Research on Stress-Strain Relationship of Sn-Rich Phase Based on Nanoindentation Method

Liu Zhigao¹, Liu Tianhan¹, Qin Hongbo¹
¹School of Mechatronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
²Guangdong Welding Institute, Guangzhou 510651, China

Abstract

In electronic packaging applications, lead-free solders are predominantly binary or ternary Sn-based eutectic or near-eutectic alloys, with the Sn-rich phase serving as the matrix phase. This study investigates the mechanical properties and stress-strain relationship of the Sn-rich phase using nanoindentation testing. The elastic modulus and hardness of the Sn-rich phase were obtained through nanoindentation experiments. Finite element inverse analysis was employed to determine the characteristic stress and characteristic strain, while dimensional analysis was used to establish the strain hardening exponent. These parameters were subsequently substituted into a power-law hardening model to calculate the yield strength and ultimately define the complete stress-strain relationship function for the Sn-rich phase.

Keywords: Sn-rich phase; nanoindentation; finite element analysis; inverse analysis; stress-strain relationship

1. Introduction

Electronic packaging broadly refers to the technology of sealing and interconnecting various components of electronic products—including semiconductor transistors or bare chips, substrates, and other packaging materials—according to design requirements to achieve environmental protection and enhanced adaptability. In a narrower sense, electronic packaging specifically denotes the process of sealing semiconductor devices. The weakest link in electronic packaging systems is the solder joint, whose mechanical failure often leads to overall product and equipment failure. With the trend toward miniaturization and multi-functionalization of electronic products, packaging density continues to increase while solder joint dimensions keep decreasing. Typical flip-chip micro-solder joints now reach diameters of 300-500 μm, while ball grid array (BGA) joints range from 700-1000 μm.

The mechanical properties and stress-strain relationships of materials or phases serve as crucial foundations for mechanical behavior characterization and failure analysis of solder joints. Conventional universal testing machines using large specimens cannot adequately represent the micromechanical behavior of materials or phases within micro-solder joints. The emergence of nanoindentation technology has provided an effective means for measuring the mechanical properties of micro-solder joints. Through load-displacement curves obtained from nanoindentation, researchers can determine hardness, elastic modulus, and creep properties of solder materials and their constituent phases.

Previous studies have demonstrated the effectiveness of nanoindentation for characterizing solder materials. Chromik et al. measured the mechanical properties of Cu₃Sn and Cu₆Sn₅ intermetallic compounds using nanoindentation. Xiao et al. investigated the hardness and elastic modulus of intermetallic compounds at Sn-3.0Ag-0.5Cu/Cu joints. Wang et al. studied the effect of Ag₃Sn and Cu₆Sn₅ compounds on the creep performance of bulk solder in micro-joints. Additionally, nanoindentation can be used to construct stress-strain relationship functions for metallic materials. Dao et al. introduced the concepts of characteristic stress and characteristic strain, establishing dimensional functions for plastic properties of metallic materials. Ma et al. subsequently utilized nanoindentation combined with finite element inverse analysis to determine yield strength and strain hardening exponent for materials like TSV-Cu.

In the electronics industry, most developed lead-free solders are based on Sn-rich phases, such as SnAg, SnCu, and SnAgCu alloys. The stress-strain relationship of the Sn-rich matrix phase largely determines the mechanical failure behavior of micro-solder joints. While numerous studies have investigated the stress-strain relationships of lead-free solder alloys, the stress-strain behavior of Sn-rich phases within the solder microstructure remains unclear, hindering microstructural mechanical analysis. Based on this research background, this study focuses on determining the stress-strain relationship function for the Sn-rich single-phase solid solution in typical lead-free solder alloys.

2. Experimental Methods

2.1 Sample Preparation and Nanoindentation Parameters

A Sn-rich single-phase solid solution alloy was prepared for this study. In solder alloys, the Sn-rich phase forms a single-phase solid solution with limited dissolution of other atoms. High-purity raw materials (99.995%) were selected and weighed using a high-precision balance. The materials were melted in a crucible at a controlled furnace temperature, using rosin as an antioxidant flux. After complete melting and stirring, the liquid metal was poured into a steel mold. Upon solidification, the cast Sn-rich solid solution alloy was removed and cut using wire electrical discharge machining.

A cubic specimen with dimensions of approximately 10 mm × 10 mm × 10 mm was extracted for nanoindentation testing. The remaining material was machined into dog-bone shaped tensile specimens according to GB/T 228.1-2010 standard, with a total length of [MATH_0] and a cross-sectional area of [MATH_1] at the gauge length center.

For nanoindentation testing, the cubic specimen was embedded in crystal epoxy resin (mass ratio of resin to hardener 2.0:5.0). After curing, the sample was ground and polished using a series of abrasive papers to prepare a metallographic surface. Nanoindentation tests were performed using an Agilent G200 nanoindenter equipped with a Berkovich diamond indenter. The instrument features a displacement resolution of 0.05 nm and a load resolution of 50 nN. Tests were conducted at five different points on the Sn-rich phase metallographic sample at a constant loading strain rate of 0.05 s⁻¹, with a maximum load of 100 mN.

2.2 Finite Element Simulation

The plastic behavior of metallic materials is commonly described using a power-law hardening model. For isotropic materials, the elastic-plastic stress-strain relationship is expressed as:

σ =
\begin{cases}
Eε & \text{for } ε ≤ ε_y \
Rε^n & \text{for } ε > ε_y
\end{cases}

where E is the elastic modulus, σ_y is the yield strength, R is the strength coefficient, ε is the plastic strain, and n is the strain hardening exponent.

From nanoindentation testing, the elastic modulus E and hardness H can be directly obtained. However, determining the strain hardening exponent n requires dimensional functions. The characteristic stress σ_r and characteristic strain ε_r, which represent a specific point on the stress-strain curve, can be determined through finite element inverse analysis. By iteratively comparing finite element simulation results with experimental nanoindentation load-displacement curves, these parameters can be refined until convergence is achieved.

A two-dimensional axisymmetric finite element model was developed in ANSYS to simulate the nanoindentation loading stage. When structural geometry is axisymmetric, 2D and 3D axisymmetric models provide equivalent accuracy while significantly improving computational efficiency. The model employed PLANE182 elements for the specimen and TARGET169/CONTACT172 elements for the indenter-specimen contact. The Berkovich indenter with a tip angle of 140.6° was equivalently represented as a conical indenter with a half-angle of 70.3°. Local mesh refinement was applied near the contact region to ensure computational accuracy while minimizing simulation time. The model consisted of [MATH_2] elements and [MATH_3] nodes. Symmetry constraints were applied along the axis, and the bottom nodes were fixed in the Y-direction. The specimen was modeled as a 10 μm × 10 μm region to match experimental conditions.

3. Results and Analysis

3.1 Nanoindentation Test Results

Five nanoindentation tests were performed on the Sn-rich phase, yielding an average elastic modulus of 51.3 GPa and an average hardness of 0.31 GPa. The load-displacement curves from all tests showed good repeatability. The average load-displacement curve was fitted to a parabolic function for subsequent analysis.

3.2 Determination of Characteristic Stress

Based on Dao et al.'s theory, when metallic materials exhibit identical characteristic strain and elastic modulus, they produce identical loading stage load-displacement curves for a given strain hardening exponent n. Using the inverse analysis method proposed by Ma et al., the characteristic stress σ_r was determined through iterative finite element simulations.

The initial characteristic stress was estimated using the relationship σ_r ≈ H/3, where H is the hardness. With the specimen's Poisson's ratio ν = 0.33, elastic modulus E = 51.3 GPa, and indenter properties (E_i = 1141 GPa, ν_i = 0.07), an initial σ_r value was input into the ANSYS model. The simulated maximum load F_FEM was compared with the experimental maximum load F_EXP. When |F_FEM - F_EXP|/F_EXP ≤ 0.5%, the iteration was terminated. The final characteristic stress was determined to be σ_r = 166.7 MPa.

3.3 Determination of Strain Hardening Exponent

The strain hardening exponent n was determined using the dimensional function established by Dao et al.:

n = (0.010 + 0.143·ln(σ_r/E)) / (1 + 0.059·ln(σ_r/E))

Substituting the determined values yielded n = 0.309.

3.4 Determination of Characteristic Strain and Stress-Strain Relationship

The characteristic strain ε_r was initially estimated using:

ε_r = σ_r / E = 166.7 / 51,300 = 0.00325

This value was substituted into the power-law model along with σ_r and n. Finite element simulations were performed with these material parameters. If the difference between simulated and experimental maximum loads exceeded 0.5%, ε_r was iteratively adjusted until convergence was achieved. The final characteristic strain was determined to be ε_r = 0.0035.

The yield strength σ_y was then calculated using the power-law relationship:

σ_y = σ_r · (ε_y/ε_r)^n

where ε_y = σ_y/E. Solving this equation yielded σ_y = 31.51 MPa.

The complete stress-strain relationship for the Sn-rich phase was thus established as:

σ =
\begin{cases}
51,300·ε & \text{for } ε ≤ 0.00061 \
166.7·(ε/0.0035)^{0.309} & \text{for } ε > 0.00061
\end{cases}

4. Tensile Test Validation

To validate the accuracy of the stress-strain relationship obtained through inverse analysis, tensile tests were conducted on the dog-bone specimens. The tests employed an extensometer gauge length of [MATH_4]. The yield strength was determined at 0.2% plastic strain offset (σ_p0.2), yielding an average value of 32.1 MPa. This result shows excellent agreement with the yield strength of 31.51 MPa obtained from nanoindentation inverse analysis, confirming the validity of the proposed methodology.

5. Conclusion

This study successfully determined the mechanical properties and stress-strain relationship of the Sn-rich phase in lead-free solder alloys through a combination of nanoindentation testing and finite element inverse analysis. The nanoindentation tests measured an elastic modulus of 51.3 GPa and hardness of 0.31 GPa. Inverse analysis yielded a characteristic stress of 166.7 MPa and characteristic strain of 0.0035, while dimensional analysis gave a strain hardening exponent of 0.309. The resulting yield strength of 31.51 MPa was validated through independent tensile testing.

The determined stress-strain relationship function for the Sn-rich phase provides essential parameters for micro-mechanical behavior analysis, mechanical failure investigation, and reliability assessment of Sn-based micro-solder joints in electronic packaging applications.

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