Creep and Constitutive Models for Near-Eutectic SnPb

Overview

One lesson learned from SnPb studies is that here is no unique constitutive model for SMT solder joints, thus the variety of models available throughout the literature. In the end, the applicability of a given model to real life assemblies and a reasonable agreement between the ensuing life predictions and test results determine whether a constitutive model will be of use to design engineers and reliability analysts.

The mechanical behavior of solder depends on the joint microstructure and is affected by many parameters such as intermetallics, joint or specimen size, cooling rate of the assembly after soldering, aging in service etc. Test factors such as specimen or load eccentricity, temperature variations and measurement errors also contribute to the scatter in the mechanical properties of solder as is well known, for example, for steady state creep. Nevertheless, simplified constitutive models have been developed to help characterize the mechanical behavior of SnPb solder and enable first-order stress/strain analysis of solder joints using methods of classical mechanics or numerical techniques such as the Finite Element Method (FEM).

What is Creep?

Figure 1: Creep curve: strain versus time under constant stress (or load) and temperature.

Under constant load or stress, solder undergoes progressive inelastic deformations over time. This timedependent deformation is called creep and the associated strains that develop over time are creep strains. When the test specimen is subjected to a constant load, the initial, instantaneous response includes elastic and time-independent plastic flow. Creep then proceeds in three stages of primary, secondary and tertiary creep (see Figure 1). During primary creep, metals strain-harden. The strain rate decreases over time, as hardening of the metal becomes more difficult. Specimen deformations keep increasing with secondary creep proceeding at a steady strain rate. Note also that the initial deformation that occurs upon loading of the test specimen includes both elastic and plastic strains. Often, these initial deformations are not reported on in the context of creep studies. However, they cannot be neglected, a priori, because of the inelastic nature of the initial plastic flow. Moreover, these initial deformations, which depend on the loading rate, may become important under service conditions with intermediate to rapid temperature ramps.

For most metals, secondary creep is the dominant deformation mode at temperatures above half the melting point, TM , in degree Kelvin. For eutectic SnPb, TM is 183°C = 456°K and 1/2 TM = 228°K = - 45°C. That is, standard SnPb solder readily creeps at, and well below, room temperature. The last stage of deformation is tertiary creep where strain rates increase rapidly until the test specimen ruptures. In SnPb solder, tertiary creep proceeds by void formation and growth along grain boundaries, micro-cracking and necking of tensile specimens.

Numerous researchers have investigated the mechanical and creep behavior of near-eutectic SnPb solder (e.g., Baker et al., 1973, Chen et al., 1971, Darveaux et al., 1995, Grivas et al., 1979, Hacke et al., 1993, Kashyap, 1981, Knecht et al., 1991, Pao et al., 1992, Shine et al., 1988, Stone et al., 1985/94, Wong et al., 1988/90). Most investigations have focused on steady-state creep, with the secondary creep rate often given as a function of stress and the absolute temperature, T:

Equation (1) is known as Dorn’s equation (Bird et al., 1969) and is often simplified as:

Equation (2) shows the strong dependence of creep rates on stress and temperature as well as grain size (in the case of SnPb solder).

Since creep properties, as well as strength and other mechanical properties, vary with specimen size, the mechanical response of tiny solder joints differs from that of bulk solder test specimens. For engineering applications dealing with surface-mount (SM) assemblies, constitutive models developed from measurements on solder joint specimens have proven to be very useful. Solder deformations, including creep, have been measured on solder joints of actual electronic assemblies (e.g., Shine and Fox, 1988, Knecht and Fox, 1991) and for several solder alloys: 60Sn-40Pb, 62Sn-36Pb-2Ag, 96.5Sn-3.5Ag, 97.5Pb-2.5Sn, 100In and 50In-50Pb (Darveaux et al., 1995). These models are presented, briefly, hereafter because they have been found to be of use to practicing engineers. The reader is referred to the original publications for additional details as well as for relevant information on the experimental techniques that were used and that could be applied to the study of lead-free solders.

Motorola / Darveaux's Constitutive Model

Darveaux and his co-workers at Motorola (Darveaux et al., 1995) conducted extensive mechanical testing of flip-chip and Ball Grid Array (BGA) solder joints and characterized the time-independent plastic flow and creep deformations of several solder alloys. Their constitutive model is described below for several alloys of electronic solder. Robert Darveaux implemented this model into two commercial finite element codes, ANSYS^TM and ABAQUS^TM. His original publication (Darveaux et al., 1995, Chapter 13 in Ball Grid Array Technology) includes detailed recommendations on how to input material constants in the preprocessor of those two programs. One important feature of Darveaux’s creep model is that it was found to apply consistently to several solder alloys: 60Sn-40Pb, 62Sn-36Pb-2Ag, 96.5Sn-3.5Ag, 97.5Pb-2.5Sn, 100In and 50In-50Pb, and over a wide range of temperatures and several orders of magnitude in strain rates.

The initial, instantaneous strain that develops at the start of a creep test includes an elastic strain and an inelastic strain that represents time-independent plastic flow. The plastic strain is described by a plastic flow or strain hardening law of the form:

The elastic constants and the plastic flow parameters for several solder alloys, including SnAg eutectic, are given in Table 1 below. Note that the elastic constants and the power-law exponent m are about the same for 60Sn-40Pb and 62Sn-36Pb-2Ag. However, the constant C₆ is about twice as low for 62Sn-36Pb-2Ag. Under equal loads, tin-lead with 2% silver will see half as much initial plastic strain than 60Sn-40Pb.

Table 1: Solder material constants for shear modulus and plastic flow rule (after Darveaux et al., 1995).

During primary or transient creep, the creep strain is given by the equation:

The primary creep constants for several alloys are given in Table 2.

Table 2: Primary creep constants for common solder alloys (after Darveaux et al., 1995).

The primary creep rate is:

Initially, at time t = 0, the primary creep rate is a factor times greater than the steady state creep rate. For 60Sn-40Pb, this factor is: = 1 + 0.026 x 403 = 11.48, i.e., the initial transient creep rate is over an order of magnitude higher than the steady creep rate. For Sn-3.5Ag, the rate factor is even larger: = 1 + 0.167 x 131 = 21.88. Thus, primary creep may not be negligible in applications with high temperature ramp rate or under thermal cycling conditions with short dwell times.

More general relationships were found to apply to steady state creep of solder in shear:

The above constants and activation energies are given for several solder alloys in Table 3.

• The creep rate versus stress curves are bi-linear, with creep regions associated with two different mechanisms:

• Grain boundary creep at low stresses and creep rates.
• Matrix creep at high stresses and creep rates.

• Solder joints undergo creep-fatigue failures associated with steady-state creep. Creep damage occurs by a combination of grain-boundary and matrix creep, with Scanning Electron Microscope (SEM) photos of fatigued solder joints showing intercrystalline voids and cracks. Isothermal fatigue life cycles have an inverse relationship to integrated matrix creep.
• The grain size of thick joints is larger than that of thin joints and solder grain size increases during fatigue testing. Under identical loads, thin joints with initially smaller grain sizes are expected to have a longer fatigue life than thick joints with larger grains.

Using the hysteresis loops and creep data from the above experiments, Knecht and Fox (Knecht et al., 1991) developed a simple constitutive model for eutectic (63Sn-37Pb) solder in shear. The constitutive equations from that study are summarized hereafter with some minor modifications. Knecht and Fox used their model to conduct finite element analysis of solder joints in SMT assemblies and to correlate fatigue life data to integrated matrix creep strains.

The average shear strain is given as the sum of an elastic strain, a time-independent plastic strain, and a secondary creep strain, that is:

The elastic strain component is:

where the temperature dependent shear modulus is:

Poisson's ratio for near-eutectic solder is = 0.4 and the temperature-dependent Young's modulus is:

The time-independent plastic strain is given by the following plastic flow rule:

temperature-dependent plasticity parameter

where the temperature-dependent plasticity parameter is obtained by curve-fitting the plasticity parameter versus temperature data in the original publication by Knecht and Fox (1991), i.e.:

Finally, creep strains are obtained by integration of the steady state creep rate equations:

The first term on the right-hand-side of equations (15) and (16) is for grain boundary creep with a stress exponent: n_GB = 2 and an activation energy: H_GB = 0.5eV. The second term on the right-hand-side of (15) and (16) is for matrix creep with a stress exponent: n_MC = 7.1 and an activation energy: H_GB = 0.84eV. As shown by Knecht and Fox, these values of stress exponents and activation energies are consistent with steady state creep parameters reported in the literature (see Table 4 below).

average activation energy for creep of eutectic SnPb solder: Q_C = 45 kJ/mole = 0.47eV

universal gas constant: R = 8.314 J/°K.mole.

The correlation band in Figure 2 has two distinct slopes that reflect different creep mechanisms:

• A dislocation glide regime with a stress exponent of 3 in the low stress region.
• A dislocation climb regime with a stress exponent of 7 in the higher stress region.

Based on the above correlation of creep data, Wong gave the following expression for average creep rates:

The equations of the lower and upper bounds of the correlation band are (Clech et al., 1988):

The above stress exponents are consistent with the steady state creep equation in DEC's steady state creep model. Note also that Wong's creep rate equations do not include grain size dependence either. Wong argued that the experimental data was inconclusive at the time (Wong et al., 1988) and this did not warrant any attempt at including grain size effects. Wong also stated that grain size dependence is not expected in either the climb or glide-controlled creep regimes. One last important aspect of Wong's steady state creep model is that it was derived based on the correlation of creep data from several independent sources, with the correlation holding over a wide range of temperatures (-60°C to 150°C), stresses and creep rates.

The above compilation of creep data has been used successfully in a solder joint life prediction model developed by Wong et al., 1988. The upper bound of the correlation band was also used as the creep rate equation in the Solder Reliability Solutions (SRS) life prediction model (Clech, 1996). The upper bound was selected in order to maximize strain rates, thus building in some conservatism in the model. The SRS model has since been validated by over sixty experiments (Clech, 2000).

Based on Sn-Pb experience, we can expect creep rates from compiled test results for lead-free solders to spread over one order of magnitude. Such spread in the data did not impede the development of useful, first-order solder joint life prediction models for Sn-Pb assemblies.

Hall's Stress / Strain Hysteresis Loop

One of the most significant contributions to the field of solder joint mechanics, and also most enjoyable reading (in this author’s opinion), is the shear strain and hysteresis loop measurements and theory developed by Peter Hall at AT&T Bell Laboratories (Hall, 1984 and 1991). Using strain gauge measurements and a simplified analysis of shear forces exerted on the solder joints of LCCC assemblies, Hall showed that the stress/strain response of solder joints during temperature cycling is a hysteresis loop (see Figure 2). The shape of the loop reflects the temperature-dependent inelastic deformations of solder and elastic deformations of the entire assembly. The thermal expansion mismatch between board and component is accommodated by shear of the solder joints and simultaneous stretching and bending of the board and component. These elastic deformations of the interconnected parts provide compliance to the assembly, suggesting practical ways to reduce solder joint stresses by designing boards and components that are more compliant.

• The loop is described clock-wise. Isothermal stress reduction lines are drawn as dashed lines between the data points corresponding to equal temperatures during the ramp-up and ramp-down phases of the thermal cycle. The stress reduction lines are shown for every 10 or 20°C temperature increment. The stress reduction lines are almost parallel to each other, with an average slope in very good agreement with the slope predicted by Hall’s assembly stiffness model.
• During the dwell periods at the temperature extremes, stresses are reduced along the stress-reduction lines (shown as solid lines) for those temperatures:

• At 125°C, where solder is very soft, shear strains are large and initial stresses are relatively low, less than 200 psi. Creep rates are very high and stress reduction is rapid. The intersect of the stress reduction line with the strain axis is the maximum available strain due the thermal expansion mismatch between the board and the LCCC component.
• At -25°C, initial stresses are much higher, of the order of -2300 psi. In spite of high stresses, there is not much stress reduction during the two-hour dwell because creep rates are rather small at cold temperatures.

• During the ramp-up phase of the thermal cycle:

• As temperature goes up, starting at -25°C, solder is relatively strong and the shear strain remains about constant. Actually, shear forces are unloaded almost elastically.
• Past about 35°C, where the shear force is zero, stresses build up due to plastic flow of solder in the opposite direction and strains start increasing with the added thermal expansion mismatch between board and component.
• Creep accelerates as temperature keeps going up. Slightly past 50°C, the creep rates are so high that stress reduction prevents any further build-up of stress. Creep strains develop at a faster rate, contributing to rapid increments in the total shear strain.
• When temperature approaches 125°C, strains keep increasing with the thermal expansion mismatch between board and component and stresses relax at a rather fast rate.

• During the ramp-down phase of the thermal cycle:

• As temperature goes down, starting at 125°C, the cycle is reversed and shear strains decrease.
• Initially, and down to about 50-60°C, shear strains decrease at a high rate. Stresses start building up in the opposite direction.
• From 50-60°C to -25°C, stresses become larger and built up at faster rate since solder becomes stronger at lower temperatures.

Figure 4: Solder joint stress/strain simulation when temperature increases from T to (T + T).

The basic algorithm that is used to generate stress/strain hysteresis loops during thermal cycling follows the stress/strain curves shown in Figure 4. During a small time-step from time t to (t + t), temperature increases from T to (T + T). At time t, the stress/strain curve intersects the stress reduction line for temperature T₁ = T. Due to the increase in temperature, stresses build-up instantaneously from ₁ to ₁₂ with a change in strain from ₁ to ₁₂ that includes an elastic strain increment _e and plastic flow with a strain increment _p. Stresses then relax from ₁₂ to ₂, and the shear strain increases by creep (increment _c) along the stress reduction line at temperature T₂ = (T + T). Knowing the stress / strain state (₁, ₁) at time t, the stress / strain state (₂, 2) at time (t + t) is obtained by solving the following system of four equations with four unknowns ₁₂, ₁₂, ₂, and ₂:

• From the stress reduction line at T2 = (T + T):

• For the strain increment due to elastic deformation and plastic flow as per, for example, DEC's plastic flow rule:

• For the creep strain developing during the short dwell of duration t at T₂:

The above algorithm can be implemented in a computer program or even in a spreadsheet. Usually, a zero stress / strain state is used, somewhat arbitrarily, to initialize the algorithm. Small enough time steps are required to follow the prescribed temperature profile closely and to generate the stress / strain response with the desired accuracy. A few cycle iterations are typically required to obtain a closed and stable hysteresis loops for stiff systems like leadless assemblies. Many more iterations are needed for compliant systems like leaded assemblies with very compliant leads. Hysteresis loops can also be obtained using finite element models that include material options for a constitutive model with elastic, plastic flow and creep (see, for example, Darveaux et al., 1995/97, Lau and Pao, 1997).