Role Of Modeling

Predictive modeling, both simulation-based and analytical, are equally important and complement each other in any important design and testing effort when there is a need to understand the physics of a phenomenon or the behavior and performance of a particular electronic product. Broad application of powerful computer programs and particular those using FEA have by no means made analytical solutions less important. Simple analytical relationships have invaluable advantages, because of the clarity and compactness of the obtained information and clear indication on interactions and dependencies of different material and design factors. FEA simulations and analytical solutions are typically based on different assumptions and use different methods. It is always advisable to obtain a solution to the problem of interest on the basis of different approaches. If the results are in a satisfactory agreement, one could be confident that they are most likely correct. It is always advisable to analytically investigate the problem before carrying out computer-aided analyses. Such a preliminary investigation helps to reduce computer time and expense, develop the most feasible and effective preprocessing model and, in many cases, avoid fundamental errors. Analytical (“mathematical”) modeling [25] enabled one to obtain closed form solutions for numerous electronic packaging problems, including stresses in solder joints configured as short cylinders [23], twist-off testing of solder joint interconnections [24], and various paradoxical situations [31] encountered in electronic packaging materials science. Analytical modeling was also effectively used to predict, quantify and, through that, to assure microelectronics reliability [28]. This type of modeling was also used in a prognostics-and-health monitoring (PHM) three-step-concept (TSC) [30], when the recently suggested physically meaningful, powerful, and flexible probabilistic Boltzmann-Arrhenius-Zhurkov (BAZ) model [29, 48] was “sandwiched” between two statistical models: Bayes’ equation, aimed at identification, on the probabilistic basis, the malfunctioning device(s), and beta- distribution (see, for example, [47]) aimed at updating reliability if a failure has been detected despite the anticipated low probability of its possible occurrence. Zhu and Marcienkiewicz [26] used advanced modeling technique based on computer simulations to model drop tests of chip scale package (CSP) and fine pitch BGA packages. Three drop tests (“board drop, board with fixture drop or shock, and system level phone drop”) were conducted and “explicit-implicit sequential modeling techniques” were used to characterize the dynamic responses of CSP/BGA packages in different board designs. Failure criteria and effects of strain rate and edge support on BGA in multicomponent boards were also investigated. A validation test with data acquisition was used to correlate the test results and computed data. Marjamaki et al. [27] and many others used FEA software to model various aspects of drop tests, and particularly for lead-free solders.

Linear Response

Whenever possible, a linear approach should be applied first to investigate/model the behavior of a dynamic system. In many cases such an approach is sufficient, unless the loading is significant and/or new phenomena that could not be captured by a linear analysis are addressed. But even when nonlinear phenomena are addressed, the linear behavior of the system should be analyzed first to capture its most important aspects that will take place at low levels of loading. Let us indicate several critical problems, in which the application of a linear approach enabled one to obtain practically important solutions in electronics packaging materials science.

• 1. There is an obvious incentive for using viscous damping to reduce the vibration amplitudes in structural elements subjected to drop testing. Application of viscous damping has proven to be effective in many engineering systems subjected to vibrations, especially when resonant or close-to-resonant conditions cannot be avoided. It is less obvious to what extent extensive damping could be advantageous in systems experiencing impact loading. The question is to what extent elevated viscous damping might be able to effectively minimize the maximum “breaking acceleration” (deceleration) during “landing” of an electronic device, as well as the “breaking distance”, such as the distance that the falling device covers from the moment when its enclosure touches the floor to the moment when the velocity of the vulnerable structural element in the device becomes zero. It has been shown [32] that the “breaking distance” always decreases with an increase in the level of the viscous damping, and that the effectiveness of such damping is higher for a low-level damping. As to the maximum acceleration (deceleration), the situation is different: there is a certain, a relatively low, level of damping that results in decelerations that are indeed somewhat lower than those in a system with no damping. However, if damping is significant, the maximum decelerations can significantly exceed the decelerations in a system with no damping at all. For this reason, viscous damping should be introduced in a system subjected to drop impact with caution, and the level of damping should be established beforehand, based on the carried out theoretical analysis [32], depending on the mass of the body to be protected and the spring constant of the protective cushion (restoring force).
• 2. The level of dynamic loading is typically expressed in electronics engineering by the magnitude of the induced accelerations (decelerations). Although such an approach is commonly accepted and is widely employed, as far as functional (electrical, optical, thermal) performance is concerned, it might be misleading when the mechanical and materials reliability is of interest [33]. In many cases, high accelerations indeed go together with high dynamic stresses, but there are also situations when these stresses are much higher in systems that experience relatively low accelerations. For instance, structural elements of the type of a low weight beam supported at the ends are characterized by high natural frequencies and, as a result of that, experience high drop induced accelerations (these are proportional to the vibration frequencies squared). On the other hand, structural elements that could be idealized as cantilever beams carrying heavy lumped masses at the end might experience high dynamic stresses despite relatively low drop induced accelerations (Figure 9.1).
• 3. In accelerated tests in electronics, there is always a temptation to replace relatively complicated, time-consuming, and expensive testing with simpler, faster, and cost-effective tests. Shock testers are widely available in the electronic industry, and so are drop testers. Shock testers are much cheaper and much simpler to operate than drop testers, and the dynamic response of specimens tested on shock testers is much easier to measure and to interpret than the drop test response. A natural question arises of whether drop test conditions could be adequately mimicked by shock conditions, and if they could, how the shock tester should be tuned so that the results of shock testing would adequately mimic drop test conditions.

FIGURE 9.1 The structural element idealized as a simply supported beam (left sketch) experiences high accelerations, but low stresses in its mid-cross section. The structural element idealized as a cantilever beam with a lump mass at the end (right sketch) experiences low accelerations, but high stress at the clamped end. (From E. Suhir, “Is the Maximum Acceleration an Adequate Criterion of the Dynamic Strength of a Structural Element in an Electronic Product?” IEEE CPMT Transactions, Part A, vol. 20, No. 4. Dec. 1997.)

It has been shown [36] that substitution of drop testing with shock testing is indeed possible, provided that the magnitude and the duration of the drop impact is preestablished by modeling, and that, based on the predicted drop test response, the shock tester is tuned accordingly. It has been particularly suggested that, assuming that the dynamic response remains linear, the shock tester should be able to produce very short impact loads (preferably shorter than one eighth of the period of vibrations) in order to adequately reflect the drop test conditions. Based on the results of the theoretical analysis [36], a shock table has been designed and built [17] to mimic real-life drop conditions. The obtained experimental data are in good agreement with the theoretical predictions. It has also been an attempt to show [22] that if the ultimate drop height, such as the height that leads to the failure, is selected as an adequate criterion of its dynamic strength, then the shear-off data for the ultimate shear-off force can be effectively used to assess the ultimate drop height. Such an assessment is expected to be conservative, because the dynamic stress-strain diagrams for a wide variety of materials indicate that that the same material is characterized by a somewhat higher level of the loading-at-failure than what could be concluded from the static stress-strain diagram.

4. As is known (see, for example, [42]), short-term loadings generate higher harmonics. An instantaneous impulse, in the absence of dissipation, generates a vibration spectrum with an infinitely large number of harmonics. How important is it that the role of these harmonics is considered? This question can be answered, using linear approach, using the reasoning in Figure 9.2 (see, for example, [5]). Consider an elongated simply supported thin plate mimicking a PCB. The board is dropped from a height H, hits the floor with its supports, and, as a result of that, experiences shock-exited

FIGURE 9.2 Role of the higher harmonics in the linear dynamic response of an elongated PCB to the drop impact applied to its ends. The formula for the kinetic energy, based on the energy balance for the case of undamped vibrations, indicates that the higher harmonics “consume” only about 23% of the initial potential energy. In the cases of damped vibrations and nonlinear vibrations, the role of the higher harmonics is even smaller. This circumstance justifies consideration of the fundamental mode only for drop impact-induced vibrations.

vibrations. Using the method of principal coordinates (see, for example, [5] or [42]), the induced deflections can be sought in the form of the series

ijix

Here, f(t) is the principle coordinate of the /'th mode, and cos-^— is the

coordinate function. This function is chosen in such a way that the zero boundary conditions for the deflection and the curvature are fulfilled at the plate’s supports. The lateral velocities can be determined by differentiation

At the initial moment of time, all the points of the board have the same velocities w(x,0) = ^/2gH. On the other hand, these velocities can be found from (9.1) as

Multiplying both parts of equation

by the coordinate function ^cosy~ and integrating along the plate, the

following formula for the initial velocity of the principal coordinate can be obtained:

Thus, while the initial velocities of all the points on the board are the same, the initial velocities of the principal coordinates are not: higher modes are characterized by lower initial velocities and, hence, by lower energies than the lower modes. The lower formula in Figure 9.2 reflects the change in the initial kinetic energy of the vibration modes. The first term is the total kinetic energy of the board. Introducing formula (9.4) into this term, the

second term could be obtained. The series in the second term are converg-

00

Y¹ 1 8

ing: > — = — .This enables one to obtain the total energy in the form

;=i.3.5....¹ 71

of the third term. The fourth term can be obtain considering that the structure of the second term and that the kinetic energy of the fundamental mode

of vibrations is Aj = Щ-magH. The fourth and the fifth terms indicate that 7t~

the total kinetic energy of the plate exceeds the energy of the fundamental mode by the factor of ^- = 1.2337. The sanity check indicates that the

total kinetic energy of the board is indeed equal to its initial potential energy. Thus, the higher modes of vibrations are “responsible” for only 23.4% of the total energy, and therefore, in an approximate and practical analysis, one can get away with considering the fundamental mode of the induced vibrations only. This is even more true for the nonlinear vibrations subsequently addressed. This is because the energy associated with the in-plane (membrane) stresses is also included into the fundamental mode.

5. When there is intent to understand the physics underlying the behavior and performance of solder joint interconnections during board-level testing, one has to be able to model and analyze, first of all, the response of the bare board itself. Figure 9.3 [6] indicates that the vibration characteristics of a PCB can be predicted with high accuracy (the FEA has been carried out by M. Vujosevic, Intel).

FIGURE 9.3 When the vibration amplitudes and the in-plane forces (stresses) are low (small) the linear theory well predicts the induced amplitudes and accelerations. But how small is “small”? This question can only be answered based on a nonlinear analysis.