Page 151 - Mechanics Analysis Composite Materials

136 Mechanics and analysis of composite materials

Iteration methods discussed above are used to solve the direct problems of stress
analysis, i.e., to find stresses and strains induced by a given load. However, there
exists another class of problems requiring us to evaluate the load carrying capacity
of the structure. To solve these problems, we need to trace the evolution of stresses
while the load increases from zero to some ultimate value. To do this, we can use the
method of successive loading. According to this method, the load is applied with
some increments, and for each s-step of loading the strain is determined as

(4.41)

where ES-l and v,-l are specified by Eqs. (4.39) and correspond to the previous
loading step. Graphical interpretation of this method is presented in Fig. 4.1 Id. To
obtain reliable results, the load increments should be as small as possible, because
the error of calculation is accumulated in this method. To avoid this effect, method
of successive loading can be used in conjunction with the method of elastic
variables. Being applied after several loading steps (black circles in Fig. 4.1 Id) the
latter method allows us to eliminate the accumulated error and to start again the
process of loading from a proper initial state (light circles in Fig. 4.1 Id).
Returning to constitutive equations of the deformation theory of plasticity,
Eq. (4.25), it is important to note that these equations are algebraic. This means that
strains corresponding to some combination of loads are determined by the stresses
induced by these loads and do not depend on the history of loading, i.e., on what
happened to the material before this combination of loads was reached.
However, existing experimental data show that, in generaI, strains should
depend on the history of loading. This means that constitutive equations should
be differential rather than algebraic as they are in the deformation theory. Such
equations are provided by the flow theory of plasticity. According to this theory,
decomposition in Eq. (4.15) is used for infinitesimal increments of stresses, Le.,

Here, increments of elastic strains are linked with the increments of stresses by
Hooke’s law, e.g., for the plane stress state

(4.43)

while increments of plastic strains

are expressed in the form of Eqs. (4.18) but include parameter A which characterizes
the loading process.