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We have investigated the possible oscillation of linear and non-aging viscoelastic functions of viscoelastic models whose governing differential equations representing the constitutive relation between stresses and strains are of second order. We have found that the Stieltjes inverse of the time-dependent Poisson ratio could be oscillatory for engineering materials whose bulk and shear relaxation moduli are monotonically decreasing.

Through sequential coupling of the differential equations for interfacial shearing and peeling stresses, concise closed-form strength-of-material solutions for these stresses in symmetric and non-symmetric sandwich structures subjected to the combined loadings of temperature, stretching, and bending have been developed. The free-edge effects of the interfacial shearing stress, the maximum magnitude of the interfacial shear stress, and the location of its occurrence are accurately modelled using a single high-frequency hyperbolic function. The analytical solutions have been validated against finite element analysis (FEA) solutions using four sandwich structures: one symmetric, two mildly asymmetric, and another severely asymmetric; and for three load cases: differential thermal expansion, differential mechanical stretching, and mechanical bending. The maximum deviation between the analytical and the FEA solutions for the interfacial shearing and peeling stresses for all the structures and load cases, including that for the severely asymmetric sandwich structure, was less than 15%.

A two-step solutions is used to model the stresses in the discrete joints of a sandwiched structure due to mismatched thermal expansion. In the first step, the discrete joints are modelled as elastic body with nil lateral stiffness, the stretch and shear stiffness of which are equivalent to that of the solder joints smearing across the entire layer. Through the introduction of a high-order frequency function, interfacial shear and peeling stresses with the correct free-edge effects are derived. In the second step, the interfacial stresses are integrated into forces and are applied to the discrete joints to obtain the mean stresses; this is followed by modification of the mean shear stress using the high-order frequency function to impose the nil-shear-stress condition at the free-edge of the discrete joints. The closed-form solutions have found good agreements with the numerical solutions.

The existing analytical solutions for the peeling and shearing stresses in polymeric adhesively bonded structures are either too inaccurate or too complex for adoption by practicing engineers. This manuscript presents a closed-form solution that is reasonably accurate yet simple and concise enough to be adopted by practicing engineers for design analysis and exploration. Analysis of these concise solutions have yielded insightful design guidelines: (i) the magnitude of peeling stress is generally higher than that of shearing stress; (ii) the peeling stress in a balanced structure may be reduced most effectively by reducing the elastic modulus of the adherends or by increasing the adhesive-to-adherend thickness ratio and less effectively by reducing the elastic modulus of the adhesive; and (iii) the peeling stress in an unbalanced structure may be reduced by increasing the in-plane compliance of the structure, which may be implemented most effectively by reducing the thicknesses of the adherends and less effectively by reducing the elastic modulus of the adherends. Published version

A new representation for viscoelastic functions, the tensor relaxation-creep duality representation, is introduced. The derivation of a tensor time-differential constitutive equation for anisotropic viscoelastic materials using this new presentation is presented. The relaxation-creep duality characteristic ingrained in the new representation enables the interconversion of viscoelastic functions, which is not possible with the conventional Prony series representation of viscoelastic functions. The new representation therefore offers a better representation of the physics of viscoelasticity leading to a reduced number of viscoelastic parameters required to describe a viscoelastic function. The new representation has been demonstrated on two anisotropic viscoelastic crystallographic systems: (i) the symmetric systems with material- and time-independent eigenvectors and (ii) the symmetric systems with material-dependent but time-independent eigenvectors.

The low-cycle-fatigue characteristics of solder joints, formed by reflowing Sn98.8/Ag1.0/Cu0.1/In0.05/Ni0.02 solder over electroless nickel immersion gold-plated copper pads, were investigated by dynamic cyclic bending of printed circuit boards (PCBs). The PCB strain amplitudes were varied from 1.2 × 10−3 to 2.4 × 10−3 and the flexural frequencies ranged from 30 Hz to 150 Hz, to simulate drop impact-induced PCB resonant frequencies. A trend of drastically decreasing fatigue life with cyclic frequency was observed, in contrast with previous reports indicating the reverse; this is attributed to the different failure mechanisms activated. A systematic procedure involving optimization followed by transformation was used to condense the strain–frequency–life data into a master curve expressed in strain–life space.

The monotonicity of the linear viscoelastic functions, namely, the shear creep compliance, the Young’s relaxation modulus, the stretch creep compliance, the P-wave relaxation modulus, the Lamé’s first function, and the time-dependent Poisson’s ratio, were examined analytically and numerically. It was shown that both the Lamé’s first function and time-dependent Poisson’s ratio can be non-monotonic. Furthermore, in contrast to the reports by other researchers, the values of the time-dependent Poisson’s ratio were found to be bounded by the limits between −1 and 0.5 after the physical constraints of the bulk and shear relaxation moduli are taken into account.

The increasing occurrence of drop-impact failure of portable electronics could be traced to the failure of the solder joints that interconnect the integrated circuit (IC) components to the printed circuit board (PCB)—collectively referred to as PCB assembly. The drop impact leads to bending of the PCB assembly within the portable electronics, and the interconnecting solder joints undergo severe deformation to accommodate the differential bending deformation between the IC component and the PCB. This manuscript presents an approximated closed-form analytical solution for the stresses in the solder joints of the PCB assembly subjected to mechanical bending. The PCB assembly is modelled as a tri-layer structure in which the IC component and the PCB are modelled as beams or plates and the solder joints as a continuous layer consisting of infinite number of beams that are capable of carrying normal force, shear force, and moment along its interfaces with the IC component and the PCB. The analytical solutions have been validated against finite element analysis. Design analysis has suggested that the robustness of the PCB assembly against mechanical bending can be increased by increasing the diameter of the solder joints; increasing the in-plane and flexural compliances of IC component and PCB; while reducing the shear and flexural stiffness of the solder joints and reducing the transverse compliant of the PCB assembly.

The stress–strain properties of eutectic Sn-Pb and lead-free solders at strain rates between 0.1 s−1 and 300 s−1 are required to support finite-element modeling of the solder joints during board-level mechanical shock and product-level drop-impact testing. However, there is very limited data in this range because this is beyond the limit of conventional mechanical testing and below the limit of the split Hopkinson pressure bar test method. In this paper, a specialized drop-weight test was developed and, together with a conventional mechanical tester, the true stress–strain properties of four solder alloys (63Sn-37Pb, Sn-1.0Ag-0.1Cu, Sn-3.5Ag, and Sn-3.0Ag-0.5Cu) were generated for strain rates in the range from 0.005 s−1 to 300 s−1. The sensitivity of the solders was found to be independent of strain level but to increase with increased strain rate. The Sn-3.5Ag and the Sn-3.0Ag-0.5Cu solders exhibited not only higher flow stress at relatively low strain rate but, compared to Sn-37Pb, both also exhibited higher rate sensitivity that contributes to the weakness of these two lead-free solder joints when subjected to drop impact loading.

Non-conductive adhesive (NCA) flip-chip interconnects are emerging as an attractive alternative to lead or lead-free solder interconnects due to their environmental friendliness, lower processing temperatures, and extendability to fine-pitch applications. The electrical connectivity of an NCA interconnect relies solely on the pure mechanical contact between the integrated circuit (IC) bump and the substrate pad; the electrical conductivity of the contact depends on the mechanical contact pressure, which in turns depends to a large extent on the cure shrinkage characteristics of the NCA. Therefore, it is necessary to monitor the evolution of the electrical conductivity which could reflect the impact of cure- and thermal-induced stresses during the curing and cooling process, respectively. In this article, in situ measurement of the development of contact resistance during the bonding process of test chips was developed by using a mechanical tester combined with a four-wire resistance measurement system. A drop of resistance induced by the cure stress during the bonding process is clearly observed. With decreasing bonding temperature, the drop of contact resistance induced by cure shrinkage becomes larger, while the cooling-induced drop of resistance becomes smaller. The evolution of contact resistance agrees well with experimental observations of cure stress build-up. It is found that vitrification transformation during the curing of the adhesive could lead to a large cure stress and result in the reduction of the contact resistance. Furthermore, no obvious changes were observed when the applied load was removed at the end of bonding.

The conductivity of a nonconductive adhesive (NCA) flip chip interconnect is completely dependent on the direct mechanical contact between the integrated circuit (IC) bump and substrate pad. Cure shrinkage of NCA is critical for the formation of the final contact force in the contacts. However, measurement of the cure shrinkage during cross-linking reaction is fairly difficult. This paper introduces a new, yet simple, approach to measure cure shrinkage of adhesives using a thermo-mechanical analyzer. Isothermal studies of shrinkage change as a function of curing show four distinct regions. First, the thickness of the epoxy decreases due to decreasing viscosity and applied load, followed by a stage where the dimension change is constant as the cross-linking reaction is yet to set in. Once cross-linking begins, the shrinkage reaches a maximum followed by a plateau where the cross-linking reaction has completed. Sharp changes of the slope of cure shrinkage versus degree of cure were observed to coincide with gelation and vitrification. After gelation, a linear relationship between the cure shrinkage and degree of cure was observed to extend until the occurrence of vitrification, which quenches the cross-linking reaction. Applied load in the range of 0.05 N was found to be optimal to minimize measurement errors.

New quadrilateral finite elements capable of continuous modelling of crack propagation are presented. The crack extension is modelled by moving the mid-node on the side of the element that coincides with the crack plane in the direction, and with the velocity of the crack propagation. Both Serendipity and Lagrangian elements are generated and prescribed with either parabolic or elliptic displacement fields, the latter capable of generating singularities. Remarkably accurate stress intensity factors for a large range of crack length/specimen width ratios are obtained using the Lagrangian elements with elliptic displacement fields. These elements have been found to be very successful in the modelling of dynamic crack propagation.

Moving node finite elements are useful in determining the stress intensity factors using strain energy concepts; in this paper the material kinetic energy is included so that the effects of the material inertia during dynamic crack propagation can be assessed. This is used to determine crack propagation speeds and these compare favourably with those quoted in the literature.