Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- a :
-
Acceleration
- b :
-
Specific body force
- [B k ]:
-
Matrix to evaluate the gradient at the integration point of element k
- \({\fancyscript{B}_0,\fancyscript{B}_t}\) :
-
Body in the reference and current placement
- da 0, da :
-
Element of area in the reference and current placement
- dm 0, dm :
-
Element of mass in the reference and current placement
- dv 0, dv :
-
Element of volume in the reference and current placement
- E :
-
Young’s modulus
- E = ln V :
-
Spatial logarithmic strain tensor
- \({{\mathbf{E}^p}^{\bullet}}\) :
-
Plastic strain rate
- \({\Delta \mathbf{E}_k^p}\) :
-
Plastic strain increment at element k
- F :
-
Total traction force in the test
- F :
-
Deformation gradient
- \({\delta\mathbf{F}^{\bullet}}\) :
-
Virtual velocity gradient
- f :
-
Resultant of the external forces
- δ D :
-
Virtual rate of deformation tensor
- I :
-
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
- n 0, n :
-
Normal vector in the reference and current placement
- p :
-
Equivalent cumulated plastic strain
- R :
-
Rotation tensor
- R :
-
Lankford parameter
- t :
-
Surface load
- S = {s ij }:
-
Deviatoric part of the Cauchy stress tensor
- T = {σ ij }:
-
Cauchy stress tensor
- T 1PK :
-
1st Piola-Kirchhoff stress tensor
- U, V :
-
Right and left stretch tensors
- u :
-
Displacement vector
- \({[\mathbf{u}_k^N ]}\) :
-
Matrix of the nodal displacement at element k
- δ v :
-
Virtual velocity vector
- \({[ \delta \mathbf{v}_k^N ]}\) :
-
Matrix of the nodal virtual velocity at element k
- x 0, x :
-
Position vector in the reference and current placement
- ν :
-
Poisson’s ratio
- ξ = {X i }:
-
Constitutive parameters vector
- σ T :
-
Equivalent stress
- σ Y :
-
Yield stress
- Φ p :
-
Yield function
- χ :
-
Motion function
- k :
-
Index referring to the element
- N cp :
-
Number of constitutive parameters
- N E :
-
Number of elements
- N t :
-
Number of time steps
- t :
-
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
- scalars:
-
Italic letters like A, B, a, b, α, β …
- vectors:
-
Small letters in bold like a, b, α, β …
- tensors:
-
Large Latin letters in bold like A, B, …
References
Bertram A (2008) Elasticity and plasticity of large deofrmations, an introduction. Springer-Verlag, Berlin
Hill R (1952) On discontinuous plastic states, with special reference to localized necking in thin sheets. J Mech Phys Solids 1(1): 19–30
Nichols FA (1980) Plastic instabilities and uniaxial tensile ductilities. Acta Metall 28: 663–673
Tvergaard N (1993) Necking in tensile bars with rectangular cross-section. Comp Meth Appl Mech Eng 103: 273–290
Cazacu O, Plunkett B, Barlat F (2006) Orthotropic yield criterion for hexagonal closed packed metals. Int J Plasticity 22: 1171–1194
Plunkett B, Lebenson RA, Cazacu O, Barlat F (2006) Anisotropic yield function of hexagonal materials taking into account texture development and anisotropic hardening. Acta Mater 54: 4159–4169
Barlat F, Yoon JW, Cazacu O (2007) On linear transformations of stress tensors for the description of plastic anisotropy. Int J Plasticity 23: 876–896
Barlat F, Lege D, Brem J (1991) A six-component yield function for anisotropic materials. Int J Plasticity 7(7): 693–712
Karafillis AP, Boyce MC (1993) A general anisotropic yield criterion using bounds and a transformation weighting tensor. J Mech Phys Solids 41: 1859–1886
Darrieulat M, Piot D (1996) A method of generating analytical yield surfaces of crystalline materials. Int J Plasticity 12: 575–610
Feigenbaum HP, Dafalias YF (2007) Directional distortional hardening in metal plasticity within thermodynamics. Int J Solids Struct 44: 7526–7542
Voyiadjis GZ, Thiagarajan G, Petrakis E (1995) Constitutive modelling for granular media using an anisotropic distortional yield model. Acta Mech 110: 151–171
Vegter H, van den Boogaard AH (2006) A plane stress yield function for anisotropic sheet material by interpolation of biaxial stress states. Int J Plasticity 22: 557–580
Bai Y, Wierzbicki T (2008) A new model of metal plasticity and fracture with pressure and Lode dependence. Int J Plasticity 24(6): 1071–1096
Hwang YM, Lin YK, Altan T (2007) Evaluation of tubular materials by a hydraulic bulge test. Int J Mach Tools Manuf 47: 343–351
Hwang Y, Wang C (2009) Flow stress evaluation of zinc copper and carbon steel tubes by hydraulic bulge tests considering their anisotropy. J Mater Process Tech 209: 4423–4428
Brunet M, Morestin F, Godereaux S (2001) Nonlinear kinematic hardening identification for anisotropic sheet metals with bending-unbending tests. J Eng Mater Technol 123: 378–383
Laws V (1981) Derivation of the tensile stress–strain curve from bending data. J Mater Sci 16: 1299–1304
Losilla G, Tourabi A (2004) Hardening of a rolled sheet submitted to radial and complex biaxial tensile loadings. Int J Plasticity 20: 1789–1816
Mohr D, Oswald M (2008) A new experimental technique for the multi-axial testing of advanced high strength steel sheets. Exp Mech 48(1): 65–77
Wierzbicki T, Bao Y, Lee Y, Bai Y (2005) Calibration and evaluation of seven fracture models. Int J Mech Sci 47(4–5): 719–743
Cooreman S, Lecompte D, Sol H, Vantomme J, Debruyne C (2007) Elasto-plastic material parameter identification by inverse methods: calculation of the sensitivity matrix. Int J Solids Struct 44: 4329–4341
Kajberg J, Lindkvist G (2004) Characterisation of materials subjected to large strains by inverse modelling based on in-plane displacement fields. Int J Solids Struct 41: 3439–3459
Mahnken R, Stein E (1994) The identification of parameters for visco-plastic models via finite-element methods and gradient methods. Model Simul Mater Sci Eng 2: 597–616
Meuwissen MHH, Oomens CWJ, Baaijens FPT, Petterson R, Janssen JD (1998) Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method. J Mater Process Tech 75: 204–211
Tao H, Zhang N, Tong W (2009) An iterative procedure for determining effective stress–strain curves of sheet metals. Int J Mech Mater Des 5: 13–27
Ling Y (1996) Uniaxial true stress-strain after necking. AMP J Tech 5: 37–48
Zhang KS, Li ZH (1994) Numerical analysis of the stress–strain curve and fracture initiation for ductile material. Eng Fract Mech 49: 235–241
Rossi M, Broggiato GB, Papalini S (2008) Application of digital image correlation to the study of planar anisotropy of sheet metals at large strains. Meccanica 43(2): 185–199
Latourte F, Chrysochoos A, Pagano S, Wattrisse B (2008) Elastoplastic behavior identification for heterogeneous loadings and materials. Exp Mech 48: 435–449
Romano AJ, Shirron JJ, Bucaro JA (1998) On the noninvasive determination of material parameters from a knowledge of elastic displacements: theory and numerical simulation. IEEE Trans Ultra Ferr 45: 751–759
Romano AJ, Bucaro JA, Ehman RL, Shirron JJ (2000) Evaluation of a material parameter extraction algorithm using MRI-based displacement measurements. IEEE Trans Ultra Ferr 47: 1575– 1581
Avril S, Bonnet M, Bretelle A-S, Grédiac M, Hild F, Ienny P, Latourte F, Lemosse D, Pagano S, Pagnacco E, Pierron F (2008) Overview of identification methods of mechanical parameters based on full-field measurements. Exp Mech 48: 381– 402
Grédiac M, Pierron F, Avril S, Toussaint E (2006) The virtual fields method for extracting constitutive parameters from full-field measurements: a review. Strain 42: 233–253
Grédiac M, Pierron F (2006) Applying the virtual fields method to the identification of elasto-plastic constitutive parameters. Int J Plasticity 22: 602–627
Avril S, Pierron F, Pannier Y, Rotinat R (2008) Stress reconstruction and constitutive parameter identification in plane-stress elasto-plastic problems using surface measurements of deformation fields. Exp Mech 48: 403–419
Pierron F, Avril S, The Tran V (2010) Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening. Int J Solids Struct 47: 2993–3010
Pierron F, Sutton MA, Tiwari V (2011) Ultra high speed DIC and virtual fields method analysis of a three point bending impact test on an aluminium bar. Exp Mech. 51: 537–563
Avril S, Pierron F, Sutton M, Yan J (2008) Identification of elasto-visco-plastic parameters and characterization of lüders behavior using digital image correlation and the virtual fields method. Mech Mater 40: 729–742
Bay BK, Smith TS, Fyhrie DP, Saad M (1999) Digital volume correlation: Three-dimensional strain mapping using X-ray tomography. Exp Mech 39: 217–226
Smith TS, Bay BK, Rashid MM (2002) Digital volume correlation including rotational degrees of freedom during minimization. Exp Mech 42: 272–278
Verhulp E, van Rietbergen B, Huiskes R (2004) A three-dimensional digital image correlation technique for strain measurements in microstructures. J Biomech 37: 1313–1320
Gates M, Lambros J, Heath MT (2011) Towards high performance digital volume correlation. Exp Mech. 51: 491–507
Franck C, Hong S, Maskarinec S, Tirrel D, Ravichandran G (2007) Three-dimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation. Exp Mech 43: 207–218
Germaneau A, Doumalin P, Dupré J (2007) 3D strain field measurement by correlation of volume images using scattered light: recording of images and choice of marks. Strain 43: 207–218
Roux S, Hild F, Viot P, Bernard D (2008) Three-dimensional image correlation from X-ray computed tomography of solid foam. Compos Part A-Appl S 39: 1235–1265
Rossi M, Chiappini G, Sasso M (2010) Characterization of aluminum alloys using a 3D full field measurement. In: 2010 SEM Annual Concerence & Exposition on Experimental & Applied Mechanics
Atay S, Kroenke C, Sabet A, Bayly P (2008) Measurement of the dynamic shear modulus of mouse brain tissue in-vivo by magnetic resonance elastography. J Biomech Eng 130: 021013
Hill R (1948) A theory of the yielding and plastic flow of anisotropic metals. Proc Roy Soc Lond Ser A, Math Phys Sci 193: 281–297
Lankford WT, Snyder SC, Bausher JA (1950) New criteria for predicting the press performance of deep drawing sheets. Trans Am Soc Metals 42: 1197–1232
Banabic D, Bunge HJ, Pöhlandt , Tekkaya A (2000) Formability of metallic materials. Springer, Berlin
Avril S, Pierron F (2007) General framework for the identification of constitutive parameters from full-field measurements in linear elasticity. Int J Solids Struct 44: 4978–5002
Guélon T, Toussaint E, Le Cam JB, Promma N, Grédiac M (2009) A new characterisation method for rubber. Polym Test 28: 715–723
Dhondt G (2004) The finite element method for three-dimensional thermomechanical applications. John Wiley & Sons, New York
Zienkiewicz OC, Taylor R (2006) The finite element method. Butterworth-Heinemann, Oxford
Avril S, Grédiac M, Pierron F (2004) Sensitivity of the virtual fields method to noisy data. Comput Mech 34(6): 439–452
Fletcher R (1987) Practical Methods of Optimization. John Wiley and Sons, New York
Constrained Nonlinear Optimization Algorithm (2011) From http://www.mathworks.com/help/toolbox/optim
Savitzky A, Golay MJE (1964) Smoothing and differentiation of data by simplified least squares procedures. Anal Chem 36: 1627–1639
Gorry A (1990) General least-squares smoothing and differentiation by the convolution (Savitzky-Golay) method. Anal Chem 62: 570–573
Author information
Authors and Affiliations
Corresponding author
Additional information
The notation used in the paper follows the indications given in Elasticity and Plasticity of Large Deformations (Bertram, 2008) [1].
Rights and permissions
About this article
Cite this article
Rossi, M., Pierron, F. Identification of plastic constitutive parameters at large deformations from three dimensional displacement fields. Comput Mech 49, 53–71 (2012). https://doi.org/10.1007/s00466-011-0627-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-011-0627-0