Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 35

When considering unidirectionally reinforced fibre composites, as will be the case in Chapter 4, the effective composite properties are often assumed to be isotropic in the plane that is normal to the fibre direction taken here to be the x₃-direction as coordinate rotations considered previously have been about the x₃-axis. It is now assumed that S11=S22, S44=S55 and S13=S23. As m2 + n2 = 1 and

it then follows from (2.184)–(2.186) that

(2.188)

It should be noted that the factor S66−2S11+2S12 appears repeatedly in these relations. When this factor is zero so that

(2.189)

it follows that

(2.190)

As it was assumed that S11=S22,​​S44=S55,S13=S23, it is clear that any rotation about the x₃-axis does not alter the value of the elastic constants on transformation. Thus, the material having the stress-strain relations (2.170) are transverse isotropic relative to the x₃-axis if the elastic constants are such that

(2.191)

For isotropic materials, the elastic constants must satisfy the relations

(2.192)

For a transverse isotropic solid the thermal expansion coefficients are such that V1=V2=V* and V3=V. It then follows from (2.187) that

(2.193)

For isotropic materials

(2.194)