Variational Principles in Plasticity and Thier Finite Element Formulas

In this paper, we first prove the elastic/viscoplastic variational principle and regard the rigid/viscoplastic, elastic-plastic and rigid-plastic va-riational principle as its special case, then derive the finite element formulas.
The constitutive equations of elastic/viscoplastic material are eq. (2-14), (2-16).
Suppose the strain-hardening function of material is H=H(\mathop \varepsilon \limits^\rm\cdot VP, \mathop \varepsilon \limits^\rm\cdot ) then have eq. (2-17).
Usually, d \mathop \varepsilon \limits^\rm\cdot is rather small, we can obtain eq. (2-18).Finally, we have eq. (2-22).
The elastic/viscoplastic variational principle says:Among all the possi-ble vi, \mathop \varepsilon \limits^\rm\cdot ij, the actual solution renders the functional (2-23) a stationary value.
From dynamic tests at different strain rate, eq. (2-27) is obtained by regression. Work rate functions are eq. (2-28) and (2-29).
The rigid/viscoplastic conventional variational principle is that among all the possible vi, \mathop \varepsilon \limits^\rm\cdot ij, the actual solution renders the functional (2-23) a stationary value.
According to the Fig. 3-1, we suppose the stress-strain relations for complicated stress state are eq. (3-5).The work rate functions are eq. (3 -6) and (3-7).The variational principle concerning non-plastic region and unloadingproblems can be discribed as follow:among all the vi, εij, the actual solu-ion renders the functional (2-23) a stationary value.
Nadai's constitutive equation (4-4) can be extended to elastic-plastic deformation case. We futher define eq. (4-5), and let eq. (4-7), then we obtain eq. (4-8).
Under the condition of εij<1,wwe obtain eq.(4-9).Since the constitutive equation (4-8) is a homogeneous function of time, the functional (2-23) can be rewritten as eq. (4-10).
The variational principle of Nadai's deformation theory says. Among all the ui,εij,the actual solution renders the functional (4-10) a stationary value.
According to the experimental results, a variational principle based on the general functional (2-23) is proved on the some proper handling of the constitutive equations for different materials. Some other variational principle which mentioned above and in our other papers can be regarded as special cases of the variational principle discussed here.