Classical laminate theory models the deformation of a laminate in response to external forces and moments under a set of assumptions. The key assumptions are: • The laminate consists of a stack of
orthotropic plies. • The overall thickness is small compared to the other dimensions and constant. • The displacements of the laminate are small compared to the overall thickness. • The in-plane strains are small compared to unity. • The transverse normal strain and shear strains can be neglected. • In-plane displacements and strains are linear functions of the through-thickness coordinate. • Each ply obeys
Hooke's law and hence all of their
stresses and
strains may be related by a
system of linear equations. This is the case for a broad range of lamina materials. • The transverse shear stresses are zero at the surfaces of the laminate, where z = \pm t/2. The laminate's deformation can be represented by the three orthogonal strains of the mid-plane/surface, \underline{\varepsilon}, and three changes in curvature, \underline{\kappa}: : \underline{\varepsilon}^0 = \begin{bmatrix} \varepsilon^0_x \\ \varepsilon^0_y \\ \tau^0_{xy} \end{bmatrix} and \underline{\kappa} = \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} where x and y define a global co-ordinate system. Because of the assumption that the strains resulting from curvature vary linearly along the z axis (in the through-thickness direction), the total in-plane strains for each ply are a sum of those derived from membrane loads and bending loads expressed as : \underline{\varepsilon} = \underline{\varepsilon}^0 + z \underline{\kappa} 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
n individual plies having a set of ply orientations : \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. In each ply's local co-ordinates, it is assumed to behave according to the stress-strain law : \underline{\sigma} = \mathbf{Q}\underline{\varepsilon} where \mathbf{Q} is the stiffness matrix for an individual ply. Transforming the local co-ordinates to the global x-y co-ordinates requires an in-plane rotation by angle \theta, which can be performed using the
rotation matrix \mathbf{T}. This gives the transformed stiffness matrix \mathbf{\bar{Q}} with modified elasticity terms: :\begin{align} \mathbf{\bar{Q}} &= \mathbf{T}^{-1}\mathbf{Q}\mathbf{T}^\mathsf{-T} \\ \bar{Q}_{11} &= Q_{11}\cos^4\theta + 2(Q_{12} + 2Q_{66})\sin^2\theta \cos^2\theta + Q_{22}\sin^4 \theta \\ \bar{Q}_{22} &= Q_{11}\sin^4\theta + 2(Q_{12} + 2Q_{66})\sin^2\theta\cos^2\theta + Q_{22}\cos^4 \theta \\ \bar{Q}_{12} &= (Q_{11} + Q_{22} - 4 Q_{66})\sin^2\theta \cos^2 \theta + Q_{12}(\sin^4 \theta + \cos^4 \theta) \\ \bar{Q}_{66} &= (Q_{11} + Q_{22} - 2 Q_{12} - 2 Q_{66})\sin^2\theta \cos^2 \theta + Q_{66}(\sin^4 \theta + \cos^4 \theta) \\ \bar{Q}_{16} &= (Q_{11} - Q_{12} - 2 Q_{66})\cos^3\theta \sin \theta - (Q_{22}-Q_{12}-2Q_{66})\cos \theta \sin^3 \theta \\ \bar{Q}_{26} &= (Q_{11} - Q_{12} - 2 Q_{66})\cos\theta \sin^3 \theta - (Q_{22}-Q_{12}-2Q_{66})\cos^3 \theta \sin \theta \end{align} Hence, in the global co-ordinate system: : \underline{\sigma} = \mathbf{\bar{Q}} \underline{\varepsilon} The external loading can be summarised by six stress resultants: three membrane forces (forces per unit length) given by N to define the in-plane loading, and three bending moments per unit length given by M to define the out-of-plane loading. These may be calculated the stresses, \underline{\sigma}, as follows: : N = \int_{-t/2}^{t/2} \underline{\sigma} \, dz = \sum_{k=1}^n \left( \int_{-z_k}^{z_{k-1}} \underline{\sigma}_k \, dz \right) \quad are the three membrane forces per unit length and : M = \int_{-t/2}^{t/2} \underline{\sigma} z \, dz = \sum_{k=1}^n \left( \int_{-z_k}^{z_{k-1}} \underline{\sigma}_k z \, dz \right) \quad are the three bending moments per unit length. The stiffness properties of composite laminates may be found by integration of in-plane
stress along the z axis, normal to the laminate's surface. Once part of a laminate, the transformed elasticity is treated as a piecewise function along the z axis (in the thickness direction), hence the integration operation may be treated as the sum of a series of
n plies, giving : \begin{bmatrix} N \\ M \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B} & \mathbf{D} \end{bmatrix} \begin{bmatrix} \underline{\varepsilon}^0 \\ \underline{\kappa} \end{bmatrix} where : A_{ij} = \sum^{n}_{k=1} (\bar{Q}_{ij})_k \left( z_k - z_{k-1} \right) \quad are the laminate extensional stiffnesses, : B_{ij} = \frac{1}{2}\sum^{n}_{k=1} (\bar{Q}_{ij})_k \left( z^2_k - z^2_{k-1} \right) \quad are the laminate coupling stiffnesses and : D_{ij} = \frac{1}{3}\sum^{n}_{k=1} (\bar{Q}_{ij})_k \left( z^3_k - z^3_{k-1} \right) \quad are the laminate bending stiffnesses. == Properties ==