Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

x Subscript i Baseline Over partial-differential x overbar Subscript upper K Baseline EndFraction StartFraction partial-differential x Subscript i Baseline Over partial-differential x overbar Subscript upper L Baseline EndFraction comma i period e period upper C equals ModifyingAbove nabla With bar x period left-parenthesis ModifyingAbove nabla With bar x right-parenthesis Superscript normal upper T Baseline period"/>(2.84)

From (2.81) and (2.82)

(2.85)

The Eulerian and Lagrange strain tensors are defined by

2 e Subscript k l Baseline equals delta Subscript k l Baseline minus c Subscript k l Baseline comma 2 upper E Subscript upper K upper L Baseline equals upper C Subscript upper K upper L Baseline minus delta Subscript upper K upper L Baseline comma (2.86)

so that (2.85) may also be written as

normal d s squared minus normal d s overbar squared equals 2 e Subscript k l Baseline d x Subscript k Baseline d x Subscript l Baseline equals 2 upper E Subscript upper K upper L Baseline d x overbar Subscript upper K Baseline d x overbar Subscript upper L Baseline period (2.87)

On using the relation u=x−x¯ for the displacement vector and (2.84), the Lagrangian strain tensor may be written in terms of the displacement vector as follows

2 upper E identical-to upper C minus upper I equals ModifyingAbove nabla With bar u plus left-parenthesis ModifyingAbove nabla With bar u right-parenthesis Superscript normal upper T Baseline plus ModifyingAbove nabla With bar u period left-parenthesis ModifyingAbove nabla With bar u right-parenthesis Superscript normal upper T Baseline comma (2.88)

where I is the symmetric fourth-order identity tensor (see (2.15)). The quantity E is the strain tensor that is used in finite deformation theory where there is no restriction on the degree of deformation provided that the deformation is continuous and the condition (2.18) is satisfied at all points in the system.

The invariants of the strain tensors in terms of principal stretches are given by the relations

upper I Subscript upper C Baseline equals lamda 1 squared plus lamda 2 squared plus lamda 3 squared comma upper I Subscript c Baseline equals StartFraction 1 Over lamda 1 squared EndFraction plus StartFraction 1 Over lamda 2 squared EndFraction plus StartFraction 1 Over lamda 3 squared EndFraction comma (2.89)

(2.90)

upper I upper I upper I Subscript upper C Baseline equals lamda 1 squared lamda 2 squared lamda 3 squared comma upper I upper I upper I Subscript c Baseline equals StartFraction 1 Over lamda 1 squared lamda 2 squared lamda 3 squared EndFraction period (2.91)

It is clear that

(2.92)

It will be very useful to introduce here the principal values CJ, J = 1, 2, 3, of Green’s deformation tensor defined using the following relations

StartLayout 1st Row det left-parenthesis upper C minus upper C Subscript upper J Baseline upper I right-parenthesis equals 0 comma 2nd Row left-parenthesis upper C minus upper C Subscript upper J Baseline upper I right-parenthesis period nu Subscript upper J Baseline equals 0 comma EndLayout right-brace upper J equals 1 comma 2 comma 3 comma (2.93)

such that the symmetric tensor C may be written in the form

upper C identical-to sigma-summation Underscript upper J equals 1 Overscript 3 Endscripts upper C Subscript upper J Baseline nu Subscript upper J Baseline nu Subscript upper J Baseline period (2.94)

The quantities νJ,J=1,2,3, are orthogonal unit vectors defining the directions of the principal values. They have the following properties

(2.95)

The polar decomposition principle (see, for example, [2, Section 1.5]) states that the deformation gradient may be expressed in the following forms (dyadic and tensor)

(2.96)

where R is the orthogonal rigid rotation tensor having the properties R.RT=RT.R=I with det(R)=±1, and where U and V are positive-definite symmetric right and left stretch tensors.

It follows from (2.84) and (2.96) that in tensor form

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