Interfacial Stresses

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 mid-cross 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 high-modulus 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

3-4%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-'

  • 29.4017x10-' for BGA
  • 107.806x10-'for CGA

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 Alr = 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„, (L-2I) = qxp(/). 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,„ (L-21) 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 low-modulus 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 high-modulus and high-bonding temperature midportion of the assembly at its boundary with the low-modulus and low-bonding 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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  • 2. E. Suhir, “Predicted Thermal Mismatch Stresses in a Cylindrical Bi-Material Assembly Adhesively Bonded at the Ends,” ASME Journal of Applied Mechanics, vol. 64, No. 1, 1997.
  • 3. E. Suhir, “Thermal Stress in a Polymer Coated Optical Glass Fiber with a Low Modulus Coating at the Ends,” Journal of Material Research, vol. 16, No. 10, 2001.
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