Anisotropic constitutive model coupled with damage for Sn-rich solder: Application to SnAgCuSb solder under tensile conditions

With the rapid development of microelectronics and nanoelectronics, Moore law has significantly slowed down and More than Moore based system in packaging (SiP) is expected to be more and more important, at least for next one to two decades. Mechanical behaviors of interconnect materials such as solders are critical for yield in processes and reliability in testing and operation. Based on the framework of crystal plastic theory and continuum damage mechanics, an anisotropic constitutive model coupled with damage was developed to describe the deformation behaviors of Sn-rich solder. In the proposed model, the inelastic shear rate function was presented by hyperbolic sinusoidal form and power law form. For the damage evolution law, the total shear strain was chosen as the damage function variable. The proposed model was implemented into the general finite element software ABAQUS by forward Euler integration procedure. Some simulation examples were performed to verify the proposed model by comparing the simulation results with the experiments at uniaxial tensile conditions with SnAgCuSb solder chosen as the Sn-rich solder. The tensile stress-strain curves of the simulation results agreed well with the experiments at small strain under different temperatures and strain rates. The simulated stress-rupture stages showed reasonable accuracy with the experiments under four representative tensile conditions. Different tensile stress-strain curves of single grains with orientation of (0-0-0)°, (0-45-0)°, and (0-90-0)° were obtained under the same loading conditions, with an inverse relationship between the tensile strength and elongation. This relationship was in accordance with a referable literature. All these results indicate that the proposed model can describe the deformation behaviors of SnAgCuSb solder well under the tensile conditions in consideration of the mechanical anisotropy and the damage evolution.

Anisotropic Constitutive Model of Intermittent Columnar Jointed Rock Masses Based on the Cosserat Theory

In this work, an anisotropic constitutive model of hexagonal columnar jointed rock masses is established to describe the distribution law of deformation and the failure of columnar joint caverns under anisotropic conditions, and is implemented to study the columnar jointed rock mass at the dam site of the Baihetan Hydropower Station on the Jinsha River. The model is based on the Cosserat theory and considers the mesoscopic bending effect on the macroscopic mean. The influences of joint plane inclination on equivalent anisotropic elastic parameters are discussed via the introduction of an off-axis transformation matrix and the analysis of an example. It is also pointed out that the six-prism columnar jointed rock mass changes from transverse isotropy to anisotropy under the influence of the angle. A numerical calculation program of the Cosserat constitutive model is developed and is applied to the simulation calculation of a Baihetan diversion tunnel to compare and analyze the respective plastic zones and stress distributions after tunnel excavation under both isotropic and anisotropic conditions. The results reveal that, compared with the isotropic model, the proposed Cosserat anisotropic model better reflects the state of stress and asymmetric distribution of the plastic zone after tunnel excavation, and the actual deformation of the surrounding rock of the tunnel is greater than that calculated by the isotropic method. The results aid in a better understanding of the mechanical properties of rock masses.

A note on the volumetric-deviatoric split on the anisotropic constitutive model for fiber-reinforced materials

In the literature, it has been suggested that for a class of anisotropic constitutive laws for fiberreinforced materials, the volumetric-deviatoric split should only be performed on the isotropic (matrix) term, but not on the anisotropic (fiber) term. In this research note, we follow up on the theoretical and numerical analyses adopted in these early publications with an intuitive example that allows us to directly analyze the effect of this split. We demonstrate that performing such split on the anisotropic term leads to non-physical volume growth of the material sample. Therefore, we consolidate the observation that the volumetric-deviatoric split should not be applied to the anisotropic (fiber) term of the total strain energy.