The system of governing equations of plastic deformation

<< Click to Display Table of Contents >>

Navigation:  Theoretical environment of QForm UK > Plastic deformation of materials >

The system of governing equations of plastic deformation

The algorithm of QForm UK implies the successive solution for problems in mechanics of deformations and thermal processes. Thus, at every step of simulation the temperature distribution through the bulk of workpiece is considered as constant. For a solution of the problem of stress-strain state determination the system of equations of mechanics of deformable bodies is used.

Click to show/hide hidden textThe system of resulting equations for visco-plastic bodies

System of equations allowing determination of body stressed and strained state at isometric plastic deformation includes:

•equilibrium equations

•compatibility conditions between velocity field and strain rate components

•Levy–Mises constitutive equations (mathematical relation between stressed and strained state)

•incompressibility condition (volume constancy law)

•essential boundary conditions on the surface Fv

•natural boundary conditions on the surfaceFp

•yield criterion

•rheological model of material (expression for flow stress)

Satisfaction of balance differential equation system, of boundary conditions with respect to Levy–Mises constitutive equations is equivalent to variational formulation that follows from the condition of Markov's functional stationary state ( δsign designates variation)

The first term in the functional right part represents the power of internal forces (the power of plastic strain in the body volume V), and the second represents the power of known external forces (Fp is the surface where the external forces are specified). At minimization of Markov's functional the balance equation system and boundary conditions are satisfied automatically.

This formulation is used for deriving of constitutive equation system of finite-element method for simulation of plastic deformations in QForm UK.
Click to show/hide hidden textThe system of governing equations for non-compact materials

For non-compact (powder) materials in the system of governing equations changes:

•the plasticity condition

•the constitutive equations
Click to show/hide hidden textElastic-plastic deformations

For elastic-plastic deformations in the system of governing equations are used the constitutive Prandtl-Reuss equations

Continuity equation is true only for plastic component of total deformation