Invariant-Based Finite Strain Anisotropic Material Model for Fiber-Reinforced Composites

Abstract

Short fibre reinforced plastic (SFRP) materials are intensively used in several engineering sectors due to their excellent mechanical properties and production rates. In this investigation, an invariant-based transversely isotropic elasto-plastic model for finite strain applications and its corresponding numerical treatment are presented. The current model is based on the multiplicative decomposition of the deformation gradient. The main characteristic of the formulation is the mathematical realization of the incompressibility assumption with regard to the plastic behaviour in anisotropic finite strain setting. The proposed model is complying with thermodynamic restrictions and allows robust reliable numerical simulations. The accuracy of the model is verified by comparison against experimental data, showing a very satisfactory level of agreement.

Appendix

This appendix addresses the weak formulation of the IBVP presented in Eq. (6) (Sect. 2), which represents the most convenient setting to formulate the corresponding numerical approximation based on FEM (Finite Element Method) through the exploitation of the standard Galerkin procedure.

Assume that the reference body boundary $\partial \mathscr {B}_{0}$ is subdivided into the disjointed parts $\partial \mathscr {B}_{0,u}\subset \partial \mathscr {B}_{0}$ and $\partial \mathscr {B}_{0,t}\subset \partial \mathscr {B}_{0}$, with $\partial \mathscr {B}_{0}=\partial \mathscr {B}_{0,u}\cup \partial \mathscr {B}_{0,t}$ and $\partial \mathscr {B}_{0,u}\cap \partial \mathscr {B}_{0,t}=\emptyset$. As customary, appropriate boundary conditions must be defined in order to guarantee the well-posedness of the IBVP. The weak form of the balance of linear momentum reads:

$$\begin{aligned}&G^{u}\left( \mathbf {u}, \delta \mathbf {u}\right) = \int _{\mathscr {B}_{0}}\left( \mathrm {DIV}\left[ \mathbf {P}\right] + \bar{ \varvec{\Upsilon }} \right) \delta \mathbf {u}\mathrm {d}\mathrm {V} = \int _{\mathscr {B}_{0}}\left( \mathrm {DIV}\left[ \delta \mathbf {u}\mathbf {P}\right] -\mathbf {P}:\nabla _{\mathbf {X}}\delta \mathbf {u}+ \bar{ \varvec{\Upsilon }} \delta \mathbf {u}\right) \mathrm {d}\mathrm {V}\nonumber \\&= \int _{\mathscr {B}_{0}}\mathbf {P}:\delta \mathbf {F}\mathrm {d}\mathrm {V}-\int _{\partial \mathscr {B}_{0}}\mathbf {T} \delta \mathbf {u}\mathrm {d}\mathrm {A}- \int _{\mathscr {B}_{0}} \bar{ \varvec{\Upsilon }} \delta \mathbf {u}\mathrm {d}\mathrm {V}=G_{int}^{u}+G_{ext}^{u}=0, \end{aligned}$$

(105)

where $\delta \mathbf {u}$ renders the virtual displacement and $\delta \mathbf {F}=\nabla _{\mathbf {X}}\delta \mathbf {u}$ and $\mathbf {T}=\mathbf {P}\mathbf {N}$ denotes the first Piola-Kirchhoff traction vector. Note that to achieve the present form of Eq. 105, the following rules are used:

$$\begin{aligned} \mathrm {DIV}\left[ \mathbf {P}\right] \delta \mathbf {u}=\mathrm {DIV}\left[ \delta \mathbf {u}\mathbf {P}\right] -\mathbf {P}:\nabla _{\mathbf {X}}\delta \mathbf {u}, \end{aligned}$$

(106)

and the Gauss-Green theorem:

$$\begin{aligned} \int _{\mathscr {B}_{0}}\mathrm {DIV}\left[ \delta \mathbf {u}\mathbf {P}\right] \mathrm {d}\mathrm {V}=\int _{\partial \mathscr {B}_{0}}\left( \mathbf {P}\mathbf {N}\right) \delta \mathbf {u}\mathrm {d}\mathrm {A}. \end{aligned}$$

(107)

The virtual internal work $G_{int}^{u}$ and the virtual work of external actions $G_{ext}^{u}$ are given by:

$$\begin{aligned} G_{int}^{u}\left( \mathbf {u}, \delta \mathbf {u}\right) =\int _{\mathscr {B}_{0}}\mathbf {P}:\delta \mathbf {F}\mathrm {d}\mathrm {V}, \end{aligned}$$

(108)

$$\begin{aligned} G_{ext}^{u}\left( \mathbf {u}, \delta \mathbf {u}\right) =-\int _{\partial \mathscr {B}_{0}}\mathbf {T}\delta \mathbf {u}\mathrm {d}\mathrm {A}-\int _{\mathscr {B}_{0}} \bar{ \varvec{\Upsilon }}\delta \mathbf {u}\mathrm {d}\mathrm {V}. \end{aligned}$$

(109)

The resulting set of nonlinear equations of the mechanical problem, Eq. 105, can be solved numerically through the use of the incremental and iterative Newton-Raphson solution scheme, which shows a quadratic convergence near the solution point. The consistent linearization of the given time integration algorithm, also called stress-update algorithm, leads to the derivation of the consistent tangent moduli, which describes in an incremental manner the stress sensitivity with respect to the deformation gradient increment. Following the directional derivative concept [16], the consistent linearization of Eq. 105 takes the following representation:

$$\begin{aligned} \mathrm {Lin}\left[ G^{u}\left( \bar{\mathbf {u}},\delta \mathbf {u},\Delta \mathbf {u}\right) \right]= & {} G^{u}\left( \bar{\mathbf {u}},\delta \bar{\mathbf {u}}\right) +DG^{u}\left( \bar{\mathbf {u}},\delta \mathbf {u}\right) \Delta \mathbf {u}. \end{aligned}$$

(110)

In Eq. 105, the term $\mathbf {P}:\delta \mathbf {F}$ has to be linearized yielding:

$$\begin{aligned} \Delta \left( \mathbf {P}:\delta \mathbf {F}\right) =\Delta \mathbf {P}:\delta \mathbf {F}, \end{aligned}$$

(111)

where $\Delta \mathbf {P}$ is derived in Sect. 4.2, with:

$$\begin{aligned} \Delta \mathbf {P}=\Delta \left( \mathbf {F}\mathbf {S}\right) =\mathbb {C}^{ep}:\Delta \mathbf {F}, \end{aligned}$$

(112)

where $\mathbb {C}^{ep}$ denotes the algorithmic elasto-plastic constitutive operator.