2-1. Introduction
Based on some aforementioned facts, the design structure involves adhesively adhesive joints, and IC chip as well as blue tape stuck together by adhesive. Strictly speaking, in IC chip pick-up process, an adhesively bonded joint includes two adherends – the IC chip (upper adherend) and the blue tape (lower adherend) bonded by an adhesive in the IC chip pick-up process. In the published articles, many methods have been applied to solve the problems of adhesively bonded joints. Generally speaking, there are mainly several basic approaches, such as finite element method (FEM), numerical method and analytical method, which are often employed to solve the problems of adhesively bonded joints. These approaches are also applied to the following literature and will be discussed in the next sections.
2-2. Adhesively Bonded Joints
2-2-1. Introduction
Adhesively bonded single-lap joints have been widely studied since the 1950s. One of the most widely quoted papers on stresses in adhesive joints is that of Goland and Reissner [2].
Goland and Reissner have developed the cemented-lap mathematical model and found the explicit solutions (closed-form solutions) to two limiting cases. One case is that the cement layer must be so thin that its effect on the flexibility of the joint may be neglected; the other case is that the joint flexibility results mainly from that of the cement layer.
Some studies that have used and extended the Goland-Reissner theory and have compared their own results with Goland-Reissner’s are described below. Oplinger [3] has released the limit of large adherend-to-adhesive layer thickness ratio to obtain the results of the
Goland-Reissner analysis. Oplinger’s model should give the most accurate results for any overlap joint length because the edge moment expression was obtained by considering the large deflections of all the components of the single lap joint structure. Carpenter [4] has verified the correctness of Goland-Reissner’s formulations by making comparisons between his finite element results and the results of Goland-Reissner’s original equations. Ojalvo and Eidinoff [5] have used a more complete shear-strain/displacement equation to solve the single-lap adhesive joints. They explained that the shear stress is the highest value at two anti-symmetrical adherend-bound interface points of the layer; the growth of joint failures originating from these points are consistent with the results obtained from actual experiments.
Carpenter [6] summarized the theories of lap joint behavior of Goland and Reissner and of Ojalvo and Eidinoff’s equilibrium of a unit width differential element in the adherend-adhesive layer.
2-2-2. Thermal Loading
Stress distributions of adhesive joint affected by thermal variation are often studied. Suhir [7-9] have investigated thermal stress in an adhesive layer subjected to temperature variation for many years. First, he [7] obtained the distribution of the stresses in the interface of the thermostat bi-metal plate subjected to uniform heating or cooling. Next, in both the longitudinal and the transverse interfacial compliances of the thermostat strips subjected to thermal or external loading, he [8] found the interfacial stresses by using the elementary beam theory. Finally, he [9] developed the thermal stress analysis model in a piecewise continuous adhesive layer. These stresses are yielded by the thermal expansion (contraction) mismatch between adhesive material and the material of adherends. In addition, Rossettos [10]
investigated thermal stresses of a single lap joint with identical adherends subjected to temperature changes.
2-2-3. Anisotropic and Orthotropic Materials
The effects of various materials on stress distribution of the joint are discussed in the following. Some authors have treated both the adherend and adhesive materials as anisotropic and orthotropic by using either a finite element analysis or theoretical analysis. Wah [11], who found stress distribution in a lap joint, considered the adherends to be anisotropic whereas the cement was treated as an isotropic material. Renton and Vinson [12] developed a mathematical model of composite materials and formulated methods of analysis for determining the behaviors of single-lap joints with orthotropic adherends.
2-2-4. Non-linear FEM and External Loading
Tsai and Morton [13] analyzed the single-lap joint by using a two-dimensional geometrically non-linear finite element and made comparisons between the solutions of FEM and those of the theoretical analysis. They analyzed the influence of large deflections of the overlap joint on the computation of the edge moments. They concluded that the influence of the deflections on the edge moments is negligible if the joint is short.
Subsequently, Luo and Tong [14] applied linear and higher order displacement theories to stress analysis of thick adhesive and validated their results through two-dimensional finite element analysis. In addition, Allman [15] stated that the elastic stresses are obtained in adhesive bonded lap joints subjected to bending, stretching and shearing of the adherends and that the effects of the shearing and tearing actions were accounted for on the stresses of the adhesive layer. Allman produced a model that allows linear variation of the peel stress through the adhesive thickness. The comparisons between analytical results and experimental data were displayed. Additionally, single-lap adhesive joints of dissimilar adherends have
been subjected to external bending moments and tensile loads [16-17], and a single-lap joint subjected to tension loading and moments induced by geometric eccentricity was studied using the finite element method [18].
2-2-5. Plastic Behavior of Adhesive Joints
Some studies have investigated the plastic behavior in adhesive joints using FEM and analytical methods; for example, a recent elastoplastic stress analysis of a single-lap joint subjected to bending moment was carried out using the finite element method [19]. The significant effects of adherend thickness and overlap length on the joint’s strength were observed. Early on, Chen and Cheng [20] analyzed an adhesively bonded single-lap joint by minimizing the functional of the variational principle of complementary energy. Subsequently, Alexandrov and Richmond [21] addressed the approaching methods to solve three-dimensional, kinematically admissible velocity fields in a flat layer of an ideally rigid plastic material subjected to tension, while Mortensen and Thomsen [22] applied the multi-segment method of integration to solve the multiple-point boundary value problem.
2-2-6. Crack Analysis and Stress Singularity
When subjected to loading or thermal loading, debonding or failure may occur at different locations in the adhesive joint. The fracture of the adhesive joint often occurs in the interface;
that is to say, debonding occurs between the adhesive and the adherent. At the scale of engineering structures, many systems are built by adhesively bonding different components, and the mechanical failure of such systems often occurs because of the failure of the bonded interfaces [23].
In fact, the stress concentrations at critical regions such as adherend-adhesive interfaces or
the fillet of an adhesive joint can be a source of damage due to interfacial shear and transverse normal stresses. Some researchers, for instance, Gleich et al. [24], Qiao and Wang [25] and Qian and Akisanya[26], have addressed cracks resulting in failure or the stress singularity in the fillet of an adhesive joint.
2-2-7. Strengthening Structures
Adhesively bonded joints are also applied to strengthen structure. Some technical studies have presented that a structure is strengthened by adhesively bonding the steel plates to the tension face of the beam [27–28]. Li et al. [27] have shown the influence of the adhesive thickness and the steel plate thickness on the behavior of strengthened concrete beam.
Taljsten [28] derived the shear and peel stresses in the adhesive layer of beam bonding by a strengthening plate whose bending stiffness was neglected. That is to say, the bending moment of the plate is neglected when the shear stress of the adhesive and the strain of the plate were during derivation. Nevertheless, the plate really had the bending moment when the peel stress of the adhesive was formulated. He simplified this issue and made it easy to be solved. However, Cornell [29], who claimed that obtaining complete theoretical solutions to this problem would be very difficult, only considered a cantilever beam consisting of the same adherends. Only if the characteristic solutions of these equations have appropriately large values can his method produce classical solutions for the differential equations.
2-3. Genetic Algorithm and Penalty Function Method
2-3-1. Introduction
Genetic algorithms are used in search and optimization, such as finding the maximum (minimum) of a function over some domain space. Genetic algorithms are less susceptible to
getting 'stuck' at local optima than gradient search methods. But they tend to be computationally expensive. Genetic algorithm with penalty function is adopted by this study because the geometrical dimensions and material properties of adhesive and adherends deeply affect stress distributions of the adhesively bonded joint in IC chip pick-up process (see Figs.
3.10 and 3.16); and choosing the most suitable adhesive among numerous types of adhesive is difficult.
2-3-2. Penalty Function Method without Any Penalty Parameters
Some authors employed genetic algorithms (GAs) and the penalty function method which does not require any penalty parameter to solve real-world search and optimization problems involving inequality and/or equality constraints.
Deb [32], for example, devised a penalty function approach by using the approach of making pair-wise comparison in a tournament selection operator. Lin and Wu [33-34]
proposed a selforganizing adaptive penalty function strategy (SOAPS) without penalty parameters, and provided a robust and efficient means for constrained genetic searches but its performance occasionally fails to reach the expectation on some highly constrained problems.
Also, SOAPS also often failed to attain the optimum when the optimization problems involve equality constraints. Subsequently, They developed a new generation of the self-organizing adaptive penalty function strategy (SOAPSII) that can be effectively applied to diverse problems with inequality and equality constraints genetic algorithms. Nanakorn and Meesomklin [35] developed a new penalty scheme that is free from the disadvantages. Those disadvantages of most penalty schemes have included that (1) some coefficients of penalty function had to be specified at the beginning of the calculation, (2) the coefficients usually had no clear physical meanings, and (3) furthermore, appropriate values of the coefficients were estimated even by experience. Nevertheless, their penalty function was able to adjust
itself during the evolution so that the desired degree of penalty was always obtained. The coefficient of their penalty scheme had a clear physical meaning.
2-3-3. Adaptive Search Techniques
Some penalty schemes and adaptive search techniques are proposed to improve the efficiency of genetic algorithm. Barbosa and Lemonge [36] proposed a parameter-less adaptive penalty scheme for genetic algorithms applied to constrained optimization problems.
They examined the performance of this scheme by using test problems from the related literature and constrained optimization problems of structural engineering. Coit and Smith [37]
presented a penalty guided genetic algorithm which identified a final, feasible optimal, or near optimal solution in effective and efficient search of promising feasible and infeasible regions of reliability optimization with the highly constrained nature. Their proposed penalty function was adaptive and responds to the search history. Bullock et al. [38] presented that increasingly efficient and cost effective hybrid approaches incorporate an adaptive search and knowledge-based techniques of genetic algorithm, and outlined design sensitivity. Hasancebi and Erbatur [39] have obtained a better efficiency of GAs by developing two new crossover techniques.
Comparative results are fully discussed between the proposed and the common crossover techniques.
The other technique methods improving genetic algorithm were also listed some literatures here. Kwon et al. [40] proposed a successive zooming genetic algorithm (SZGA) for identifying global solutions by using continuous zooming factors. The algorithm was that the search space was zoomed around the design point with the best fitness per 100 generations and compared with a simple genetic algorithm and a micro-genetic algorithm for their ability to minimize multi-modal continuous functions and simple continuous functions. The results showed that the SZGA significantly improved the ability of a GA to identify a precise global
minimum and identified a more exact optimum value than the conventional GAs. Wu and Chow [41] applied genetic algorithms to a constrained nonlinear optimization problem with a mix of discrete sizing and continuous configuration variables. The discrete sizing variable was formed by mapping relationships between binary digit strings and discrete values by the medium or unsigned decimal integers.
2-3-4. Genetic Algorithm Application to Adhesively Bonded Joints
Genetic algorithm was applied to the subjects related to the studies of adhesively bonded joints. Govindaraj and Ramasamy [42] applied Genetic Algorithms to optimize the design of reinforced concrete continuous beams, which satisfied the strength, serviceability, ductility, durability and other constraints. Their optimum design considered the cross-sectional dimensions of the beam alone as the design variables and design results are compared with those in the available and related literature. Cho and Rhee [43] optimized the maximum interlaminar stresses of laminated composites with free edges under extension, bending, and twisting loads by using genetic algorithm (GA) in which a repair strategy was adopted to satisfy given constraints. Moreover, uncertainties were taken into account in lightweight design of laminated composite structures.
2-4. Methods Applied to Solve the Issues of Adhesively Bonded Joints
Three basic approaches presented by the aforementioned literatures including direct numerical method, finite element method (FEM) and analytical method are often employed to solve the issues of adhesively bonded joint. These approaches are discussed as follows.
In the first approach, solutions of differential equations with boundary conditions are
obtained by iteration methods or finite difference methods. Nevertheless, the use of numerical methods in real applications is under many limitations because these methods are based on a very limited number of geometries. Furthermore, it is easily divergent to solve the coupled differential equations by using direct numerical method.
The second approach employs finite element method which is widely used in many scientific and engineering fields including fluid flow, heat conduction, and structural analysis.
The finite element method is often applied to the determination of stresses in adhesively bonded joint structures. The continuum model is firstly discretized and represented by a discrete model. (i.e. a discretization procedure is to divide the structure into small parts and to formulate the model of each one of these parts and then to re-assemble those small parts to model the whole structure.) Subsequently, a system of algebraic equations is derived, commonly from energy functionals. Consequently, no general expressions are obtained for the solution and, therefore, stresses are given at specific points, such as Gauss points. The rapid development of computers has made the use of numerical techniques more appealing and feasible. Finite element methods can be used to analyze models with arbitrary geometries and loading conditions. They are suitable for the analysis of structures comprised of different materials. However, if one dimension value in the geometrical model is much greater than the others (i.e. dimension values have the great differences in the geometrical shape), numerical solutions (such as values of peel and shear stresses) become much more difficult to be accurately achieved by FEM because of the mesh problem. In other words, it is a little bit difficult to generate the finer mesh of adhesive and adherends if either the ratio of adhesive’s thickness to joint length or the ratio of adherend’s thickness to joint length is very large.
Additionally, because the stresses of this joint are obtained more accurate solutions of FEM, the much finer mesh is required. However, if this joint with the much finer mesh is accurately solved, much more CPU run time of the computer is required and taken.
In the last approach, a set of differential equations and boundary conditions is formulated.
The solutions of these equations are analytical expressions which give values of stresses at any point of the joint. Analytical solutions (closed-form solutions), such as those presented here for single lap joint, provide a good insight into the behavior of adhesively bonded joints.
They are also useful for analysis and planning of tests and for parametric analysis which can lead to the establishment of design criteria. However, the use of the method in real applications is very much limited because they are based on restrictive assumptions and a very limited number of geometries. In addition, the closed-form solutions are difficult to be found. Especially, as the governing equations are coupled differential equations, the closed-form solutions are still more difficult to be obtained.
2-5. Concluding Remarks
In this present study both adhesively bonded adherends are subjected to a concentrated force and the peel and shear stress distributions in the adhesive layer joining the two adherends are examined. Such stress distributions are affected by geometric conditions, including the thicknesses of adherends and the length and thickness of the adhesive layer, as well as by the action point of the concentrated force.
These preceding advantages are the reasons why the close-formed solutions are adopted in this research while the aforementioned disadvantages of coupled differential equations die out by the application of symbolic manipulation. Additionally, FEM is not suitable to solve this issue because of the mesh problem described before. That is to say, if the ratio of adhesive’s thickness to joint length is large and if the stresses of the joint are accurately solved, the joint must have much finer mesh and much more CPU run time of the computer is required and taken.
Under some limited conditions, close-formed solutions may be derived by some literatures described before. For examples, two adherends have to have the same material properties.
Furthermore, many literatures only investigate the relations of force (or moment) to stresses.
However, in this present study, the relations of the displacements to force (or moment) have to be derived because of boundary and constraint conditions.
Cornell [29] claimed that obtaining complete theoretical solutions to this problem would be very difficult. As obtaining analytical solutions is even more difficult here than in the work of Cornell, the model uses symbolic manipulation to solve the coupled differential equations in the Mathematica package, thereby enabling to find complete and complicated solutions that are not limited to solving only the characteristic solutions with large values (i.e. the characteristic solutions had to have large values [29]). In this analysis, 31 constraint and boundary conditions are imposed on the analytical solutions. Thus, the numerical solutions can be found by singular value decomposition (SVD) [31] employed as the basis for finding the inverse matrix of a matrix in which the magnitude of the matrix elements varies much.
Nevertheless, it is still somewhat difficult to converge and directly solve the coupled differential equations by using the numerical method.
This theoretical model can be easily linked with genetic algorithm with penalty function and be applied to solve the IC chip pick-up problem. This method also can decrease the CPU run time of this problem.