Thermal Stress Modeling In Microelectronics and Photonic Structures, and the Application of the
Probablistic Approach: Review and Extension
Thermal Stress Modeling In Microelectronics
and Photonic Structures, and the Application of
the Probablistic Approach: Review and
Extension
E. Suhir
Distinguished Member of Technical Staff,
Physical Sciences and Engineering Research Division,
Bell Laboratories, Lucent Technologies, Inc.,
600 Mountain Ave
Room 1D-443
Murray Hill, New Jersey 07974, USA
Phone: 908-582-5301
Fax: 908-582-5106
e-mail: suhir@lucent.com
Abstract
Elevated thermal stresses and deformations can result in a mechanical (structural), as well as in a functional (electrical or optical),
failure of an electronic or photonic material, structure or package. Predictive modeling, whether analytical (“mathematical”) or
numerical (typically, Finite Element), is an effective research/engineering tool for the prediction of the level and the adverse consequences of thermal stresses and deformations. Accordingly, in this review, the author discusses the major attributes of the predictive
modeling of the thermal stresses in microelectronics and photonics packaging. In addition, the role that a probabilistic approach
might play for understanding the effect of the variability/uncertainty in materials properties, structural geometry, and loading conditions on the thermal stress has been indicated. Based on several practical examples, one can demonstrate that, in some practically
important problems of packaging engineering, when such a variability/uncertainty cannot be ignored, the application of a probabilistic approach can be very helpful in the analysis and design of a viable and reliable structure. The review is based primarily on the
author’s research conducted at Bell Laboratories during the last decade.
Key words:
Thermal Stress Modeling, Probablistic Approach, Microelectronics, and Photonics.
1. Background: Thermal Stress
Failures
Thermal (“internal”) loading, and the resulting stresses and
deformations can be defined as those that are associated with the
change in temperature, and/or as those, which depend on
thermomechanical properties of the employed materials. Thermal stresses occur during fabrication, testing, storage, and operation of the microelectronic and photonic equipment. Thermally induced stresses and strains can be due to dissimilar materials that expand/contract at different rates during temperature
excursions, and/or to the nonuniform distribution of temperature
(temperature gradient).
Elevated thermal stresses are the major contributor to the finite service life of microelectronics and photonic structures, packages, and systems. Although the most serious consequences of
the elevated thermal stresses are usually associated with mechanical (structural) failures (ductile rupture, brittle fracture, failures
due to fatigue, creep, thermal relaxation, thermal shock, stress
corrosion, excessive deformation, among other factors), thermal
stresses and strains can result also in the functional failures, such
as, in the loss of the electrical or optical performance of a compo-
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and structural engineering, including the area of microelectronics and photonics6. This should be attributed to the availability
of powerful and flexible computer programs, which enable one
to obtain, within a reasonable time, a solution to almost any stressstrain-related problem.
Based on the comparison of the Finite Element and analytical
evaluations, it has been found, Mishkevich and Suhir11, that the
approach previously used in application to thermal stress thin
film structures, Suhir17, can be employed to further simplify the
interfacial thermal stress calculations in bi-material assemblies
with a certain thickness ratios of the constituent materials. This
approach suggests that the interfacial shearing stress can be evaluated from a simplified equation obtained under an assumption
that the interfacial shearing stress can be evaluated without con2. Thermal Stress Modeling
sidering the effect of the “peeling” stress. Such an assumption is
conservative, resulting in a reasonable overestimation of the
maximum shearing stress, in comparison with the more accurate
Pioneering work in modeling of thermal stress in bodies comvalue, obtained from the coupled equations for the interfacial
prised of dissimilar materials has been performed by
stresses15. The simplified solution suggests that the maximum
20
21
1
Timoshenko , Völkersen , and Aleck . Timoshenko and
value of the shearing stress always takes place at the assembly
Völkersen based their treatment of the problem on a structural
edge, while the solution, based on the coupled equations for the
analysis (strength-of-materials) approach, while Aleck applied
interfacial stresses, indicates that the maximum shearing stress
theory-of-elasticity method. Both approaches were then extended
occurs at some distance from the edge. After the shearing stress
and substantially modified in application to modeling of the meis determined, the “peeling” stress can be found from an equachanical behavior of assemblies and structures employed in varition similar to the equation of bending of a beam lying on a
ous fields of engineering, including the area of microelectronics
continuous elastic foundation.
and photonics packaging. The structural analysis approach was
In connection with the wide use of computational methods in
used by Chen and Nelson3, Chang2, Glascock and Webster5,
thermal stress analyses for microelectronics and photonics packSuhir12,13,14,15, Hall et al.7, Lau10, and many other investigators.
ages, it should be pointed out that broad application of computers has, by no means, made analytical solutions unnecessary or
The theory-of-elasticity approach was used by Zeyfang23, Eischen4,
even less important, whether exact, approximate, or asymptotic14.
Kuo8, Yamada22 and other researchers.
The structural analysis (engineering) approach enables one to
Simple and easy-to-use analytical relationships have invaluable
determine, often with sufficient accuracy, the magnitude and the
advantages, due to of the clarity and “compactness” of the obdistribution of the interfacial shearing and through-thickness
tained information and clear indication of the role of various
(“peeling”) stresses, as well as the stresses acting in the cross
factors affecting the given phenomenon or the behavior of the
sections of the constituent materials. This approach results in
given system. These advantages are especially significant when
simple and easy-to-use formulas, and can be (and, actually, has
the parameter under investigation depends on more than one
been) successfully employed, as a part of a physical design provariable. As to the asymptotic techniques and formulas, they can
cess of a component or device. It can be used to select materials,
be successful in those cases in which there are difficulties in the
establish the dimensions of the structural elements, and compare
application of computational methods, such as, in problems condifferent designs from the standpoint of the stress level. As to
taining singularities. But, even when the application of numerithe theory-of-elasticity method, it is based on rather general ascal methods encounters no difficulties, it is always advisable to
sumptions and equations of the elasticity theory and provides a
investigate the problem analytically before carrying out computerrigorous mechanical treatment of the problem. This approach is
aided analyzis. Such a preliminary investigation helps to reduce
advisable, when there is a need for the most accurate evaluation
computer time and expense, to develop the most feasible and
of the induced stresses and strains. Applied within the frameeffective preprocessing model and, in many cases, to avoid funwork of linear elasticity, this approach leads, in the majority of
damental errors. It is noteworthy that the Finite Element method
cases, to a stress singularity at the assembly edge. Its application
was originally developed for structures with complicated geomhas been found particularly useful when there is an intention to
etry and/or with complicated boundary conditions (such as, avifurther proceed with fracture analysis. The engineering and the
onics structures), when it might be difficult to apply analytical
theory-of-elasticity approaches should not be viewed, of course,
approaches. As a consequence, this method is especially widely
as “competitors”, but rather as different research tools, which
used in those areas of engineering in which structures of comhave their merits and shortcomings, and their areas of applicaplex configuration are typical: aerospace, marine, and offshore
tion.
structures, and some civil engineering structures. In contrast, a
Finite Element modeling has become, since the mid-1950s,
relatively simple geometry and simple configurations usually
the major research tool for theoretical evaluations in mechanical
characterize microelectronics and fiber optics structures. ThereThe International Journal of Microcircuits and Electronic Packaging, Volume 23, Number2, Second Quarter 2000 (ISSN 1063-1674)
nent or device. For instance, transistor junction failure can occur due to of the elevated thermal stress in the integrated circuit,
if the heat produced by the chip cannot readily escape9. Optical
performance failure (such as, loss in coupling efficiency) occurs19,
when the lateral thermally induced displacement in the gap between two lightguides exceeds the allowable limit (this limit can
be as low as 0.5 m). In laser packages, such displacements can
be due also to thermal stress relaxation in a laser weld. The
ability to understand the sources of thermal stresses and displacements, predict their distribution and the maximum values, and
possibly minimize them is of clear practical importance.
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Thermal Stress Modeling In Microelectronics and Photonic Structures, and the Application of the
Probablistic Approach: Review and Extension
fore, they can be easily and quite accurate idealized as beams,
flexible rods, plates, and various composite structures of relatively simple geometry. Therefore, there is a clear incentive for a
broad application of analytical stress modeling for such structures.
3. Probabilistic Approach
and contain the corresponding deterministic models as first approximations. It is important to emphasize that the use of probabilistic methods and approaches is due not so much to the fact
that the available information is insufficient for a deterministic
analysis, but, first of all, to the fact the variability and uncertainty are inherent in the very nature of many physical phenomena, materials characteristics, engineering designs, and application conditions.
Probabilistic models enable one to establish the scope and the
limits of the application of deterministic theories. These models
provide a solid basis for a substantiated and goal-oriented accumulation, and effective use of empirical data. Realizing and
emphasizing the fact that the probability of failure of an engineering product is never zero, probabilistic methods enable one
to quantitatively assess the degree of uncertainty in various factors, which determine the performance of a product, and to design on this basis a product with a low probability of failure.
Probabilistic methods, underlying all the modern methods of forecasting and decision making, allow one to extend the accumulated experience on new products and new designs, which may
differ from the existing ones by type, dimensions, materials, and
operating conditions.
Although probabilistic approaches and models are able to account for a substantially larger number of different factors than
the deterministic methods, they should not be viewed as a sort of
a panacea that is able to cure all the engineering troubles. These
methods cannot perform miracles and have their limitations. For
instance, probabilistic methods cannot be applied in situations,
where the conditions of an experiment or a trial are not reproducible or when the events are very rare. Quite often a serious
obstacle for applying probabilistic methods is the difficulty of
obtaining the necessary input information. In such situations,
the designer considers the “worst case”, uses a quasi- deterministic approach, or a more or less consistent combination of probabilistic and deterministic reasoning. However, when the application of probabilistic methods is possible, justified, and is supported by reliable enough input information, these methods provide a powerful, effective, and well-substantiated resource for
engineering analyses and designs.
Packaging/reliability engineer uses predictive modeling and,
particularly, modeling of thermal stresses and other thermal phenomena, at all the stages of the analysis, design, testing, manufacturing, operation, and maintenance of a product or system.
The traditional approach in predictive modeling can be referred
to as deterministic. Such an approach does not pay sufficient
attention to the variability of parameters and criteria used. This
approach is acceptable and can be justified in many cases, when
the deviations (“fluctuations”) from the mean values are small,
when the design parameters are known, or can be predicted with
reasonable accuracy, and when the processes and procedures that
the engineer deals with are “stochastically stable”, that is, when
“small causes” result in “small effects”. There are, however,
numerous situations, in which the “fluctuations” from the mean
values are significant and in which the variability, change and
uncertainty play a vital role. In the majority of such situations,
the product will most likely fail, if these uncertainties are ignored. Therefore, understanding the role and significance of the
“laws of chance”, and the causes and effects of variability in
material properties, structural dimensions, tolerances, bearing
clearances, loading conditions, stresses and strength, applications, and environments, and a multitude of other design parameters, is critical for the creation and successful operation of a
viable and reliable product or structure. In the majority of practical cases, the random nature and various uncertainties in the
design characteristics and parameters can be described on the
basis of the methods of the theory of probability.
Probabilistic methods proceed from the fact that uncertainties
are an inevitable and essential feature of the nature of an engineering system or design and provide ways of dealing with quantities that cannot be predicted with absolute certainty. Unlike
4. Example: Solder Glass Attachment
deterministic methods, probabilistic approaches address more
general and more complicated situations, in which the behavior
in a Ceramic Electronic Package
of the given characteristic or parameter cannot be determined
with certainty in each particular experiment or in a particular
situation. However, for products, which are manufactured in large
As an example of probabilistic modeling of thermal stresses
quantities, and for experiments, which are repeated many times
and a probabilistic structural design in microelectronics, one can
in identical conditions, this behavior can be described by probaexamine a solder (“seal”) glass attachment in a ceramic (“Cerdip/
bilistic/statistical relationships. These relationships manifest
Cerquad”) package of an integrated circuit (IC) device. The
themselves as trends in a large number of random events.
mechanical performance of solder glass in the package, subjected
Probabilistic models reflect the physics of phenomena and the
to the temperature change during its fabrication, testing, or storvariability of the behavior and performance of an engineering
age, is affected primarily by the stresses occurring due to of the
product much better than the deterministic ones. It would not be
thermal expansion (contraction) mismatch of the seal glass and
an exaggeration to say that all the fundamental theories and apthe body of the package. Solder glass is a brittle material and can
proaches of modern physics and engineering are probabilistic,
break when stretched or bent.
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The thermally induced stresses arising in the solder glass can
be minimized, if all the ceramic components of the package (the
window frame, the base, and the lid) have the same coefficients
of expansion. In such a case, the shearing stresses, which concentrate at the end portions of the glass seal, are expected to be
low, and the normal stresses in the glass layer will be due to the
thermal expansion (contraction) mismatch of the seal glass with
the ceramics. If, for some reason, ceramics with different coefficients of expansion are used, it is imperative that these coefficients, being temperature dependent, match well at least at lowtemperature conditions, when the expected thermally induced
stresses are the highest. Being brittle materials, solder glasses
are able to withstand much higher stresses in compression than
in tension. Therefore, it is desirable, for low compressive stresses
in the glass layer, that the glass has a somewhat smaller coefficient of expansion than the ceramics.
The maximum interfacial shearing stress in a thin solder glass
layer can be computed by the formula13,
(1)
where h1 is the layer thickness, and K is the parameter of the
assembly compliance. K is given as follows,
(2)
and
(3)
λ is the longitudinal compliance of the assembly, h0 is the total
thickness of the ceramics parts, E0 and E1 are Young’s moduli of
the ceramics and the glass, respectively, ν0 and ν1 are Poisson’s
ratios of these materials,
(4)
κ is the interfacial compliance of the assembly,
are these coefficients for the given temperature t, t0 is the annealing (“zero stress”, “setting up”) temperature, and α0(t) and α1(t)
are the “instantaneous” (“actual”) values of the coefficients of
expansion. In an approximate analysis, one can assume that the
axial compliance, λ, of the assembly is due to the seal glass only,
that is,
(7)
and therefore, the maximum normal stress, σmax, in the glass can be
computed by a simplified formula,
(8)
While the geometrical characteristics of the assembly, the temperature change, and the elastic constants of the ceramics and
the glass can be determined with quite high accuracy, the reliability in the prediction of the difference in the coefficients of
thermal expansion of the glass and the ceramics is not as good.
This is due, first of all, to the fact that these coefficients exhibit
strong temperature dependence and are very sensitive to variations in the glass composition, and thermal treatment. However,
what is even more important, is that, due to the clear incentive to
minimize the thermal expansion (contraction) mismatch of the
glass and the ceramics, such a mismatch is characterized by a
small difference of close and not necessarily very small numbers.
This leads to an elevated uncertainty in the evaluation of this
difference, thereby, justifying the application of a probabilistic
approach.
Treating the coefficients of thermal expansion as random variables, one can evaluate the probability, P, that the stress in the
seal glass is compressive and does not exceed the allowable value,
σ*, as the probability that the difference,
ψ = α0 - α1
(9)
falls within the interval
(10)
(5)
σmax is the maximum normal stress in the midportion of the glass
layer, ∆t is the change in temperature, ∆α = α 0 − α1 is the
difference in the “effective” coefficients of thermal expansion for
the ceramics and the glass,
Let us assume that the random variables α0 and α1 follow the
normal law of distribution, such that their probability density
functions are expressed by the formulas,
.
(11)
Then, one can conclude that the function, y, of nonfailure (safety
(6)
margin) is also normally distributed,
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Probablistic Approach: Review and Extension
(12)
= 1.2726 and γ* = 7.8201, and the probability of nonfailure predicted on the basis of the formula (13), is as follows,
P = φ1(6.5475) + φ1(1.2726) - 1 = 0.898.
Here, the mean, <ψ>, and the variance, Dψ, of the random variable
ψ are <ψ> = <α0> - <α1> and Dψ = D0 + D1, respectively. The
probability that the variable y falls within the interval (0, ψ*) can be
computed as follows,
(13)
where
(14)
and the reliability (safety) indices, γ and γ*, are expressed by the
formulas,
.
If the standard deviations D 0 and D1 of the coefficients of
thermal expansion were only 0.1 × 10-6 1/°C, then the reliability
indices would become γ = 3.1825 and γ* = 19.5559, respectively,
and the probability of nonfailure would increase to P = 0.999. Hence,
the standard deviations, which reflect the degree of uncertainty in
the prediction of the coefficients of expansion, have a strong effect on the probability criterion, and must be kept sufficiently small
for higher reliability of the package. In this connection, it is important to point out that it is advisable that materials vendors provide
information not only about the mean values of the coefficients of
expansion, but about their variances (standard deviations) as well.
This would enable a designer to establish the adequate allowable
stress level, so as not to compromise the package reliability.
Let us examine now several additional problems to illustrate the
effectiveness of the application of the probabilistic approach in
the thermal stress.
5. Some Other Problems
(15)
If the probability P of nonfailure is close to unity, then, one can
be confident that the normal stresses in the glass layer are compressive and do not exceed the allowable level σ*, while the shearing stresses (which concentrate at the ends of the glass layer) are
small and do not exceed the τ* = kh1σ* level.
Let, for instance, the elastic constants of the solder glass be E1
= 0.66 x 106 kg/cm2 and ν1 = 0.27, the seal temperature be 485°C, the
lowest (testing) temperature be -65°C (so that the change in temperature is ∆t = 550°C), the calculated “effective” coefficients of
expansion at this temperature be α1 = 6.75 × 10 −6 1 / $ C and
α 0 = 7.20 × 10 −6 1 / $ C , the standard deviations of these coefficients be D 0 = D1 = 0.25 × 10 −6 1 / $ C , and the (experimentally
obtained) compressive strength of the glass be σu = 5,500 kg/cm2.
With the safety factor of, say, η = 4, we have σ* = σu/4 = 1,375 kg/
cm2. The required level of the function ψ of nonfailure, as predicted by equation (10), is as follows,
Problem # 1. (The use of Bayes statistics). A bi-material
assembly is subjected to temperature change and, hence, experiences thermally induced loading. It has been predicted (for instance, on the basis of accelerated testing) that the probabilities
that the constituent materials, #1 and #2, will not fail during the
lifetime of the assembly are p1 and p2, respectively. The field
report indicated that the assembly failed. The details were not
reported, however, and are not available. What is the probability
that it was the material #1 that failed?
The following four hypotheses were possible prior to the obtained field report: H0 = {none of the materials will fail}; H1 =
{only the first material will fail}; H2 = {only the second material
will fail}; H3 = {both materials will fail}. The probabilities of
these hypotheses are as follows,
P0 = p1p2; P1 = (1-p1)p2; P2 = p1(1-p2); P3 = (1-p1)(1-p2) .
(1.1)
The conditional probabilities, associated with the observed event
A, “the assembly failed”, are as follows,
,
the mean value of this function is of the form,
P(A/H0) = 0,
P(A/H1) = P(A/H2) = P(A/H3) = 1
(1.2)
The Bayes formula,
,
(1.3)
and the variance is of the form,
-12 1
2
Dψ = D0 + D1 = 0.125 × 10 ( /°C) .
Then, the reliability indices, calculated by the formulas (15), are γ
The formula indicates, how the posterior probability, P(Hi/A), of
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the event A can be determined from the known prior probabilities,
P(Hi), and, hence, makes it possible to “revise” the probabilities of
the initial hypotheses on the basis of the new information.
Applying the formula (1.3), one can obtain the following expression for the probability that only the first material failed,
(1.4)
might be still faulty, but only in two cases out of a thousand.
Problem # 3. (This problem shows, how the probability distribution function for a quantity of interest can be obtained from
the known probability distribution functions for the quantities it
depends upon). A solder joint in a Flip Chip design of an integrated circuit (IC) package experiences thermally induced shear
strain due to the thermal expansion (contraction) mismatch of
the chip and the substrate materials. This strain can be assessed
by the following simplified formula,
If, for instance, p1 = 2p2 = p = 0.99 (the probability on nonfailure
of the first material is twice as large as the probability of nonfailure
of the second material), then,
If the probability of nonfailure is the same for both materials (p1
= p2 = p), then, with p = 0.99, one can have,
(3.1)
Here, ε = ∆α∆t is the thermal mismatch strain, ∆α is the difference
in the coefficients of thermal expansion (CTE) of the soldered components (the chip and the substrate), ∆t is the change in temperature, h is the solder bump’s height (stand-off), and l is the distance
of the bump from the center of the chip. The strain, ε, and the
stand-off, h, are normally distributed random variables with the
probability density functions,
(1.5)
For very reliable materials (p ~
= 1 ), the probability that only the
first material failed is P(H1/A) = 12 . On the other hand, for very
unreliable materials (p ~
= 0), this probability is zero, quite likely
that both materials failed.
Problem # 2. (Another example of the use of Bayes statistics).
The estimated probability that the thermally induced bow of a
Printed Circuit Board (PCB), manufactured at the given factory,
meets the specification requirement is p. A series of control tests
was carried out to determine whether the given batch of PCB’s
meets the specification. However, due to the test equipment employed, the test results are not absolutely certain. It has been
established that the probability that the tests give the correct answer is p1, and the probability of an erroneous answer is p2. The
given PCB passed the control tests. What is the probability that
the PCB meets the specification?
The probability of the hypothesis H1 = {the given PCB meets
the specification}, is P(H1) = p. The probability of the hypothesis
H2 = {the given PCB does not meet the specification}, is P(H2) =
1-p. If, in reality, the hypothesis H1 takes place, than the probability that the PCB passes the tests is, clearly, P(A/H1) = p1. If
the hypothesis H2 is fulfilled, then the probability that the PCB
passes the tests is P(A/H2) = p2. The posterior (“actual”) probability of the event A, “the PCB meets the specification”, as predicted by the Bayes formula (1.3), is follows,
.
,
(3.2)
respectively. Find the distribution of the shear strain, g (this
strain is deemed to be responsible for the reliability of the joint).
The distribution of the shear strain, g, can be found, based on the
formula (3.1), as follows,
,
(3.3)
where z = ε/h is the ratio of the random variables e and h. Using the
formula for the probability density function of the ratio of two
random variables18, one can obtain,
(3.4)
where the following notation is used,
.
(2.1)
The integral in the formula (3.4) can be represented as follows,
If, for instance, p = P(H1) = 0.96, P(A/H1) = p1 = 0.98, and P(A/H2) =
p2 = 0.05, then P(A) = 0.998. Thus, if the PCB passes the tests, it
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Thermal Stress Modeling In Microelectronics and Photonic Structures, and the Application of the
Probablistic Approach: Review and Extension
,
(3.10)
where
(3.5)
where the parameter β is expressed by the formula,
(3.11)
is the mean value of the shear (angular) strain and,
(3.6)
(3.12)
and
is its variance.
Note that the above distribution for the shear strain, g, could
have been obtained also directly from the probability density function, fε(ε), and the relationship ε = (h/l)γ. Indeed,
is the Laplace function (probability integral).
The variance, Dh, of the solder joint standoff, h, is typically
much smaller than the variance, Dε, of the thermal mismatch strain,
and therefore the formula (3.6) for the β value can be simplified as
follows,
(3.13)
(3.7)
The β value is rather large, since the mean value of the standoff is,
as a rule, substantially larger than its standard deviation. Then,
the following approximate formula can be used for the evaluation of the function Φ(β),
,
(3.8)
The expression (3.13) is similar to the formula (12) and the probability P that the strain γ can be found within the interval (0, γ*) can
be evaluated by the formula,
P = P{0 ≤ γ ≤ γ*} = Φ1(δ* - δ) + Φ1 (δ) - 1,
where the function Φ1(t) = erf t is expressed by the formula (14)
and the parameters δ and δ*,
and the expression (3.3) can be written as follows,
Problem # 4. (The application of the extreme value statistics).
An electronic component is operated in thermal cycling conditions for n cycles. Assuming that the amplitude R(t) of the maximum thermally induced stress, when a single cycle is applied, is
distributed in accordance with the Rayleigh law,
.
,
(4.1)
(3.9)
determine the most likely extreme stress value for a large n number.
The extreme response, Yn, expected during a certain number,
n, of observations is a random variable. Its probability density
function, g(yn), and the probability distribution function, G(yn),
can be obtained from the “basic” distributions, f(x) and F(x), for
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If only the randomness in the thermal expansion mismatch
were considered, then, substituting in the obtained expression Dh
= 0 and <h> = h, one would obtain,
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the random response, X, in the case of n=1, by the formulas18,
g(yn) = n{f(x)[F(x)}n-1}x=yn ,
(4.2)
G(yn) = {[F(x)]n.}x=yn .
(4.3)
From the initial (“basic”) distribution (4.1), using the formula
(4.2), one can obtain,
,
· Application of the probabilistic approach enables one to quantitatively assess the role of various uncertainties in the materials
properties, geometrical characteristics and loading conditions,
and, owing to that, to design and manufacture a viable and reliable product.
· Future work should include research aimed at the accumulation of the information about the probabilistic characteristics of
materials, technologies and anticipated loading conditions.
(4.4)
References
where
(4.5)
1. B. J. Aleck, “Thermal Stresses in a Rectangular Plate Clamped
Along an Edge”, ASME Journal of Applied Mechanics, Vol.
16, pp. 118-122, 1949.
and Dx is the variance of the ordinates X(t) of the induced stress.
2. F.-V. Chang, “Thermal Contact Stresses of Bi-Metal Strip
The condition,
Thermostat”, Applied Mathematics and Mechanics, Vol. 4,
No. 3, pp. 363-376, Tsing-hua University, Beijing, China,
g’(yn) = 0
(4.6)
1983.
3. W. T. Chen and C. W. Nelson, “Thermal Stresses in Bonded
yields the following,
Joints”, IBM Journal, Research and Development, Vol. 23,
No. 2, pp. 178-188, 1979.
4. J. W. Eischen, C. Chung, and J. H. Kim, “Realistic Modeling
(4.7)
of the Edge Effect Stresses in Bimaterial Elements”, ASME
Journal of Electronic Packaging, Vol. 112, No. 1, 1990.
In the case of large n, the second term in this equation is signifi5. H. H. Glascock and H. J. Webster, “Structural Copper: a Plicantly smaller than the first term, and the most likely value yn*
able High Conductance Material for Bonding to Silicon Power
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.
(4.8)
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As evident from the obtained formula, the most likely extreme
8.
A.
Y. Kuo, “Thermal Stress at the Edge of a Bi-Metallic Ther*
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stress D x in the case of single loading. If, for instance, n =
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n
IEEE
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793,
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© International Microelectronics And Packaging Society
Thermal Stress Modeling In Microelectronics and Photonic Structures, and the Application of the
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The International Journal of Microcircuits and Electronic Packaging, Volume 23, Number2, Second Quarter 2000 (ISSN 1063-1674)
© International Microelectronics And Packaging Society
223