# Modeling the Rate- and Temperature-Dependent Behavior of Sintered Nano-Silver Paste Using a Variable-Order Fractional Model

## TL;DR

This paper introduces a new model to predict how sintered nano-silver paste behaves under different temperatures and strain rates.

## Contribution

The novelty is a variable-order fractional model capturing temperature and strain rate effects on nano-silver paste.

## Key findings

- The model accurately predicts mechanical behavior under varying temperatures and strain rates.
- The variable-order fractional model outperforms classical models at lower temperatures.
- Quantitative descriptions of elastic modulus and relaxation time are provided.

## Abstract

Sintered nano-silver paste is widely used in electronic packaging due to its excellent thermal and electrical conductivity. A phenomenological variable-order fractional constitutive model has been developed to characterize the evolution of its mechanical properties, incorporating dependencies on both temperature and strain rate. Based on the Weissenberg number and classical Arrhenius equation, a formulation for relaxation time with temperature and strain rate dependence has been proposed. A temperature- and rate-sensitive fractional order is introduced to capture the coupled influences of thermal and strain rate effects. Furthermore, the effects of temperature and the strain rate on the elastic modulus and relaxation time are quantitatively described through established coupling criteria. Simulation results demonstrate that the proposed model offers high accuracy and strong predictive capability. Comparisons with the classical Anand model highlight the effectiveness of the variable-order fractional model, particularly at lower temperatures.