Properties for Design of Composite Structures. Neil McCartney

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Properties for Design of Composite Structures - Neil McCartney


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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

      Start 6 By 1 Matrix 1st Row epsilon 1 2nd Row epsilon 2 3rd Row epsilon 3 4th Row epsilon 4 5th Row epsilon 5 6th Row epsilon 6 EndMatrix equals Start 6 By 6 Matrix 1st Row 1st Column upper S 11 2nd Column upper S 12 3rd Column upper S 13 4th Column upper S 14 5th Column upper S 15 6th Column upper S 16 2nd Row 1st Column upper S 21 2nd Column upper S 22 3rd Column upper S 23 4th Column upper S 24 5th Column upper S 25 6th Column upper S 26 3rd Row 1st Column upper S 31 2nd Column upper S 32 3rd Column upper S 33 4th Column upper S 34 5th Column upper S 35 6th Column upper S 36 4th Row 1st Column upper S 41 2nd Column upper S 42 3rd Column upper S 43 4th Column upper S 44 5th Column upper S 45 6th Column upper S 46 5th Row 1st Column upper S 51 2nd Column upper S 52 3rd Column upper S 53 4th Column upper S 54 5th Column upper S 55 6th Column upper S 56 6th Row 1st Column upper S 61 2nd Column upper S 62 3rd Column upper S 63 4th Column upper S 64 5th Column upper S 65 6th Column upper S 66 EndMatrix Start 6 By 1 Matrix 1st Row sigma 1 2nd Row sigma 2 3rd Row sigma 3 4th Row sigma 4 5th Row sigma 5 6th Row sigma 6 EndMatrix plus Start 6 By 1 Matrix 1st Row upper V 1 2nd Row upper V 2 3rd Row upper V 3 4th Row upper V 4 5th Row upper V 5 6th Row upper V 6 EndMatrix upper Delta upper T comma(2.168)

      and the corresponding orthotropic form is

      Start 6 By 1 Matrix 1st Row epsilon 1 2nd Row epsilon 2 3rd Row epsilon 3 4th Row epsilon 4 5th Row epsilon 5 6th Row epsilon 6 EndMatrix equals Start 6 By 6 Matrix 1st Row 1st Column upper S 11 2nd Column upper S 12 3rd Column upper S 13 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column upper S 21 2nd Column upper S 22 3rd Column upper S 23 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column upper S 31 2nd Column upper S 32 3rd Column upper S 33 4th Column 0 5th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column upper S 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column upper S 55 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column upper S 66 EndMatrix Start 6 By 1 Matrix 1st Row sigma 1 2nd Row sigma 2 3rd Row sigma 3 4th Row sigma 4 5th Row sigma 5 6th Row sigma 6 EndMatrix plus Start 6 By 1 Matrix 1st Row upper V 1 2nd Row upper V 2 3rd Row upper V 3 4th Row 0 5th Row 0 6th Row 0 EndMatrix upper Delta upper T period(2.169)

      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.

      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

      x equals x 1 i 1 plus x 2 i 2 plus x 3 i 3 equals x prime Subscript 1 Baseline i prime Subscript 1 Baseline plus x prime Subscript 2 Baseline i prime Subscript 2 Baseline plus x prime Subscript 3 Baseline i prime Subscript 3 Baseline period(2.171)

      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

      StartLayout 1st Row x 1 equals x prime Subscript 1 Baseline cosine phi minus x prime Subscript 2 Baseline sine phi comma x prime Subscript 1 Baseline equals x 1 cosine phi plus x 2 sine phi comma 2nd Row x 2 equals x prime Subscript 1 Baseline sine phi plus x prime Subscript 2 Baseline cosine phi comma x prime Subscript 2 Baseline equals minus x 1 sine phi plus x 2 cosine phi comma 3rd Row x 3 equals x prime Subscript 3 Baseline comma x prime Subscript 3 Baseline equals x 3 period EndLayout(2.173)

      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)


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