Computational Mechanics

, Volume 49, Issue 1, pp 53–71 | Cite as

Identification of plastic constitutive parameters at large deformations from three dimensional displacement fields

  • Marco Rossi
  • Fabrice Pierron
Original Paper


The aim of this paper is to provide a general procedure to extract the constitutive parameters of a plasticity model starting from displacement measurements and using the Virtual Fields Method. This is a classical inverse problem which has been already investigated in the literature, however several new features are developed here. First of all the procedure applies to a general three-dimensional displacement field which leads to large plastic deformations, no assumptions are made such as plane stress or plane strain although only pressure-independent plasticity is considered. Moreover the equilibrium equation is written in terms of the deviatoric stress tensor that can be directly computed from the strain field without iterations. Thanks to this, the identification routine is much faster compared to other inverse methods such as finite element updating. The proposed method can be a valid tool to study complex phenomena which involve severe plastic deformation and where the state of stress is completely triaxial, e.g. strain localization or necking occurrence. The procedure has been validated using a three dimensional displacement field obtained from a simulated experiment. The main potentialities as well as a first sensitivity study on the influence of measurement errors are illustrated.


Constitutive modelling Inelastic and finite deformation Inverse methods Numerical algorithms 

List of symbols





Specific body force


Matrix to evaluate the gradient at the integration point of element k


Body in the reference and current placement

da0, da

Element of area in the reference and current placement

dm0, dm

Element of mass in the reference and current placement

dv0, dv

Element of volume in the reference and current placement


Young’s modulus

E = lnV

Spatial logarithmic strain tensor


Plastic strain rate

\({\Delta \mathbf{E}_k^p}\)

Plastic strain increment at element k


Total traction force in the test


Deformation gradient


Virtual velocity gradient


Resultant of the external forces


Virtual rate of deformation tensor


Unit tensor

\({\widehat{\mathbf{N}}_{P\,\,k}^{\,\,\,\,\,\,(t)}=\{ \widehat{n}_{ij}^P \}}\)

Normalised tensor of the plastic flow

\({\widehat{\mathbf{N}}_{S\,\,k}^{\,\,\,\,\,(t)}=\{ \widehat{n}_{ij}^S \}}\)

Normalised tensor of the deviatoric stress

n0, n

Normal vector in the reference and current placement


Equivalent cumulated plastic strain


Rotation tensor


Lankford parameter


Surface load

S = {sij}

Deviatoric part of the Cauchy stress tensor

T = {σij}

Cauchy stress tensor


1st Piola-Kirchhoff stress tensor

U, V

Right and left stretch tensors


Displacement vector

\({[\mathbf{u}_k^N ]}\)

Matrix of the nodal displacement at element k


Virtual velocity vector

\({[ \delta \mathbf{v}_k^N ]}\)

Matrix of the nodal virtual velocity at element k

x0, x

Position vector in the reference and current placement


Poisson’s ratio

ξ = {Xi}

Constitutive parameters vector


Equivalent stress


Yield stress


Yield function


Motion function



Index referring to the element


Number of constitutive parameters


Number of elements


Number of time steps


Index referring the time step



Inner or scalar product

| |

Modulus of a tensor or absolute value of a scalar

Grad, grad

Material and spatial gradient operator

Generic notations


Italic letters like A, B, a, b, α, β


Small letters in bold like a, b, α, β


Large Latin letters in bold like A, B, …


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.LMPF, Arts et Métiers ParisTechChâlons-en-ChampagneFrance

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