<1

- 0.5

Cl = k()I.,.11-(ll2·1l

k(ll,.1) .(llz .1)ß =

k(rt,-ll- (llz -1)

k(ll,"1). (ll2 ·1)k = ~z/~t

Figure 1. ParaJlelogram of Dundurs' composite parameters for physically relevant materialcombinations under plane strain condition.

each pair of values a, ß within the parallelograrn is a measure for the elasticanisotropy of the corresponding material combination. The a-ß-diagram can bedivided into two regions by a straight linethrough thtf origin, along which thecondition /ll = /lz, i.e., k = 1 holds. The region on the left side containscombinations with /ll > /lz, i.e., k > 1, while the other corresponds 10 combinations with /ll < /lz, i.e., k < 1. The vertical sides Qf the a-ß-diagram,a '= ± 1, represent combinations with a rigid body, k = 0 or 00, and the othersides correspond to t'MJ extreme cases v, = 0, Vz = 0.5 and v, = 0.5, Vz = O.When the index of the materials 1 and 2 changes places with another, the sign ofthe parameters a and ß changes without change of their absolute values.

The stress field in a composite of ideally bonded isotropie elastic materials inastate of plane deformation depends only on the composite parameters a and ßif certain restrictions of the loading and connectivity of the regions are obeyed asproved by Dundurs [2].

The effective modulus of elasticity E* is defined for a pair of materials 1 and2 by [6]

_1_ = _1 ( 1 + x, + 1 + Xz )E* 16 /ll /lz(4)

By the use of the parameter E* solutions of problems related 10 elastic strainenergies at the interface can be simply described.


APPLICATION OF COMPOSlTE PARAMETERS

Stress Singularity at aBimaterial Wedge

The difference in the elastic properties of the components in a compositecauses high stress concentrations at the edges of the interface, which may resultin partial debonding, or in an initiation or acceleration of an interfacial failure[6]. The stress field at the vertex of the edge or the corner of the elastic bimaterialwedge (Figure 2a) possesses a singularity, the nature of which depends on thecomposite parameters of the material combination if the bimaterial undergoes aplane deformation [4,7,8,9]. The local stress field can be characterized by theasymptotic behaviour of the complex stress functions 4>iz) and fj(z) for eachmedium j (j = 1,2) which are represented in series

(5)

where Q is an arbitrary complex constant which depends on the stress and displacement field remote of the vertex of the wedge. The eigenvalue A and thecoefficient ajt, aj2, bjt and bj2 of the stress functions can be determined by theboundary conditions valid for the wedge surfaces. For a traction-free wedge theeigenvalue A is given by the solution of the characteristic equation:

y

~ 81 8

(a)

x

y

1r

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

~~~~~~~.,2 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.................................... .............................................................................................................................................................................................................. ~ .

(b)

x

Figure 2. Geometries and notations tor (a) abimaterial wedge and (b) an interface crack.

Composite Parameters and Mechanical Compatibility of Material Joints 921

Ä(A) = I ~)(A)Ä2(A)

(6)

where

Ä)(A) = -A2(a - ß)AoBo + [(a - l)A - (a - ß)AB + (a + l)B}

Ä2(A) = A{(a - l)Ao - (a - ß)(AoB + ABo) + (a + l)Bo}

(7)

A = A(A) = 1 - e2iA8)

B = B(A) = 1 - e2iA82

A = A(A) = 1 - e-lil.9)

The coefficients of the complex stress functions are

if Ä)(A) * 0 or ~2(A) * 0

if Ä)(A) = ~2(A) = 0

if Ä)(A) * 0

if ~2(A) * 0

if Ä)(A) = ~2(A) = 0(8)

with

aj!(A) = Cj!(A)al1 + Cj2(A)a12

aj2(A) = ~2(A)al1 + ~l(A)a12

bj)(A) = Dj)(A)al1 + Dj2(A)al2

bj2(A) = i5j2(A)al1 + i5j)(A)a12

t

(a-ß) (a+l)C2!(A) = ~ A - ~

(a-ß)C22(A) = a = 1 AAo

, Cl1 =

, C12 = 0


DuO.) = A(Ao - I)Cu + (A - I)C12

(9)

D12(A) = A(Ao - I)C12 + (A - I)Cu D21(A) = A(Bo - I)C21 + (B - 1)C22D22(A) = A(Bo - 1)C22 + (B - l)C21

Obviously the complex stress functions, and therefore the stresses in the vicinityof the bimaterial wedge, depend only on the composite parameters, a and ß.

Fracture Mechanics Parameters for an Interface Crack

The mechanical problem of interface cracks is a subject of major importancefür the strength of bonded materials. Several continuum mechanics models of aninterface crack in elastic dissimilar materials have been proposed. The singularstress field in the vicinity of the crack tip is characterized by the stress intensityfactor K = K[-iKII defined as [6]

K = (1 + ')')..J2;lim Z-()o,(ll-1l</>;(Z):-0

(10)

where </>I(Z) is the complex stress function ofmaterial1 and A(1) is the eigenvalueof the stress function containing the minimum positive real part. Although thestress fie1d of the loaded composite is influenced by the e1astic constants of thebonded materials the stresses at the interface ahead of the interface crack can be

described independently of the elastic composite parameters by the stress intensity factor K:

(11)

This fact provides a theoretical base for the use of the critical stress intensity factor as a measure of the fracture resistance of interfaces.

The stress functions for the conventional traction-free interfuce crack model

can be derived directly from the solution of the bimaterial wedge given in theprevious section. The eigenvalue of the stress functions A(1) is expressed; by

with

1 .A(1) = 2" + lE

1

E = 211" In ')'

(12)


The presence of the imaginary part of the eigenvalue leads to oscillating singularities of the stress field at the crack tip and the physically unrealistic featureof interpenetrations of the crack surfaces [10]. Another disadvantage of the conventional model is that the ratio Ku/ Kl depends not only on the externaiload conditions but also on the choice of the dimension of length, e.g., meter orcentimeter.

To avoid such unsatisfactory properties several crack tip models with a contactzone have been proposed, e.g., the slip model by Comninou [12] and the interlock model by Mak et al. [13]. Because additional boundary conditions for the tipregion of the crack surfaces are introduced, these models lead to non-oscillatingsolutions. However, one of the components of the stress intensity factor, Kl orKu, is always zero, so that the loading mode ofthe crack tip cannot generally bedescribed by the stress intensity factor K. A new model proposed by the presentauthors [6,14] combines these t\\O models to describe more rationally the loadingmode of the crack tip. At the very tip of the crack, surface tractions are assumedin this model which are proportional to the opening displacements of the cracksurfaces:

where

The stress functions which satisfy these boundary conditions are given by thestress functions for the conventional model with the parameter f replaced byzero. The eigenvalue of the stress functions für this model is real as in the ca seof homogeneous materials:

1

:\(1) = 2

Therefore, the anomalies of the local stress field in the conventional model andthe undesirable feature of a vanishing component of the stress-intensity factor inthe other contact zone models are removed in the model introduced here.

The stresses in the vicinity of the interfacial crack plane are independent of theelastic properties of the material and given by

(15)

Obviously the stress intensity factor of mode I, Klo corresponds to the tensile


stress and the stress intensity factor of mode II, Ku, to the shear stress in theinterface. However, the boundary conditions given in Equation (13) are in thesame way arbitrary as those of the other interface crack models.

Since the stress functions of above models can be derived from those of the

bimaterial wedge, the dependence of the overall stress field of these interfacecrack models on the elastic properties is also described by the compositeparameters a and ß.

Fracture Resistance and Fracture Energy of Interfaces

If the stress and strain field in the vicinity of a crack tip are known, the energyrelease rate G can be calculated in terms of the stress intensity factor K. It is important to note that a unique relationship holds between the energy release rateG and the absolute value of the complex stress intensity factor K = ...;K; + K;rof an interfacial crack which is independent of the chosen crack tip model andgiven by:

(16)

According to Equation (16) a critical stress intensity factor Kc can be derivedfrom the critical energy release rate Ge:

(17)

This definition of the critical stress intensity factor Ke is generally valid for afailure in the transition region of real material combinations, since the criticalenergy release rate Ge is based on an energy balance conceptfor the crack extension which needs no special modelling of the crack tip. Thus, two competitiveparameters are available to characterize the strength of interface or interfaCialregions in bonded materials. The interfacial fracture energy given by the criticalenergy release rate Ge can be related to the structure of the interface and the adjacent regions and compared to the thermodynamical work of adhesion and toenergy contributions due to dissipation processes by dislocation movement ormicrocrack formation [15]. The interfacial fracture resistance given by the critical stress intensity factor Ke allows a comparison of the interfacial strength of different material combinations with the strength of homogeneous materials withoutany consideration of the elastic properties of the materials. This concept has beenapplied successfully to characterize the bond strength of several ceramic-metaljoints [16] and plasma-sprayedceramic coatings [17].

Correction Functions for the Energy Release Rate

An essential requirement for the experimental determination of the interfacial

fracture energy is the knowledge of the relationship between the applied load F

Composite Parameters and Mechanical Compatibility 01 Material JoinJs 925

and the energy release rate G represented by the so-called correction function YG,

wh ich must be calculated for each specimen geometry and each combination ofmaterials [6]. Ifthis relationship is expressed for a given specimen geometry by

G = P YGE*

(18)

the correction function YG will depend only on the composite parameters a andß for the two materials bonded together since the stresses in the specimens arealso a function of these parameters:

(19)

Figure 3 shows an example for the dependence of the correction function YG onthe specimen geometry and the composite parameters. The bend test specimensin Figure 3 consist of layered material combinations 1/2/1 and 2/1/2 which arenotched or pre-cracked with a constant depth either at one bonded interface orparallel to the interface. Under these assumptions the correction function YG

depends on the thickness d of the midlayer, the distance of the notch to the interface, h, and the composite parameters a and ß. d and h are normalized by theheight W of the specimens. Yb Y2, and Y12 denote correction functions for thecrack in material I, material 2, and in the interface 1/2, respectively. For the extreme cases h/W - 0, ± ClOor d/W - 0, ClOthe values of the correction functions are bounded by the correction function Yiso for homogeneous materials [18]

0---- 0·--- h/W ----

Y,,,,•

_."v. fa",,"••-- IIY~ Io

Id/W

I

Id/W

Io

I

lnt.,.tcaolfalh ••••

----h/W--- 0-

r:cohHlve tOllur. In rnotW1Q1 2

Cl •••••••• a.'1 l.«.)y 1".aIY .., """,., I (r-ä: ••• Iiiiiiic::J JolcltrlOl 2 ••••

o I I YI ' I (,-:a-IY'2..1r--.(;-:O:)Y'2 Y'2 •......•~-~--..~j I I~~Y Y2 [

~ll"" I 11 .- I 11~)Y'2Y2 (,-!-aJY,2 Y,2 ~Y''''I n. 1 LA-.! ~d/W~ .. _

Io

I

Figure 3. Correction funetions of materialjoints as funetions of the normalized eraek positionhIW, and the ratio of layer thiekness to speeimen height dlW.


or by the correction function Ybi for abimaterial multiplied by a factor of 1/(l ± a), (1± a), or (1± a)/(l =t= a).

Misfitdislocations

Von der Merwe [19] formulated an analytical solution of the elastic stress tieldand the energy of an array of mistitdislocations at the coherent phase boundarybetween two semi-intinite isotropie crystals (Figure 4). He used the PeierlsNabarro model of a sine force law to describe the equilibrium of interfacial shearforces at the phase boundary. Nakahara [20] incorporated in this solution of theproblem the interfacial normal stress, which was neglected in the original theory.

Under the assumption of a non-gliding interface the solution can also be expressed by a function of the composite parameters and the etfective modulus ofelasticity. The corresponding parameter* in the paper of Nakahara is substitutedin our notation by

(20)

where fJ.o is the etfective shear modulus of the interface or the coefficient of thePeierls-Nabarro potential and a, and a2 are the lattice constants of the crystal 1and 2, respectively.

y

0,"

I

II

I I.L .L,J.XI

I 0 II

I~a~ III

, I IIi! TI

iI .ii 'HI

I 'p iIIi I

Figure 4. Quasi-continuum mechanicaf model tor a coherent phase boundary with misfit dislocations.

*This parameter is given by Equation (30) in Reference [20].

Composite Parameters and Mechanical Compatibility of Material Joints

y

9Tl

x

Figure 5. Partially bonded dissimilar material.

Thermally Induced Stress Intensity

The mechanical compatibility of the components of a material joint subjectedto thermal stresses is often characterized by the difference in the thermal expansion coefficients between the bonded materials. However, this simple approachseems to be inadequate since an interfacial damage or failure may occur by theinitiation and propagation of microcracks due to stress concentrations in theinterface region. Therefore it is more appropriate to consider the thermallyinduced stress intensity as a measure of compatibility. The problem of twobonded and uniformly heated dissimilar semi-infinite planes containing interfacial cracks has been solved by Erdogan [21]. If we take the case of partiallybonded dissimilar semi-infinite planes with a ligament length 2a. as shown inFigure 5, the complex stress function for the interface crack model with stress"free crack surfaces is given by

.J:y f (2al: - ;z) (~)i' J$;(z) = -8-~aT~TE*l-1 + -.Ja2 _ Z2 ,a + z,(21)

'Nhere t.aT is the difference of the coefficients of the thermal expansion and ~T ,the temperature difference. The stress intensity factor at the bonded edgex = - a can be determined from

K = (l + y)-.J27r!im (z + a) "2 + ;, cf>;(z)::--ll

(22)

928

which gives the solution


where

K = KT....[ia (2E + i)(2a)i. t:.T (23)

(24)

In the case of boundary conditions (13) along the crack surfaces Equation (23)must be written

K = i Kr & t:.T (25)

From this equation the important conclusion can be drawn that thermal stressesinduce failure in a pure shear mode. In real material combinations ß does not exceed 0.25 and its inftuence on Kr is therefore less than 3.3 %. For most practicalmaterial combinations this number is less than 0.5 % and is thus negligible.

The coefficient KT of the thermally induced stress intensity can be used as arepresentative parameter of the mechanical compatibility of materials withdifferent thermal expansion behaviour.

COMPOSITE PARAMETERS AND THE MECHANICALCOMPATIBILITY OF TYPICAL MATERIAL COMBINATIONS

The composite parameters a and ß, and the coefficient of the thermally induced stress intensity Kr as a compatibility parameter were calculated for variousmaterial combinations used in composites; joints, and coating-substrate systems.The elastic constants necessary für the calculations are cümpiled in Table 1. The

Table 1. Elastic properties of severaJ engineering materials.

ThermalYoung's

Poisson'sExpansionModulus

RatioCoefficientNo.

Material E (GPa)va (10·8/K)

1

Epoxy 3.90.34040.02

Thermoplast (Nylon) 2.40.35080.Q3

Silica (SiO,) 73.50.1701.34

Soda lime glass 70.30.2408.65

Borosilicate (Pyrex) 66.50.2003.26

Leadsilicate (EK7) 81.50.2088.37

E-glass fiber 72.10.2504.88

Polymerfiber (Kevlar) 133.00.30075.0

(continued)

Table 1. (continued).

ThennalYoung's

Poisson'sExpansionModulus

RatioCoefficientNo.

Material E (GPa)va 10·6/K

9

Boron fiber (BIW, B/C) 420.00.2701.510

Carbon Fiber (HM) 570.00.3905.011

B4C 390.00.1504.512

AI,O, (Poros. 0.1 %) 375.00.2708.213

Sapphire fiber 425.00.2207.914

SiC-PLS, fiber 417.00.1654.815

SiC-RS (KT, Refel) 332.00.1273.416

SiC-whisker, CVD 482.00.1905.517

Si,N,-HP 314.00.2802.718

Si,N,-whisker, CVD 379.00.2002.319

MgO 295.00.36013.620

ZrO, Mg-PSZ 192.00.3027.621

TiC 318.00.1877.722

MoSi, 380.00.1658.523

AI 70.60.34523.524

Cu 129.80.34317.025

Ni 210.00.31013.326

Ti 120.20.3618.927

Zr 103.00.3505.928

Hf 137.00.3706.029

V 127.60.3658.330

Nb 104.90.3977.231

Ta 185.70.3426.532

Cr 270.00.2106.533

Mo 324.80.2935.134

W 411.00.2804.535

AI-alloy 71.00.33022.536

Steel 215.30.28311.537

Ti-4AI-6V 110.00.3105.838

Ni-, Co-alloy 210.00.29011.539

Zr-alloy (Zircaloy) 95.00.3205.740

WC-Co 600.00.1504.341

Si 115.00.4407.642

Be 241.00.30012.043

Plasmasprayed zrO, 46.00.3607.644

Plasmasprayed NiCrA/Y 128.00.30012.845

Cr,O, 274.00.3088.446

Fe,O, 212.00.14012.347

Y,O, 171.50.2989.348

MsA/,OA 238.00.2949.749

FeCr,O, 215.00.2809.450

NiO 101.00.40017.151

CuO 107.00.3809.3

929

930

ß

0.3


0.5

o.

1.0

0.8

a

-0.1

'0.20.2

0••

0.8 o AI / Ceramic .Mo.W.B.Cc CUt Ceromic .Mo.W.B.C• Ni / Ceromic Yo.W.B.C• Ti.Zr.Hf.V.r-b/Cercmic:.Mo.W• To.cr.""'.W / Ceromic• •.•• / Plastics

/Others

Figure 6. Carrelation between the Dundurs ' composite parameters a and 13tor typical combinatians. Numbers denate material combinations (see Tables 2 and 1).

results are given in Figures 6 and 7 where ß and KT are plotted against cx. Thecode numbers dose by the points refer to Table 2 where the different materialcombinations are listed. The sequence of the material land 2 was changed forsome combinations, so that the sign of the parameter a is always positive. Absolute values are used for the stress intensity coefficient Kr.

Adhesive epoxy joints are specified by code numbers 1 to 28, fibre reinforcedplastics by 29 to 44, and fibre reinforced metals by 45 to 43, respectively. Metalmetal and ceramic-metal composites and coating-metal substrate combinationsare represented by code numbers 74 to 88, 89 to 144, and 145 to 166, respectively.

As shown by Figure 6, the ß values of the combinations are arranged in a narrow band between -0.05 and 0.24, whereas the cx values are distributed over thewhole possible region. Combinations with plastics are characterized by a stronganisotropy. Their cx values are restricted to 0.88 to 0.99 and their ß values arearranged between 0.20 and 0.24. Among the meta! composites, aluminium composites are considerably anisotropie with a values of 0.6 to 0.8 and ß values of 0.1to 0.2. The other metal composites have cx values of 0 to 0.6 and ß values of-0.05 to 0.26.

The Kr values lie between 0 and I MPa/K. According to Figure 7 the materialcombinations may be subdivided into six groups. Group I comprises combinations with plastics components which are characterized by a high anisotropy andlow Kr values. The group 2 of aluminium composites is also strongly anisotropieand, furthermore, exhibits high K; values. High Kr values are also found, for the


copper composites of group 3 associated with a medium anisotropy and for thenickel composites of group 4 combined with a low anisotropy. The eombinationswith Ti, Zr, Hf, V, or Nb in group 5 show low Kr values and medium anisotropywhereas material transitions of group 6 with Ta, Cr, Mo, or W exhibit both lowelastie anisotropy and low interfucial thermal stresses.

Ceramie-metal eombinations with especially low Kr values are Ab03/V (0.005MPa/K), Zr02/Nb (0.016 MPa/K), SiC/Mo (0.029 MPa/K), Ab03/Ti (0.036MPa/K), and Ab03/Nb (0.048 MPa/K). Combinations between the structuralcerarnie silicon nitride Si3N4 and refractory metals have relatively high Kr valuesof the order of 0.1 to 0.2 MPa/K. Investigations on the bond strength of eeramicmetal joints demonstrated that a good meehanieal compatibility of the components leads to high values of the interfacial fraeture energy and fraeture resistance, if a ehemical bond is developed and the chemical eompatibility is sufficient[1]. Nb-Ab03 joints of a measured interfaeial fracture resistance of Kc = 2.9MN/m3/2 are eombinations with both exeellent meehanical and ehernieal eompatibility of its eomponents. Niobium and alumina are direetly bonded without anymicroseopieally deteetable intermediate reaetion layer. By a proper selection ofthe metallayer thiekness in sandwich-layered Si3N4/Zr/Si3N4 joints and by a reduetion of the internal stresses via an allotropie transformation of the zirconiumlayer the mechanical compatibility of the eomponents of the system is considerably enhanced so that the fraeture resistanee exeeeds values of 4 MN/m3/2

1.01 I

0.8

0.4 .~ ""

o·

o.

o·0"o•

o.

..

(Po.

o "ö.00"

0••"0••

GM0••·••0•• "HO ••o.

1.0.-

......~:.·..if

0.80.60.4

.,u•. ne,.J.."..,

0.2

•• u.,., '10 •• n

•••• 107 •••••• ,. ••••

..• • "a_

•....".

•..•..

';;;.~ ",:' .')II:'~$eu:s ., ••.••J•..•• 'l1 .••,.,n0.0

Q

Figure 7. Gorrelat/on between the thermally /nduced stress /ntens/ty coeff/e/ent KT and theDundurs' compos/te parameter a. Numbers denote mater/al eomb/nations whieh are givenin Tables 2 and 1.

Table 2. Index of material combination for Figures 6 and 7.Numbers put in parentheses denote indices of material components

1 and 2 given in Table 1.

Material Combination (MateriaI1/Material 2)

1 (1/ 3)2 (1/ 4)3 (1/ 5)4 (1/ 6)5 (1/12)6(1/14)7(1/15)8(1/17)9(1/19)

10 (1/20)11 (1/23)12(1/24)

13 (1/25)14(1/26)15 (1/27)16(1/28)17 (1/29)18(1/30)

19 (1/31)20 (1/32)21 (1/33)

22 (1/34)23 (1/35)24 (1/36)25 (1/37)

26 (1/38)27 (1/39)28 (1/40)29 (1/ 7)30 (1/ 8)31 (1/ 9)32 (1/10)33(1/11)

34 (1/13)35 (1/14)36 (1/36)37 (2/ 7)

38 (2/ 8)39 (2/ 9)

40 (2/10)

41 (2/11)42 (2/13)

932

43 ( 2/14)

44 ( 2/36)45 ( 9/23)46 ( 9/24)47 ( 9/25)48 ( 9/26)49 (10/23)50 (10/24)

51 (10/25)52 (10/26)53 (11/23)54 (11/24)55 (11/25)56 (11/26)57 (11/35)58 (11/37)

59 (11/38)60 (13/23)61 (13/24)62 (13/25)63 (13/26)64 (13/35)65 (13/37)66 (13/38)

67 (14/23)68 (14/24)

69 (14/25)70 (14/26)71 (14/35)72 (14/37)

73 (14/38)74 (42/26)75 (36/23)

76 (36/24)77 (36/25)78 (36/26)79 (36/35)

80 (24/23)81 (33/24)82 (33/35)83 (33/37)84 (33/38)

85 (34/24)86 (34/35)87 (34/37)88 (34/38)89 (12/23)

90 (12/24)91 (12/25)

92 (12/26)93 (12/27)94 (12/28)95 (12/29)96 (12/30)97 (12/31)

98 (12/32)99 (12/33)

100 (12/34)101 (14/26)102 (14/27)

103 (14/28)104 (14/29)105 (14/30)106 (14/31)107 (14/32)108 (14/33)109 (14/34)

110 (14/38)111 (14/39)112 (15/26)113 (15/27)114 (15/28)

115 (15/29)116 (15/30)117 (15/31)118 (15/32)119 (15/33)120 (15/34)121 (15/38)122 (15/39)123 (17/26)

124 (17/27)125 (17/28)126 (17/29)

127 (17/30)128 (17/31)129 (17/32)130 (17/33)

131 (17/34)132 (17/38)133 (17/39)134 (20/26)135 (20/27)

136 (20/28)137 (20/29)138 (20/30)139 (20/31)140 (20/32)141 (20/33)142 (20/34)143 (20/38)

144 (20/39)145 (21/40)146 (33/38)147 (22/38)

148 (22/30)149 (22/31)150 (16/38)

151 (18/38)152 (43/36)153 (43/38)154 (44/36) 155 (44/38)156 (12/23)157 (12/35)158 (12/25)159 (12/38)160 (45/38)

161 (20/27)162 (20/39)163 (46/36)

164 ( 3/36)165 (51/24)

166 (50/25)


[16]. On the other side, Sie-Mo combinations showed poor adherence despite ofthe good mechanical compatibility of their components since a porous and lowstrength intermediate reaction layer is formed during the diffusion bonding process.

CONCLUSIONS

The interface crack model introduced in this paper obeys the physically significant equivalence of stresses on both crack surfaces in the contact zone at thecrack tip. The incorporation of the composite parameters into the analysis of elastic bimaterial problems simplifies the description and generalization of the solutions as can be seen by the examples given in this paper. The narrow range of possible values for the composite parameters of technologically important materialcombinations facilitates their application. An estimate of the mechanical compatibility of material components by means of the composite parameters and athermally induced stress intensity coefficient leads to a classification of materialcombinations into six groups. However, plastic deformation in ductile components will take place du ring bonding at the interface and under externally applied loads at the tip of an interface crack. Additionally, chemical compatibilityof the material components must also be taken into account to establish asoundfoundation for the prediction of mechanical bonding properties of composites.

REFERENCES

1. Elssner, G. and G. Petzow. "Compatibility Between the Material Components in Metal-toCeramic Composites,- Z. Metallkunde, 64:280 (1973).

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6. Suga, T. Dissertation, University of Stuttgart (1983).

7. England, A. H. "On Stress Singularities in Linear Elasticity,- 1m. 1. Engng. Sei., 9:571 (1971).

8. 'Hein, V. L. and F. Erdogan. "Stress Singularities in a Two-Material Wedge;' Int. J. Fract. Mech.~7:317 (1971).

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11. Rice, 1. R. and G. C. Sih. "Plane Problems of Cracks in Dissimilar Media," 1. App/. Mech.,32:418 (1965).

12. Comninou, M. "The Interface Crack," J. Appl. Mech., 44:631 (1977).

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15. Petzow, G., T. Suga, G. Eissner and M. Turwitt, "Nature and Structure of Metal-Ceramic Interfaces," Proc. 1m. Schoo/ on Simered Meta/-Ceramic Composites, New Dehli (December 1983).


16. Diem, W., G. Elssner, T. Suga and G. Petzow. "Bond Strength Characterization of Metal-toCeramic and Adhesive Joints by Critical Energy ReleaseR!ites," Proc. Int. Conf. on AdhesiveJoints, Kansas City (September 1982).

17. Elssner, G., T. Suga and I. Kvemes. "Mechanical and Metallographie Techniques for the Characterization of Thermal BaITier Coatings,"Proc. NATO Advanced Workshop on Coatings jor HeatEngines, Acquafredda de Maratea, ltaly (April 1984).

18. Tada, H., P. C. Paris and G. R. lrwin. The Stress Analysis oj Cracks Handbook, Dei ResearchCorp., Pennsylvania (1973),

19. van der Merwe, J. H. "Structure of Epitaxial Crystal Interfaces," Surface Sei., 31:198 (1972).

20. Nakahara, S. "van der Merwe Misfitdislocation Theory: Reconsiderationof the Peierls-NabarroModel;' Thin Solid Films, 72:171 (1980).

21. Erdogan, F. "Stress Distribution in Bonded Dissimilar Materials with Cracks," J. Appl. Mech.,32:403 (1965).

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