Bonded Tri-Material Specimen Subjected to Shear-Off Testing: Predicted Interfacial Stresses

A physically meaningful and easy-to-use analytical model based on the concept of the interfacial compliance is developed for the evaluation of the interfacial shearing stresses at failure from the measured shear off force. The level of these stresses characterizes the adhesive strength of the bonding material of interest. A tri-material test specimen rigidly attached to an ideally strong immovable base is considered. The shear-off force is applied to its upper (“free”) component. It is assumed that the shear-off mechanical stresses are significantly higher than the residual thermally induced stresses, and therefore the latter do not have to be considered. The objective of the analysis is to determine the interfacial strength of the bonding material. It is shown that the distributions of the shearing stresses at the interfaces of this material with the bonded components mimic the shearing stress at the immovable base. The general concept is illustrated by a numerical example carried out for electronic silicon-copper assembly bonded using sintered silver. In this example the predicted stresses are higher at the interface of the bonding material with the upper, “free”, component than at its interface with the component attached to the immovable base. The suggested model can be used when there is an intent to select, during product development testing, the most feasible bonding material from the standpoint of its interfacial strength. Future work should include, first of all, experimental data and finite-element confirmation of the analytical suggested analytical model. Citation: Suhir E (2020) Bonded Tri-Material Specimen Subjected to Shear-Off Testing: Predicted Interfacial Stresses. J Aerosp Eng Mech 4(1):201-205 Suhir. Dermatol Arch 2020, 4(1):201-205 Open Access | Page 202 | this parameter is zero. The shearing stress τ2(x) can be found from (1) by differentiation: ( ) ( ) ( ) 1 2 = = cosh + sinh x T x k C kx C kx τ ′ (2) For large enough x values one can put cosh sinh kx kx ≈ . It is clear also that the shearing stress is zero for cross-sections remote from the left end of the assembly, where the force Ť is applied. Then the equation (2) yields: C1 = -C2, and the formulas (1) and (2) result in the following relationships for the total axial force acting in the assembly cross-sections and for the corresponding shearing stress: ( ) ( ) 0 1 = + sinh cosh T x C C kx kx , ( ) ( ) ( ) 1 = = cosh sinh x T x kC kx kx τ ′ (3) The force T(x) should satisfy the boundary conditions: ( ) 0 = T T 


Introduction
The objective of the analysis that follows is to develop a simple, physically meaningful and easy-to-use analytical model for the prediction of stresses from the measured shear-off force. This is a typical product-development test in microelectronic packaging engineering (see, e.g., [1,2]). The model is based on the concept of the interfacial compliance. This concept was first applied in [3] and recently addressed in detail in [4]. The numerical example is carried out for the case of a sintered silver bond [5][6][7][8][9][10][11][12][13][14]. Sintered silver is, as is known, a good candidate for die bonding as an alternative to lead alloys. Little is known, however, about its mechanical properties, and shear off testing can shed important light on the macroscopic characteristics of this material and, first of all, on its bonding strength. It should be emphasized that actual experimentation, which is considered as future work, should be based on the data obtained on the basis of the suggested analytical model. It is desirable; of course, that the data obtained using the developed model is confirmed by finite-element-analysis (FEA). Another point that should be made in connection with the suggested model and the numerical analysis below is that the suggested tri-material body considers that the axial compliance of the adhesive layer (zero compo-nent in Figure 1) does not have to be by orders of magnitude larger than the axial compliances of the two other components (in our numerical analysis this compliance is about 7-9 times larger than the compliances of the two other components). This means that the suggested model can be used to decide if a simplified bi-material model (of the type suggested in Ref.3 in application to thermal stresses) could be used in the addressed problem.

Analysis
The distributed longitudinal force T(x) acting in the cross-sections of the tri-material test-specimen (assembly) in

Abstract
A physically meaningful and easy-to-use analytical model based on the concept of the interfacial compliance is developed for the evaluation of the interfacial shearing stresses at failure from the measured shear off force. The level of these stresses characterizes the adhesive strength of the bonding material of interest. A tri-material test specimen rigidly attached to an ideally strong immovable base is considered. The shear-off force is applied to its upper ("free") component. It is assumed that the shear-off mechanical stresses are significantly higher than the residual thermally induced stresses, and therefore the latter do not have to be considered. The objective of the analysis is to determine the interfacial strength of the bonding material. It is shown that the distributions of the shearing stresses at the interfaces of this material with the bonded components mimic the shearing stress at the immovable base. The general concept is illustrated by a numerical example carried out for electronic silicon-copper assembly bonded using sintered silver. In this example the predicted stresses are higher at the interface of the bonding material with the upper, "free", component than at its interface with the component attached to the immovable base. The suggested model can be used when there is an intent to select, during product development testing, the most feasible bonding material from the standpoint of its interfacial strength. Future work should include, first of all, experimental data and finite-element confirmation of the analytical suggested analytical model. this parameter is zero.
The shearing stress τ 2 (x) can be found from (1) by differentiation: For large enough x values one can put cosh . It is clear also that the shearing stress is zero for cross-sections remote from the left end of the assembly, where the force Ť is applied. Then the equation (2) yields: C 1 = -C 2 , and the formulas (1) and (2) result in the following relationships for the total axial force acting in the assembly cross-sections and for the corresponding shearing stress: The force T(x) should satisfy the boundary conditions: Then the expression (1) results in the following equations for the constants C 0 and C 1 : so that The maximum shearing stress takes place at the origin: Figure 1 can be sought in the form where k is the parameter of the sought shearing stress. This parameter is loading independent and, for a stiff enough assembly that does not experience bending deformation can be evaluated as follows [15].
where the following notation is used: The partial parameters k 01 and k 02 refer to bi-material assemblies comprised of "zero" (bonding) and #1 components or of "zero" and #2 components; λ 0 , λ 1 and λ 2 are axial compliances of the assembly components; к 0 , к 1 and к 2 are their interfacial compliances; G 0 , G 1 and G 2 are the shear moduli of the component materials; E 0 , E 1 and E 2 are their Young's moduli; v 0 and v 1 are their Poisson's ratios; and δ is the parameter that characterizes the role of the relative axial compliances of the assembly components: when the "zero" component is considerably more compliant than the two outer components, this parameter is equal to 1. If all the three components have the same axial compliance, this parameter is equal to 0.25. When the "zero" component is very stiff, Bonding layer (material under test, Shear-off force Immovable base Figure 1: Tri-material test specimen subjected to shear-off force. with the "zero" component; T 0 (x), T 1 (x) and T 2 (x) are the forces acting in the cross-sections of the assembly components; τ 0 (x) is the shearing stress acting at the interface of the "zero" component with the component #2; τ 1 (x) is the shearing stress acting at the interface of the "zero" component with the component #1; terfacial compliance of the "zero" component;  This stress changes from infinity to -kŤ, when the length L of the specimen changes from zero to a large enough value.
Let us determine now the shearing stresses acting at oth-er interfaces of the assembly.
The longitudinal interfacial displacements can be sought, in accordance with the concept of the interfacial compliance [3], as Where u 01 (x) is the displacement of the "zero" compo-nent at its interface with the component #1; u 10 (x) is the dis-placement of the component #1 at its interface with the "zero" component; u 02 (x) is the displacement of the "zero" component at its interface with the component #2; u 20 (x) is the displacement of the component #2 at its interface Here some of the notations (3) were used. As one could see from the solutions (17)  compliances of the components. These formulas consider the two-dimensional state of stress.
The first terms in the formulas (9) are based on the Hooke's law and reflect an assumption that the longitudinal (axial) displacements are uniformly distributed over the cross-sections of the given assembly component. The second terms are corrections to this assumption. They consider that the interfacial displacements are somewhat larger than the displacements of the inner points of the cross-sections (Figure 2).
The structure of these terms reflects an assumption that the corrections of interest can be sought as products of the stress-independent interfacial compliances and thus far unknown interfacial shearing stresses acting in this cross-section.
The conditions u 01 (x) = u 10 (x) and u 02 (x) = u 20 (x) of the displacement compatibility result in the equations: Obviously, Then the equations (10) results in following system of equations: The equations of equilibrium require that ( ) ( ) and therefore ( ) ( ) Then the equations (12) Assuming that a similar relationship holds for the sought shearing stresses τ 0 (x) and τ 1 (x) the following system of algebraic equations for the shearing stress functions τ 0 (x) and τ 1 (x) could be obtained: