Numerical Example
- Bimaterial Assembly Subjected to Thermal Stress: Propensity to Delamination Assessed Using the Interfacial Compliance Model
- Background/Incentive
- Strain Energy Release Rate (SERR) Computed Using the Interfacial Compliance Approach
- Adequate SERR Specimen’s Length
- Numerical Example #1
- Numerical Example #2
- Probabilistic Approach: Application of the Extreme Value Distribution
- Probabilistic Approach: Numerical Example
Input Data:
Assembly length: L = 2a = 20 mm; Temperature change: At = 275°C;
Component |
CTE, a, 1/°C |
Young's Modulus, E, kg/mm/sq. |
Poisson's Ratio, v |
Thickness, h, mm |
#1 |
2.4E-6 |
12.500 |
0.24 |
0.5 |
#2 |
7.0E-6 |
36.000 |
0.33 |
1.0 |
0 |
- |
12.000 |
0.33 |
0.05 |
#3 |
16.5E-6 |
13.000 |
0.34 |
2.5 |
Computed Data:
Axial compliances:
Parameter of axial compliances Shear moduli:
Shearing compliances:
Total interfacial compliance for a bimaterial assembly is:
For a trimaterial assembly, the longitudinal interfacial compliances are
The parameter of the interfacial shearing stresses is
for the bimaterial assembly. For a trimaterial assembly, we find:
and the parameter of the interfacial shearing stresses computed for a trimaterial assembly in accordance with formula (2.54) is
This parameter is by 6.8% larger than the parameter of the interfacial shearing stress in a bimaterial assembly.
Through-thickness compliances evaluated for a bimaterial assembly in accordance with formulas (2.27) are:
For the trimaterial assembly the through-thickness compliances are:
Through-thickness stiffness for a bimaterial assembly is:
Through-thickness stiffnesses for a trimaterial assembly are:
Flexural rigidities:
Parameter of the flexural rigidities in trimaterial assemblies evaluated by formula (2.67) is as follows:
If all the assembly components have the same flexural rigidities, then this parameter would be equal to 0.5. If the component #2 were significantly more rigid than the other two, then this parameter would be very small.
Parameters of the peeling stresses in a bi material assembly computed by formulas (2.29) and (2.34) are:
and
For a trimaterial assembly, formulas (2.56) and (2.57) yield:
Auxiliary parameters for the peeling stress in a trimaterial assembly predicted by formulas (2.66) are:
Then, formulas (2.62) yield:
The thermally induced forces acting in the midportion of the bimaterial assembly components are given by formulas (2.8):
The component #1 is in compression, and the component #2 is in tension. For the trimaterial assembly, formulas (2.43) yield:
Clearly, the sum of these forces is zero. The components #1 and #2 are in compression, and the component #3 is in tension.
The component #1 in the bimaterial assembly experiences compressive
7i° 9 0221
stresses of the magnitude
/?i 0.5
component #2 of this assembly experiences tensile stresses of the magnitude
- 7)° 9.0221 , _{2}
- 02= — =-= 9.0221 kg/mm^{-};
h_{2} 1.0 ^{c}
The components #1 and #2 in the trimaterial assembly experience compressive stresses
and
respectively. The component #3 in the trimaterial assembly experiences tensile stress
The maximum interfacial shearing stress in the bimaterial assembly, as predicted by formula (2.19), is
The maximum interfacial shearing stresses in the trimaterial assembly, computed by formulas (2.55), are as follows:
and
These stresses are considerably higher than the maximum interfacial stress in a bimaterial assembly, especially at the interface between the substrate and the heat sink. This stress is almost by an order of magnitude higher than the maximum interfacial shearing stress in a bimaterial assembly comprised of the chip and the ceramic substrate.
Formula (2.33) for the factor of the peeling stress at the end of a bimaterial assembly yields:
The product (3o = 2.8452 x 10 = 28.452 is significant, so that formula (2.45) can be used to evaluate the maximum value of the peeling stress in the bimaterial assembly:
Thus, the calculated peeling stress in a bimaterial assembly is by about 60% higher than the predicted maximum interfacial shearing stress.
For the factors A and A_{2} of the peeling stresses acting at the interfaces of a trimaterial assembly, formulas (2.63), with
result in the following values of these factors:
Then, with r| = 0.9366, pi =6.5375, p_{2} = 4.1408, Yi = 2.4371, and y_{2} = 1.3412, we obtain the following values of the maximum peeling stresses at the interfaces #1 and #2: /?!<«) = —13.2102 kg/mm^{2}; and p_{2}(a) = -229.5843 kg/mm^{2}. While the maximum peeling stress p_{t}(a) at the die-substrate interface is only about 40% of the maximum interfacial shearing stress at the interface #1 between the die and the substrate, the maximum peeling stress p_{2}(a) at the interface #2 between the substrate and the heat sink is by a factor of 1.8 higher than the maximum shearing stress at this interface. Comparing the calculated maximum peeling stresses in the trimaterial and bimaterial assemblies, we conclude that the maximum peeling stress at the interface #1 between the die and the substrate in the trimaterial assembly is only about 60% of the peeling stress in the bimaterial assembly. This means that adding a robust heat sink to the assembly resulted in an appreciable stress relief at the die-substrate interface. The “bad news,” however, is that the maximum peeling stress at the interface #2 between the substrate and the heat sink is by an order of magnitude higher than the maximum peeling stress in a bimaterial assembly.
Bimaterial Assembly Subjected to Thermal Stress: Propensity to Delamination Assessed Using the Interfacial Compliance Model
Background/Incentive
One way to make a design for reliability (DfR) decision for a bonded assembly [39-42], as far as its interfacial fracture toughness is concerned, is by comparing the anticipated failure criterion with a critical load factor. Based on this criterion, the adhesive and the cohesive strength of the bonding material could be judged upon, and the adequate bonding material and its thickness could be selected. There are numerous proposed theories and predictive models to understand the physics behind the material failure in bonded joints (see, for example, [43-45]). The majority of models use, in one way or another, fracture mechanics concepts [46-56], and the most popular models proceed from the strain energy release rate (SERR) [57-58]. In the analysis that follows, we suggest using the interfacial compliance model [59, 60] and the probabilistic concept [61-67]; simple and physically meaningful engineering models for the assessment of the SERR in shear for a bonded assembly subjected to the change in temperature. Both the actual and the critical SERR values are random variables, and the loading (such as temperature cycling) is a step-wise process that can be described best by the extreme value distribution (EVD) model, in which both the level and the number of loadings are important. Although the analysis is carried out for the case of thermal loading and is geared in application to electronic materials and assemblies, it can be used also, with some modifications, for mechanical loading (such as the one in shear-off testing) and in numerous applied science problems beyond the electronics materials field.
Strain Energy Release Rate (SERR) Computed Using the Interfacial Compliance Approach
Consider a bonded bimaterial assembly manufactured at an elevated temperature and subsequently cooled down to a low (room or testing) temperature. The thermally induced stresses that arise because of the dissimilar adherend materials can possibly result in the interfacial cracking (delamination) of the assembly. The SERR is defined, as is known, as the energy dissipated during fracture (crack propagation) per unit surface length of a newly created fracture (see, for example, [39-41]):
Here, U is the potential strain energy available for crack growth and a is the crack length. The SERR-based failure criterion states that a crack will grow when the available (actual) SERR, G_{e}, exceeds its critical value (threshold) G_{c}:
The critical fracture energy is considered to be a material property and should be evaluated experimentally based on the specially designed and conducted failure- oriented accelerated tests (FOATs) (see, for example, [65]). The adequate geometry of the FOAT test specimen is certainly important.
The interfacial thermally induced shearing stress in a bonded assembly can be evaluated, in an approximate analysis, based on the interfacial compliance approach [59, 60], by the formula:
where
are the thermally induced forces acting in the adherend cross sections,
is the total longitudinal (axial) compliance of the assembly, a, and a_{2} are the coefficients of thermal expansion (CTE) of the adherend materials, £,° = -—— and 1 - v
£? =—— are the effective Young’s moduli of the materials, E_{{} and E_{2} are their Ei
actual Young’s moduli, v, and v_{2} are the Poisson’s ratios, /?, and hi are the thicknesses of the adherends, At is the change in temperature,
is the parameter of the interfacial shearing stress,
is the total longitudinal interfacial compliance of the assembly,
is the interfacial compliance of the bonding layer, h_{0} is its thickness, G_{0} is its shear modulus,
are the interfacial compliances of the adherends [59], and G| and G_{2} are the shear moduli of the adherend materials. The interfacial shearing stress x(x) is related to the forces T(x) acting in the adherend cross sections as
where / is half the assembly length. The origin of the coordinate x is in the mid-cross section of the assembly.
Introducing (2.77) into (2.84) we find:
This formula meets the zero boundary condition T(l) = 0 at the assembly ends, where the maximum shearing stress
takes place.
The elastic strain energy (work), needed to deform a unit volume of the bonding material, associated with the distortion of its form and caused by the induced shearing stresses in it, can be evaluated by the formula (see, for example, [60])
where
is the octahedral shearing stress, and x,, x_{2}, and x_{3} are the principal shearing stresses. Assuming x, = 0, and x, = x_{2} = x(x), we have
and formula (2.87), when applied to an elementary length (segment) dx of the bond, yields
The thermally induced interfacial stresses and strains are antisymmetric with respect to the mid-cross section of the assembly. The strain energy (per unit assembly width) contained in each half of the assembly length can be evaluated, using formula (2.77) and assuming that the stress is uniform over the bond thickness h_{0}, as
where the function
TABLE 2.1
Function Reflecting the Effect of the Assembly Length on the Strain Energy
kl |
0 |
0.5 |
1.0 |
1.010 |
2.0 |
3.0 |
4.0 |
5.0 |
00 |
XoW) |
0 |
0.0689 |
0.3416 |
0.3480 |
0.8227 |
0.9655 |
0.9940 |
0.9990 |
1.0000 |
(anh kl |
0 |
0.4621 |
0.7616 |
0.7658 |
0.9640 |
0.9950 |
0.9994 |
0.9995 |
1.0000 |
Xi (kD |
0 |
0.1817 |
0.3198 |
0.3199 |
0.1362 |
0.0292 |
0.00536 |
0.00091 |
0 |
reflects the effect of the assembly length on the strain energy. This function changes from zero (very short assemblies and/or assemblies with very compliant bonds) to one (long assemblies and/or assemblies with stiff bonds). The function %oШ) is tabulated in the second line of Table 2.1. As evident from the calculated data, the strain energy increases with an increase in the kl value, but does not practically change with an increase in the kl product, if this product reaches and exceeds the kl = 5.0 level.
The hyperbolic tangent tanhk/ reflects, as evident from formula (2.77), the effect of the kl product on the maximum shearing stress. Table 2.1 indicates that the “saturation” of this stress starts at about kl = 4. Thus, the interfacial shearing stress increases faster with an increase in the product kl than the strain energy level does.
From (2.91), we find, by differentiation where the function
reflects the effect of the assembly size on the derivative of the strain energy w'ith respect to the change in the assembly length. For the interfacial delamination crack that propagates from the assembly end (where the stress level is the highest) inward the assembly, formula (2.93) also determines the SERR, since the crack length a can be found as the difference between the constant length of the assembly (specimen) and the variable remaining length / of the still undamaged assembly, so thatrfa = -dl. The function Xi(&0 increases from zero to its maximum value of = 0.3199 at kl ~ 1.010 and then decreases with the further increase in the kl value. It becomes next-to-zero for kl values exceeding kl = 4.0.
The maximum SERR takes place for kl values that could be found from the equation = 0, which yields
dx
This equation is fulfilled for kl = 1.0096. Thus, the maximum SERR takes place for not very long assemblies, because in such assemblies, although the stress level is high, the strain energy does not change significantly with an increase in the crack (delamination) length. The maximum SERR does not occur for very small size assemblies either, because, although the SERR is appreciable, the shearing stress level is low. This conclusion is important, particularly in connection with choosing the most appropriate test specimen size (see Numerical Example #1), when the critical value of the SERR is sought based on a FOAT experiment.
Adequate SERR Specimen’s Length
Numerical Example #1
Input data:
Component#!: Young’s modulus: E_{{} = 12300 kg/mm^{2}; Poisson’s ratio: v, =0.24;
CTE: a, = 2.2 x 10^{_6}1/°C; Thickness: /?i = 0.5 mm;
Component#!: Young’s modulus: E_{2} = 2000 kg/mm^{2}; Poisson’s ratio: v_{2} =0.30;
CTE: a_{2} = 13.2 x 10~^{6}1/°C; Thickness: /?, = 1.5 mm;
Bonding layer (zero component): Young’s modulus: E_{0} = 2000 kg/mm^{2}; Poisson’s ratio: v_{0} = 0.40; CTE: a_{0} = 13.2 x 10^{4i}l/^{o}C; Thickness: /to = 0.05 mm; Change in temperature: At = 100°C
Computed data:
“External” thermal strain: AaAt = 11 x 10^{6} x 100 = 0.0011;
Axial compliances of the assembly components:
Total axial compliance of the assembly:
Shear moduli of the materials:
Interfacial shearing compliances:
Total interfacial compliance of the assembly:
Note that the axial compliance of the assembly is due mostly to the adherends, while the interfacial compliance is due to both the adherends and the adhesive. Parameter of the interfacial shearing stress:
Length of the test specimen (half of the undamaged assembly length) with the highest SERR:
The actual test specimen should be a little longer.
Numerical Example #2
Shear-off testing is considered for an assembly with characteristics in Numerical Example #1. How high should the measured shear-off force be at failure so that the high enough fracture toughness is assured?
The minimum shear-off force that results in the same maximum interfacial shearing stress as the thermally induced loading can be determined by the formula [61]
With AocAf = (13.2 —2.0.0012)1CT^{6} x 100 = 0,00110, X = 3.5691 x КГ^{4} mm/kg and a short specimen with kl = 1.010 this formula yields: T = 2.2785 kg/mm. If a long specimen is tested, then the required minimum shear-off force at failure would be
Probabilistic Approach: Application of the Extreme Value Distribution
When applying the probabilistic concept [62-66] for the assessment and assurance of the adequate interfacial fracture toughness of an assembly of interest, we assume that the loading process can be approximated by the extreme value distribution (EVD) (see, for example, [67]) for the SERR and that the SERR threshold (level) can be assumed to be a regular normal process. In accordance with the Appendix A results, and assuming that the process Z(t) represents the process G_{c}(t) of the SERR, and that the process X(t) represents the critical SERR level G_{c}(t) the probability that the actual random SERR G„ remains below its critical value G_{c} after the action of the /Vth loading cycle, can be sought in the form of an integral
where
Here g_{a} and g_{c} are the mean values of the actual and the critical SERR levels, respectively, and D„ and Ц are variances of the random stationary processes G_{a}(t) and G_{c}(t). The process G„(t) is characterized by its EVD, which depends on the number N of loadings. Equation (2.96) determines the probability that the random difference W(t) = G_{c}(t) - G*,(t) of the random critical SERR value 6', and the extreme value G'_{i: }of the random actual G_{a} level remains below a certain threshold w = . To apply
the integral (2.93), one has to first calculate the safety factors ^ and
which is the lower limit of the integral (2.93), the variance ratio 6 = —, and assumes a certain П level of the safety factor based on the difference W(t).
Probabilistic Approach: Numerical Example
Input data:
Mean value of the actual SERR process G_{a}: g„ = 0.0229 kg/mm
Mean value of the critical SERR process G_{c}: g_{c} = 0.0100 kg/mm
Variances of the above processes: D_{a} = 1.3110x 10 ^{4} kg^{2}/mm^{6},
Д =0.0400x1 O'^{4} kg^{2} /mm^{6 }Number of loadings N = 10
Computed data:
Safety factor
Safety factor [lower limit of integration in the integral (2.96)
Variance ratio Formula (2.93) yields
The calculated values of the integral (2.93) for different dimensionless SERR values are shown in Table 2.2. Clearly, the probability that the random difference between the critical and the actual SERR is below a certain level increases with an increase in this level. An example of the calculation procedure is shown in Appendix В for B = 7.
Thus, the application of the interfacial compliance approach and the probabilistic DfR concept is suggested for the assessment of the adhesive and cohesive strength of the bonding material in a bimaterial assembly subjected to the temperature change. The SERR is used as a suitable criterion of the level of the fracture toughness of the bond. The shearing mode of failure is considered. As far as the PDfR approach is concerned, it has been accounted for the random nature of both the actual and the critical SERR values. The results of the analysis can be used also beyond the field of electronic materials.
TABLE 2.2
Calculated Values of the Integral Р_{Я}(Г|)
1 |
0 |
3 |
4 |
5 |
6 |
7 |
8 |
F_{n}(r) |
0 |
0.0750 |
0.2926 |
0.5299 |
0.7046 |
0.8384 |
0.9194 |