Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 74

Consider the displacement field having the form

(4.93)

where kT and μt are the transverse bulk and shear moduli, respectively.

On using the strain–displacement relations (2.142), the corresponding strain field is given by

(4.94)

When ΔT=0, the stress-strain relations (4.14)–(4.17) are written as

(4.95)

(4.96)

(4.97)

(4.98)

where ET=2μt(1+νt). It follows directly from (4.94) and (4.98) that

(4.99)

From (4.94) and (4.97) it is clear that

(4.100)

and on summing (4.95) and (4.96)

(4.101)

On substituting for σzz using (4.100), it then follows that

(4.102)

On subtracting (4.95) and (4.96),

(4.103)

It follows from (4.94), on addition and subtraction, that

(4.104)

(4.105)

Relations (4.102) and (4.104) then assert that

(4.106)

Relations (4.103) and (4.105) assert that

(4.107)

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

(4.108)

(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))

(4.111)

(4.112)

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