indices I and J ranging from 1 to 6. For orthotropic materials the stress-strain relations have the simpler matrix form
The stress-strain relations (2.163) may be written, using a repeated summation convention for uppercase indices over the range 1, 2, …, 6, as
The inverse of the matrix CIJ is denoted by the symmetric matrix SIJ such that
where δIK is the Kronecker delta symbol having the value 1 when I=J and the value 0 otherwise. On multiplying (2.165) on the left by SLI and on using (2.166), it can be shown that
The quantities VI are the components of the vector V which is associated with the thermal expansion tensor αij. The matrix form of (2.167) is given by
and the corresponding orthotropic form is
When expanded using the stress and strain tensor components and the symmetry of SIJ, the stress-strain relations may be written as
2.16 Tensor Transformations
When considering laminated composite materials, where each ply is reinforced with aligned straight fibres that are inclined at various angles to a global set of coordinates, there is a need to define a set of local coordinates aligned with the fibres in each ply. There is also a need to determine the properties of each ply referred to the global coordinates. For a right-handed set of global coordinates x1, x2 and x3, i 1, i 2 and i 3 are unit vectors for the directions of the x1-, x2- and x3-axes, respectively. For laminate models, the fibres are usually assumed to be in the x1-direction and coordinate transformations involve rotations about the x3-axis. When modelling unidirectional plies as transverse isotropic materials the rotations would need to be taken about the x1-axis if the fibres are in the x1-direction. Coordinate transformations involving rotations about the x3-axis are now considered.
A right-handed second set of local coordinates x′1,x′2andx′3 is obtained by rotating the reference set of coordinates about the x3-axis by an angle ϕ as shown in Figure 2.1. The rotation is clockwise when viewing along the positive direction of the x3-axis. The unit vectors in the directions of the x′1,x′2andx′3 axes are denoted by i′1,i′2andi′3, respectively. Rotating about the x3-axis enables account to be taken of the effects of off-axis plies in laminates, as considered in Chapters 6 and 7.
