3. Results and Discussions
3.1 High-speed thermal cycling test
Figure 3 shows X-ray transmission images of the solder layer after the specified number of cycles under each test condition. The white parts in the images correspond to cracks. Images of different specimens were observed. As the cycling progressed, the number of cracks increased, and cracks propagated and connected with each other to form crack networks. At higher temperatures, cracks emerged after fewer cycles and the fatigue crack networks formed in shorter periods. These X-ray transmission images were used to investigate the relationship between the number of cracks, crack length, and number of cycles. The number of cracks and crack length were measured with image processing software (ImageJ).
3.2 Formulation of damage rate equation
3.2.1 Formation mechanisms of fatigue crack networks in the solder layer
To predict the life of fatigue crack networks, a damage rate equation based on fracture mechanisms is required. Therefore, the mechanism of fatigue crack network formation that was clarified in our previous report2) is described here again.
Figure 4 shows the contours of axial stresses in the x and y directions parallel to the solder joint surface during the high-speed thermal cycling test, which was obtained by FEM analysis. As the temperature increased, the solder layer was put into a biaxial compressive stress state in the direction parallel to the joint surface, while the solder layer was put into a biaxial tensile stress state as the temperature decreased. Figure 5 shows the stress-creep strain hysteresis loops in the x, y, z, and xy (shear) directions at the center of the solder layer. The biaxial stresses in the x and y directions were dominant and the hysteresis loops coincided with each other. Stress did not occur in the z direction and both stress and strain in the shear direction were nearly zero. Thus, equibiaxial tensile-compressive deformation occurred over the entire area of the solder layer during the thermal cycling. Because the specimen used in the study had a structure in which the solder was sandwiched between silicon plates, there are no differences in the thermal expansion coefficients between the top and the bottom layers, and no shear deformation occurs. However, because thermal expansion in the solder layer was constrained by the Si, the solder layer was in a biaxial compressive stress state in the direction parallel to the joint surface as the temperature increased, resulting in stress leading to compressive deformation. As the temperature decreased, the solder layer contracted and deformed in the direction of the compressive deformation when the temperature increased to return to the previous state, and the solder layer was in a biaxial tensile stress state and deformed under tension in the direction parallel to the joint surface. Figure 6 shows fatigue crack networks reproduced by FEM analysis. Tensile deformation occurred as the temperature dropped, opening cross-shaped cracks in the area where equibiaxial stresses were generated and cracks propagated in the crisscross direction, thereby expanding the damaged area. Concurrent with the propagation of existing cracks, new cracks also initiated in the undamaged areas, and all these cracks became interconnected and further propagated, ultimately forming a fatigue crack network. During the fatigue crack network development process, crack connections and decreased strain energy density near the cracks caused the saturation of the fatigue crack network development. To consider the fracture life of fatigue crack networks, it is necessary to incorporate crack initiation, propagation, and saturation of damage by crack interconnection in the area where equibiaxial stresses occur. Therefore, crack initiation and propagation behaviors were closely examined as factors that dominate the damage rate and an equation for damage rate was studied based on the mechanisms of the fatigue crack network formation.
The crack initiation rate was calculated from the relationship between the number of cracks and the number of cycles observed from the x-ray transmission images. Figure 7 shows the relationship between the number of cracks and the number of cycles under each test condition obtained from the x-ray transmission images. In each test condition, the number of cracks increased progressively and became saturated at a certain number of cycles. In the fatigue crack network fracture, an initial crack emerged after some incubation period and then the number of crack initiations increased. When the number of cracks became sufficient, the strain energy density near the cracks decreased, leading to a decrease in the number of crack initiations and eventually resulting in saturation of the number of cracks. Thus, the relationship between the number of cracks and the number of cycles became sigmoidal (S-shaped). Because the crack initiation rate depends on the inelastic strain energy density range, the strain energy density range that caused fractures increased and the crack initiation rate increased at higher temperatures.
The behaviors of the S-shaped curve were quantified by using the logistic model3) described in eq. (1). Functionalizing the rate of crack initiation was explored by investigating the dependence of each constant of the logistic model on the driving force of crack initiation.
| \begin{equation}
y = \frac{M}{1 + \exp (- aN - b)}
\end{equation}
| (1) |
Here,
y is the number of cracks,
M is the maximum value of the number of cracks,
N is the number of cycles, and
a is a constant for the crack initiation frequency. The crack initiation rate is described by the differential equation shown in
eq. (2).
| \begin{equation}
\frac{dy}{dN} = \frac{ay(M - y)}{M}
\end{equation}
| (2) |
shows the relationship between the crack initiation rate and the number of cycles, as obtained by
eq. (2). The use of the logistic model enabled reproduction of the increase in the crack initiation rate to the point that the maximum value in the number of cracks under each temperature condition was attained and a decrease in the crack initiation rate due to the decreased strain energy density in the vicinity of the cracks.
The constants a, b, and M in eq. (1) and eq. (2) have the following qualities. The value of the crack initiation frequency constant a is related to the crack initiation rate. As a becomes smaller, the logistic curve becomes more gradual and the angle of the curve until the number of cracks becomes the maximum decreases. The value of b is found from aN + b = 0, that is, when N = −b/a, it represents the cycle with the maximum rate of crack initiation. When the driving force of crack initiation caused by the equibiaxial stress is large, the cycle with the maximum crack initiation rate becomes short. Therefore, −b/a depends on the driving force of crack initiation. Furthermore, it can be predicted from the logistic approximation in Fig. 7 that the maximum number of cracks M also depends on the driving force of crack initiation. Therefore, FEM was used to analyze how the constants a, −b/a, and M in the area where equibiaxial tensile-compressive deformation occurs depend on the driving force of crack initiation. The inelastic strain energy density range ΔWin was employed as the driving force of crack initiation.
First a and M were considered. In the testing of corrosion fatigue in metals where numerous cracks occur randomly, the relationship between the number of cracks and the number of cycles is expressed by an S-shaped function similar to the one in the present study. It has been reported that the crack initiation frequency factor and the maximum number of cracks form a relationship that is semi-logarithmic in terms of stress amplitude.4) Therefore, the crack initiation frequency factor of the logistic curve a and the maximum number of cracks M were also allowed to take a semi-logarithmic relationship with ΔWin and the relationships found between each constant and ΔWin are indicated by eq. (3) and eq. (4) respectively.
| \begin{equation}
a = 7.00 \times 10^{-4}\exp\ (5.29\Delta W_{\text{in}})
\end{equation}
| (3) |
| \begin{equation}
M = 24.4\exp\ (8.84\Delta W_{\text{in}})
\end{equation}
| (4) |
Thus, the constant of crack initiation frequency
a and the maximum number of cracks
M are provided as functions of Δ
Win.
Next −b/a was considered. When low cycle fatigue fractures in a metal are dominated by the inelastic strain energy density range, Morrow’s power law between ΔWin and the fatigue life Nf holds.5) Because multiple cracks occur on the surface of a smooth test specimen during high temperature fatigue, the lifetime given by Morrow’s law is interpreted as the number of cycles until the maximum crack initiation rate. In the fatigue crack network fracture, because multiple cracks occur in the damaged area, the cycle b/a when the crack initiation rate becomes the maximum is assumed to be equivalent to the lifetime given by Morrow’s law. Thus, the constant of Morrow’s law was determined to be eq. (5) from −b/a and ΔWin at each level obtained from the logistic curve.
| \begin{equation}
\Delta W_{\text{in}} \cdot \left(- \frac{b}{a} \right)^{1.0}{} = 623
\end{equation}
| (5) |
Here, the fatigue exponent becomes 1, which indicates that crack initiation in the study is governed by creep. From the above, the calculation of Δ
Win by using FEM gives
a, −
b/
a and
M under all conditions.
3.2.3 Driving force of fatigue crack propagation
It was assumed that ΔWin around a fatigue crack tip represents the mechanical environment of the fracture propagation area, based on which ΔWin was used as the driving force of crack propagation. As described previously, cracks propagate by the opening of crisscross-shaped micro cracks caused by equibiaxial tensile deformation, which occurs as the temperature decreases. Therefore, the FEM model in Fig. 9 having a crisscross-shaped crack inserted in the die attach center of the joint specimen was used for calculation of ΔWin. An area of 60 µm2 around the crack tip was divided into 10 µm square elements and ΔWin was calculated as the volume average of each element.6,7)
| \begin{equation}
\Delta W_{\text{in}} = \frac{\displaystyle\sum \Delta W_{\text{in}}^{\text{element}} \times V^{\text{element}}}{\displaystyle\sum V^{\text{element}}}
\end{equation}
| (6) |
Here, Δ
Winelement is the elemental solution of the inelastic strain energy density range and
Velement is the element volume. In terms of material properties, Si was treated as an elastic body and the solder alloy as an elasto-creep body. The creep constitutive equation obtained from a separate experiment
8) followed Norton’s law as shown in
eq. (7).
| \begin{equation}
\dot{\varepsilon}_{\text{ss}} = 1.26 \times 10^{-8}\sigma^{8.77}\exp \left(\frac{-60\,\text{kJ/mol}}{RT} \right)
\end{equation}
| (7) |
Here,
$\dot{\varepsilon }_{\text{ss}}$ is the steady-state creep rate (s
−1), σ is the stress (MPa),
R is the gas constant (J mol
−1K
−1) and
T is the temperature (K). Paris’ fatigue crack propagation law shown in
eq. (8) was used to calculate the fatigue crack propagation rate.
| \begin{equation}
\frac{da}{dN} = A_{1}\Delta W_{\text{in}}^{A_{2}}
\end{equation}
| (8) |
where
A1 and
A2 are the material constants, and
da/
dN is the fatigue crack propagation rate.
3.2.4 Damage rate equation of fatigue crack network
Hereafter, derivation of a damage rate equation based on the mechanism of fatigue crack network formation is considered. Based on the fatigue crack network formation mechanism, it is assumed that fatigue damage develops based on the mechanisms 1) to 3) below. A schematic illustration for defining damage in the fatigue crack network is shown in Fig. 10.
-
1)
The solder layer is in the equibiaxial stress state and cracks propagate in the crisscross-shape by the equibiaxial stress as in Fig. 10, and the square surrounding the cross-shaped crack is defined as a damaged area.
-
2)
Because the thickness of the solder layer is very thin compared with the area of the joint surface, only the spread in the planar direction viewed from the top surface of the solder layer was counted as the extent of the damaged area.
-
3)
New cracks occur in undamaged areas, and cross-shaped cracks propagate at a constant rate until damaged areas (cracks) collide with each other, and then after the collisions, the crack propagation is arrested.
-
Based on the assumption that the fatigue crack networks are formed as described above, it is thought that the extent of the damaged areas and the arrest of crack growth by collision between damaged areas can be expressed by extended volume theory
9–11) and a damage rate equation was investigated.
The damage ratio in the case where the damaged areas extended with no consideration given to collisions between damaged areas is described by eq. (9), which is defined as the extended damage ratio Se.
| \begin{equation}
S_{e} = \sum_{i}S_{i}
\end{equation}
| (9) |
Here,
Si is the damage ratio per single crack. The probability
U that the locations where new cracks occur are not included in the area that is already damaged (or the probability of crack occurring in undamaged locations) can be described by
eq. (10).
| \begin{equation}
U = \prod_{i}(1 - S_{i})
\end{equation}
| (10) |
is then obtained by taking the logarithms of both sides of
eq. (10).
| \begin{equation}
\ln U = \ln \prod_{i}(1 - S_{i}) = \sum_{i}\ln (1 - S_{i})
\end{equation}
| (11) |
When the damage ratio
Si is sufficiently smaller than 1,
Si is described by
eq. (12). Therefore, ln
U is described by
eq. (13) using the extended damage ratio
Se.
| \begin{equation}
\ln (1 - S_{i}) = - S_{i}
\end{equation}
| (12) |
| \begin{equation}
\ln U = \sum_{i} - S_{i} = - S_{e}
\end{equation}
| (13) |
When the existing damage ratio is taken to be
S, the probability
U that new crack initiation sites are not included in the area already damaged is given by 1 −
S. As ln
U can be described by
eq. (14), the relationship between the damage ratio and the extended damage ratio is given by
eq. (15).
| \begin{equation}
\ln U = \ln (1 - S) = - S_{e}
\end{equation}
| (14) |
| \begin{equation}
S = 1 - \exp (- S_{e})
\end{equation}
| (15) |
The specific equation of extended damage ratio Se is also discussed. Because the damaged area is formed by the propagation of fatigue cracks in a crisscross-shape, when the half-length of a crack is assumed to be a as in Fig. 10 and the crack propagation rate da/dN is used, then the damaged area s formed by the crack propagation from the incubation cycle number Ni to a given cycle number N is obtained by eq. (16).
| \begin{equation}
s = (2a)^{2} = 4\left(\frac{da}{dN} \right)^{2}(N - N_{i})^{2}
\end{equation}
| (16) |
When the crack initiation rate at
Ni is assumed to be
Y(
Ni), the extended damage ratio at
N is given by
eq. (17).
| \begin{equation}
S_{e} = \int_{N_{i}}^{N}Y(N_{i})4 \left(\frac{da}{dN} \right)^{2}(N - N_{i})^{2}dN_{i}
\end{equation}
| (17) |
Although the crack initiation rate can be expressed by a logistic function of
eq. (2), the logistic curve includes the rate of decrease caused by collisions between cracks. However, because
eq. (17) needs the damaged area to continuously extend regardless of collisions between damaged areas,
Y(
Ni) needs to be constant without including the decreased rate caused by collisions between cracks. It is assumed that
Y(
Ni) is the average of the crack initiation rates derived in 3.2.2 under individual test conditions. When
eq. (17) is integrated, the extended damage ratio
eq. (18) is then obtained.
| \begin{equation}
S_{e} = \frac{4}{3}Y\left(\frac{da}{dN} \right)^{2}N^{3}
\end{equation}
| (18) |
Finally, the damage rate is given by
eq. (19).
| \begin{equation}
S = 1 - \exp \left\{-\frac{4}{3}Y\left(\frac{da}{dN} \right)^{2}N^{3} \right\}
\end{equation}
| (19) |
The relationship between the damage rate based on the fatigue crack network formation and the number of cycles was found by using the derived damage rate equation eq. (19). Figure 11 shows the relationships between the damage ratios under each test condition and the number of cycles obtained by eq. (19) and comparison of experimentally obtained damage ratios for the number of respective cycles. Because the fatigue crack propagation law of Sn–3.0Ag–0.5Cu derived using ΔWin is not available, the fatigue crack propagation law constant was chosen such that eq. (19) best reproduces the experimental values. Equation (20) shows the fatigue crack propagation law determined by the method.
| \begin{equation}
\frac{da}{dN} = 0.075\Delta W_{\text{in}}^{1.2}
\end{equation}
| (20) |
The relationship between the damage rates and the number of cycles under individual test conditions are reproduced by the damage rate equation
eq. (19) as shown in
Fig. 11.
Figure 12 shows a comparison between
eq. (20) and the fatigue crack propagation speed of Sn–5.0Sb derived in the past.
12) The fatigue crack propagation law of Sn–5.0Sb and
eq. (20) mostly coincided with each other. Considering that the fatigue crack propagation law of solder alloys in low-cycle fatigue is relatively insensitive to the kinds of material,
eq. (20) is thought to be valid as a fatigue crack propagation law for Sn–3.0Ag–0.5Cu and the proposed damage rate equation in
eq. (19) allows for a simplified prediction of the damage developed by fatigue crack network formation.
