Predicted thermal stress in a circular bonded assembly with identical adherends
Motivation
The objective of the analysis that follows is to develop an easy-to-use, simple, and physically meaningful analytical (“mathematical”) stress model for the evaluation of the thermally induced stresses in a circular adhesively bonded assembly with identical adherends. The assembly is fabricated at an elevated temperature and is subsequently cooled down to a low (e.g., room) temperature. Thermal stresses arise because of the different coefficients of thermal expansion (contraction) of the dissimilar materials of the two adherends witli the adhesive layer. The developed model can be helpful in stress-strain analyses and physical design of electronic and photonic assemblies of the type in question.
Assumptions
The following assumptions are used in our study.
- • The adherends can be treated as circular plates experiencing small deflections.
- • The engineering theory of bending of plates (see, for example, [51]) can be used to predict their mechanical behavior.
- • The “peeling” stresses (i.e., the interfacial normal stresses acting in the through-thickness direction of the assembly) do not affect the interfacial shearing stresses and therefore do not have to be accounted for when evaluating the shearing stresses.
- • The “peeling” stresses can be determined based on the evaluated shearing stresses.
- • The interfacial compliance of the assembly in its plane is due to the joint interfacial compliance
of the two adherends and the compliance
of the bonding layer [6]. In these formulas, h_{0} is the thickness of one of the
adherends, /г, is the thickness of the bonding layer, G_{0} =--— is the
l 2(1 +Vo)
shear modulus of the adherend material, G, =-is the shear modulus
2(1 + V|)
of the bonding material, E_{0} is the Young’s modulus of the adherend material, £, is the Young’s modulus of the bonding material, and v„ and v, are the Poisson’s ratios of the adherend and the bonding materials.
• The interfacial radial displacements n,(r) of the bonding material in the radial direction can be evaluated as the sum of the stress-free radial displacements c/.|A?r; the radial displacements u(r) due to the thermally induced forces caused by the thermal contraction mismatch of the bonding material and the material of the adherends; and the displacements K[T_{0}(r) of the interfacial point at the given radius r with respect to the displacements к(г) of the inner points of the cross section:
In this formula, м,(г) are the total interfacial radial displacements of the bonding material, a, is the coefficient of thermal expansion (contraction) of the bonding material, At is the change in temperature, r is the current radius, u(r) are the (stress-dependent) radial displacements of the bonding layer, x_{0}(r) is the interfacial shearing stress in the given cross section, and к, is the interfacial compliance of the bonding layer defined by formula (3.54).
- • The displacements u(r) in formula (3.55) can be evaluated based on the Hooke’s law, and are considered the same for all the points of the given circumferential cross section.
- • The third term in (3.55) considers the deviation of the given cross section from planarity: the interfacial radial displacements are somewhat larger than the displacements of the inner points of the cross section.
- • The interfacial radial displacements, u_{0}(r), in the adherends can be evaluated as
where a_{0} is the coefficient of thermal expansion (contraction) of the adherend material, т_{0}(r) is the interfacial shearing stress, and k_{()} is the interfacial compliance of the adherend. This compliance can be computed by formula (3.53).
- • The first term in the right part of formula (3.56) is the stress-free thermal contraction of the adherends. The second term is due to the interfacial shearing stress. This term reflects an assumption that the interfacial radial displacements of the adherends, caused by their interaction with the bonding layer, are proportional to the interfacial shearing stress in the given cross section and are not affected by the stresses and strains in the adjacent cross sections.
- • Formula (3.56) also reflects an assumption that the radial displacements of the inner portions of the given cross section of the adherend are not affected by the displacements of a substantially thinner and low modulus bonding layer.
Basic Equation
Formulas (3.55) and (3.56) for the interfacial radial displacements in the adherends and in the bond, and the condition u_{0}(r) = u,(r) of the compatibility of these displacements result in the following formula for the radial displacements in the bonding layer:
Here, Да = а, - а_{0} is the difference in the CTE (contraction) of the bonding material and the material of the adherends, and к = к_{0} + K| is the total interfacial compliance of the assembly. Expression (3.57) and the Cauchy formulas [54]
for the normal radial, e_{r} strains and the normal circumferential (tangential), e_{e}, strains yield:
The corresponding radial, a„ and circumferential, c_{e}, normal stresses in the bonding layer can be evaluated, using Hooke’s law equations [54]
as follows:
Since the thickness, h_{t}, of the bonding layer is small, and the interfacial shearing stress should be symmetric with respect to the horizontal midplane of the assembly, the gradient ^{1}— of the interfacial shearing stress, x_{r}., in the through-thickness
dz
direction, z, can be represented using the following approximate formula: Introducing formulas (3.61) and (3.62) into the equilibrium equation [54] we obtain the following basic differential equation for the shearing stress function,
where
is the parameter of the interfacial shearing stress, and
is the radial compliance of the bonding layer.
Solution to the Basic Equation
Equation (3.64) has the following solution:
where C_{0} and C, are the constants of integration, a is the assembly radius, к is the parameter of the interfacial shearing stress, and I_{0}(kr) and I_{t}(kr) are the modified Bessel functions of the first kind of zero and first order, respectively (see, for examples, [55, 56]). These functions obey the following rules of differentiation:
Introducing the sought solution (3.67), with consideration of formula (3.68), into the basic equation (3.64), we find that the following equation should be fulfilled for any radius, r.
Hence, one should put C_{0} = 0, and solution (3.67) can be simplified:
Note that the modified Bessel function I_{t}(kr) plays the role of the hyperbolic sine in the solution obtained earlier for an elongated rectangular plate [12].
Substituting solution (3.70) into the first equation (3.61), we obtain the following expression for the radial stress:
Since the edge r = a of the assembly is stress-free, the condition
should be fulfilled. Then we obtain:
and solution (3.70) results in the following formula for the interfacial shearing stress: where the maximum value
of this stress takes place at the end r = a of the assembly.
Large and/or Stiff Assemblies
With a small bond thickness, h_{{}, the radial in-plane compliance, X,, defined by formula (3.66), is large, and so is the parameter, k, of the interfacial shearing stress, expressed by formula (3.65). For large arguments z, the modified Bessel function of the order n can be evaluated by the approximate formula [55]:
Then, with
we obtain solution (3.73) in the form: where
is the maximum interfacial shearing stress. As evident from formula (3.78), the interfacial shearing stress, T_{0}(r), concentrates around a narrow peripheral ring, and is next to zero for the inner radii of the bonding layer (r «. a). Note that for large and/or stiff enough assemblies, the maximum interfacial shearing stress is assembly size independent.
Normal Stresses in the Bonding Layer
Introducing formula (3.74) for the interfacial shearing stress into formulas (3.61), we obtain the following expressions for the normal radial, o_{r}, and the normal circumferential, a_{e}, stresses in the bonding layer:
where
is the normal stress in the midportion of the bonding layer. Formula (3.80) defines the stress in a thin film fabricated on a thick substrate formed by the two adherends. The expressions in the brackets in formula (3.80) are, in effect, factors, which consider the role of the finite radius, a, of the assembly. These factors indicate the change in the normal stresses in the bonding layer, when the current radius r changes from zero to the radius a of the assembly.
In the case of a large size (large a values) and/or stiff (large к values) assemblies, formulas (3.80) can be simplified, considering the relationships (3.77), as follows:
Formulas (3.82) indicate that the normal stresses, o_{r} and o_{e}, in the bonding layer are uniformly distributed over the inner portion of the assembly (r
is by the factor of—!— lower than the stress, a,, in the inner portion of the film.
1 - V|
Bow
The assembly as a whole does not experience any bowing. Each of the adherends, however, can bow with respect to the horizontal midplane of the assembly. We seek the angles of rotation, w'(r), of the adherend cross sections in the form:
where Л, and A_{2} are thus far unknown constants. Then we obtain, by differentiation: and the radial bending moment, Af_{r}, acting on the adherend, can be evaluated as [51]
where
is the flexural rigidity of one of the adherends treated as a thin plate. On the other hand, the radial bending moment, M_{n} can be determined, based on the first formula in (3.79) for the radial normal stress, c_{r}, as follows:
In an approximate analysis, one can assume that Poisson’s ratio, v_{0}, of the adherend material in the expression in the brackets in formula (3.85) can be substituted with Poisson’s ratio, v_{h} of the material of the film. Then, comparing the expressions (3.84) and (3.85), we conclude that the constants A, and A_{2} can be evaluated by the formulas:
where
are the effective Young’s moduli of the adherend and the bonding materials, respectively.
Introducing formulas (3.88) into expression (3.84), we obtain the following formula for the adherend curvature:
The rotation angles, expressed formula (3.82), are:
The deflection function, w(r), can be found by integration:
where the constant of integration, A_{0}, is the displacement of the adherend as a non- deformable rigid body. Since we are interested in the elastic displacements only, this constant can be chosen in an arbitrary fashion. Choosing it, for instance, in such a way that w(a) = 0, one can obtain formula (3.92) in the form:
The maximum deflection takes place at the center of the substrate (/ =0) and is where
is the maximum deflection in the case of a large and/or stiff structure. The term in the brackets in formula (3.94) reflects the effect of the finite size of the assembly on the maximum deflection.
In the case of a large and/or a stiff assembly, formula (3.93) yields and the induced radial curvature can be evaluated as
This formula indicates that the thermally induced curvatures of the adherends are proportional to the ratio, £*I El, of the effective Young’s moduli of the adhesive and the adherend materials; to the thickness, h_{{}, of the bonding layer; and to the thermal mismatch strain А с/.At between the materials of the adhesive and the adherends; and is inversely proportional to the thickness, h_{0}, of the adherend squared. For thick enough adherends this curvature is next-to-zero.
Bending Stresses in the Adherends
The radial, M_{r}, and the circumferential, M_{0}, bending moments in the adherends can be evaluated, using formula (3.90) for the curvature and formula (3.91) for the angles of rotation, as follows:
The corresponding maximum bending stresses in the adherends are
where the stress a, is expressed by formula (3.81). For sufficiently large and/or stiff assemblies,
Comparing these formulas with formula (3.82) for the normal stresses in the bonding material, we conclude that, in an approximate analysis, the normal bending stresses in the adherends can be assumed to be proportional to the corresponding in-plane normal stresses in the bonding layer (this layer does not experience, of course, any bending), at the same current radius, r, and can be obtained by multiplying these
stresses by the reduction factor of 3y-.
'to
Peeling Stress
Since the bonding layer does not experience bending, the total radial bending moment acting in its cross sections must be zero. The lateral load, q_{Q}(r), acting on the bond, is due to the interfacial “peeling” stress, p_{0}(x), and the interfacial shearing stress, T_{0}(r), and can be found as
The total bending moment will be zero, if this load is zero, such as if where
is the “peeling” stress at the edge r = a. Considering formula (3.75) for the maximum shearing stress, T_{max}, one can write formula (3.102) as follows:
In the case of sufficiently large and/or stiff assemblies, formulas (3.102), (3.103) and (3.104) yield:
The second formula in (3.104) indicates that for thin enough bonding layers, the maximum peeling stress can be very low, even if the factor of the interfacial shearing stress is significant.
Numerical Example
Input Data
Component |
Adherends |
Adhesive |
Young’s modulus |
E„ = 7384 kg/mm^{2} |
E_{t} = 500 kg/mm^{2} |
Poisson’s ratios |
v_{0} = 0.25 |
v, =0.45 |
CTE |
a_{0} = 0.5xl0^{_6}l/°C |
a, =60.5x10 ^{6}1/°C |
Thickness |
A„= 1.5 mm |
h, = 0.05 mm |
Assembly radius a = 50.8 mm; Temperature Change Al = 40°C
Calculated Data
Axial compliance of the bonding layer: /1, = -—— = ——— = 0.0319mm/kg ^{y} Eh 500x0.05
Interfacial compliance of the two adherends:
Interfacial compliance of the bonding layer:
Total к = Ко + к, = 3.3857 x 1 O'^{4} + 0.2417 x 10^{-4} = 3.6274 x 1 O'^{4} mm^{J} /kg
Parameter of the interfacial shearing stress: к = = /—°'^{0638}— = 13.2621 mm^{4}
V к V 3.6274 xlO~*
Parameter ka = 13.2621 x 50.8 = 673.7152 is significant, and therefore simplified formulas for large and stiff assemblies can be used to evaluate stresses.
Maximum interfacial shearing stress:
Maximum longitudinal displacement (at the meniscus at the assembly edge)
Normal tensile stress in the mid-portion of the assembly
Effective Young’s moduli
Maximum deflections of the adherends:
kh
Maximum peeling stress: p_{mM} = —^-X_{max} = 0.3316x0.7234 = 0.2398kg/mm^{2}
Thus, a simple, easy-to-apply and physically meaningful analytical (“mathematical”) stress model is developed for the prediction of the stresses in a circular adhesively bonded assembly with identical adherends. The developed model can be helpful for stress-strain analyses and physical design of electronic and photonic assemblies of the type in question, and particularly holographic memory assemblies.
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