A novel large-strain kinematic framework for fiber-reinforced laminated composites and its application in the characterization of damage

TL;DR

The paper addresses modeling large-strain damage in fiber-reinforced laminated composites by introducing a three-term multiplicative kinematic framework that combines multiple natural configurations with Bedford–Stern multicontinuum theory, yielding $\mathbf{F}=\mathbf{F}^e\mathbf{F}^r_\alpha\mathbf{F}^d_\alpha$ ($\alpha\in\{m,f\}$). It defines damage contents for matrix cracking, fiber breakage, debonding, interfacial slip, and delamination through configuration incompatibilities and relative motions, producing measures ${\mathbf G}_m$, ${\mathbf G}_f$, $\boldsymbol{\lambda}_m$, $\boldsymbol{\Sigma}$, and $\hat{\boldsymbol{\Sigma}}$, with geometric interpretation via torsion. The approach extends to laminates by treating each lamina as a coupled pair of matrix and fiber configurations, enabling a consistent description of dissipative processes and interface phenomena. A key contribution is the true, invariant delamination damage density $\hat{\boldsymbol{\Sigma}}$ that remains consistent under compatible changes in the reference configuration, providing a robust basis for constitutive modeling and damage evolution. Overall, the framework offers a principled path to capture large-deformation damage in complex laminated composites and to inform constitutive theories for multiphase materials.

Abstract

In this paper, a novel kinematic framework for fiber-reinforced composite materials is presented. For this purpose, we use the multiple natural configurations in conjunction with the multi-continuum theory of Bedford and Stern~(1972). Keeping the underlying physics of the proposed kinematics consistent. The proposed kinematics results in a three-term decomposition of the deformation gradient i.e. $\mathbf{F}=\mathbf{F}^e\mathbf{F}^r_α\mathbf{F}^d_α$, where $α$ represents either the matrix or the fiber. After discussing the kinematic framework in detail, we use this new kinematic framework to characterize the damage contents associated with four damage mechanisms. These damage mechanisms are matrix cracking, fiber breakage, interfacial slip or debonding, and delamination. While the first two are derived by measuring the incompatibility of the pertinent configuration occupied by individual constituents, the latter two involve a relative displacement between either the constituents or the laminæ. The geometric interpretation corresponding to these damage mechanisms is also presented using tools from differential geometry. The derived damage contents can be used in developing an appropriate constitutive model for laminated composites undergoing damage.

Figures (7)

• Figure 1: A multiplicative decomposition of the deformation gradient and the relevant configurations.
• Figure 2: A motion corresponding to $\mathbf{F}^d_\alpha$ maps the undeformed configuration of the body to its deformed configuration where a (micro-) crack exists. Discontinuity in the displacement field results $\text{P}$ being no longer a single-valued point, and thus the circuit is not closed.
• Figure 3: Interface $S$ in a region of the continuum $\Omega$. $\Gamma$ represents a curve that represents the intersection between the interface and the bulk of the continuum, i.e., $\Gamma=S~\cap~\Omega$. The body can be divided into two parts across the interface, denoted by the signs '+' and '-'. $\tilde{t}_1$ and $\tilde{t}_2$ are mutually orthogonal base vectors on the 2-D interface whereas $\tilde{N}$ is normal to these base vectors.
• Figure 4: The configurations describing the deformation of a lamina of a fiber-reinforced composite material undergoing damage and the associated tangent maps.
• Figure 5: Different damage mechanisms in a lamina of a fiber-reinforced composite material. The damage mechanisms shown here are fiber breakage, matrix cracking, and interfacial debonding.