Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 66

The boundary and interface conditions that must be satisfied are

(4.28)

On differentiating the displacement field, it follows, on using (2.139) on setting uθ≡0, that.

(4.29)

(4.30)

(4.31)

(4.32)

It follows directly from (4.17), (4.30) and (4.32) that for fibre and matrix

(4.33)

On subtracting (4.14) and (4.15), it follows that for fibre and matrix

(4.34)

(4.35)

Relations (4.29) and (4.31) then assert that

(4.36)

On substituting (4.33) and (4.34) into the equilibrium equations (4.20)–(4.22) it follows that

(4.37)

(4.38)

On integrating (4.38)₁ subject to the condition (4.28)₁,

(4.39)

and from (4.36) it follows that

(4.40)

On integrating (4.37)₁ using (4.39) together with the continuity condition (4.28)₃

(4.41)

and from (4.36) it follows that

(4.42)

The substitution of (4.29)₃, (4.41) and (4.42) into (4.16) leads to

(4.43)

It is clear from (4.41) that σzzf is a constant, automatically satisfying (4.37)₂. The substitution of (4.31)₃, (4.39) and (4.40) into (4.16), applied to the matrix, leads to

(4.44)

thus automatically satisfying (4.38)₂. The substitution of (4.29), (4.31), (4.39)–(4.44) into relation (4.14), applied to the fibre and matrix, leads to

(4.45)

(4.46)

where kTf is the plane strain bulk modulus for the transverse isotropic fibre and kTm is the plane strain bulk modulus of the isotropic matrix, defined by (see (2.202))

(4.47)

(4.48)

It should be noted that

(4.49)

where km is the bulk modulus of the matrix defined by (see (2.205))

(4.50)

It now only remains to determine the constant ϕ, which can be specified on applying the remaining condition (4.28)₅, because the conditions (4.28)₂, (4.28)₄ and (4.28)₆ are automatically satisfied by (4.25) and (4.33). It follows from (4.26), (4.27), (4.28)₅, (4.45) and (4.46) that

(4.51)

where

(4.52)

The displacement distribution is specified by (4.25)–(4.27), and the corresponding stress distribution is specified by (4.33), (4.39)–(4.44). The stress-strain relations (4.14)–(4.17) and the equilibrium equations (4.20)–(4.22) are satisfied exactly. The boundary and interface conditions (4.28) are also satisfied exactly.