Properties for Design of Composite Structures. Neil McCartney

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Название Properties for Design of Composite Structures
Автор произведения Neil McCartney
Жанр Техническая литература
Серия
Издательство Техническая литература
Год выпуска 0
isbn 9781118789780
Properties for Design of Composite Structures - Neil McCartney
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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 follows that

      StartLayout 1st Row upper S prime Subscript 11 Baseline equals upper S 11 comma upper S prime Subscript 12 Baseline equals upper S 12 comma upper S prime Subscript 22 Baseline equals upper S 11 comma 2nd Row upper S prime Subscript 13 Baseline equals upper S 13 comma upper S prime Subscript 23 Baseline equals upper S 13 comma 3rd Row upper S prime Subscript 33 Baseline equals upper S 33 comma upper S prime Subscript 44 Baseline equals upper S 44 comma upper S prime Subscript 55 Baseline equals upper S 44 comma upper S prime Subscript 66 Baseline equals upper S 66 comma 4th Row upper S prime Subscript 45 Baseline equals 0 comma upper S prime Subscript 16 Baseline equals 0 comma upper S prime Subscript 26 Baseline equals zero width space zero width space 0 comma upper S prime Subscript 36 Baseline equals 0 period EndLayout(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

      StartLayout 1st Row upper S 11 equals StartFraction 1 Over upper E Subscript upper A Baseline EndFraction comma upper S 12 equals minus StartFraction nu Subscript upper A Baseline Over upper E Subscript upper A Baseline EndFraction comma upper S 13 equals minus StartFraction nu Subscript a Baseline Over upper E Subscript upper A Baseline EndFraction comma 2nd Row upper S 21 equals minus StartFraction nu Subscript upper A Baseline Over upper E Subscript upper A Baseline EndFraction comma upper S 22 equals StartFraction 1 Over upper E Subscript upper T Baseline EndFraction comma upper S 23 equals minus StartFraction nu Subscript t Baseline Over upper E Subscript upper T Baseline EndFraction comma 3rd Row upper S 31 equals minus StartFraction nu Subscript a Baseline Over upper E Subscript upper A Baseline EndFraction comma upper S 32 equals minus StartFraction nu Subscript t Baseline Over upper E Subscript upper T Baseline EndFraction comma upper S 33 equals StartFraction 1 Over upper E Subscript t Baseline EndFraction comma 4th Row upper S 44 equals StartFraction 1 Over mu Subscript t Baseline EndFraction comma upper S 55 equals StartFraction 1 Over mu Subscript a Baseline EndFraction comma upper S 66 equals StartFraction 1 Over mu Subscript upper A Baseline EndFraction comma 5th Row upper V 1 equals alpha Subscript upper A Baseline comma upper V 2 equals alpha Subscript upper T Baseline comma upper V 3 equals alpha Subscript t Baseline comma EndLayout(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

      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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