Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

k Subscript upper T Baseline Over k Subscript upper T Baseline plus mu Subscript t Baseline EndFraction StartFraction upper B Over r squared EndFraction minus StartFraction 2 k Subscript upper T Baseline Over k Subscript upper T Baseline minus mu Subscript t Baseline EndFraction 3 upper C r squared plus StartFraction 6 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta period"/>(4.105)

Relations (4.102) and (4.104) then assert that

(4.106)

Relations (4.103) and (4.105) assert that

$sigma subscript r r end subscript minus sigma subscript theta theta end subscript equals 2 mu subscript text t end text end subscript left parenthesis epsilon subscript r r end subscript minus epsilon subscript theta theta end subscript right parenthesis space space space space space space space space space space space space space space space equals 4 mu subscript text t end text end subscript open parentheses negative A plus fraction numerator k subscript text T end text end subscript over denominator k subscript text T end text end subscript plus mu subscript text t end text end subscript end fraction B over r squared minus fraction numerator k subscript text T end text end subscript over denominator k subscript text T end text end subscript minus mu subscript text t end text end subscript end fraction 3 C r squared plus fraction numerator 3 D over denominator r to the power of 4 end fraction close parentheses cos 2 theta.$ (4.107)

The addition and subtraction of (4.106) and (4.107) leads to the results

sigma Subscript r r Baseline equals 2 mu Subscript t Baseline left-parenthesis negative upper A plus StartFraction 2 k Subscript upper T Baseline Over k Subscript upper T Baseline plus mu Subscript t Baseline EndFraction StartFraction upper B Over r squared EndFraction plus StartFraction 3 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta comma (4.108)

sigma Subscript theta theta Baseline equals 2 mu Subscript t Baseline left-parenthesis upper A zero width space zero width space plus StartFraction 6 k Subscript upper T Baseline Over k Subscript upper T Baseline minus mu Subscript t Baseline EndFraction upper C r squared minus StartFraction 3 upper D Over r Superscript 4 Baseline EndFraction right-parenthesis cosine 2 theta period (4.109)

From (4.94) and (4.98), it follows that

(4.110)

It is easily shown that the stress field given by relations (4.108)–(4.110) satisfies automatically the following equilibrium equations for any values of the parameters A, B, C and D (see (2.125)–(2.127))

StartFraction partial-differential sigma Subscript r r Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential theta EndFraction plus StartFraction sigma Subscript r r Baseline minus sigma Subscript theta theta Baseline Over r EndFraction equals 0 comma (4.111)

StartFraction partial-differential sigma Subscript r theta Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction StartFraction partial-differential sigma Subscript theta theta Baseline Over partial-differential theta EndFraction plus StartFraction 2 sigma Subscript r theta Baseline Over r EndFraction equals 0 comma (4.112)

where use has been made of (4.99) and the fact that σzz is independent of z.

4.5.2 Stress Field in the Absence of Fibre

It follows from (4.93) and (4.108)–(4.110) that, when the matrix occupies the whole of space and is subject to a transverse shear stress τ at infinity, the resulting displacement and stress field is given by

(4.113)

4.5.3 Displacement and Stress Fields in Fibre

The displacement and stress fields within the fibre must be bounded as r→0 and it follows from (4.93) and (4.108)–(4.110) that

(4.114)

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