Properties for Design of Composite Structures. Neil McCartney

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Properties for Design of Composite Structures - Neil McCartney


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and The inverse form is

      When the stress field is hydrostatic so that σij=−pδij, and because σkk≡σ11+σ22+σ33, it follows from (2.157) that

      epsilon Subscript i j Baseline equals left-parenthesis minus StartFraction 1 minus 2 nu Over upper E EndFraction p plus alpha upper Delta upper T right-parenthesis delta Subscript i j Baseline period(2.159)

      The inverse form (2.158) may be written as

      sigma Subscript i j Baseline equals 2 mu epsilon Subscript i j Baseline plus lamda epsilon Subscript k k Baseline delta Subscript i j Baseline minus left-parenthesis 3 lamda plus 2 mu right-parenthesis alpha upper Delta upper T delta Subscript i j Baseline comma(2.160)

      where λ and μ are Lamé’s constants, μ being the shear modulus, which can be calculated from Young’s modulus and Poisson’s ratio as follows:

      2.15 Introducing Contracted Notation

      The components of the stress and strain components are now assembled in column vectors of length six so that

      Start 6 By 1 Matrix 1st Row sigma 11 2nd Row sigma 22 3rd Row sigma 33 4th Row sigma 23 5th Row sigma 13 6th Row sigma 12 EndMatrix identical-to 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 comma Start 6 By 1 Matrix 1st Row epsilon 11 2nd Row epsilon 22 3rd Row epsilon 33 4th Row 2 epsilon 23 5th Row 2 epsilon 13 6th Row 2 epsilon 12 EndMatrix identical-to 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 period(2.162)

      It should be noted that a factor of two has been applied only to the shear terms of the relation involving the strains so that the quantities 2εij for i≠j correspond to the widely used engineering shear strain values. General linear elastic stress-strain relations, including thermal expansion terms, have the contracted matrix form

      where CIJ are symmetric elastic constants, which are components of the second-order matrix C, and where UI are thermoelastic constants associated with the tensor βij, which are components of the vector U, the uppercase indices I and J ranging from 1 to 6. For orthotropic materials the stress-strain relations have the simpler matrix form

      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 equals Start 6 By 6 Matrix 1st Row 1st Column upper C 11 2nd Column upper C 12 3rd Column upper C 13 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column upper C 21 2nd Column upper C 22 3rd Column upper C 23 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column upper C 31 2nd Column upper C 32 3rd Column upper C 33 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column upper C 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column upper C 55 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column upper C 66 EndMatrix 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 minus Start 6 By 1 Matrix 1st Row upper U 1 2nd Row upper U 2 3rd Row upper U 3 4th Row 0 5th Row 0 6th Row 0 EndMatrix upper Delta upper T period(2.164)

      The stress-strain relations (2.163) may be written, using a repeated summation convention for uppercase indices over the range 1, 2, …, 6, as

      The inverse of the matrix CIJ is denoted by the symmetric matrix SIJ such that

      where δIK is the Kronecker delta symbol having the value 1 when I=J and the value 0 otherwise. On multiplying (2.165) on the left by SLI and on using (2.166), it can be shown that

      The


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