Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

and The inverse form is

sigma Subscript i j Baseline equals StartFraction upper E Over 1 plus nu EndFraction left-parenthesis epsilon Subscript i j Baseline plus StartFraction nu Over 1 minus 2 nu EndFraction epsilon Subscript k k Baseline delta Subscript i j Baseline right-parenthesis minus StartFraction upper E alpha upper Delta upper T Over 1 minus 2 nu EndFraction delta Subscript i j Baseline period (2.158)

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:

lamda equals StartFraction upper E nu Over left-parenthesis 1 plus nu right-parenthesis left-parenthesis 1 minus 2 nu right-parenthesis EndFraction comma mu equals StartFraction upper E Over 2 left-parenthesis 1 plus nu right-parenthesis EndFraction period (2.161)

2.15 Introducing Contracted Notation

The general formulation for describing the elastic constants of anisotropic materials involves fourth-order tensors that are difficult to apply in many practical situations where analytical methods can be used. A simplified contracted notation is usually used for such analyses where the fourth-order tensors of elastic constants are replaced by a second-order matrix formulation that is now described. The matrix formulation makes use of the fact that the stress and strain tensors are symmetric. These symmetry properties enabled the derivation of the relationships (2.149)–(2.151).

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

(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

(2.163)

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

(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

sigma Subscript upper I Baseline equals upper C Subscript upper I upper J Baseline epsilon Subscript upper J Baseline minus upper U Subscript upper I Baseline upper Delta upper T comma upper I equals 1 comma 2 comma ellipsis comma 6 period (2.165)

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

upper C Subscript upper I upper J Baseline upper S Subscript upper J upper K Baseline equals upper S Subscript upper I upper J Baseline upper C Subscript upper J upper K Baseline equals delta Subscript upper I upper K Baseline comma (2.166)

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

epsilon Subscript upper I Baseline equals upper S Subscript upper I upper J Baseline sigma Subscript upper J Baseline plus upper V Subscript upper I Baseline upper Delta upper T comma upper V Subscript upper I Baseline equals upper S Subscript upper I upper J Baseline upper U Subscript upper J Baseline comma upper I equals 1 comma 2 comma ellipsis comma 6 period (2.167)

The

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