Solder fatigue

Last updated June 18, 2023

Solder fatigue is the mechanical degradation of solder due to deformation under cyclic loading. This can often occur at stress levels below the yield stress of solder as a result of repeated temperature fluctuations, mechanical vibrations, or mechanical loads. Techniques to evaluate solder fatigue behavior include finite element analysis and semi-analytical closed-form equations.^[1]

Overview

Solder is a metal alloy used to form electrical, thermal, and mechanical interconnections between the component and printed circuit board (PCB) substrate in an electronic assembly. Although other forms of cyclic loading are known to cause solder fatigue, it has been estimated that the largest portion of electronic failures are thermomechanically^[2] driven due to temperature cycling.^[3] Under thermal cycling, stresses are generated in the solder due to coefficient of thermal expansion (CTE) mismatches. This causes the solder joints to experience non-recoverable deformation via creep and plasticity that accumulates and leads to degradation and eventual fracture.

Historically, tin-lead solders were common alloys used in the electronics industry. Although they are still used in select industries and applications, lead-free solders have become significantly more popular due to RoHS regulatory requirements. This new trend increased the need to understand the behavior of lead-free solders.

Much work has been done to characterize the creep-fatigue behavior of various solder alloys and develop predictive life damage models using a Physics of Failure approach. These models are often used when trying to assess solder joint reliability. The fatigue life of a solder joint depends on several factors including: the alloy type and resulting microstructure, the joint geometry, the component material properties, the PCB substrate material properties, the loading conditions, and the boundary conditions of the assembly.

Thermomechanical solder fatigue

During a product's operational lifetime it undergoes temperature fluctuations from application specific temperature excursions and self-heating due to component power dissipation. Global and local mismatches of coefficient of thermal expansion (CTE) between the component, component leads, PCB substrate, and system level effects^[4] drive stresses in the interconnects (i.e. solder joints). Repeated temperature cycling eventually leads to thermomechanical fatigue.

The deformation characteristics of various solder alloys can be described at the microscale due to the differences in composition and resulting microstructure. Compositional differences lead to variations in phase(s), grain size, and intermetallics. This affects susceptibility to deformation mechanisms such as dislocation motion, diffusion, and grain boundary sliding. During thermal cycling, the solder's microstructure (grains/phases) will tend to coarsen^[5] as energy is dissipated from the joint. This eventually leads to crack initiation and propagation which can be described as accumulated fatigue damage.^[6]

The resulting bulk behavior of solder is described as viscoplastic (i.e. rate dependent inelastic deformation) with sensitivity to elevated temperatures. Most solders experience temperature exposures near their melting temperature (high homologous temperature) throughout their operational lifetime which makes them susceptible to significant creep. Several constitutive models have been developed to capture the creep characteristics of lead and lead-free solders. Creep behavior can be described in three stages: primary, secondary, and tertiary creep. When modeling solder, secondary creep, also called steady state creep (constant strain rate), is often the region of interest for describing solder behavior in electronics. Some models also incorporate primary creep. Two of the most popular models are hyperbolic sine models developed by Garofalo^[7] and Anand^[8]^[9] to characterize the steady state creep of solder. These model parameters are often incorporated as inputs in FEA simulations to properly characterize the solder response to loading.

Fatigue models

Solder damage models take a physics-of-failure based approach by relating a physical parameter that is a critical measure of the damage mechanism process (i.e. inelastic strain range or dissipated strain energy density) to cycles to failure. The relationship between the physical parameter and cycles to failure typically takes on a power law or modified power law relationship with material dependent model constants. These model constants are fit from experimental testing and simulation for different solder alloys. For complex loading schemes, Miner's linear superposition damage law^[10] is employed to calculate accumulated damage.

Coffin–Manson model

The generalized Coffin–Manson^[11]^[12]^[13]^[14] model considers the elastic and plastic strain range by incorporating Basquin's equation^[15] and takes the form:

${\frac {\Delta \epsilon }{2}}={\frac {\sigma _{f}^{'}-\sigma _{m}}{E}}(2N_{f})^{b}+\epsilon _{f}^{'}(2N_{f})^{c}$

Here ∆ε ⁄ 2 represents the elastic-plastic cyclic strain range, E represents elastic modulus, σ_m represents means stress, and N_f represents cycles to failure. The remaining variables, namely σ_f,ε'_f,b,and c are fatigue coefficients and exponents representing material model constants. The generalized Coffin–Manson model accounts for the effects of high cycle fatigue (HCF) primarily due to elastic deformation and low cycle fatigue (LCF) primarily due to plastic deformation.

Engelmaier model

In the 1980s Engelmaier proposed a model,^[16] in conjunction with the work of Wild,^[17] that accounted for some of the limitations of the Coffin–Manson model, such as the effects of the frequency and temperature. His model takes a similar power law form:

$N_{f}(50\%)={\frac {1}{2}}({\frac {\Delta \gamma }{2\epsilon '_{f}}})^{\frac {1}{c}}$

$c=-0.442-6\cdot 10^{-4}T_{s}+1.74\cdot 10^{-2}\ln(1+f)$

Engelmaier relates the total shear strain (∆γ) to cycles to failure (N_f). ε'_f and c are model constants where c is a function of mean temperature during thermal cycling (T_s) and thermal cycling frequency (f).

$\Delta \gamma =C({\frac {L_{D}}{h_{s}}})\Delta \alpha \Delta T$

∆γ can be calculated as function of the distance from the neutral point (L_D) solder joint height (h_s), coefficient of thermal expansion (∆α), and change in temperature (ΔT). In this case C is empirical model constant.

This model was initially proposed for leadless devices with tin-lead solder. The model has since been modified by Engelmaier and others^{[ who? ]} to account for other phenomena such as leaded components, thermal cycling dwell times, and lead-free solders. While initially a substantial improvement over other techniques to predict solder fatigue, such as testing and simple acceleration transforms, it is now generally acknowledged ^{[ citation needed ]} that Engelmaier and other models that are based on strain range do not provide a sufficient degree of accuracy.

Darveaux model

Darveaux^[18]^[19] proposed a model relating the quantity of volume weighted average inelastic work density, the number of cycles to crack initiation, and the crack propagation rate to the characteristic cycles to failure.

$N_{0}=K_{1}\Delta W_{avg}^{K_{2}}$

${\frac {da}{dN}}=K_{3}\Delta W_{avg}^{K_{4}}$

$N_{f}=N_{0}+{\frac {a}{da/dN}}=K_{1}\Delta W_{avg}^{K_{2}}+{\frac {a}{K_{3}\Delta W_{avg}^{K_{4}}}}$

In the first equation N₀ represents the number of cycles to crack initiation, ∆W represents inelastic work density, K₁ and K₂ are material model constants. In the second equation, da/dN represents the crack prorogation rate, ∆W represents inelastic work density, K₃ and K₄ are material model constants. In this case the crack propagation rate is approximated to be constant. N_f represents the characteristic cycles to failure and a represents the characteristic crack length. Model constants can be fit for different solder alloys using a combination of experimental testing and Finite Element Analysis (FEA) simulation.

The Darveaux model has been found to be relatively accurate by several authors.^[20]^[21] However, due to the expertise, complexity, and simulation resources required, its use has been primarily limited to component manufacturers evaluating component packaging. The model has not received acceptance in regards to modeling solder fatigue across an entire printed circuit assembly and has been found to be inaccurate in predicting system-level effects (triaxiality) on solder fatigue.^[22]

Blattau model

The current solder joint fatigue model preferred by the majority of electronic OEMs worldwide ^{[ citation needed ]} is the Blattau model, which is available in the Sherlock Automated Design Analysis software. The Blattau model is an evolution ^{[ citation needed ]} of the previous models discussed above. Blattau incorporates the use of strain energy proposed by Darveaux, while using closed-form equations based on classic mechanics to calculate the stress and strain being applied to the solder interconnect.^[23] An example of these stress/strain calculations for a simple leadless chip component is shown in the following equation:

$(\alpha _{1}-\alpha _{2})\cdot \Delta T\cdot L_{D}=F\cdot ({\frac {L_{D}}{E_{1}A_{1}}}+{\frac {L_{D}}{E_{2}A_{2}}}+{\frac {h_{S}}{A_{s}G_{s}}}+{\frac {h_{c}}{A_{c}G_{c}}}+({\frac {2-\nu }{9G_{b}a}}))$

Here α is the CTE, T is temperature, L_D is the distance to the neutral point, E is elastic modulus, A is the area, h is the thickness, G is shear modulus, ν is Poisson's ratio, and a is the edge length of the copper bond pad. The subscripts 1 refer to the component, 2 and b refer to the board, and s refer to the solder joint. The shear stress (∆τ) is then calculated by dividing this calculated force by the effective solder joint area. Strain energy is computed using the shear strain range and shear stress from the following relationship:

$\Delta W={\frac {1}{2}}\Delta \gamma \Delta \tau$

This approximates the hysteresis loop to be roughly equilateral in shape. Blattau uses this strain energy value in conjunction with models developed by Syed^[24] to relate dissipated strain energy to cycles to failure.

Other fatigue models

The Norris–Landzberg model is a modified Coffin–Manson model.^[25]^[26]

Additional strain range and strain energy based models have been proposed by several others.^[24]^[27]^[28]

Vibration and cyclic mechanical fatigue

While not as prevalent as thermomechanical solder fatigue, vibration fatigue and cyclic mechanical fatigue are also known to cause solder failures. Vibration fatigue is typically considered to be high cycle fatigue (HCF) with damage driven by elastic deformation and sometimes plastic deformation. This can depend on the input excitation for both harmonic and random vibration. Steinberg^[29] developed a vibration model to predict time to failure based on the calculated board displacement. This model takes into account the input vibration profile such as the power spectral density or acceleration time history, the natural frequency of the circuit card, and the transmissibility. Blattau developed a modified Steinberg model^[30] that uses board level strains rather than displacement and has sensitivity to individual package types.

Additionally, low-temperature isothermal mechanical cycling is typically modeled with a combination of LCF and HCF strain range or strain energy models. The solder alloy, assembly geometry and materials, boundary conditions, and loading conditions will affect whether fatigue damage is dominated by elastic (HCF) or plastic (LCF) damage. At lower temperatures and faster strain rates the creep can approximated to be minimal and any inelastic damage will be dominated by plasticity. Several strain range and strain energy models have been employed in this type of a case, such as the Generalized Coffin–Manson model. In this case, much work has been done to characterize the model constants of various damage models for different alloys.

Related Research Articles

A eutectic system or eutectic mixture is a homogeneous mixture that has a melting point lower than those of the constituents. The lowest possible melting point over all of the mixing ratios of the constituents is called the eutectic temperature. On a phase diagram, the eutectic temperature is seen as the eutectic point.

In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size, which occurs when the stress intensity factor of the crack exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure.

In materials science, creep is the tendency of a solid material to undergo slow deformation while subject to persistent mechanical stresses. It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increase as they near their melting point.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

Thermal shock is a phenomenon characterized by a rapid change in temperature that results in a transient mechanical load on an object. The load is caused by the differential expansion of different parts of the object due to the temperature change. This differential expansion can be understood in terms of strain, rather than stress. When the strain exceeds the tensile strength of the material, it can cause cracks to form and eventually lead to structural failure.

Flat no-leads packages such as quad-flat no-leads (QFN) and dual-flat no-leads (DFN) physically and electrically connect integrated circuits to printed circuit boards. Flat no-leads, also known as micro leadframe (MLF) and SON, is a surface-mount technology, one of several package technologies that connect ICs to the surfaces of PCBs without through-holes. Flat no-lead is a near chip scale plastic encapsulated package made with a planar copper lead frame substrate. Perimeter lands on the package bottom provide electrical connections to the PCB. Flat no-lead packages usually, but not always, include an exposed thermally conductive pad to improve heat transfer out of the IC. Heat transfer can be further facilitated by metal vias in the thermal pad. The QFN package is similar to the quad-flat package (QFP), and a ball grid array (BGA).

<span class="mw-page-title-main">Coble creep</span>

Coble creep, a form of diffusion creep, is a mechanism for deformation of crystalline solids. Contrasted with other diffusional creep mechanisms, Coble creep is similar to Nabarro–Herring creep in that it is dominant at lower stress levels and higher temperatures than creep mechanisms utilizing dislocation glide. Coble creep occurs through the diffusion of atoms in a material along grain boundaries. This mechanism is observed in polycrystals or along the surface in a single crystal, which produces a net flow of material and a sliding of the grain boundaries.

Damage mechanics is concerned with the representation, or modeling, of damage of materials that is suitable for making engineering predictions about the initiation, propagation, and fracture of materials without resorting to a microscopic description that would be too complex for practical engineering analysis.

A deformation mechanism, in geology, is a process occurring at a microscopic scale that is responsible for changes in a material's internal structure, shape and volume. The process involves planar discontinuity and/or displacement of atoms from their original position within a crystal lattice structure. These small changes are preserved in various microstructures of materials such as rocks, metals and plastics, and can be studied in depth using optical or digital microscopy.

The Larson–Miller relation, also widely known as the Larson–Miller parameter and often abbreviated LMP, is a parametric relation used to extrapolate experimental data on creep and rupture life of engineering materials.

Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.

Thermo-mechanical fatigue is the overlay of a cyclical mechanical loading, that leads to fatigue of a material, with a cyclical thermal loading. Thermo-mechanical fatigue is an important point that needs to be considered, when constructing turbine engines or gas turbines.

Dislocation creep is a deformation mechanism in crystalline materials. Dislocation creep involves the movement of dislocations through the crystal lattice of the material, in contrast to diffusion creep, in which diffusion is the dominant creep mechanism. It causes plastic deformation of the individual crystals, and thus the material itself.

Pad cratering is a mechanically induced fracture in the resin between copper foil and outermost layer of fiberglass of a printed circuit board (PCB). It may be within the resin or at the resin to fiberglass interface.

Physics of failure is a technique under the practice of reliability design that leverages the knowledge and understanding of the processes and mechanisms that induce failure to predict reliability and improve product performance.

Sherlock Automated Design Analysis is a software tool developed by DfR Solutions for analyzing, grading, and certifying the expected reliability of products at the circuit card assembly level. Based on the science of Physics of Failure, Sherlock predicts failure mechanism-specific failure rates over time using a combination of finite element method and material properties to capture stress values and first order analytical equations to evaluate damage evolution. The software is designed for use by design and reliability engineers and managers in the electronics industry. DfR Solutions is based in Beltsville, Maryland, USA, and was acquired by ANSYS, Inc. in May 2019.

Polymer fracture is the study of the fracture surface of an already failed material to determine the method of crack formation and extension in polymers both fiber reinforced and otherwise. Failure in polymer components can occur at relatively low stress levels, far below the tensile strength because of four major reasons: long term stress or creep rupture, cyclic stresses or fatigue, the presence of structural flaws and stress-cracking agents. Formations of submicroscopic cracks in polymers under load have been studied by x ray scattering techniques and the main regularities of crack formation under different loading conditions have been analyzed. The low strength of polymers compared to theoretically predicted values are mainly due to the many microscopic imperfections found in the material. These defects namely dislocations, crystalline boundaries, amorphous interlayers and block structure can all lead to the non-uniform distribution of mechanical stress.

Low cycle fatigue (LCF) has two fundamental characteristics: plastic deformation in each cycle; and low cycle phenomenon, in which the materials have finite endurance for this type of load. The term cycle refers to repeated applications of stress that lead to eventual fatigue and failure; low-cycle pertains to a long period between applications.

Digital image correlation analyses have applications in material property characterization, displacement measurement, and strain mapping. As such, DIC is becoming an increasingly popular tool when evaluating the thermo-mechanical behavior of electronic components and systems.

A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of fatigue cracks can result in catastrophic failure, particularly in the case of aircraft. A crack growth equation can be used to ensure safety, both in the design phase and during operation, by predicting the size of cracks. In critical structure, loads can be recorded and used to predict the size of cracks to ensure maintenance or retirement occurs prior to any of the cracks failing.

References

↑ Serebreni, M., Blattau, N., Sharon, G., Hillman, C., Mccluskey, P. "Semi-analytical fatigue life model for reliability assessment of solder joints in qfn packages under thermal cycling". SMTA ICSR, 2017. Toronto, ON, https://www.researchgate.net/publication/317569529_SEMI-ANALYTICAL_FATIGUE_LIFE_MODEL_FOR_RELIABILITY_ASSESSMENT_OF_SOLDER_JOINTS_IN_QFN_PACKAGES_UNDER_THERMAL_CYCLING
↑ G. Sharon, "Temperature Cycling and Electronics", https://www.dfrsolutions.com/hubfs/Resources/services/Temperature-Cycling-and-Fatigue-in-Electronics-White-Paper.pdf
↑ Wunderle, B.; B. Michel, "Progress in Reliability Research in Micro and Nano Region", Microelectronics and Reliability, V46, Issue 9-11, 2006.
↑ https://www.dfrsolutions.com/hubfs/Resources/System_Level_Effects_on_Solder_Joint_Reliability.pdf ^{[ bare URL PDF ]}
↑ Crina Rauta, Abhijit Dasgupta, Craig Hillman, "Solder Phase Coarsening, Fundamentals, Preparation, Measurement and Prediction", https://www.dfrsolutions.com/hubfs/Resources/services/Solder-Phase-Coarsening-Fundamentals-Preparation-Measurement-and-Prediction.pdf?t=1514473946162
↑ "Solder Joints in Electronics: Design for Reliability". CiteSeerX 10.1.1.115.7354 .
↑ Garofalo, F., 1965, "Fundamentals of Creep and Creep-Rupture in Metals", Macmillan, New York.
↑ Anand, L., 1985, "Constitutive Equations for Hot Working of Metals", J. Plasticity, 1(3), pp. 213–231
↑ Brown, S. B.; Kim, K. H.; Anand, L., 1989, "An Internal Variable Constitutive Model for Hot Working of Metals," Int. J. Plasticity, 5(2), pp. 95–130
↑ M. A. Miner, "Cumulative damage in fatigue", Journal of applied mechanics, vol. 12, pp. 159-164, 1945
↑ L. F. Coffin, "The Problem of Thermal Stress Fatigue in Austenitic Steels", Special Technical Publication 165, ASTM, 1954, p. 31
↑ L. F. Coffin, "A study of the Effects of Cyclic Thermal Stresses on a Ductile Metal", Trans. ASME, 76, 931–950 (August 1954).
↑ S. S. Manson, "Behavior of materials under conditions of thermal stress", Proceedings of the Heat Transfer Symposium, University of Michigan Engineering Research Institute, Ann Arbor, Mich, pp. 9-75, 1953
↑ Dowling, N. E., "Mechanical Behavior of Materials", 2nd Edition, Upper Saddle River, New Jersey, 1999.
↑ Basquin, O. H. (1910). "The exponential law of endurance test". Proceedings of the American Society for Testing and Materials. 10: 625–630.
↑ Engelmaier, W., "Fatigue Life of Leadless Chip Carrier Solder Joints During Power Cycling", Components, Hybrids, and Manufacturing Technology, IEEE Transactions on, vol.6, no.3, pp. 232-237, September 1983
↑ Wild, R. N., "Some Fatigue Properties of Solders and Solder Joints", IBM Tech. Rep. 73Z000421, January 1973.
↑ Darveaux, R., 1997, "Solder Joint Fatigue Life Model", in Design & Reliability of Solder and Solder Interconnections, Proceedings of the TMS, The Minerals, Metals & Materials Society, Orlando, Florida, February 1997.
↑ Darveaux, R. (2000) Effect of simulation methodology on solder joint crack growth correlation. Electronic Components and Technology Conference, 2000 IEEE, pp 158–169
↑ Ye, Yuming, et al. "Assessment on reliability of BGA package double-sided assembled". High Density Packaging and Microsystem Integration, 2007. HDP'07. International Symposium on. IEEE, 2007
↑ Meifunas, M., et al. "Measurement and prediction of reliability for double-sided area array assemblies". Electronic Components and Technology Conference, 2003. Proceedings. 53rd. IEEE, 2003
↑ https://www.dfrsolutions.com/hubfs/Developing%20Damage%20Models%20for%20Solder%20Joints%20Exposed%20to%20Complex%20Stress%20States.pdf, Hillman, C., "Developing Damage Models for Solder Joints Exposed to Complex Stress States: Influence of Potting, Coating, BGA Mirroring, and Housing on Solder Joint Fatigue", Proceedings of the EMPC, Warsaw, Poland, September, 2017
↑ https://www.dfrsolutions.com/hubfs/DfR_Solutions_Website/Resources-Archived/Publications/2005-2007/2006_Blattau_IPC_working.pdf ^{[ bare URL PDF ]}
1 2 Syed, A., "Accumulated Creep Strain and Energy Density Based Thermal Fatigue Life Prediction Models for SnAgCu Solder Joints", ECTC 2004, pp. 737-746 - corrected.
↑ Norris, K C, and AH Landzberg. "Reliability of Controlled Collapse Interconnections" IBM Journal of Research and Development 13, no. 3 (1969): 266-271
↑ "Enabling More than Moore: Accelerated Reliability Testing and Risk Analysis for Advanced Electronics Packaging" (PDF). 2014.
↑ S. Knecht; L. Fox, "Integrated matrix creep: Application to accelerated testing and lifetime prediction", in Solder Joint Reliability Theory and Applications, J. H. Lau, Ed. New York: Van Nostrand Reinhold, 1991, ch. 16.
↑ Lee, W. W.; Nguyen, L. T.; Selvaduray, G. S., "Solder joint fatigue models: review and applicability to chip scale packages". Microelectronics Reliability 40 (2000) 231-244, 1999.
↑ Steinberg, D. S. "Vibration analysis for electronic equipment". John Wiley & Sons, 2000.
↑ https://www.dfrsolutions.com/hubfs/Resources/Guarantee-Reliability-with-Vibration-Simulation-and-Testing.pdf ^{[ bare URL PDF ]}

External links

Solder joint fatigue calculators

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Serebreni, M., Blattau, N., Sharon, G., Hillman, C., Mccluskey, P. "Semi-analytical fatigue life model for reliability assessment of solder joints in qfn packages under thermal cycling". SMTA ICSR, 2017. Toronto, ON, https://www.researchgate.net/publication/317569529_SEMI-ANALYTICAL_FATIGUE_LIFE_MODEL_FOR_RELIABILITY_ASSESSMENT_OF_SOLDER_JOINTS_IN_QFN_PACKAGES_UNDER_THERMAL_CYCLING

[2] G. Sharon, "Temperature Cycling and Electronics", https://www.dfrsolutions.com/hubfs/Resources/services/Temperature-Cycling-and-Fatigue-in-Electronics-White-Paper.pdf

[3] Wunderle, B.; B. Michel, "Progress in Reliability Research in Micro and Nano Region", Microelectronics and Reliability, V46, Issue 9-11, 2006.

[4] ttps://www.dfrsolutions.com/hubfs/Resources/System_Level_Effects_on_Solder_Joint_Reliability.pdf ^{[ bare URL PDF ]}

[5] Crina Rauta, Abhijit Dasgupta, Craig Hillman, "Solder Phase Coarsening, Fundamentals, Preparation, Measurement and Prediction", https://www.dfrsolutions.com/hubfs/Resources/services/Solder-Phase-Coarsening-Fundamentals-Preparation-Measurement-and-Prediction.pdf?t=1514473946162

[6] "Solder Joints in Electronics: Design for Reliability". CiteSeerX 10.1.1.115.7354 .

[7] Garofalo, F., 1965, "Fundamentals of Creep and Creep-Rupture in Metals", Macmillan, New York.

[8] Anand, L., 1985, "Constitutive Equations for Hot Working of Metals", J. Plasticity, 1(3), pp. 213–231

[9] Brown, S. B.; Kim, K. H.; Anand, L., 1989, "An Internal Variable Constitutive Model for Hot Working of Metals," Int. J. Plasticity, 5(2), pp. 95–130

[10] M. A. Miner, "Cumulative damage in fatigue", Journal of applied mechanics, vol. 12, pp. 159-164, 1945

[11] L. F. Coffin, "The Problem of Thermal Stress Fatigue in Austenitic Steels", Special Technical Publication 165, ASTM, 1954, p. 31

[12] L. F. Coffin, "A study of the Effects of Cyclic Thermal Stresses on a Ductile Metal", Trans. ASME, 76, 931–950 (August 1954).

[13] S. S. Manson, "Behavior of materials under conditions of thermal stress", Proceedings of the Heat Transfer Symposium, University of Michigan Engineering Research Institute, Ann Arbor, Mich, pp. 9-75, 1953

[14] Dowling, N. E., "Mechanical Behavior of Materials", 2nd Edition, Upper Saddle River, New Jersey, 1999.

[15] Basquin, O. H. (1910). "The exponential law of endurance test". Proceedings of the American Society for Testing and Materials. 10: 625–630.

[16] Engelmaier, W., "Fatigue Life of Leadless Chip Carrier Solder Joints During Power Cycling", Components, Hybrids, and Manufacturing Technology, IEEE Transactions on, vol.6, no.3, pp. 232-237, September 1983

[17] Wild, R. N., "Some Fatigue Properties of Solders and Solder Joints", IBM Tech. Rep. 73Z000421, January 1973.

[18] Darveaux, R., 1997, "Solder Joint Fatigue Life Model", in Design & Reliability of Solder and Solder Interconnections, Proceedings of the TMS, The Minerals, Metals & Materials Society, Orlando, Florida, February 1997.

[19] Darveaux, R. (2000) Effect of simulation methodology on solder joint crack growth correlation. Electronic Components and Technology Conference, 2000 IEEE, pp 158–169

[20] Ye, Yuming, et al. "Assessment on reliability of BGA package double-sided assembled". High Density Packaging and Microsystem Integration, 2007. HDP'07. International Symposium on. IEEE, 2007

[21] Meifunas, M., et al. "Measurement and prediction of reliability for double-sided area array assemblies". Electronic Components and Technology Conference, 2003. Proceedings. 53rd. IEEE, 2003

[22] ttps://www.dfrsolutions.com/hubfs/Developing%20Damage%20Models%20for%20Solder%20Joints%20Exposed%20to%20Complex%20Stress%20States.pdf, Hillman, C., "Developing Damage Models for Solder Joints Exposed to Complex Stress States: Influence of Potting, Coating, BGA Mirroring, and Housing on Solder Joint Fatigue", Proceedings of the EMPC, Warsaw, Poland, September, 2017

[23] ttps://www.dfrsolutions.com/hubfs/DfR_Solutions_Website/Resources-Archived/Publications/2005-2007/2006_Blattau_IPC_working.pdf ^{[ bare URL PDF ]}

[Syed-24] 1 2 Syed, A., "Accumulated Creep Strain and Energy Density Based Thermal Fatigue Life Prediction Models for SnAgCu Solder Joints", ECTC 2004, pp. 737-746 - corrected.

[25] Norris, K C, and AH Landzberg. "Reliability of Controlled Collapse Interconnections" IBM Journal of Research and Development 13, no. 3 (1969): 266-271

[26] "Enabling More than Moore: Accelerated Reliability Testing and Risk Analysis for Advanced Electronics Packaging" (PDF). 2014.

[27] S. Knecht; L. Fox, "Integrated matrix creep: Application to accelerated testing and lifetime prediction", in Solder Joint Reliability Theory and Applications, J. H. Lau, Ed. New York: Van Nostrand Reinhold, 1991, ch. 16.

[28] Lee, W. W.; Nguyen, L. T.; Selvaduray, G. S., "Solder joint fatigue models: review and applicability to chip scale packages". Microelectronics Reliability 40 (2000) 231-244, 1999.

[29] Steinberg, D. S. "Vibration analysis for electronic equipment". John Wiley & Sons, 2000.

[30] ttps://www.dfrsolutions.com/hubfs/Resources/Guarantee-Reliability-with-Vibration-Simulation-and-Testing.pdf ^{[ bare URL PDF ]}

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]