Bi-material assembly subjected to thermal stress: propensity to delamination assessed using interfacial compliance model

Abstract

It is shown that an engineering stress model suggested about 30 years ago for the approximate evaluation of the interfacial stresses in adhesively bonded or soldered bi-material assemblies experiencing thermal loading and based on the concept of interfacial compliance can be employed also for the assessment of the assembly’s propensity to delamination. The analysis is limited to the shearing mode of failure and to the elastic stresses. A probabilistic extreme value distribution (EVD) approach can be used to consider the random nature of both the actual and the critical stress-energy-release-rates (SERR). The step-wise nature of the typical failure oriented accelerated test (FOAT) loading in the electronics reliability field (such as, e.g., temperature cycling) is addressed. The general concepts are illustrated by numerical examples.

Appendices

Appendix A: convolution of extreme value distribution (EVD) with a normally distributed variable

The objective of the analysis that follows is to obtain a convolution of the extreme value distribution (EVD)

$$ F_{{z^{*} }} (z^{*} ) = \exp \left[ { - N\exp \left( { - \frac{{(z^{*} - \bar{z})^{2} }}{{2D_{z} }}} \right)} \right] $$

(25)

for a stationary normal random process Z(t), which is the basic distribution for this EVD (see, e.g., [26]), with a stationary normally distributed random variable X(t), whose probability density distribution function is

$$ f_{x} (x) = \frac{1}{{\sqrt {2\pi D_{x} } }}\exp \left[ { - \left( {\frac{{(x - \bar{x})^{2} }}{{2D_{x} }}} \right)} \right] $$

(26)

In these formulas, the basic random process Z(t) is a homogeneous (the probability that the given level $z^{*}$ is exceeded depends only on the duration of the time interval and is independent of the initial moment of time) and ordinary (none of the events Z ≻ $z^{*}$ can possibly occur simultaneously with another similar event) stationary normal random process; $Z^{*}$(t) are the extreme values of the process Z(t); N is the number of oscillations during the time interval between two adjacent upward crossings Z ≻ $z^{*}$of the level $z^{*}$ by the process Z(t) (in such a situation the flow of the events Z ≻ $z^{*}$ is a Poisson’s process), $\bar{z}$ is the mean value of the process Z(t), $\bar{x}$ is the mean value of the process X(t) and D _z and D _x are variances of the processes Z(t) and X(t). The events Z ≻ $z^{*}$ are assumed to be statistically independent. The number Nin the formula (25) is supposed to be not very small.

The probability density and the probability distribution functions of the random difference W(t) = X(t) − $Z^{*}$(t) are as follows:

$$ f_{w} (w) = \int\limits_{ - \infty }^{\infty } {f_{x} } (x)f_{{z^{*} }} (x - w)dx $$

(27)

$$ F_{w} (w) = \int\limits_{ - \infty }^{\infty } {f_{x} } (x)dx\int\limits_{0}^{x - w} {f_{{z^{*} }} } (z^{*} )dz^{*} = \int\limits_{ - \infty }^{\infty } {f_{x} } (x)F_{{z^{*} }} (x - w)dx $$

(28)

In these formulas the limits of integration for the variable X(t) are defined by the range, within which the function f _x (x) is positive. With the distributions (25) and (26) we have:

$$ \begin{aligned} F_{w} (w) & = \frac{1}{{\sqrt {2\pi D_{x} } }}\int\limits_{ - \infty }^{\infty } {\exp \left[ { - \frac{{(x - \bar{x})^{2} }}{{2D_{x} }}} \right]\exp \left[ { - N\exp \left( { - \frac{{(w - x + \bar{z})^{2} }}{{2D_{z} }}} \right)} \right]} dx \\ = \frac{1}{\sqrt \pi }\int\limits_{{\gamma_{z} }}^{\infty } {\exp [ - (\gamma - \xi )^{2} } - N\exp ( - \delta \xi^{2} )]d\xi \\ \end{aligned} $$

(29)

where a new variable $\xi = \frac{{w - x + \bar{z}}}{{\sqrt {2D_{x} } }}$ of integration is introduced and notation

$$ \gamma = \eta + \gamma_{z} - \gamma_{x} ,\quad \eta = \eta (w) = \frac{w}{{\sqrt {2D_{x} } }},\quad \gamma_{x} = \frac{{\bar{x}}}{{\sqrt {2D_{x} } }},\quad \gamma_{z} = \frac{{\bar{z}}}{{\sqrt {2D_{x} } }},\quad \delta = \frac{{D_{x} }}{{D_{z} }} , $$

(30)

is used. The $\gamma_{x} = \frac{{\bar{x}}}{{\sqrt {2D_{x} } }}$ ratio is the safety factor for the process X(t), and the safety factor $\gamma_{z} = \frac{{\bar{z}}}{{\sqrt {2D_{x} } }}$ for the process Z(t) can be found as $\gamma_{z} = \frac{{\bar{z}}}{{\sqrt {2D_{x} } }} = \gamma_{z} \sqrt \delta$. The integral (29) determines the probability that the difference W = X(t) − $Z^{*}$(t) is below the $w = \eta \sqrt {2D_{x} }$ value. When N → ∞, F _w (w) → 0: in a long run the process $Z^{*}$(t) will always exceed the X(t) values. When $\bar{z} \to \infty$, then γ _z  → ∞, γ → ∞,  and F _w (w) → 0: when the mean value of the process Z(t) is significant, the process $Z^{*}$(t) will always exceed the X(t) values.

When the variance D _z of the process Z(t) is significantly greater than the variance D _x of the process X(t), so that the variance ratio $\delta = \frac{{D_{x} }}{{D_{z} }}$ can be put equal to zero, then the integral (29) can be simplified:

$$ F_{w} (w) = \frac{1}{2}e^{ - N} [1 + \varPhi (\gamma - \gamma_{z} )] = \frac{1}{2}e^{ - N} \left[ {1 + \varPhi \left( {\frac{{w - \bar{x}}}{{\sqrt {2D_{x} } }}} \right)} \right] $$

(31)

where

$$ \varPhi \left( \alpha \right) = \frac{2}{\sqrt \pi }\int\limits_{0}^{\alpha } {e^{{ - t^{2} }} } dt $$

(32)

is the tabulated Laplace function.

Appendix B: Numerical integration: example

The example below is given in Table 3 for the case η = 7. The integrand is as follows:

Table 3 Computed integrand for different values of the variable ξ of integration

$$ f\left( \xi \right) = \exp [ - (11.5608 - \xi )^{2} - 10\exp ( - 0.0305\xi^{2} )] $$

and the corrected sum ∑ _corrected is computed as the sum ∑minus half of the sum of the extreme ordinates.

$$ F = \frac{1}{\sqrt \pi }\Delta \xi \sum {_{corrected} } = 0.5642 \times 0.25 \times 5.9442 = 0.8384 $$

Thus, the probability that the difference between the critical value of the SERR and its actual value will be found below the (rather high) level of the probability that the difference between the critical value of the SERR and its actual value will occur below the (rather high) level of $w = \eta \sqrt {2D_{x} } = 7.0\sqrt {2 \times 0.0400 \times 10^{ - 4} } = 0.01980{\text{ kg/mm}}^{3}$ is as high as 0.8384.