# Bimaterial Assembly: Interfacial Peeling Stresses

To determine the interfacial peeling stress *p(x),* we proceed from the following equations of equilibrium:

Here w,(x) and *w _{2}(x)* are the deflections of the assembly components,

are their flexural rigidities (the components are treated here as elongated rectangular thin plates), and *T _{l 2}(x)* are the thermally induced forces acting in their cross sections and related to the interfacial shearing stress by formula (2.12). The left part of equation (2.20) is the bending moment caused by the peeling load

*p(x).*The first terms in the right parts of these equations are bending moments caused by the elastic bending forces. The second terms in the right parts are the moments caused by the thermally induced forces 7i(x) and

*T*acting in the cross sections of the assembly components.

_{2}(x)We assume that the peeling stress *p(x)* is related to the deflection vv, (x) and *w _{2}* (x) as

where *К* is the through-thickness interfacial spring constant. In an approximate analysis, this spring constant can be evaluated by the formula:

where

are the through-thickness compliances of the assembly components. Formula (2.25) indicates that no peeling stress can possibly occur in a cross section, where both components of the assembly have the same deflections. From (2.23) and (2.25) we obtain the following differential equation for the peeling stress function, *p(x):*

where

is the parameter of the interfacial peeling stress,

is the effective thickness of the assembly, and the derivative x[(x) of the interfacial shearing stress function, X|(x), is expressed as

In the taken approach, the interfacial shearing stress is evaluated assuming that it is not affected by the interfacial peeling stress and that the peeling stress can be evaluated based on the determined magnitude and the distribution of the shearing stress. The validity of this assumption has been confirmed by the FEA carried out for assemblies with comparative thickness of its components.

The equation of the type (2.28) is known in the engineering theory of beams lying on continuous elastic foundations (see, for example, [18, 38]), where a similar equation was obtained, however, for the deflection functions, and not for the interfacial peeling stress.

The particular solution to the inhomogeneous equation (2.28) can be sought in the form

Introducing this solution into equation (2.28) we find where the ratio

considers the relative role of the parameters of the interfacial shearing and peeling stresses. The general solution to equation (2.28) can be sought as follows:

where C_{0} and C_{2} are constants of integration, and the functions l(,(p.v) and V_{2}(px) are expressed as

These functions obey the following simple rules of differentiation:

The functions V,(p.v) and VjCP-X) are

The first two terms in solution (2.35) represent the general solution to the homogeneous equation corresponding to equation (2.28).

Since there are no external forces acting on the assembly, the peeling stress function, *p(x),* must be self-equilibrated, and, since this function should be also symmetric with respect to the origin, the following conditions of equilibrium should be fulfilled:

The first condition indicates that the bending moment at the ends of the assembly should be zero, and the second condition indicates that the shearing force acting in the through-thickness direction of the assembly should be zero as well. These conditions result in the following equations for the constants of integration:

These equations yield:

The peeling stress is therefore expressed as follows:

For sufficiently large (3a values, such as for long assemblies with stiff (in the through-thickness direction) interfaces (say, (3a > 2.5), formulas (2.41) can be simplified as

and solution (2.39) can be simplified as

Hence, the maximum value of the peeling stress that occurs at the ends of a long assembly with a stiff interface is

# Trimaterial Assembly: Interfacial Shearing Stresses

In the case of a trimaterial assembly, formulas (2.5) and (2.6) yield:

and the interfacial displacements can be sought in the following approximate forms:

Here, Mi (jc) are the interfacial displacements of the component #1 at its boundary with the intermediate component #2; n_{2},(x) are the interfacial displacements of the component #2 at its boundary with the component #1; *u _{2}*3(x) are the interfacial displacements of the component #2 at its boundary with the component #3; и

_{3}(х) are the interfacial displacements of the component #3 at its boundary with the component #2;

*T*and r

_{[}(x),T_{2}(x),_{3}(x) are the thermally induced forces acting in the cross sections of the components #1, #2, and #3, respectively; T|(x) is the interfacial shearing stress at the boundary between the components #1 and #2; T

_{2}(x) is the interfacial shearing stress at the boundary between the components #2 and #3; and к

_{ь}к

_{2}, and K

_{3}are the interfacial compliances of the components. In these expressions for the interfacial displacements of the component #2, we have considered that the three thermally induced forces have to be in equilibrium, and therefore the condition

must take place. We seek the forces 7] (x), *T _{2}* (x), and 7

_{3}(x) in the form of equation (2.10), such as

and require that the conditions

of the displacement compatibility are fulfilled. If there are compliant layers employed at the interfaces between the assembly components, then the conditions (2.50) should be replaced by the conditions of the condition (2.16) type.

The conditions (2.50) of the displacement compatibility require, considering the relationships (2.18) and (2.19), that the following homogeneous equations for the forces 7i° and Г_{3}° are fulfilled:

The forces 7]° and *T"* cannot be zero, and therefore the determinant of equations (2.51) must be zero. This leads to the following biquadratic equation for the parameter *к* of the interfacial shearing stress:

where the following notation is used:

Equation (2.52) results in the following expression for the interfacial shearing stress parameter:

When *k _{2}*з =0 (the interface #2 is infinitely compliant, i.e., simply does not exist),

*к = k*When

_{i2}.*k*0 (the interface #2 is infinitely compliant, i.e., does not exist),

_{i2}=*к = к*When all the assembly components are identical (only the CTEs might be

_{2Ъ}.*[3*

different), formula (2.54) yields: *к = k, _{2}J— = l.2247k_{l2}.* Thus, trimaterial assemblies are characterized by higher parameters of the interfacial shearing stress than bimaterial assemblies.

After the parameter *к* of the interfacial shearing stresses is determined, the interfacial shearing stresses T|(x) and *T _{2}(x)* can be evaluated as

# Trimaterial Assembly: Interfacial Peeling Stresses

In the case of a trimaterial assembly, the equations of equilibrium for the peeling stresses *p(x)* and *pi(x)* acting in the interfaces #1 and #2, are as follows:

Note that the curvatures of the assembly components depend on both the interfacial peeling stresses and the axial forces, such as on the interfacial shearing stresses. Even if the peeling stresses *p _{{}(x)* and

*p*are the same, the curvature of the component #2 might not be zero, but will depend on the level of the axial forces (and interfacial shearing stresses).

_{2}(x)We assume that the peeling stresses *p _{f}(x)* and

*p*are related to the deflection functions w,(x), w

_{2}(x)_{2}(x), and w

_{3}(x) of the assembly components by the equations:

where *K _{{}* and

*K*are spring constants of the interfaces in the through-thickness direction. Solving equations (2.56) for the corresponding deflection functions, substituting the obtained expressions into the relationships (2.57) and differentiating the obtained relationships twice with respect to the coordinate

_{2}*x,*we obtain the following equations for the peeling stress functions acting at the interfaces #1 and #2:

where

are the parameters of the interfacial peeling stresses at the interfaces #1 and #2, respectively, and the following notation is used:

We seek the particular solutions to equations (2.58) as

Introducing these solutions into equations (2.55), we obtain the following equations for the constants A, and A_{2}

Equations (2.62) yield:

We seek the solutions to equations (2.58) as follows:

Introducing the homogeneous parts of these solutions into equations (2.58) and requiring that the constants of integration in these solutions are nonzero, we obtain the following formulas for the parameters Yi and *У У*

where

and

is the parameter of the flexural rigidities. Note that in the case of a bimaterial assembly (D| = 0 or = 0) the parameter 8 is zero. In the case when one of the components (say, #1 or #3) is significantly more rigid than the other two (this is the case of a thin film system fabricated on a thick substrate, or a substrate/IC system attached to a heat sink), the parameter 8 becomes independent of the rigid component’s flexural rigidity. This parameter is either 8 = /——— or 8 = I———. If the flexural **° **^{V}** VA+£>2 ***D _{2}+D;*

rigidity of all the assembly components is the same, then 8 = -^. Note also that, for a

bimaterial assembly, when the parameters is zero, p, =2p^{2}, p_{2} =0. andy, =y_{2} = p, as it is supposed to be.

The constants of integration in solutions (2.64) can be determined from the selfequilibrium conditions for the peeling stress functions. We use the following formulas that can be easily obtained by integration by parts:

Applying these formulas to the solution for the peeling stress function, *p(x),* in (2.64), we obtain the following equations for the constants *C _{m}* and C

_{21}of integration:

Here «i = Yi*a* and *щ =* y_{2}a. The determinant of equations (2.69) is

Then equations (2.69) yield:

Similar expressions can be obtained for the constants C_{02} and C_{22} of integration in the expression for the peeling stress *p _{2}(x*): the factor A, should be simply replaced in formulas (2.69) by the factor A

_{2}.

For sufficiently long (large *a* values) and/or stiff (large *к* and y, values), formulas (2.69) and solutions (2.65) can be simplified, and these solutions yield:

where

The maximum stresses act at the assembly ends and are as follows: