Properties for Design of Composite Structures. Neil McCartney
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.
Figure 2.1 Transformation of right-handed Cartesian coordinates.
Any point in space can be represented by the vector x (a first-order tensor) the value of which is wholly independent of the coordinate system that is used to describe its components so that
It then follows on resolving vectors that
Transformation of a set of Cartesian coordinates (x1,x2,x3) to (x′1,x′2,x′3) by a rotation of the x1- and x2-axes about the x3-axis through an angle ϕ (as shown in Figure 2.1) leads to
These relationships can be established from the geometry shown in Figure 2.1 on making use of the various constructions shown as dotted lines.
The displacement vector u is a physical quantity that is wholly independent of the coordinate system that is used to describe its components. This vector may be written as (where summation over values 1, 2 and 3 is implied by repeated lowercase suffices)
where uk and u′k are the displacement components referred to the two coordinate systems being considered. It follows on using (2.172)