Page 306 - Mechanics Analysis Composite Materials

Chapter 6. Failure criteria and strength Of'laminates 29 I

Following Gol'denblat and Kopnov (1968) consider material strength under
tension in the 1'-direction and in shear in plane (1'. 2'). Taking first
oil' lh
II = 8(/,,oL2= 0, ~;"1= 0 and then $F2 = Tlh$ ofl = O? 0%-_= 0 we get from Eq.
(6.37)

or in the explicit form

These equations allow us to calculate material strength in any coordinate frame
whose axes make angle 4 with the corresponding principal material axes. Taking
into account Eqs. (6.34) and (6.38) we can derive the following relationship from
Eqs. (6.39):

(6.40)

So, Z, is indeed the invariant of the strength tensor whose value for a given material
does not depend on (6.
Thus, tensor-polynomial strength criteria provide universal equations that can be
readily written in any coordinate frame, but on the other hand, material mechanical
characleristics, particularly material strength in different directions, should follow
the rules of tensor transformation, i.e., compose invariants (like Is)that are the same
for all coordinate frames.
To demonstrate the difference between the tensor-polynomial and approximation
polynomial criteria, consider again Eq. (6.IS) and write it for the special orthotropic
material described above (see Fig. 6.18). Then, Eq. (6.15) formally reduces to Eqs.
(6.25) or Eq. (6.27) in which

(6.41)

Matching these results with Eqs. (6.28), we can see that coefficient RY2 is different
here. Moreover, because this coefficient was selected to provide the proper
approximation of experimental results (see, e.g., Fig. 6.8), we can hardly expect
that Eqs. (6.41) specify the components of a strength tensor. To show that this is
really not the case, apply to the approximation criterion under study transformation
resulting in Eq. (6.30). The coefficients of thus constructed approximation criterion
in coordinates (l', 2') become