Page 308 - Mechanics Analysis Composite Materials

Chapter 6. Failure criteria and strength of laminates 293

first time proposed by M.T. Huber in 1904. However, this fact became widely
known only in 1924 at the International Congress on Applied Mechanics in Delft,
The Netherlands. Before this Congress, this criterion was associated with R. Mises'
paper published in 1913 in which it was introduced as an approximation criterion.
The original plasticity criterion of maximum shear stress being widely recognized
but having not an easy-to-use hexagon form on the plane of principal stresses
(a:02, 212 = 0) was approximated by R. Mises with Eq. (6.45) as shown in Fig.
6.19.
Thus, for isotropic materials, the quadratic approximation strength criterion
under discussion has an invariant form. However, this is not true for orthotropic
materials. Using transformation in Eq. (6.37) for $J # 45" we arrive at constraints
similar to Eq. (6.40) that do not coincide, in contrast to the tensor criterion. with
Eqs. (6.44) for 5, = 45". From this it follows that approximation polynomial criteria
can be used only in coordinates in which they approximate experimental results.
In general, comparing tensor-polynomial and approximation strength criteria
we can conclude the following. Tensor criteria should be used if our purpose is to
develop a theory of material strength, because a consistent physical theory must be
covariant, i.e., constraints that are imposed on material properties within the
framework of this theory should not depend on a particular coordinate frame. For
practical applications, approximation criteria are more suitable, but in the forms
they are presented here they should be used only for orthotropic unidirectional plies
or fabric layers in coordinates whose axes coincide with the fibers' directions.
To evaluate the laminate strength, we should first determine the stresses acting in
the plies or layers (see Section 5.10), identify the layer that is expected to fail first
and apply one of the foregoing strength criteria. The fracture of the first ply or layer
may not necessarily result in the failure of the whole laminate. Then, simulating the
failed element with a proper model (see, e.g., Section 4.4.2) the strength analysis is
repeated and continued up to the failure of the last ply or layer.
In principle, failure criteria can be constructed for the whole laminate as a quasi-
homogeneous material. Being not realistic for design problems, to solve which we

Fig. 6.19. Maximum shear stress criterion (-) and its elliptic approximation with Eq. (6.45)
(---- ) on the plane of principal stresses.