Rate independent plasticity occurs at low temperatures, roughly in the temperature range below 0.57^. The deformation is due to

dislocation glide and strain rate due to thermal fluctuations plays no significant role. The relaxation time is zero. In rate independent plasticity the total strain rate is decomposed into elastic and plastic strain rate.

Introduce the identities:

tp = *J(2/3)DPDP > 0 (5-13)

is the effective plastic strain rate or equivalent plastic strain rate.

The plastic stretching or deformation rate tensor is:

Dp=4bil£pNp (5-14)

where Np is the unit tensor defining the direction of plastic flow.

Introduce the outward unit normal to the yield surface at the current stress point:

N = 4bT2T'lo (5-15)

At this point we adopt the classic associated plasticity (or normality) flow rule:

NP=N (5-16)

The rule (5-16) means that the principal axes of plastic stretching are the same as the principal axes of deviatoric stress or, in other words, the direction of plastic flow is the outward normal to the yield surface at the current stress point.

Consider the derivative of the plastic potential function (5-4) with respect to effective stress <7 :

P=e0{-)Vm (5-17)

s

The yield phenomenon suggests that there is a switching from

єp = 0 if d <s to an unbounded plastic strain rate if <7 > s. That is the case if m—>0. Physically this means that an unbounded plastic strain rate immediately leads to material hardening (for a hardening material) that, in turn, moves the yield surface up to the stress point and satisfies the constraint s = a at any time moment. Then define

the evolution equations for deformation resistance (isotropic hardening):

s = hep (5-18)

For rate independent isotropic plasticity, the time integration of (5-9) and (5-10) is implemented by [35]. Full details are presented in [7].

Steady-State Formulation

It has long been recognized that a quasi-steady state temperature field exists in certain long welds. If only thermal loads from a weld heat source are present in such welds, due to this quasi-steady state, all materials points on one flow line or trajectory in these welds will experience the same thermal cycle. Microstructures and mechanical properties in a weldment are expected to change along flow lines during welding. These observations have motivated the analysis of stresses in such steady state welds.

The notable points of steady state formulation for stress analyses of welds are the following:

-The kinematic model differs from the usual Lagrangian or Eulerian models.

-History-related quantities at a material point are obtained from streamlines in a reference frame after some mapping.

-The solution includes history dependent effects such as plasticity and microstructure evolution. It includes also the strain caused by phase transformation.

-Tests have been done on an edge weld. The distortion in the welded bar is clearly demonstrated.

-Satisfactory results are obtained. Computed longitudinal stresses have been compared with the data calculated in the Lagrangian formulation and measured in experiments.

-The convergence rate is similar to that of a Lagrangian formulation.

-Significant savings of computing time and memory usage have been achieved.

This method has several distinguishing features:

(a) The thermal strain creates the load in the weldment. If there is no constraint on a weldment, it is the only load.

(b) The mechanical properties of most materials are both temperature and (plasticity and microstructure) history dependent

(c) Heat sources, and therefore the loads, are remarkably localized.

(d) The resulting thermal stress is high enough to cause plastic deformation.

(e) Phase transformations can either increase or release stresses.

The steady state analysis differs from the usual Lagrangian

formulation for calculating the stiffness matrix and residual vector in that the current formulation takes Gauss point data from flow lines for computation. Every Gauss points knows the element it resides in, which makes it easier for a Gauss point to make contributions to the element, or to the nodes of the element. Since the Gauss points are object-oriented, pointers from Gauss points to elements are simple and natural.

The steady state formulation uses flow lines to trace the history and thus the evolution of all internal variables. Before welding, the mesh has no distortion, and the elements of the mesh are regular, Figure 5-11. The Gauss points in elements are aligned to form flow lines by the methods and data structure described by Gu M. in references [6, 25 and 50].

The Gauss points are initially aligned to form straight lines in the undeformed state. Since the mesh deforms with respect to a Lagrangian frame, these straight lines distort continuously with the deformation. They are the exact flow lines for the current state when the convergence is reached. The whole nonlinear approach is to compute these flow lines.

Constitutive equations describe the material response to the strains at the current state. In a numerical approach, it is customary to assume initially that the deformation is elastic. In Gu et al [6], the elasto-plastic constitutive model uses a modified effective-stress - function (ESF) algorithm [15, 16 and 17].

The trial stress, 'a*, at each integration point is assumed to be:

/

'<T*=M(7+ ^Dde (5-19)

/-1

where D is a modified material property tensor. The superscript і

indicates a material point x (at time t) on flow line containing a

spatial point x.

The i-1 indicates the same material point at time (t - At) on the flow line. The incremental trial stress, therefore,

і

А'ст* = }Dde (5-20)

/-1

Including thermal, elastic, plastic, transformation and creep deformation, constitutive equations can be written in the form:

T = -^-(е-,єр-ієс-ієТгр) (5-21)

1+' v

‘E

(em-ieth-ieTrv) (5-22)

1-2'v where at point i:

'T = deviatoric stress tensor

/ _ /

— G— (J

^ HI

'e = deviatoric strain tensor

= ‘є-є

ПІ

'єр = plastic strain tensor

'єс - creep strain tensor

'eTrp = transformation plastic strain tensor

1 оm = stress tensor

'em = strain tensor

‘e‘h = thermal strain tensor

і „Trv

є = strain due to volume change in phase transformation 'E,' v = Young's modulus and Poisson's ratio at point і corresponding to temperature at point і

The ESF algorithm has been adopted to increase the efficiency of solving for the incremental plastic strain, Aep, as well as the incremental creep strain, Aec. A brief description of ESF used in the current case is given by Gu et al [6 and 50]. Details of the ESF can be found in Kojic and Bathe [9 and 10]. The transformation plasticity formulation is due to Oddy et al [17 and 42].