ORIGINAL ARTICLE On the identification of kinematic hardening with reverse shear test A. F. G. Pereira • P. A. Prates • N. A. Sakharova • M. C. Oliveira • J. V. Fernandes Received: 24 April 2014 / Accepted: 21 June 2014 Ó Springer-Verlag London 2014 Abstract An inverse analysis methodology for deter- mining the parameters of the kinematic law of sheet metals is proposed. The sensitivity of the load versus displacement curves, obtained by reverse shear tests of rectangular and notched specimens, to the kinematic law parameters are studied following a forward analysis, based on finite ele- ment simulations. Afterwards, an inverse analysis meth- odology using a gradient-based Levenberg–Marquardt method is established, by evaluating the relative difference between numerical and experimental results of the shear test, i.e. the load evolution in function of the displacements of the grips. The use of a notched specimen is proposed in order to allow an easy and suitable numerical representa- tion of the boundary conditions of the shear experimental test. This methodology has proven to be appropriate for determining the parameters of the kinematic hardening law. Keywords Reverse shear test  Kinematic hardening  Inverse analysis  Parameters identification 1 Introduction The mechanical behaviour of metal sheets is usually described by constitutive equations that allow the approx- imate representation of experimental tests. This behaviour is often described by a law of elasticity, the generalized Hooke’s law, and for the plastic behaviour, by a yield criterion and an isotropic hardening law, which is generally suitable for the deep drawing of components with simple geometries and requirements. The production of compo- nents with complex geometries and tight requirements that imposes successive changes of strain path (multi-stage stamping, etc.) requires the accurate characterization of the material behaviour. Under these conditions, additional characterization using a kinematic hardening law, which describes the material behaviour under reverse strain path, is required. Mechanical tests with reverse strain path are used to identify the parameters of the kinematic law of metal sheets [1–4], which characterizes the Bauschinger effect of the material. The shear test is widely used for this identi- fication [1, 5], since it allows characterizing the behaviour of the material up to large plastic deformations under for- ward and reverse paths, which cannot be achieved in ten- sile testing of sheet metals, for example. Different geometries of the shear specimens were suggested [6, 7]. In short, the shear specimens can have a rectangular shape, and be clamped by fixed and moving grips [6], or two symmetrical rectangular shear zones, and be acted by axial and lateral loads through an inner and two outer grips to equilibrate each other [7]. The determination of the shear stress–shear strain curve is not always properly achieved. This is mainly due to the heterogeneity of the strain and stress fields, namely due to the end effects and the occur- rence of plastic deformation in the regions of the specimens under the grips [5]. The occurrence of heterogeneous deformation, on a greater or lesser degree, leads to inac- curacies in determining the average shear stress and strain. To overcome some of these problems it is recommended to use specimens with appropriate dimensions and measuring the local shear strain using digital image correlation A. F. G. Pereira (&)  P. A. Prates  N. A. Sakharova  M. C. Oliveira  J. V. Fernandes CEMUC, Department of Mechanical Engineering, University of Coimbra, Polo II, Rua Luı ´s Reis Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal e-mail: andre.pereira@dem.uc.pt 123 Engineering with Computers DOI 10.1007/s00366-014-0369-7