# Nonlinear elastic moduli of composite materials with nonlinear spherical inclusions dispersed in a nonlinear matrix

## TL;DR

This paper develops a theoretical method to evaluate the nonlinear elastic properties of composite materials with nonlinear spherical inclusions in a nonlinear matrix, providing analytical formulas and numerical verification.

## Contribution

It introduces an analytical approach to determine both linear and nonlinear elastic moduli of composites with nonlinear inclusions, including explicit formulas for effective Murnaghan moduli.

## Key findings

- Effective Murnaghan moduli depend linearly on constituent moduli.
- Analytical formulas for composite moduli are derived.
- Numerical modeling confirms theoretical results.

## Abstract

A theory is developed for evaluation of nonlinear elastic moduli of composite materials with nonlinear inclusions dispersed in another nonlinear material (matrix). We elaborate a method aimed for determination of elastic parameters of a composite: its linear elastic moduli (second-order elastic constants) and nonlinear elastic moduli, which are known as the Murnaghan moduli (third-order elastic constants). We find an analytical form for the effective Murnaghan moduli of a composite with spherical inclusions. The effective moduli depend linearly on Murnaghan moduli of constituents. The results obtained have been verified through numerical modeling using the finite element method.