Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

Subscript 55 Baseline equals upper S 44 comma zero width space zero width space zero width space upper S prime Subscript 45 Baseline equals 0 comma 5th Row upper S prime Subscript 16 Baseline equals minus upper S prime Subscript 26 Baseline equals zero width space zero width space m n left-parenthesis m squared minus n squared right-parenthesis left-parenthesis upper S 66 minus 2 upper S 11 plus 2 upper S 12 right-parenthesis comma 6th Row upper S prime Subscript 36 Baseline equals 0 comma zero width space zero width space upper S prime Subscript 66 Baseline equals upper S 66 minus 4 m squared n squared left-parenthesis upper S 66 minus 2 upper S 11 plus 2 upper S 12 right-parenthesis period EndLayout"/>(2.188)

It should be noted that the factor S66−2S11+2S12 appears repeatedly in these relations. When this factor is zero so that

upper S 66 equals 2 left-parenthesis upper S 11 minus upper S 12 right-parenthesis comma (2.189)

it follows that

(2.190)

As it was assumed that S11=S22,S44=S55,S13=S23, it is clear that any rotation about the x₃-axis does not alter the value of the elastic constants on transformation. Thus, the material having the stress-strain relations (2.170) are transverse isotropic relative to the x₃-axis if the elastic constants are such that

upper S 11 equals upper S 22 comma zero width space zero width space upper S 44 equals upper S 55 comma upper S 13 equals upper S 23 comma upper S 66 equals 2 left-parenthesis upper S 11 minus upper S 12 right-parenthesis period (2.191)

For isotropic materials, the elastic constants must satisfy the relations

upper S 11 equals upper S 22 equals upper S 33 comma zero width space zero width space upper S 44 equals upper S 55 equals upper S 66 comma upper S 13 equals upper S 23 equals upper S 12 comma upper S 66 equals 2 left-parenthesis upper S 11 minus upper S 12 right-parenthesis period (2.192)

For a transverse isotropic solid the thermal expansion coefficients are such that V1=V2=V* and V3=V. It then follows from (2.187) that

upper V prime Subscript 1 Baseline equals upper V prime Subscript 2 Baseline equals upper V Superscript asterisk Baseline comma upper V prime Subscript 3 Baseline equals upper V comma upper V prime Subscript 6 Baseline equals 0 period (2.193)

For isotropic materials

upper V prime Subscript 1 Baseline equals upper V prime Subscript 2 Baseline equals upper V prime Subscript 3 Baseline equals upper V comma upper V prime Subscript 6 Baseline equals 0 period (2.194)

2.17.2 Introducing Familiar Thermoelastic Constants

It is useful to express the elastic constants SIJ in terms of more familiar physical quantities such as the elastic constants, for linear elastic media, known as Young’s moduli, shear moduli and Poisson’s ratios. Consider a thin rectangular plate made of an orthotropic fibre reinforced material where the in-plane directions x1 and x2 are parallel to the edges of the plate and where the through-thickness direction is parallel to the x₃-axis. The straight fibres in the plate are all parallel to the x₁-axis. For this situation, the elastic constants SIJ in (2.170) are written in the form

(2.195)

where Young’s moduli are denoted by E, shear moduli by μ, Poisson’s ratios by ν and thermal expansion coefficients by α. The stress-strain relations (2.170) may then be written as

(2.196)

The subscripts ‘A’ and ‘T’ refer to axial and transverse thermoelastic constants, respectively, involving in-plane stresses and deformations. The subscripts ‘a’ and ‘t’ refer to axial and transverse constants, respectively, associated with out-of-plane stresses and deformations. The parameter ΔT is the difference between the current temperature of the material and the reference temperature for which all strains are zero when the sample is unloaded.

It is clear that when the plate is uniaxially loaded in the x₁-direction, the parameter νA is the Poisson’s ratio determining the in-plane transverse deformation in the x₂-direction whereas νa is Poisson’s ratio determining the transverse through-thickness

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