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Properties for Design of Composite Structures. Neil McCartney

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Properties for Design of Composite Structures - Neil McCartney

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m squared upper S 44 plus n squared upper S 55 comma 2nd Row upper S prime Subscript 45 Baseline equals m n left-parenthesis upper S 44 minus upper S 55 right-parenthesis comma 3rd Row upper S prime Subscript 55 Baseline equals n squared upper S 44 plus m squared upper S 55 comma EndLayout"/>(2.185)

      and where

      2.17.1 Transverse Isotropic and Isotropic Solids

      When considering unidirectionally reinforced fibre composites, as will be the case in Chapter 4, the effective composite properties are often assumed to be isotropic in the plane that is normal to the fibre direction taken here to be the x3-direction as coordinate rotations considered previously have been about the x3-axis. It is now assumed that S11=S22, S44=S55 and S13=S23. As m2 + n2 = 1 and

m Superscript 4 Baseline plus n Superscript 4 Baseline equals 1 minus 2 m squared n squared comma left-parenthesis m squared minus n squared right-parenthesis squared equals m Superscript 4 Baseline plus n Superscript 4 Baseline minus 2 m squared n squared equals 1 minus 4 m squared n squared comma

      it then follows from (2.184)–(2.186) that

      StartLayout 1st Row upper S prime Subscript 11 Baseline equals upper S prime Subscript 22 Baseline equals upper S 11 plus left-parenthesis upper S 66 minus 2 upper S 11 plus 2 upper S 12 right-parenthesis m squared n squared comma 2nd Row upper S prime Subscript 12 Baseline equals upper S 12 minus left-parenthesis upper S 66 minus 2 upper S 11 plus 2 upper S 12 right-parenthesis m squared n squared comma zero width space zero width space zero width space zero width space 3rd Row upper S prime Subscript 13 Baseline equals upper S prime Subscript 23 Baseline equals upper S 13 comma zero width space zero width space zero width space upper S prime Subscript 33 Baseline equals upper S 33 comma 4th Row upper S prime Subscript 44 Baseline equals upper S prime 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

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