Measurements and FEM Analyses of Strain Distribution in Small Sn Specimens with Few Crystal Grains

Abstract

Soldering is used to bond a semiconductor chip to a print circuit board (PCB). It is known that Sn, which is the base metal of Pb-free solder, shows remarkable crystal anisotropy. Clarifying the effect of Sn anisotropy on strain distribution is important for lifetime evaluation. The strain distribution in a micro specimen was measured under a tensile test by a digital image correlation method (DICM) with a microscope. Strain distributions were also analyzed by the finite element method with Hill’s anisotropic yield criterion and the crystal plasticity finite element analysis (CPFEA) with considering the critical resolves shear stress (CRSS) of each slip system. The deformation of the crystal structure of β-Sn depends on the size, number, and orientation of crystal grains. The CRSS was noticeably different for each slip system, and the yield stress varied with the orientation of crystal grains. Although the CPFEA without considering strain hardening was effective for predicting deformation within crystal grains, it is necessary to consider the strain hardening of crystals to predict the stress-strain curve of a micro specimen.

Fig. 10 Microphotographs taken by an optical microscope (top), distribution of equivalent strain measured by the DICM (middle), and calculated by the CPFEA (bottom) of seven specimens.¹⁵⁾

1. Introduction

Electronic devices must be mounted on printed circuit boards to connect outer circuits.¹⁾ For many years, Sn–Pb solder alloys were widely used for bonding printed circuit board and electronic components. However, the use of specific hazardous substances in the production of electronic devices is currently limited. Pb is prohibited for use in electronic devices with a few exceptions. Now, Pb-free solder alloys based on Sn have largely replaced Sn–Pb solder alloys. The reliability of the solder joint is one of the main issues in electronic equipment. In recent years, Pb-free solder alloys such as Sn–Ag–Cu, Sn–Zn–Bi, Sn–Cu and Sn–Bi have been used as solder materials.²⁾ β-Sn is the main component of these solders. Since β-Sn has a complex tetragonal structure, its crystal properties have not been sufficiently clarified. Several active slip systems have been reported by experiments, but not all have been elucidated.³⁾ In addition, few studies have proposed the critical resolved shear stress (CRSS) for each sliding system.

The structural reliability of solder joints is currently evaluated by assuming a homogeneous material. However, we cannot ignore the anisotropy of respective crystal grains in a small solder ball, which has only a few grains. It is well known that Sn crystal exhibits remarkable anisotropy.⁴⁾ Therefore, it is important to clarify the influence of the anisotropy of respective crystal grains on the strength of the small solder structure.

Kariya et al. performed experiments investigating the anisotropy of Sn crystals. The slip systems of Sn crystal are very complicated, but they proposed a simplification using Hill’s orthotropic theory.⁵⁾ We analyzed the strain distribution in miniture Sn solder specimens using the finite element method (FEM) with Hill’s orthotropic theory. Furthermore, we analyzed the specimens using the crystal plasticity model with the finite element analysis (CPFEA). Some researchers have used the CPFEA to analyze the strain distribution in crystal grains of aluminum.^6–9) In this study, we used the CPFEA for miniature specimens of β-Sn with few crystal grains.

We performed tensile tests of small specimens of β-Sn using a micro-tester (Saginomiya Co., Ltd.) under a confocal laser scanning microscope (CLSM), and measured the distribution of strain in miniature specimens using the digital image correlation method (DICM).¹⁰⁾ We determined the CRSS of respective slip systems of the β-Sn crystal, and used these CRSS for the CPFEM.

We compared the analyzed strain distributions of miniature specimens using the FEM with Hill’s anisotropic elastoplasticity theory, and the CPFEA with measures determined experimentally using the DICM with a CLSM.

2. Experimental Procedure

2.1 Specimen

A miniature specimen was produced from Sn wire with a purity of 99.995%. The size of the specimen was 2 mm in gauge length, 0.5 mm in width, and 0.3 mm in thickness. Specimen dimensions are shown in Fig. 1.

Fig. 1

Schematic diagram of a miniature specimen.

The Sn wire was inserted into the metal mold and pressurized at room temperature. The heat treatment was then carried out for 3 h at 85% of the melting point to remove residual stress. The fabrication flow of a miniature specimen is shown in Fig. 2.

Fig. 2

Fabrication flow of a miniature specimen.

2.2 Experimental method

The specimen surface was mirror polished and the crystal orientation was measured using electron backscatter diffraction (EBSD) with a scanning electron microscope (SEM). Tensile tests of miniature specimens were conducted under a CLSM as shown in Fig. 3. We stretched miniature specimens up to 7.5% strain under a strain rate of 1.0 × 10⁻⁴/s, taking microphotographs of the surfaces of specimens every 10 s. The distribution of the strain on the surfaces of the specimens was measured using the DICM from the microphotographs. We also observed miniature specimens to detect the slip lines in respective crystal grains before and after the tensile tests using an optical microscope.

Fig. 3

Loading test machine under a CLSM.

3. FEM Analyses

3.1 Analysis model

We performed FEM analyses of the miniature specimens using commercial FEM software (MSC.marc2014.2). We modeled respective crystal grains of β-Sn from EBSD images of miniature specimens using tetrahedral meshes. In these models, we assumed homogeneous crystal orientation for the thickness directions of these specimens due to the thinness of samples.

Figures 4 and 5 show the boundary condition of the FE model and a sample of the FE mesh, respectively.

Fig. 4

Schematic diagram of an analysis model of a β-Sn specimen.

Fig. 5

FE mesh of specimen E.

3.2 Hill’s criterion of anisotropic elastic plasticity¹¹⁾

We used the yield function proposed by Hill¹¹⁾ to perform anisotropic elastic plastic analysis. This theory is well-known as the yield criterion for orthotropic materials. The yield criterion for general materials is expressed as   

\begin{equation} f(\sigma_{11},\sigma_{22},\sigma_{33},\tau_{12},\tau_{23},\tau_{31}) = k = \frac{1}{\sqrt{3}}\sigma_{Y}, \end{equation} (1)

where f is the function of the stress tensor, and k is the material constant determined through the experiment. Von Mises yield criterion is given as   

\begin{align} 2f(\sigma_{ij})&\equiv(\sigma_{33} - \sigma_{11})^{2} + (\sigma_{11} - \sigma_{22})^{2} \\ &\quad + (\sigma_{22} - \sigma_{33})^{2} + 6(\tau_{23}^{2} + \tau_{31}^{2} + \tau_{12}^{2})^{2} = 2\sigma_{Y}^{2} \end{align} (2)

Dividing both sides with $2\sigma _{Y}^{2}$, we obtain Hill’s yield criterion.   

\begin{align} 2f &\equiv F(\sigma_{22} - \sigma_{33})^{2} + G(\sigma_{33} - \sigma_{11})^{2} \\ &\quad + H(\sigma_{11} - \sigma_{22})^{2} + 2L\tau_{23}^{2} + 2M\tau_{31}^{2} + 2N\tau_{12}^{2} = 1 \end{align} (3)

where F, G, H, L, M, and N are the Hill’s anisotropic constants. The yield stress ratios are defined as follows.   

\begin{equation} F = G = H = \frac{L}{3} = \frac{M}{3} = \frac{N}{3} = \frac{1}{2\sigma_{Y}^{2}}. \end{equation} (4)

If $\sigma _{1}^{Y}$, $\sigma _{2}^{Y}$, $\sigma _{3}^{Y}$ are the tensile yield stresses for respective orthotropic crystal axes, we obtain   

\begin{align} \frac{1}{(\sigma_{1}^{Y})^{2}} &= G + H,\quad F = \frac{1}{2}\left[\frac{1}{(\sigma_{2}^{Y})^{2}} + \frac{1}{(\sigma_{3}^{Y})^{2}} - \frac{1}{(\sigma_{1}^{Y})^{2}}\right]\\ \frac{1}{(\sigma_{2}^{Y})^{2}} &= H + F,\quad G = \frac{1}{2}\left[\frac{1}{(\sigma_{3}^{Y})^{2}} + \frac{1}{(\sigma_{1}^{Y})^{2}} - \frac{1}{(\sigma_{2}^{Y})^{2}} \right]\\ \frac{1}{(\sigma_{3}^{Y})^{2}} &= F + G,\quad H = \frac{1}{2}\left[\frac{1}{(\sigma_{1}^{Y})^{2}} + \frac{1}{(\sigma_{2}^{Y})^{2}} - \frac{1}{(\sigma_{3}^{Y})^{2}} \right] \end{align} (5)

Similarly, if $\tau _{12}^{Y}$, $\tau _{23}^{Y}$, $\tau _{31}^{Y}$ are the shear yield stresses, we have   

\begin{equation} L = \frac{1}{2(\tau_{23}^{Y})^{2}},\quad M = \frac{1}{2(\tau_{31}^{Y})^{2}},\quad N = \frac{1}{2(\tau_{12}^{Y})^{2}}. \end{equation} (6)

The analysis uses the above equations. However, instead of $\sigma _{1}^{Y}$, $\sigma _{2}^{Y}$, $\sigma _{3}^{Y}$, $\tau _{12}^{Y}$, $\tau _{23}^{Y}$, $\tau _{31}^{Y}$ we used R, as determined from the yield stress ratio in each direction. R_xx, R_yy and R_zz are the direct stress yield ratio, and R_xy, R_yz and R_zx are the shear stress yield ratio as   

\begin{align} R_{xx} &= \frac{\sigma_{1}^{Y}}{\sigma_{0}^{Y}},\quad R_{yy} = \frac{\sigma_{2}^{Y}}{\sigma_{0}^{Y}},\quad R_{zz} = \frac{\sigma_{3}^{Y}}{\sigma_{0}^{Y}},\\ R_{xy} &= \sqrt{3} \frac{\tau_{xy}^{Y}}{\sigma_{0}^{Y}},\quad R_{yz} = \sqrt{3} \frac{\tau_{yz}^{Y}}{\sigma_{0}^{Y}},\quad R_{zx} = \sqrt{3} \frac{\tau_{zx}^{Y}}{\sigma_{0}^{Y}} \end{align} (7)

Here, $\sigma _{0}^{Y}$ is a yield stress that can be decided arbitrarily. We used the parameters (shown in Table 1) as measured by Kariya et al.⁵⁾

Table 1 Material parameters for orthotropic plasticity of Hill’s theory.⁵⁾

3.3 Crystal plasticity model⁹⁾

Plastic deformation occurs by sliding in a specific direction along the sliding direction, as shown in Fig. 6. The crystal plasticity model is a constitutive equation built on the basis of this mechanism. In this constitutive equation, the crystal orientation can be considered directly as the variable expressing the internal state of the material.

Fig. 6

Slip system and slip deformation.

We assumed the micro deformation theory and used the following basic formula. Hook’s law is   

\begin{equation} \dot{\boldsymbol{{\sigma}}} = \mathbf{D}^{e}:(\dot{\varepsilon} - \dot{\varepsilon}^{p}), \end{equation} (8)

where $\dot{\boldsymbol{{\sigma}}}$ is the stress velocity, D^e is the elastic coefficient, $\dot{\varepsilon }$ is the strain rate, and $\dot{\varepsilon }^{p}$ is the inelastic strain rate. The inelastic strain rate is expressed by the following equation.   

\begin{equation} \dot{\varepsilon}^{p} = \sum_{\alpha}\dot{\gamma}^{\alpha}\boldsymbol{{\mu}}^{\alpha}, \end{equation} (9)

where $\boldsymbol{{\mu}}^{\alpha }$ is called the Schmidt tensor and can be expressed by the slip plane m^α and the slip direction s^α of each slip system, and $\dot{\gamma }^{\alpha }$ is the sliding velocity for each slip system.   

\begin{equation} \boldsymbol{{\mu}}^{\alpha } = \frac{1}{2}(\mathbf{s}^{\alpha} \otimes \mathbf{m}^{\alpha} + \mathbf{m}^{\alpha} \otimes \mathbf{s}^{\alpha}). \end{equation} (10)

Sliding speed is driven by decomposition shear stress, τ^α, using time-dependent constitutive formulas:   

\begin{equation} \boldsymbol{{\tau}}^{\alpha} = \boldsymbol{{\sigma:}}\boldsymbol{{\mu}}^{\alpha} \end{equation} (11)

  

\begin{equation} \dot{\gamma}^{\alpha} = \dot{\gamma}_{0}\frac{\boldsymbol{{\tau}}^{\alpha}}{\boldsymbol{{\tau}}_{0}}\left|\frac{\boldsymbol{{\tau}}^{\alpha}}{\boldsymbol{{\tau}}_{0}}\right|^{(n - 1)}, \end{equation} (12)

where $\dot{\gamma }_{0}$ is the standard sliding velocity, τ₀ is the critical resolves shear stress (CRSS), n is the stress index, and α is the slip number. The slip system of β-Sn is shown in Table 2. The CRSS of the five slip systems obtained from the tensile test shown in Fig. 7 is taken into account in the CPFEA.

Table 2 Slip system of β-Sn (slip plane: $\mathbf{m}_{0}^{\alpha }$, slip direction: $\mathbf{s}_{0}^{\alpha }$).

Fig. 7

Critical resolved shear stress (CRSS) measured by the experiment.

4. Results and Discussion

4.1 CRSS of each slip system

For each specimen subjected to the tensile test, we detected the crystal grain with the initial activated slip system from the distribution of strain measured by the DICM. The orientation of the crystal grain was decided by the Schmidt factor measured by the EBSD. The active slip system in the crystal grain was identified from the slip line observed in the microphotograph of the specimen. The Schmidt factor is one of the indicators for finding the main slip system. If all CRSS for slip systems are identical, the slip system with the largest Schmidt factor would be the most slippery.¹²⁾ However, the CRSSs of respective slip systems in β-Sn were different, and the slip system with the largest Schmidt factor might not slip first.²⁾ Therefore, the CRSS of each slip system had to be considered separately.

Figure 7 and Table 3 show the active slip system and CRSS. The CRSS of each slip system was close to the corresponding CRSS reported by Martin et al.¹³⁾ as shown in Table 3. The CRSS of each slip system in Table 3 was different. This is why the yield stress of miniature specimens was scattered over a wide range. The yield stress of miniature specimen is dominated by the slip system that is activated first.

Table 3 CRSS of each slip system calculated from the experiment.

4.2 Finite element analyses with Hill’s elastic plastic yield criterion

Seven specimens from A to G were analyzed using the FEM with Hill’s theory. Figure 8 shows the microphotographs obtained by an optical microscope after the tensile test, the equivalent strain distribution measured by the DICM with a CLSM, and the equivalent strain distribution calculated by the FEM with Hill’s theory. Comparison of the strain distributions obtained by the DICM and FEM with Hill’s theory showed that they are similar in the distributions of specimens D and G. On the other hand, specimens A, B, C, E and F were different. Hill’s anisotropic elasto-plastic theory assumes orthotropic anisotropy. Although β-Sn is a crystal structure close to orthotropic anisotropy, some slip systems are completely different from orthotropic anisotropy. Hill’s theory could predict only the distributions of strain limited crystal grains of β-Sn.

Fig. 8

Microphotographs taken by an optical microscope (top), distribution of equivalent strain measured by the DICM (middle), and calculated by the FEM (bottom) of seven specimens.¹⁵⁾

Figure 9 presents the stress-strain curves measured by the experiment and calculated by Hill’s theory. In the experimental data, the yield stress of specimen F is twice of that of specimen E. The yield stresses of respective specimens obviously depend on the orientations of crystal grains in respective miniature specimens. Assumption of isotropic material is not correct in the small structure of β-Sn with few crystal grains.

Fig. 9

Stress-strain curves measured by the experiment and calculated by the FEM with Hill’s theory.¹⁵⁾

The stress-strain curves calculated by the FEM with Hill’s theory do not correspond to those measured by the experiment. Hill’s theory could not predict the stress-strain curves in small structures of β-Sn with few crystal grains.

4.3 Crystal plasticity model with the finite element analysis (CPFEA)

The elastic constants and material parameters of β-Sn used for the CPFEA are shown in Tables 4 and 5, respectively. The CRSS for each slip system in Table 5 was obtained by the experiment shown in Fig. 7. In this study, five slip systems were considered. Figure 10 shows the microphotographs taken by an optical microscope after the tensile test, the distribution of equivalent strain measured by the DICM with a CLSM, and the distribution of equivalent strain analyzed by the CPFEA. Strain distributions obtained by the CPFEA corresponded qualitatively with those measured by the DICM in the cases of specimens A, C, D, F and G.

Table 4 Elastic constants of β-Sn used in crystal plasticity finite element method.¹⁴⁾

Table 5 Material properties for the crystal plasticity finite element method (FPFEM).

Fig. 10

Microphotographs taken by an optical microscope (top), distribution of equivalent strain measured by the DICM (middle), and calculated by the CPFEA (bottom) of seven specimens.¹⁵⁾

In specimen B, slip lines can be observed in the middle and right grains. In the case of the CPFEA analysis, only the middle grain was deformed. Only five slip systems were considered in the CPFEA, although other slip systems might have existed. However, the CPFEA could predict the distribution of strain in crystal grains more accurately than Hill’s anisotropic theory. The CPFEA is effective in forecasting the deformation of β-Sn crystals, because it can take more slip systems into account than Hill’s anisotropic theory.

Figure 11 shows the stress-strain curves measured by experiments and those analyzed by the CPFEA. The stress analyzed by the CPFEA was much lower than that measured by the experiments. In the experiments, we considered that the work hardening occurred after reaching the CRSS. We did not consider the work hardening in the CPFEA currently. Consideration of more slip systems and their work hardening will improve the accuracy of the CPFEA.

Fig. 11

Stress-strain curves measured by the experiment and calculated by the CPFEM.¹⁵⁾

5. Conclusions

In this study, a tensile test was performed for miniature specimens of β-Sn with few crystal grains. The strain distribution in respective crystal grains was measured using the DICM with a CLSM. We also analyzed the distribution using Hill’s anisotropic theory¹¹⁾ with the FEM and the CPFEA. We summarize our conclusions as follows.

1. (1)    The deformation of the miniature specimens with few crystal grains strongly depends on the deformation of each crystal grain. The difference of yield stresses of miniature specimens is very large. For example, the maximum yield stress is more than twice the minimum yield stress.
2. (2)    Hill’s anisotropic theory with the FEM can predict a few cases of strain distribution in respective crystal grains in miniature specimens under tensile stress. The theory assumes orthotropic anisotropy; however, the anisotropy of the crystal of β-Sn is more complicated than that of the orthotropic crystal.
3. (3)    We measured the CRSS of the main five slip systems of β-Sn, and performed the CPFEA of crystal grains in miniature specimens of β-Sn under tensile stress. CPFEA could describe the distribution of strain in crystal grains in more miniature specimens than Hill’s anisotropic theory with the FEM. The CPFEA is more effective for analyzing the deformation of miniature specimens with few crystal grains for β-Sn.
4. (4)    The CPFEA could not predict the stress-strain curves of miniature specimens under tensile stress currently. Analyzed yield stress was much lower than that obtained by the experiments. The CPFEA in this study did not consider the work hardening of β-Sn slip systems. If we took those into account, we expect it would improve the accuracy of the analyses, including the stress-strain curves.

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