Composite laminate

Composite laminates may be regarded as a type of plate or thin-shell structure, and as such their stiffness properties may be found by integration of in-plane stress in the direction normal to the laminates surface. The broad majority of ply or lamina materials obey Hooke's law and hence all of their stresses and strains may be related by a system of linear equations. Laminates are assumed to deform by developing three strains of the mid-plane/surface and three changes in curvature

$${\displaystyle \varepsilon ^{0}={\begin{bmatrix}\varepsilon _{x}^{0}&\varepsilon _{y}^{0}&\tau _{xy}^{0}\end{bmatrix}}^{T}}$$

and

$${\displaystyle \kappa ={\begin{bmatrix}\kappa _{x}&\kappa _{y}&\kappa _{xy}\end{bmatrix}}^{T}}$$

where ${\displaystyle x}$ and ${\displaystyle y}$ define the co-ordinate system at the laminate level. Individual plies have local co-ordinate axes which are aligned with the materials characteristic directions; such as the principal directions of its elasticity tensor. Uni-directional ply's for example always have their first axis aligned with the direction of the reinforcement. A laminate is a stack of individual plies having a set of ply orientations

$${\displaystyle {\begin{bmatrix}\theta _{1},&\theta _{2},&\dots &\theta _{N}\end{bmatrix}}}$$

which have a strong influence on both the stiffness and strength of the laminate as a whole. Rotating an anisotropic material results in a variation of its elasticity tensor. If in its local co-ordinates a ply is assumed to behave according to the stress-strain law

$${\displaystyle [\sigma ]=\mathbf {Q} [\varepsilon ]}$$

then under a rotation transformation (see transformation matrix) it has the modified elasticity terms

$${\displaystyle {\begin{aligned}Q_{11}^{*}&=Q_{11}\cos ^{4}\theta +2(Q_{12}+2Q_{66})\sin ^{2}\theta \cos ^{2}\theta +Q_{22}\sin ^{4}\theta \\Q_{22}^{*}&=Q_{11}\sin ^{4}\theta +2(Q_{12}+2Q_{66})\sin ^{2}\theta \cos ^{2}\theta +Q_{22}\cos ^{4}\theta \\Q_{12}^{*}&=(Q_{11}+Q_{22}-4Q_{66})\sin ^{2}\theta \cos ^{2}\theta +Q_{12}(\sin ^{4}\theta +\cos ^{4}\theta )\\Q_{66}^{*}&=(Q_{11}+Q_{22}-2Q_{12}-2Q_{66})\sin ^{2}\theta \cos ^{2}\theta +Q_{66}(\sin ^{4}\theta +\cos ^{4}\theta )\\Q_{16}^{*}&=(Q_{11}-Q_{12}-2Q_{66})\cos ^{3}\theta \sin \theta -(Q_{22}-Q_{12}-2Q_{66})\cos \theta \sin ^{3}\theta \\Q_{26}^{*}&=(Q_{11}-Q_{12}-2Q_{66})\cos \theta \sin ^{3}\theta -(Q_{22}-Q_{12}-2Q_{66})\cos ^{3}\theta \sin \theta \end{aligned}}}$$

Hence

$${\displaystyle [\sigma ]^{*}=\mathbf {Q} ^{*}[\varepsilon ]^{*}}$$

An important assumption in the theory of classical laminate analysis is that the strains resulting from curvature vary linearly in the thickness direction, and that the total in-plane strains are a sum of those derived from membrane loads and bending loads. Hence

$${\displaystyle \varepsilon =\varepsilon ^{0}+\kappa \cdot z}$$

Furthermore, a three-dimensional stress field is replaced by six stress resultants; three membrane forces (forces per unit length) and bending moments per unit length. It is assumed that if these three quantities are known at any location (x,y) then the stresses may be computed from them. Once part of a laminate the transformed elasticity is treated as a piecewise function of the thickness direction, hence the integration operation may be treated as the sum of a finite series, giving[1]

$${\displaystyle {\begin{bmatrix}\mathbf {N} \\\mathbf {M} \end{bmatrix}}={\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {B} &\mathbf {D} \end{bmatrix}}{\begin{bmatrix}\varepsilon ^{0}\\\kappa \end{bmatrix}}}$$

where

$${\displaystyle \mathbf {A} =\sum _{j=1}^{N}\mathbf {Q} ^{*}\left(z_{j}-z_{j-1}\right)}$$

$${\displaystyle \mathbf {B} ={\frac {1}{2}}\sum _{j=1}^{N}\mathbf {Q} ^{*}\left(z_{j}^{2}-z_{j-1}^{2}\right)}$$

$${\displaystyle \mathbf {D} ={\frac {1}{3}}\sum _{j=1}^{N}\mathbf {Q} ^{*}\left(z_{j}^{3}-z_{j-1}^{3}\right)}$$

• Carbon-fiber-reinforced polymer
• Composite material
• High-pressure laminate
• Laminate
• Lay-Up process
• Void (composites)

• Advanced Composites Centre for Innovation and Science