Nonlinear Elastoplastic Response and Damage Modeling in Power Electronics Packages Under Thermal Cycling ^†

Abstract

One of the common reliability tests performed on power modules for automotive applications is passive thermal cycling, which is conventionally representative of the highly demanding thermomechanical loads typical of steady-state operating conditions. The mechanical response of the electronics devices subjected to such testing procedures, in terms of stress-strain response and of damage, is usually predicted by finite elements analyses where the remarkable nonlinearities intrinsic in the phenomena need to be properly addressed. This work regards the FEM modeling of the thermomechanical behavior of a power electronics package subjected to thermal cycles, focusing on the critical importance of modeling the complete elastoplastic behavior of materials, in contrast to the conventional elastic approach. By incorporating the full elastoplastic properties, the study aims to accurately evaluate the actual irreversible deformations and resulting stresses that develop within the package subjected to a representative passive thermal cycle and to compare the outcomes to those from purely elastic simulations. Additionally, damage models are compared for predicting the local detachment of the encapsulating resin from other layers. The predictions of the cohesive zone model (CZM) applied to a conventional interface layer are compared to those of a modified Tresca (MT) stress-dependent damage model applied to the resin bulk material. In addition, the estimate of linear-nonlinear evolutions of plastic strain and of damage at increasing numbers of cycles is investigated in the attempt to identify procedures for guessing the long-term mechanical response from short-term simulations.

1. Introduction

Power electronics packages are critical components in modern electronic systems, playing a crucial role in energy conversion and control applications. They typically consist of multiple layers made from metallic and ceramic materials [1]. The primary components of these systems are semiconductor dies, usually made from silicon (Si), silicon carbide (SiC), or gallium nitride (GaN). These dies are attached and interconnected to the copper–ceramic–copper substrate, which forms electrical circuits on the top layer and facilitates effective thermal exchange on the bottom layer, using solder alloys and aluminum-based wires. To insulate the electronic components and protect the layers from oxidation and corrosion, the entire assembly is typically encapsulated in an epoxy resin [2].

These packages are engineered to manage high power densities and operate in challenging environments and are typically subjected to harsh thermal conditions, where significant temperature fluctuations can induce complex thermomechanical stresses. As electronic devices operate in cycles of heating and cooling, the materials within these packages experience different repeated thermal expansion and contraction, leading to the accumulation of stress, strain, and potential damage over time.

The reliability of power electronics is heavily influenced by the ability of the package to withstand these thermal cycles without failure. The materials used in these packages, including metals, ceramics, and polymers, exhibit different thermomechanical properties. As a result, these components are prone to warpage due to stresses from temperature fluctuations during operation [3]. One of the most critical failure mechanisms is delamination [4], which occurs when layers separate at their interfaces. This leads to performance degradation, reduced reliability, and potentially catastrophic failure.

Traditional experimental methods for studying delamination are often time-consuming, expensive, and limited by the difficulty of accessing critical regions within the package. Consequently, finite element simulations have become a valuable tool for investigating and analyzing damage and delamination more efficiently and cost-effectively [5,6]. Traditional modeling approaches often assume purely elastic behavior for simplicity. For the delamination simulation, one of the most common approaches is cohesive zone modeling (CZM) which offers a phenomenological approach to failure by describing traction-separation relationships along delamination interfaces [7,8,9,10,11]. With a different approach, some researchers have tried to model damage in resins with the modified Tresca (MT) criterion [12,13].

This study aims to develop a novel comprehensive FEM modeling approach to analyze the nonlinear elastoplastic response and damage evolution in a multi-layered power electronics package subjected to thermal cycling. In particular, the work novelties are related to the inclusion of cyclic plasticity in the materials’ response and to the comparison between different approaches to analyze and simulate damage in elastic and elastoplastic materials. Then, by comparing elastic and elastoplastic simulations and exploring these different damage modeling approaches, this research seeks to provide a more accurate assessment of the package’s performance under realistic operating conditions, highlighting the critical aspects to be faced.

2. Materials and Methods

This study thoroughly investigates the thermomechanical behavior of a multi-layered power electronics package subjected to passive thermal cycling. The temperature range for the thermal cycling was set between −65°C and 150 °C, reflecting the realistic environmental conditions that these components are likely to encounter during reliability passive testing, with a stress-free starting condition. The study encompassed both single thermal cycles and a series of consecutive cycles to capture the evolution of stress and strain responses within the package. By focusing on critical interfaces and critical points within the package, the research discusses detailed insights into how stress and strain distributions develop and evolve over time, highlighting the key focal points for assessing the overall reliability of the package.

2.1. Materials

The considered component is a SiC-MOSFET power electronics package for automotive applications for which a not-in-scale scheme with dimensions is shown in Figure 1. The materials composing the package are, from bottom to top, copper, solder (PbSnAg), silicon carbide (SiC), tetraethyl orthosilicate (TEOS), silicon nitride (Si₃N₄ or SiN briefly), polyimide, and the encapsulating epoxy resin. All the materials are always considered elastic (characteristics shown in Figure 2a according to literature data [14,15,16,17,18,19]), with the exception of copper, solder, and polyimide, which will be considered in some simulations to be elastoplastic, with their hardening curves shown in Figure 2b, according to literature data [19,20,21,22]. Only elastic and fully elastoplastic simulations of a single thermal cycle were carried out, focusing on the obtained differences in stress distributions, particularly at material interfaces. Then, a series of thermal cycles were simulated only with more accurate elastoplastic model to investigate the accumulation of plastic deformation and its impact on material damage over time. In the simulations, the isotropic thermal expansion of all materials was taken into account, using the coefficients shown in Figure 2a. On the other hand, all material properties were assumed to be temperature independent.

This simplifying assumption does not affect the generality of the proposed approach; furthermore, it is expected to introduce minor approximations due to the relatively narrow temperature range at hand.

2.2. Finite Element Model

A 3D finite element model of the power electronics package was developed using the commercial FEM software Hexagon Marc^® v2023 (Figure 3). The model was constructed based on a quarter-symmetry approach, exploiting the geometric symmetry of the package to reduce computational costs. It was created with detailed multi-layer representation.

The selected mesh type and size for different materials depend on the thickness of the corresponding layer. Specifically, the thicker layers (resin, SiC, copper) are modeled using HEX-8 brick elements (element type #7), with a minimum of five through-the-thickness subdivisions and an aspect ratio below 5 throughout the model. In contrast, in order to avoid excessively small mesh sizes, the thinner layers (solder, TEOS, SiN, polyimide, resin cavity) are modeled using 8-noded HEX-8 solid shell elements (element type #185), which are shell elements with eight-node brick topology that can replace classical shell elements in applications that require double-sided contact. The representative size for the brick elements ranges between 0.2 × 0.2 × 0.04 mm³ (finer mesh, SiC) and 0.2 × 0.25 × 0.38 mm³ (coarser mesh, resin). In order to ensure the adhesion between different layers with non-coherent mesh sizes, the “glued contact” option from Hexagon Marc^® is adopted. Simulations are conducted within the framework of associative plasticity, assuming that all elastoplastic materials are governed by von Mises plasticity. Furthermore, the constitutive behavior of all materials is described exclusively by isotropic hardening.

2.3. Damage

Considering damage, the objective of the study is twofold: to analyze the fundamental damage parameters in elastoplastic materials and to model the purely elastic damage in the resin. Resin is taken as a reference for a general damageable elastic material; the same approach can be followed for the other elastic materials.

In the case of the resin, two different approaches to damage modeling are considered: the modified Tresca (MT) criterion, to be applied to the resin bulk material, and the cohesive zone model (CZM), implemented for simulating interface damage and delamination between the resin and the rest of the package.

2.3.1. Modified Tresca (MT) Criterion

The MT criterion is used to assess material damage in the resin bulk material under multiaxial stress states, focusing on the principal stresses and the hydrostatic stress component. The relationship representing the condition of failure of this criterion is shown in Equation (1), where ${\sigma }_1<{\sigma }_2<{\sigma }_3$ are the principal stresses, ${\sigma }_H$ is the hydrostatic stress, ${\tau }_{cr}$ is the failure shear stress with ${\sigma }_H=0$, and $\mathrm{\mu }$ is the coefficient of internal friction. Note that in the case of ${\sigma }_H=0$, it is the classic Tresca criterion.

$${\displaystyle \frac{\left|{\sigma }_1-{\sigma }_3\right|}{2}}={\tau }_{cr}+\mathrm{\mu }{\sigma }_H$$

(1)

The two parameters to calibrate are $\mathrm{\mu }$ and ${\tau }_{cr}$, which govern the shape and the size of the failure locus, respectively.

To calibrate the shape parameter $\mathrm{\mu }$ for the resin at hand, the works of Morelle et al. [13] and Hu et al. [12] have been taken into consideration. In particular, Equation (2) has been considered.

$$\left\{\begin{array}{c}\left({\displaystyle \frac{\left|{\sigma }_1\right|}{2}}-{\tau }_{cr}\right)·{\displaystyle \frac{3}{+\left|{\sigma }_1\right|}}={\mu }_{tension } Tensile load{ \sigma }_1>0\\ \left({\displaystyle \frac{\left|{\sigma }_1\right|}{2}}-{\tau }_{cr}\right)·{\displaystyle \frac{3}{-\left|{\sigma }_1\right|}}={\mu }_{compression} Compression load{ \sigma }_1<0\end{array}\right.$$

(2)

In principle, these two equations could apply to two data sets from the literature (a total of four equations) if such experimental results perfectly agreed with the modified Tresca approach for the same material. However, this is not the case, and a single value of $\mathrm{\mu }$ cannot satisfy all four equations. Therefore, a value of $\mathrm{\mu }$ is found that minimizes the approximation error by considering all the experimental data available from Morelle et al. [13] and Hu et al. [12], according to the Excel^® Solver minimization (Generalized Reduced Gradient, Nonlinear). The obtained value is reported in

Table 1. Shape parameter $\mathbf{\mu }$ from Morelle et al. [13], from Hu et al. [12], and used in this work.

	Morelle et al. [13]	Hu et al. [12]	Present Work
$\mathrm{\mu }$	−0.441	−0.200	−0.275

together with the ones from the considered literature references. Figure 4 shows all the considered failure data normalized with respect to the uniaxial tensile critical value${\sigma }_{cr}$, i.e., the experimental failure data from Morelle et al. (red squares) and Hu et al. (blue circles) and the MT criterion failure locus according to Morelle et al. (dashed red line), Hu et al. (blue dotted line), and the present work by Mirone et al. (black solid line).

Having defined the shape parameter $\mathrm{\mu }$, only the size parameter ${\tau }_{cr}$ remains as the single tunable parameter. Moreover, given $\mathrm{\mu }$, a value of the failure shear stress ${\tau }_{cr}$ corresponds uniquely to a single value of uniaxial tensile critical value${\sigma }_{cr}$. Consequently, the size of the failure locus can be defined as needed by the value of ${\tau }_{cr}$ or ${\sigma }_{cr}$. The calibration of the size of the failure locus will be discussed in the following sections.

The MT criterion has been implemented in the commercial FEM software Hexagon Marc^® via Fortran subroutine.

2.3.2. Cohesive Zone Model (CZM)

The CZM is commonly employed to simulate delamination at material interfaces, a common failure mode in power electronics packages. According to CZM, the interface element reacts as a non-linear spring in which the reaction force per unit interface area, also called traction, depends on the relative displacements between the upper and the lower edges of the element, also called separation. In this work, the implemented model uses a bi-linear traction-separation law to represent the interaction between the resin and all the rest of the package. Brick elements with 8 nodes have been placed at the interfaces between the resin and the materials in contact with it, acting as cohesive elements, in which the desired CZM characteristics can be set. The CZM is natively implemented in the FEM software Hexagon Marc^®. All the calibration parameters related to Failure Mode I and II for the interface at hand between the resin and all the rest of the package have been set according to the work of Krieger et al. [23] and are shown in

Table 2. CZM parameters according to Krieger et al. [23], where ${\sigma }_{max}$ is the maximum traction stress, ${\delta }_{C }$ is the maximum opening displacement, and ${\delta }^{*}$ is the critical opening displacement.

	Mode I	Mode II
${\sigma }_{max} \left[\mathrm{M}\mathrm{P}\mathrm{a}\right]$	30	400
${\delta }_{C }\left[\mathsf{\mu }\mathrm{m}\right]$	2.32	11.08
${\delta }^{*}\left[\mathsf{\mu }\mathrm{m}\right]$	0.232	1.108

.

3. Results and Discussion

3.1. Single Thermal Cycle: Elastic vs. Elastoplastic Simulations

The analysis of a single thermal cycle was performed, comparing the results obtained from the simulation with all purely elastic materials with the results from the one with copper, solder, and polyimide considered as elastoplastic materials.

3.1.1. Critical Interfaces

In Figure 5a, the critical interfaces analyzed are identified, specifically those where at least one of the two interfacing materials was also simulated as elastoplastic. In Figure 5b, the stresses of an interface element are schematically shown. In particular, the three stresses that the two interfacing materials have in common are indicated in blue, while the three different ones are shown in black. Consequently, for a complete analysis of each interface, a total of nine stresses should be displayed, namely the three common ones and the three different ones for the two materials. In our analysis, for better readability, we reduced the displayed data by considering only ${\sigma }_{11}$ between the normal stresses ${\sigma }_{11}$ and ${\sigma }_{33}$, as they are very similar due to the system geometry. In particular, ${\sigma }_{11}$ will be called in the text “in-plane stress,” as it is parallel to the horizontal plane 1–3 of the global frame of reference. Moreover, we neglected the shear stress ${\tau }_{13}$ because it is very small compared to the others.

In Figure 6, the obtained results are shown in terms of stresses path plots in the considered interfaces at the end of the heating phase and of the heating + cooling entire cycle. The blue arrows in the graphs have been added to give a spatial reference along a generic path. The term “Frame” in the graphs refers to the assembly TEOS-SiN-polyimide. The analysis of stress distributions during a single thermal cycle reveals significant differences between elastic and elastoplastic material models.

The purely elastic model tends to overestimate the stresses at the end of the heating phase with higher peaks, as it does not account for the plastic deformation that materials undergo at elevated temperatures.

On the other hand, the elastic model obviously predicts no residual stresses after cooling, which is the main limitation in the adoption of linear-elastic analyses for the study of such problems. This is evident across all the analyzed interfaces.

In contrast, the elastoplastic model provides a more accurate representation of the materials’ behavior. At the end of the heating phase, accounting for plastic deformation, this model leads to lower stress levels compared to the elastic model. Moreover, the elastoplastic model captures residual stresses that remain after the cooling phase, which are critical for assessing the long-term reliability of the package.

It is useful to underline that the stress overestimation of the elastic model can lead to peak values higher than the actual admissible tension of the material at hand, as in the case of the SiC in the SiC–solder interface. These results highlight the importance of incorporating elastoplastic material characteristics to predict the actual stress state of power electronics packages under thermal cycling.

3.1.2. Critical Points

From the previous analysis of the interfaces, the critical points of each of them were identified and are shown in Figure 7. At these points, the histories during the entire thermal cycle of von Mises, maximum shear, and in-plane stresses were analyzed, once again comparing the results obtained from the simulation with all materials considered elastic to those obtained by considering copper, polyimide, and solder as elastoplastic.

The obtained results are shown in Figure 8. The analysis of stress evolution during a single thermal cycle, focusing on critical points in the package, reveals key insights into the differences between elastic and elastoplastic material behavior. The elastic model consistently overestimates maximum stresses, resulting in all cases in a triangular stress pattern with no residual stress once the cycle is completed, following the triangular trend of the imposed temperature. The stress overestimation is due to the model’s assumption that the material does not undergo any plastic deformation, which is not representative of real-world conditions.

In contrast, the elastoplastic model shows that stresses do not follow the triangular temperature trend. Instead, the stresses in the package accumulate with various trends and leave residual stresses after the cooling phase, in the elastic materials resin and SiC as well as in the elastoplastic materials. These residual stresses are crucial because they can lead to progressive damage in the package over multiple thermal cycles. The presence of residual stresses is particularly evident in the resin–polyimide and SiN–polyimide interfaces.

The agreement between the elastic and elastoplastic analyses is maintained up to very low plastic strains, beyond which Figure 8 shows the onset of divergence between the elastic and elastoplastic responses.

Given these results, it is once again clear that elastoplastic modeling provides a more realistic assessment of the package’s durability under operational conditions, correctly predicting residual stresses and understanding their evolution over time. The following analyses referring to a series of cycles are then conducted considering only the elastoplastic model.

3.2. Multiple Thermal Cycles: Elastoplastic Simulation

The analysis of plastic strain accumulation over a series of thermal cycles reveals significant insights into the long-term behavior of materials used in power electronics packages. Recalling that all simulations have been performed under the hypothesis of isotropic hardening, in Figure 9, the histories of plastic strain at the critical points of the elastoplastic bodies considering ten consecutive thermal cycles are shown. This shows that plastic deformation evolves differently across various critical points in the package, with some regions experiencing more pronounced deformation than others.

In the case of copper, the three critical points exhibit practically the same evolution of plastic strain, as the critical area appears to be quite uniformly deformed, which tends to saturate from the beginning. Moreover, the maximum strain is relatively low, around 0.02, and far from concerning levels in terms of damage.

In the case of polyimide, in all the three critical points, the plastic strain increases sharply in the first cycle before gradually approaching saturation. In this case, there is a significant difference between the two most deformed points and the third one, highlighting a marked gradient in the deformation from the corner inward. In this case, the maximum plastic deformation seems to saturate at a consistent level of about 0.25, which is to be monitored regarding damage initiation.

The solder material is highlighted as the critical component due to its substantial plastic deformation. The analysis demonstrates that the deformation is not uniform across the solder joint, leading to very consistent strain gradients. The maximum strain achieved in the corner point is about 1.1, and the history curve is still growing at a remarkable pace at the end of the tenth cycle. It must be noted that this trend can be due to an approximate constitutive curve of the solder, as will be demonstrated in the following.

Figure 10 shows the path plots of plastic strain, cycle by cycle, along the diagonal of the elastoplastic bodies. By analyzing the distribution of plastic deformation across multiple thermal cycles, it can be seen that there are significant differences in how deformation accumulates in various regions of the power electronics package. In all the considered bodies, plastic deformation is not uniformly distributed.

In the case of copper, there are very low strains concentrated in the central area of the package. The growth is saturating with increasing cycles. Solder is confirmed to be the critical body, with very high strains still showing significant growth and with a very consistent strain gradient. In the case of polyimide, there are high deformations, but they are almost fully saturated, with a large deformation gradient between the first two nodes and the rest. The findings from this analysis suggest that certain materials within the package, particularly the solder, are more susceptible to plastic deformation and are therefore more prone to failure under repeated thermal cycling.

Influence of the Solder Constitutive Curve

Since solder is the critical body and given the uncertainty related to its constitutive curve, three more simulations of the ten consecutive cycles were carried out, implementing three alternative constitutive curves of the solder material. The comparison between the different alternative curves could also, in some way, provide indications of a hypothetical effect of temperature on the behavior of the material. The alternative curves were obtained by applying small changes to the original literature curve in terms of amplitude and slope, ensuring that they remained plausible curves.

Moreover, given the particular nature of the material, similar to a paste depending on the temperature, the possibility that its behavior becomes perfectly plastic beyond a certain deformation was considered. These curves, compared in Figure 11a to the reference one (REF), are depicted as flat (FL), increased slope + flat (SL + FL), and shifted (SH). The FL curve is obtained by flattening the reference curve from a strain of about 0.23. The SL + FL curve is obtained by increasing the slope of the reference curve and flattening it from a strain of about 0.23. The SH curve is obtained by simply shifting the reference curve upwards by about 15 MPa. The results of the alternative simulations are first analyzed in terms of history of plastic strain in the critical point of the solder (Figure 11b) and path plot of plastic strain, cycle by cycle, along the diagonal of the solder (Figure 11c).

This analysis emphasizes the importance of the solder material’s constitutive curve. Its shape and characteristics significantly impact how plastic deformation accumulates over repeated thermal cycles.

A flatter constitutive curve (FL) leads to a more pronounced and rapid accumulation of plastic deformation, which continues to grow at increasing speed. Then, a material with a similar curve can significantly increase the risk of damage initiation and propagation, making it a critical factor in the reliability of the package.

The SL + FL curve is characterized by a steeper first part in respect to the reference one. This factor induces a more gradual increase in plastic deformation. However, when the SL + FL curve becomes flat, the strain starts to grow faster and faster and, at the end of the tenth cycle, with a higher slope than the reference one.

Looking at the SH curve results, it is possible to see that materials with higher yield strength but the same curve slope induce less severe deformation and lower potential for damage accumulation.

These findings highlight the importance of selecting solder materials with appropriate constitutive properties to enhance the reliability of power electronics packages. By choosing materials that exhibit less plastic deformation under thermal cycling, it is possible to reduce the risk of damage and improve the long-term performance of these packages. On the other hand, this analysis also highlights the need for accurate material characterization in the design phase to ensure that FEM simulations can predict real-world behavior effectively.

3.3. Damage: Elastoplastic Materials

The analysis presented in Figure 12 focuses on the evolution of plastic strain and triaxiality factor (TF), which is the measure of the ratio of hydrostatic stress to equivalent stress, i.e., $TF={\sigma }_H/{\sigma }_{eq}$, in copper and polyimide within the power electronics package during thermal cycling, considering the reference solder constitutive curve. TF provides useful insight into the materials’ susceptibility to damage. Understanding how TF and plastic deformation evolve is essential for predicting the long-term reliability of the package, as it highlights the areas most vulnerable to failure.

For copper, the TF exhibits a behavior that fluctuates around zero, with TF values becoming negative during the heating phase and positive during the cooling phase. This pattern indicates that copper experiences compressive stresses during heating, which can reduce the likelihood of immediate damage, but these stresses shift to tensile during cooling, which increases the risk of damage initiation. Despite these fluctuations, the maximum and minimum levels of TF in copper remain relatively stable across the thermal cycles, with significant variations only occurring at specific peaks.

In the case of polyimide, the TF remains consistently positive, with the highest peaks occurring during the cooling phases of the thermal cycles. These peaks tend to increase with the number of cycles. This behavior is particularly significant because it may suggest that even polyimide can accumulate damage over time due to thermal cycling.

The analysis shown in Figure 13 focuses on the impact of different solder constitutive curves on the history of TF and plastic deformation at the critical point of the solder body.

Considering the reference curve, TF fluctuates around zero, with its value becoming negative during the heating phases and positive during the cooling phases. The maximum and minimum values, after an initial decreasing period, tend to increase with the number of cycles, posing the problem of the accumulation of damage with time.

The alternative flat solder curve results in significantly higher plastic strain, as previously shown, but with TF peaks that are slightly lower compared to the reference curve ones and that seem to remain stable with the cycles. However, the greatly increased plastic deformation can lead to a greater likelihood of damage initiation.

Solder with the slope + flat constitutive curve exhibits lower and higher plastic strain growth compared to the reference curve in the first and second simulation parts, respectively, as already shown before. Conversely, in the corresponding simulation parts, the TF peaks are higher and lower compared to the reference curve ones.

The shifted curve presents both plastic strain levels and TF peaks lower than the reference curve.

This study also reveals useful insights into the relationship between the solder material’s constitutive curve, plastic strain, and TF. At a given curve height, modifications that lead to lower strain growth tend to result in higher peaks of TF, and, conversely, those that increase strain tend to reduce TF peaks. Moreover, increasing the overall height of the constitutive curve not only reduces the strain growth but also diminishes the TF peaks.

Figure 14 shows the TF vs. plastic strain histories in the critical points of solder, copper, and polyimide considering different solder constitutive curves. Such graphs are meaningful because, as a general reference concept, when TF is greater than zero, the area under the TF vs. plastic strain curve is proportional to plastic damage that accumulates in the material.

The results indicate that the levels of plastic deformation vary greatly between different materials and that these levels depend significantly on the shape of the solder’s constitutive curve for all the materials considered. For instance, the flat curve in the solder tends to increase the deformation in polyimide and reduce it in copper, while increasing it in the solder itself. Conversely, the shifted curve does exactly the opposite. It is worth noting that in the solder material, low or negative TF values may justify the very high strain observed, which may be tolerated without damage.

The study also shows that the solder’s constitutive curve has a profound effect on the strain distribution across all materials, demonstrating how the mechanical behavior of one material can directly influence another within the same package. The interdependence observed between the different materials emphasizes the need for a holistic approach to material selection and package design and the importance of having an accurate description of the mechanical behavior of the involved materials, including their plastic behavior, to have reliable simulations. Moreover, the results can provide useful information for solder manufacturers, as for these applications, it might be important to adjust the material properties in order to induce significant changes in the evolution of plastic deformation and damage-related variables in the solder itself and in the entire package.

3.4. Damage: Elastic Materials

The study now focuses on the modeling of elastic damage of the resin within the power electronics package, specifically utilizing and comparing the modified Tresca (MT) criterion and the cohesive zone model (CZM). As already pointed out, resin is taken as a reference for a general damageable elastic material; the same approach can be followed for the other elastic materials.

Considering the MT criterion first, as mentioned previously, of the two calibration parameters, the shape parameter $\mathrm{\mu }$ and the size parameter ${\tau }_{cr}$, the first one was defined by minimizing the approximation error while considering all the experimental data provided by Morelle et al. [13] and Hu et al. [12], whereas their experimental stresses were normalized by the respective ${\tau }_{cr}$. The value obtained for $\mathrm{\mu }$ according to this procedure is 0.275, as reported in Table 1.

For the size parameter ${\tau }_{cr}$, we initially adopted the value by Hu et al. [12], that is, a value of ${\tau }_{cr}$ equal to 43.5 MPa, associated with a ${\sigma }_{cr}$ equal to 73.5 MPa.

The resulting failure locus is shown in Figure 15, together with the ones by Morelle et al. [13] and Hu et al. [12].

Considering the path shown in the left part of Figure 16, the obtained damage path plot at the end of the first cycle is shown in the right part of Figure 16. The damage variable is calculated by normalizing the actual shear stress from Equation (1) with respect to the critical shear stress ${\tau }_{cr}=43.5\mathrm{M}\mathrm{P}\mathrm{a}$ previously discussed. Therefore, the material failure is achieved for values of damage equal or greater than 1. As shown in the figure, considering these calibration parameters, resin failure occurs after a single thermal cycle, but only around the corners corresponding to the changes in interfaces. It should be noted that the resin body mesh is not highly refined around these corners, unlike the TEOS, SiN, and polyimide meshes. Therefore, a mesh dependency issue may be present other than the usual critical aspect of modeling sharp edges and materials discontinuities [24].

The damage assessments are based on the stresses over different materials at each interface, always delivered by the glued contact. In order to check the accuracy of the contact algorithm, normal stress to the interface (${\sigma }_{22}$) and two shear stresses lying on the interface plane (${\sigma }_{12}$, and ${\sigma }_{23}$), are compared in Figure 17 for both materials along the reference path of the previous Figure 16 belonging to the resin–sic and resin–polyimide interfaces, at the moment of initial failure (30 °C heating phase) and at the end of the first cycle, according to the fully elastoplastic approach. In fact, according to continuity and equilibrium considerations, at the interface, the above stresses should be identical for both materials, whereas the three remaining stress components (in-plane normal stresses ${\sigma }_{11}$ and ${\sigma }_{33}$ and out-of-plane shear stress ${\sigma }_{13}$) may be discontinuous.

Figure 17 shows a reasonable continuity of stresses ${\sigma }_{22}$ ${\sigma }_{12}$, and ${\sigma }_{23}$ across the resin–sic interface, while remarkable discontinuities are found across the resin–poly one. This is likely attributed to the pronounced difference of tangent stiffness between the two latter materials and suggests that improvements in the modeling of contact might be pursued in the future.

Despite the above considerations, we now want to evaluate what value ${\sigma }_{cr}$ should have, i.e., which size should be the failure locus, to prevent the resin from failing after one cycle, meaning that the damage value does not reach one at any node. For this purpose, Figure 18 shows the historical plots of damage in four different nodes of the resin located in the previously identified critical area, varying ${\sigma }_{cr}$. It can be observed that for all points and all ${\sigma }_{cr}$ values, the damage increases only during the heating phase of the first cycle and then remains constant at that value as the cycles increase. Furthermore, a critical sigma value of 181 MPa can be identified, as for this value, the most critical point reaches a damage value exactly equal to one. This information could be useful for resin manufacturers, as it would indicate the minimum critical stress needed to prevent the resin from breaking in this type of application.

Having identified the ${\sigma }_{cr}$ limit value related to the MT criterion, we now want to compare the implementation of this damage model in the resin bulk material with the cohesive zone model applied to the interface between the resin and the rest of the package, calibrated according to Krieger et al. [23], as mentioned previously.

Figure 19 shows the comparison between the two approaches in terms of damage contour plots at the end of the first heating and at the end of the first cycle in both elastic and elastoplastic simulations. Analyzing the figure, it can be observed that with both models, and in both elastic and elastoplastic simulations, there are no substantial differences between the contour plots at the end of the first heating phase and the end of the first cycle, indicating that all the damage occurs during the initial heating. On the other hand, the two models yield qualitatively different results in two aspects. First, in the case of the MT criterion, a higher damage value is reached in the elastoplastic simulation compared to the elastic one, contrary to the results obtained with the CZM. Additionally, it can be noted that the location of maximum damage differs between the two cases. Quantitatively, the differences are evident, with the maximum damage values being 0.875/1 for the MT criterion and 0.02/0 for the CZM in the elastic/elastoplastic simulations. It is important to highlight that the CZ model addresses the resistance to failure at the resin interface, while the MT model governs the failure properties of the resin bulk material.

In fact, while the outcome of CZM is just limited at discontinuous interfaces, the MT damage prediction is not taken into account at such interfaces where material properties and stresses are discontinuous but is used for predicting damage of bulk material within layers.

These considerations, together with the knowledge that qualification tests for similar electronic packages aim at ensuring electrical operativity up to a few thousands of thermomechanical cycles, imply that the above predictions of the CZM and MT approaches give insight into their soundness.

In fact, the failure of electrical operativity of the package reflects mechanical failure occurring independently of it either at the interface between two layers or within the bulk material of a single layer. Then, the zero-damage prediction of the elastic + CZM approach stands out as the less reliable option among the four evaluated.

Among the three remaining combinations of modeled stress-strain response and failure prediction (elastic + MT, elastoplastic + CZM, and elastoplastic + MT), the most sounding one should be that predicting failure around 3000 or 4000 cycles, no matter whether it occurs at an interface or within a layer.

Reliable numerical estimates at such high numbers of cycles would require finite elements simulation extended up to at least hundreds of cycles, much heavier than those discussed here and exceeding the demonstrative scope of this work.

When commenting on these differences in results, it is important to consider that the two models were calibrated independently of each other, even if done realistically, and not with respect to the same experimental references. Moreover, as previously mentioned, using the CZM requires the insertion of a layer of interface elements between the resin and the rest of the package. Consequently, the mesh used to implement the two different models is not exactly the same. Despite this, the differences in the results are evident both qualitatively and quantitatively, highlighting the critical challenges related to simulating damage phenomena in these particularly complex applications. The results are strongly influenced by the modeling choices made. Therefore, to determine the approach that most closely aligns with reality for a given application, it would be necessary to have experimental data that is as detailed as possible.

In order to analyze the damage evolution, however small, in the simulations with the CZM, Figure 20 shows the historical plots of damage in two critical points of the resin in both the elastic and elastoplastic simulations. Analyzing the figure, it can be seen that in the elastic analysis, the damage remains exactly zero in both points, which is supposed to be slightly unrealistic.

In the case of the elastoplastic analysis, at the point previously identified as the most critical, the damage reaches a value of around 0.02, which is still very low but more physically meaningful. It is noteworthy that after the first heating cycle, the damage value already reaches approximately 0.015, once again demonstrating that with this approach, as in the case of the MT criterion, the damage occurs almost entirely during the first heating cycle.

4. Conclusions

This study focused on the accurate simulation of the thermomechanical behavior and damage inception of power electronics packages under thermal cycling.

The results demonstrated that even when considering a single thermal cycle, purely elastic models tend to significantly overestimate stress levels at the end of the heating phase. This overestimation can lead to unrealistic predictions of stress peaks, particularly at critical interfaces, which may result in inaccurate assessments of the package’s reliability. On the other hand, the elastoplastic model provided a more accurate representation by capturing both lower stresses at the end of the heating phase due to the plastic deformation and the resulting residual stresses after the cooling phase. These residual stresses, absent in the elastic model predictions, are essential for understanding the potential for cumulative damage over multiple thermal cycles. In addition, the ability of the elastoplastic model to predict these residual stresses is crucial for accurately assessing the long-term durability and reliability of power electronics packages.

Considering multiple consecutive thermal cycles, the study analyzed the plastic strain distribution and evolution in the elastoplastic materials, highlighting very different trends in plastic strain accumulation, with significant implications for their durability. Copper showed relatively low and uniform plastic strain (maximum strain 0.02), indicating a stable response over time with low damage risk. In polyimide, the maximum plastic strain increased sharply during the initial cycles before gradually approaching saturation at a considerable level (around 0.25), with a significant spatial gradient in strain, particularly from the corner areas inward. These findings suggest that polyimide may be susceptible to damage over time. Nevertheless, solder emerged as the most critical material due to its substantial and uneven plastic deformation (maximum strain around 1.2), particularly at specific critical points. These results highlight the vulnerability of solder to damage propagation under continued thermal cycling, raising concerns about its long-term reliability. However, the solder results may be due to its approximate constitutive curve. Indeed, it was demonstrated that the shape and characteristics of its constitutive curve have a profound impact on how plastic deformation accumulates. Modifications to the solder curve can either exacerbate or mitigate the rate of strain growth and the corresponding damage potential. This finding underscores the need for accurate characterization of the materials’ mechanical behavior, including the plastic curve, in order to have reliable simulations and, in turn, damage predictions to enhance the overall reliability of power electronics packages.

This study also analyzed the distribution and evolution of the triaxiality factor TF in the elastoplastic materials, which is, together the plastic strain, one of the most important parameters related to ductile damage. For copper, the TF was observed to fluctuate around zero, becoming negative during the heating phases and positive during the cooling phases. Despite these fluctuations, the TF levels in copper remained relatively stable across cycles, with significant variations occurring only at specific peaks. This stability, together with the low plastic strain level, suggests that copper is resilient under thermal cycling. Polyimide, on the other hand, consistently exhibited positive TF values, with the highest peaks occurring during the cooling phases of the thermal cycles. These peaks increased with the number of cycles, indicating a growing susceptibility to damage over time. This behavior underscores the importance of monitoring the evolution of TF and plastic deformation in polyimide to predict potential failure points. Regarding solder, the influence of its constitutive curve on TF and plastic strain distribution and evolution was examined, not only in the solder material but in the entire package. Considering the solder itself, it was found that at a given curve height, modifications that led to lower strain growth tended to result in higher TF peaks, and conversely, those that increased strain growth resulted in lower TF peaks. Moreover, increasing the overall height of the constitutive curve effectively reduced both strain growth and TF peaks, providing a strategy for optimizing material properties to minimize damage risks. Additionally, it was found that the solder’s constitutive curve has a profound effect on the strain distribution across all materials, demonstrating how the mechanical behavior of one material can directly influence another within the same package. Moreover, the results can offer valuable insights for solder manufacturers, as adjusting material properties in these applications could significantly impact the evolution of plastic deformation and damage variables in both the solder and the entire package.

Finally, the study focused on the modeling of elastic damage within the resin of power electronics packages, comparing the usage of the modified Tresca (MT) criterion applied to the bulk material and the cohesive zone model (CZM) applied to an ad hoc layer of interface elements interposed between the resin and the rest of the package. Resin was taken as a reference for a general damageable elastic material; the same approach can be followed for the other elastic materials. Making use of MT criterion, the provided information could benefit resin manufacturers by highlighting the minimum critical stress required to avoid resin failure in these types of applications. Moreover, the findings underscored the complexities and critical considerations involved in accurately predicting damage in elastic materials under thermal cycling. Both damage models indicated that significant damage occurs during the initial heating phase, with minimal additional damage in subsequent cycles. However, the differences in the results between the two approaches were evident both qualitatively and quantitatively, being strongly influenced by the modeling choices made. The analysis also pointed out potential issues related to mesh dependency, where stress concentrations at sharp corners and material discontinuities might be artificially amplified, influencing the damage predictions.

In conclusion, the framework provided by this study offers a valuable foundation for optimizing the design and enhancing the longevity of power electronics, highlighting the fundamental pillars and critical aspects in the modeling and simulation of these complex devices.