Hypoelastic, hyperelastic, discrete and semi-discrete approaches for textile composite reinforcement forming

Abstract

The clear multi-scale structure of composite textile reinforcements leads to develop continuous and discrete approaches for their forming simulations. In this paper two continuous modelling respectively based on a hypoelastic and hyperelastic constitutive model are presented. A discrete approach is also considered in which each yarn is modelled by shell finite elements and where the contact with friction and possible sliding between the yarns are taken into account. Finally the semi-discrete approach is presented in which the shell finite element interpolation involves continuity of the displacement field but where the internal virtual work is obtained as the sum of tension, in-plane shear and bending ones of all the woven unit cells within the element. The advantages and drawbacks of the different approaches are discussed.

Appendix A

γ is the shear angle, γ = θ₁–θ₂ (Fig. 3). It is shown that a shear angle increment dγ gives stresses proportional to Gdγ in the stress computation scheme presented “Continuous approach 1: a hypoelastic model” (Eq. 3 to 11). That is important because the in-plane shear behaviour of a textile material is function of the shear angle (G is not a constant and depends on γ).

The polar rotation tensor and deformation gradient tensor are respectively:

$$ \underline{\underline {\mathbf{R}}} = {\underline {\mathbf{e}}_\alpha } \otimes {\underline {\mathbf{e}}_{\alpha 0}}\quad \underline{\underline {\mathbf{F}}} = {\lambda_\beta }{\underline {\mathbf{f}}_\beta } \otimes {\underline {\mathbf{e}}_{\beta 0}} $$

(28)

${\lambda_\beta }$ is the deformed length of an initially unit fibre in the direction β. The right stretch tensor $\underline{\underline {\mathbf{U}}}$ is given by the polar decomposition:

$$ \underline{\underline {\mathbf{U}}} = {\underline{\underline {\mathbf{R}}}^{\text{T}}} \cdot \underline{\underline {\mathbf{F}}} = \left( {{{\underline {\mathbf{e}} }_{\alpha 0}} \otimes {{\underline {\mathbf{e}} }_\alpha }} \right) \cdot \left( {{\lambda_\beta }{{\underline {\mathbf{f}} }_\beta } \otimes {{\underline {\mathbf{e}} }_{\beta 0}}} \right) = \left( {{\lambda_\beta }{{\underline {\mathbf{f}} }_\beta } \cdot {{\underline {\mathbf{e}} }_\alpha }} \right)\left( {{{\underline {\mathbf{e}} }_{\alpha 0}} \otimes {{\underline {\mathbf{e}} }_{\beta 0}}} \right) $$

(29)

The symmetry of $\underline{\underline {\mathbf{U}}}$ imposes

$$ {\lambda_1}{\underline {\mathbf{f}}_2} \cdot {\underline {\mathbf{e}}_1} = {\lambda_2}{\underline {\mathbf{f}}_1} \cdot {\underline {\mathbf{e}}_2} $$

(30)

In the case of pure in plane shear (λ₁ = λ₂ = 1) or in the case of equal fibre elongations in warp weft directions, this equation becomes

$$ {\underline {\mathbf{f}}_2} \cdot {\underline {\mathbf{e}}_1} = {\underline {\mathbf{f}}_1} \cdot {\underline {\mathbf{e}}_2}\quad {\text{or}}\quad {\underline {\mathbf{h}}_2} \cdot {\underline {\mathbf{e}}_1} = {\underline {\mathbf{g}}_1} \cdot {\underline {\mathbf{e}}_2} $$

(31)

In the case of most of the composite reinforcements, the fibre elongations are small and (31) can be considered.

Because the frames (${\underline {\mathbf{e}}_1},{\underline {\mathbf{e}}_2}$), (${\underline {\mathbf{g}}_1},{\underline {\mathbf{g}}_2}$), (${\underline {\mathbf{h}}_1},{\underline {\mathbf{h}}_2}$) are orthonormal

$$ {\underline {\mathbf{g}}_1} \cdot {\underline {\mathbf{e}}_1} = {\underline {\mathbf{g}}_2} \cdot {\underline {\mathbf{e}}_2}\quad {\underline {\mathbf{g}}_1} \cdot {\underline {\mathbf{e}}_2} = - {\underline {\mathbf{g}}_2} \cdot {\underline {\mathbf{e}}_1} $$

(32)

$$ {\underline {\mathbf{h}}_1} \cdot {\underline {\mathbf{e}}_1} = {\underline {\mathbf{h}}_2} \cdot {\underline {\mathbf{e}}_2}\quad {\underline {\mathbf{h}}_1} \cdot {\underline {\mathbf{e}}_2} = - {\underline {\mathbf{h}}_2} \cdot {\underline {\mathbf{e}}_1} $$

(33)

Considering a shear increment ${\text{d}}\gamma = {\text{d}}{\theta_1} - {\text{d}}{\theta_2} = {\text{d}}\varepsilon_{12}^{\text{g}} - {\text{d}}\varepsilon_{12}^{\text{h}}$, the Eqs. 31, 32, 33 lead to the specific form of the stress calculation Eq. 13:

$$ {\text{d}}\sigma_{\alpha \beta }^{\text{e}} = {\text{G}}\left( {{\text{d}}\varepsilon_{12}^{\text{g}} - {\text{d}}\varepsilon_{12}^{\text{h}}} \right)\left( {{{\underline {\text{e}} }_\alpha }.{{\underline {\text{g}} }_1}} \right)\left( {{{\underline {\text{e}} }_\beta }.{{\underline {\text{g}} }_2}} \right) = {\text{Gd}}\gamma \left( {{{\underline {\text{e}} }_\alpha }.{{\underline {\text{g}} }_1}} \right)\left( {{{\underline {\text{e}} }_\beta }.{{\underline {\text{g}} }_2}} \right) $$

(34)