Interfacial Stresses
 Numerical Example
 Optimized Design
 Optimization Condition
 Peripheral Material with a Low Fabrication Temperature
 Peripheral Material with a Low Parameter of the Interfacial Shearing Stress
 Peripheral Material with a Low Parameter of the Interfacial Shearing Stress and Low Fabrication Temperature
 Conclusions
 References
Introducing formula (6.36) for the induced force at the boundary between the midportion and the peripheral portions of the assembly into formulas (6.32) and (6.34) and assuming a long enough midportion, we obtain the following expressions for the interfacial shearing stresses:
at the assembly’s midportion at its boundary with the peripheral portion,
at the peripheral portion at its boundary with the midportion, and
at the assembly end (free end of the peripheral portion). The shearing stress increases from zero at the midcross section of the assembly to the magnitude determined by formula (6.39) at the end of the midportion (at its boundary with the peripheral portion), then drops to the magnitude defined by formula (6.40) at the same cross section, then increases again, and reaches the magnitude (6.41) at the assembly ends.
Numerical Example
The numerical example is carried out for a typical package (component #1)/PCB (component #2) assembly with either BGA or CGA highmodulus solder joint interconnection at the midportion of the assembly, and with an epoxy adhesive at its peripheral portion(s), so that both the fabrication/curing temperature and Young’s modulus of the peripheral material are low compared to the material in the midportion of the assembly. Our objective is to assess the level of the induced interfacial shearing stresses.
Input data:
Structural Element 
Package 
PCB 
34%Ag0.5 1 %Cu Solder at the Midportion 
Epoxy Adhesive at the Peripheral Portions 
Young’s modulus, £.kg/mnr 
8775.5 
2321.4 
5510.0 
415.3 
Poisson’s ratio, V 
0.25 
0.40 
0.35 
0.35 
CTE ce.l/°C 
6.5 x 10"* 
15.0x10" 
Not important 
Not important 
Thickness. A, mm 
2.0 
1.5 
0.60 for BGA 2.20 for CGA 
0.60 in the case of BGA 2.20 in the case of CGA 
Shear modulus. G.kg/mnr 
3367.3 
892.7 
2040.7 
153.8 
Axial compliance, X.mm/kg 
3.9884 xlO’^{5} 
20.1028x10' 
Not important 
Not important 
Interfacial compliance, K,mm'/kg 
19.7982x10'’ 
56.0099x10' 

390.1170x10' in the case of BGA 1430.4291x10'in the case of CGA 
Estimated yield stress of the solder material in shear is: Xy = 1.85 kg/mnr for the solder; temperature change Дt,„ = 200°C for solder, and Al_{r} = 100°C for the epoxy, so that their ratio is = 0.5. Half
assembly length is L = 15 mm. Lengths of the peripheral portions are 21 = 2.0 mm. Thermally induced
force in a long midportion of the assembly is ^^{a}^^{lm} =^ 0017— _ 2.3490 kg/mm.
^{c н 3} X, 72.3701 x 10 ^{w}
Calculated data
Axial compliances of the assembly components:
Interfacial compliances of the assembly components:
Interfacial compliances of the solder: in the case of BGA, and
in the case of for CGA. Interfacial compliances of the epoxy: in the case of BGA in its midportion, and
in the case of for CGA in the midportion. Total interfacial compliance at the midportion of the assembly:
in the case of BGA, and
in the case of CGA. Total interfacial compliance at the peripheral portions of the assembly:
in the case of BGA, and
in the case of CGA. Flexural rigidities of the assembly components:
Total axial compliance of the assembly with consideration of the finite flexural rigidities of its components:
The parameter of the interfacial shearing stress at the midportion of the assembly: in the case of BGA, and
in the case of CGA. The parameter of the interfacial shearing stress at the peripheral portions of the assembly:
in the case of BGA, and
in the case of CGA. The ratio of these parameters of the interfacial shearing stress: in the case of BGA, and
in the case of CGA. The force at the boundary is in the case of BGA, and
in the case of CGA. The interfacial shearing stress in the midportion of the assembly at its boundary with the peripheral portion is
in the case if BGA and
in the case of CGA. The interfacial shearing stress in the peripheral portion of the assembly at its boundary with the midportion is
in the case if a BGA and
in the case of a CGA. The stress at the assembly end is
in the case of BGA and in the case of CGA.
The calculated data are summarized in the following table:
Solder System 
BGA 
CGA 
Stress in the midportion at its boundary with the peripheral portion, kg/mnr 
1.6674 
1.3464 
Stress in the peripheral portion at its boundary with the midportion, kg/mnr 
0.1338 
0.0293 
Stress at the assembly end, kg/mnr 
0.5258 
0.2430 
When a homogeneous bonding layer with the characteristics of the midportion in the carried out example were applied, the maximum interfacial shearing stress would be
in the case of BGA, and
in the case of CGA. Thus, the application of the epoxy adhesive at the assembly ends resulted in this example in 14.42% stress reduction in the case of BGA and in 8.70% reduction in the case of CGA.
Optimized Design
Optimization Condition
Let us define an optimized design as the one in which the interfacial shearing stress at the ends of the midportion at its boundaries with the peripheral portions is equal to the stress at the assembly ends. It is anticipated that by doing that, a significant stress relief could be obtained. By equating expressions (6.39) and (6.41), we obtain the following condition of optimization that could be written in the form of a quadratic equation for the t value:
This equation has the following solution:
After the t value is determined, the lengths of the peripheral zones can be evaluated as
Peripheral Material with a Low Fabrication Temperature
For an assembly in which the parameter of the interfacial stress is the same for the midportion and the peripheral portions (Г) = 1), the condition (6.42) yields (8 = t). This results in the following interfacial shearing stresses:
I
The function /(8) = —^ ^{+} ^ has its minimum /_{mi}„ = ^=5 = 0.6569 at
 1 + 8 л/2
 8 = V2 — 1 = 0.4142. Then, as follows from formulas (6.45), the shearing stresses at the ends of the midportion and the peripheral portions are
Hence, stress relief as significant as 34.3% can be obtained in this case.
Peripheral Material with a Low Parameter of the Interfacial Shearing Stress
For an assembly in which only the difference in the parameters of the interfacial shearing stresses is considered (8 = t), the condition (6.43) yields
This equation and, hence, requirement (6.42) cannot be fulfilled for q values below q = 0.8350. The relationship (6.48) is tabulated for this and higher q values in Table 6.1:
TABLE 6.1
0.8350 
0.8500 
0.9000 
0.9500 
1.0000 

0.3626 
0.5053 
0.6975 
0.8512 
1.0000 

0.8028 
0.8352 
0.8970 
0.9496 
1.0000 

0.9614 
0.9826 
0.9967 
0.9997 
1.0000 
The calculated values
of the dimensionless stresses at the end of the midportion and at the end of the assembly are also shown in this table. Clearly, x„, (L2I) = qx_{p}(/). The table data show particularly that the length of the peripheral area should be small enough so that the induced stresses become sufficiently low. The data for the x,„ (L21) stresses indicate that depending on the q ratio, the stress relief could be as high as 19.7%. Note that this relief, although significant, is lower than in the case of a peripheral material with a low fabrication temperature.
Peripheral Material with a Low Parameter of the Interfacial Shearing Stress and Low Fabrication Temperature
From (6.42), we have Then, formula (6.39) yields
For assemblies, in which the peripheral portions are characterized with the same parameter of the interfacial stress as the assembly midportion (q = 1), formula (6.50) yields
For an assembly with very short peripheral areas (t = 0) or long enough peripheral areas (kl > 2.5), when E = tanhW can be put equal to one, formula (6.51) leads to the same result: x„, = 1, which means that no relief in the interfacial stress could be expected. But what about the intermediate lengths?
dr
Equation (6.50) and the condition = 0 lead to the following equation for the sought t value that minimizes the stress (6.50):
In the extreme case q = 1, this condition yields
Its solution is t = v/2 1 = 0.4142. Then, formula (51) yields x,„ = 0.6569, so stress relief as high as 34.3% can be expected in this case.
The general minimization condition (6.52) could and should be used to determine the optimum t value for the given (accepted) q ratio. Then, the appropriate 6 ratio could be determined on the basis of formula (6.49). It should be pointed out that one cannot simplify formula (6.50) first and then use the minimization condition
dx
—— = 0 to determine the corresponding t value. Using as an example the simpli dt
fled relationship (6.51), let us show that this approach is erroneous. Applying the dx
condition —— = 0 to formula (6.51), we obtain the following equation for the t value: dt
Г  — — = 0. Than we have: 1 = ^ ^{+}  = 0.6899. This result is quite different of
5 5 5 ^{1}
the result t = 0.4142 obtained using the general stress minimization condition (6.52),
and so is the stress relief of x,„ = 0.8525 predicted by formula (6.51) for the t = 0.6899
value. This relief is only 14.7%, while the actual stress relief predicted based on the
general stress minimization condition (6.52) is as high as 34.3%.
As another suitable example, let us consider a hypothetical case of q = 0.8350.
Then, the condition (6.52) yields
This equation has the following solution: t = 0.3210, and formulas (6.49) and (6.50) yield 6 = 1 and x„, = 0.7987. Hence, stress relief of about 20.1% is expected in this case.
Conclusions
The following conclusions can be drawn from the carried out analysis:
• Simple and physically meaningful predictive analytical models are developed for the assessment of thermal stresses in a BGA or in a CGA with a lowmodulus solder material at the peripheral portions of the assembly.
 • It is shown that significant thermal stress relief can be achieved if the bonding system is designed in such a way that the interfacial shearing stress at the ends of the highmodulus and highbonding temperature midportion of the assembly at its boundary with the lowmodulus and lowbonding temperature peripheral portion is made equal to the stress at the assembly ends. It is shown that stress relief as high as 34.3% can be obtained in the optimized design.
 • If such a design is employed, there is a possibility that no inelastic stresses and strains in the solder joints will occur, and the fatigue life of the vulnerable solder material will be dramatically increased.
 • Further work will include systematic computations to evaluate the role of the material and geometric characteristics of the design on the level of the induced stresses; FEAs and experimental investigations.
References
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