Triangular distribution of in-plane stresses

II.1.6.2 Triangular distribution of in-plane stresses

In most expressions below, the components of tensors are expressed in laminate axes. Therefore, the “lam” underscore is often added to the different quantities used in the equations.

First, the components of the Cauchy stress tensor are calculated. However, a few simplifying assumptions must be made. The strain tensor components are calculated from the laminate average strain tensor and curvature as follows:

{ϵ}_{lam} = {ϵ^{0}}_{lam} + z {κ}_{lam},

{ϵ^{0}}_{lam} = {[a]}_{lam} \cdot {N}_{lam} + {[b]}_{lam} \cdot {M}_{lam},

{κ}_{lam} = [b] T_{lam} \cdot {N}_{lam} + {[d]}_{lam} \cdot {M}_{lam} .

(The thermo-elastic contributions have been neglected.) In most out-of-plane shear theories presented in the literature, a decoupling between in-plane load response and out-of-plane shear response is assumed, allowing several terms in the equations to be neglected:

{ϵ^{0}}_{lam} \approx {[b]}_{lam} \cdot {M}_{lam},

{κ}_{lam} \approx {[d]}_{lam} \cdot {M}_{lam} .

A simple expression of the in-plane laminate deformation tensor is then written:

{ϵ}_{lam} = {ϵ^{0}}_{lam} + z {κ}_{lam} .

Then, the components of Cauchy stress tensor are given by:

{σ}_{lam} (x, y, z) = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) \cdot {M}_{lam} (x, y) .

(II.1.38)

In this last expression, the $3 \times 3$ matrix ${[C]}_{lam} (z)$ corresponds to the plies in-plane moduli expressed in laminate axes. It depends on $z$ because the components generally change from one ply to another. However, it is assumed that the components of the moduli matrix are constant in each ply.

Note that, in the local and global equilibrium relations (II.1.34) to (II.1.37), only partial derivatives of bending moments and Cauchy stress tensor components appear. The decoupling between the out-of-plane shear behavior and the absolute bending in laminate is assumed. However, as shown by expressions (II.1.36) and (II.1.37), the out-of-plane shear is related to the gradient of bending moment. Equation (II.1.38) is then differentiated with respect to $x$ and $y$ :

{\begin{matrix} σ_{x x, x} (x, y, z) \\ σ_{y y, x} (x, y, z) \\ τ_{x y, x} (x, y, z) \end{matrix}}_{lam} = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) \cdot {\begin{matrix} M_{x x, x} (x, y) \\ M_{y y, x} (x, y) \\ M_{x y, x} (x, y) \end{matrix}}_{lam},

{\begin{matrix} σ_{x x, y} (x, y, z) \\ σ_{y y, y} (x, y, z) \\ τ_{x y, y} (x, y, z) \end{matrix}}_{lam} = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) \cdot {\begin{matrix} M_{x x, y} (x, y) \\ M_{y y, y} (x, y) \\ M_{x y, y} (x, y) \end{matrix}}_{lam} .

At this point, it is no longer necessary to assume a dependence of the gradient of bending moments with respect to $x$ and $y$ . The same is true for the gradient of the Cauchy stress tensor. A new notation is also introduced:

{[F]}_{lam} (z) = {[C]}_{lam} (z) \cdot ({[b]}_{lam} + z {[d]}_{lam}) .

Then, the components of Cauchy stress tensor gradient are obtained from the components of bending moments gradient with the following expression:

{\begin{matrix} σ_{x x, x} (z) \\ σ_{y y, x} (z) \\ τ_{x y, x} (z) \\ σ_{x x, y} (z) \\ σ_{y y, y} (z) \\ τ_{x y, y} (z) \end{matrix}}_{lam} = [\begin{matrix} {[F]}_{lam} (z) & \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} & {[F]}_{lam} (z) \end{matrix}] \cdot {\begin{matrix} M_{x x, x} \\ M_{y y, x} \\ M_{x y, x} \\ M_{x x, y} \\ M_{y y, y} \\ M_{x y, y} \end{matrix}}_{lam} .

(II.1.39)

[next] [prev] [prev-tail] [front] [up]