Modelling of Failure of Woven Composites. Part 2: Experimental and Numerical Justification of the Interzone Concept

Abstract

The failure of woven composites has been examined. This study is presented in two parts:

• Modelling of failure of woven composites. Part 1: nomenclature defining the interzone concept;

• Modelling of failure of woven composites. Part 2: experimental and numerical justification of the interzone concept.

In the first part, the concepts of the interzone and the geometry of an interzone have been defined in a general way for a large panel of woven composites. In the second part, it has been shown that the failure of woven composites is well described by using the interzone concept. The load transfer between intact interzones and broken interzones has been evaluated for two types of loadings (tensile loading and loading in bending). The analysis of these load transfers explains why in the case of a tensile loading the failure is of a sudden-death type whereas in the case of bending loading the failure is progressive. The concept of failure of an interzone has been also defined.

Appendix - Notations:

• local frame associated with the cell: R=(O,B), B=(x ₁ = x,x ₂ = y,x ₃ = z), with, for the point M, the following parameters projected in B:

• .  cartesian coordinates: (x _i ) _i=1,2,3

• .  displacement: u(M)=(u _i (M)) _i=1,2,3

• .  tensor of stress: σ(M)=(σ _{i j} (M)) _i,j=1,2,3

• .  tensor of strain: ε(M)=(ε _{i j} (M)) _i,j=1,2,3

• .  principal basis of stresses: b _σ (M)=(p ₁(M),p ₂(M),p ₃(M))

• .  principal stress (in decreasing order): (σ _I (M),σ _{I I} (M),σ _{I I I} (M))

• .  usual Euler angles allowing B and b _σ (M) to coincide: (ψ ^σ(M),𝜃 ^σ(M),φ ^σ(M))

• .  principal basis of strain: b _ε (M)=(q ₁(M),q ₂(M),q ₃(M))

• .  principal strains (in decreasing order): (ε _I (M),ε _{I I} (M),ε _{I I I} (M))

• .  if the material constituents of the composite are linearly elastic its linear elastic behaviour can be written as σ(M) = a(M)×ε(M) or in an equivalent manner ε(M) = A(M)×σ(M), in which respectively the stiffness and compliance tensors are a(M)=(a _{i j k h} (M)) _{i,j,k,h=1,2,3} and A(M)=(A _{i j k h} (M)) _{i,j,k,h=1,2,3}, with the usual symmetries

• the yarn identified by I d F=(W a Z−i,p) is the p-th warp yarn of the i-th warp plane (W a Z−i)

• the yarn designated by I d F=(W e Z−j,q) is the q-th weft yarn of the j-th weft plane (W e Z−j)

• local frame of the warp yarn I d F at a point M of the mean line: R ^(warp/IdF)(M)=(M,B ^(warp/IdF)(M)) with:

• .  $B^{(warp/IdF)}(M) = (\textbf {x}_{1}^{(warp/IdF)}(M), \textbf {x}_{2}^{(warp/IdF)}(M), \textbf {x}_{3}^{(warp/IdF)}(M))$

• .  $\textbf {x}_{1}^{(warp/IdF)}(M)$ is in M, the unit vector which orientates the local tangent of the mean line

• .  $\textbf {x}_{2}^{(warp/IdF)}(M) = \textbf {x}_{2}$

• .  $(\textbf {x}_{2}^{(warp/IdF)}(M), \textbf {x}_{3}^{(warp/IdF)}(M))$ is the basis of the plane of the transverse section of the yarn

• .  $\alpha ^{(warp/IdF)}(M) = \angle ({\textbf {x}_{1}, \textbf {x}_{1}^{(warp/IdF)}(M)})$

• .  usual Euler angles allowing B to coincide with B ^(warp/IdF)(M): (ψ ^(warp/IdF)(M),𝜃 ^(warp/IdF)(M),φ ^(warp/IdF)(M)). We obtain:

  $$\begin{array}{@{}rcl@{}} \psi^{(warp/IdF)}(M) = \frac{\pi}{2} \,, \theta^{(warp/IdF)}(M) &=& 2 \pi - \alpha^{(warp/IdF)}(M) \,, \\ &&\varphi^{(warp/IdF)}(M)\ = \frac{3\pi}{2} \end{array} $$

• .  $\sigma ^{(warp/IdF)}(M) = (\sigma ^{(warp/IdF)}_{ij}(M))_{i,j = 1, 2, 3}$ is the stress tensor at the point M of the mean line of the warp yarn I d F expressed in B ^(warp/IdF)(M)

• .  $\varepsilon ^{(warp/IdF)}(M) = (\varepsilon ^{(warp/IdF)}_{ij}(M))_{i,j = 1, 2, 3}$ is the strain tensor at the point M of the mean line of the warp yarn I d F expressed in B ^(warp/IdF)(M)

• .  the linear elastic behaviour law is σ ^(warp/IdF)(M) = a ^(warp/IdF)(M)×ε ^(warp/IdF)(M) or in an equivalent manner ε ^(warp/IdF)(M) = A ^(warp/IdF)(M)×σ ^(warp/IdF)(M), in which the stiffness and compliance tensors are respectively $a^{(warp/IdF)}(M) = (a^{(warp/IdF)}_{ijkh}(M))_{i,j, k, h = 1, 2, 3}$ and $a^{(warp/IdF)}(M) = (a^{(warp/IdF)}_{ijkh}(M))_{i,j, k, h = 1, 2, 3}$, with the usual symmetries

• local frame of the weft yarn I d F at the point M of the mean line: R ^(weft/IdF)(M)=(M,B ^(weft/IdF)(M)) with:

• .  $B^{(weft/IdF)}(M) = (\textbf {x}_{1}^{(weft/IdF)}(M), \textbf {x}_{2}^{(weft/IdF)}(M), \textbf {x}_{3}^{(weft/IdF)}(M))$

• .  $\textbf {x}_{1}^{(weft/IdF)}(M)$ is, in M, the unit vector which orientates the local tangent of its mean line

• .  $\textbf {x}_{2}^{(weft/IdF)}(M) = - \textbf {x}_{1}$

• .  $(\textbf {x}_{2}^{(weft/IdF)}(M), \textbf {x}_{3}^{(weft/IdF)}(M))$ is the basis of the plane of the transverse section of the yarn

• .  $\alpha ^{(weft/IdF)}(M) = \angle ({\textbf {x}_{2}, \textbf {x}_{1}^{(weft/IdF)}(M)})$

• usual Euler angles allowing B to coincide with B ^(weft/IdF)(M): (ψ ^(weft/IdF)(M),𝜃 ^(weft/IdF)(M),φ ^(weft/IdF)(M)). We obtain:

  $$\psi^{(weft/IdF)}(M) = 0 \,, \theta^{(weft/IdF)}(M) = \alpha^{(weft/IdF)}(M) \,, \varphi^{(weft/IdF)}(M) = \frac{\pi}{2} $$

• .  $\sigma ^{(weft/IdF)}(M) = (\sigma ^{(weft/IdF)}_{ij}(M))_{i,j = 1, 2, 3}$ is the stress tensor at the point M of the mean line of the weft yarn I d F expressed in B ^(weft/IdF)(M)

• .  $\varepsilon ^{(weft/IdF)}(M) = (\varepsilon ^{(weft/IdF)}_{ij}(M))_{i,j = 1, 2, 3}$ is the strain tensor at the point M of the mean line of the weft yarn I d F expressed in B ^(warp/IdF)(M)

• .  the linear elastic behaviour law is written as σ ^(weft/IdF)(M) = a ^(weft/IdF)(M)×ε ^(weft/IdF)(M) or in an equivalent manner ε ^(weft/IdF)(M) = A ^(weft/IdF)(M)×σ ^(weft/IdF)(M), in which the stiffness and compliance tensors are respectively: $a^{(weft/IdF)}(M)= (a^{(weft/IdF)}_{ijkh}(M))_{i,j, k, h = 1, 2, 3}$ and $a^{(weft/IdF)}(M)= (a^{(weft/IdF)}_{ijkh}(M))_{i,j, k, h = 1, 2, 3}$, with the usual symmetries