Analytic study of the onset of plastic necking instabilities during biaxial tension tests on metallic plates

• We model the development of plastic necking instabilities during dynamic tension tests on plates biaxially loaded in their plane.
• We determine the instabilities growth rate with a linear stability analysis.
• The model applies whatever the thickness.
• The material is an elastoviscoplastic incompressible homogeneous metal satisfying Von Mises' criterion and normality flow rule.
• We retrieve Dudzinski and Molinari's results for very thin statically loaded plates and rate-dependent materials.

In this article, we study the onset of the development of plastic necking instabilities during dynamic tension tests on metallic plates biaxially loaded in their plane. This model applies whatever the thickness. The material is supposed to be homogeneous, isotropic, incompressible, elastoviscoplastic, to satisfy Von Mises' plasticity criterion and normality flow rule (damage and heat conduction are neglected). As Dudzinski and Molinari did in 1988 for static tests on very thin plates in the framework of generalized plane stress theory (thickness is supposed to vary slowly and slightly along loading directions) (Instabilité de la déformation viscoplastique en chargement biaxial, 1988, Compte Rendu à l'Académie des Sciences de Paris307, série 2, pages 1315–1321), we carry out a linear stability analysis. The flow is viewed as the sum of the mean homogeneous flow of the perfect plate, and of small perturbations δv→ of the velocity field, periodic along the x1− and x2− loading directions, growing exponentially. We search for the most unstable pair of wavelengths (λ1(d),λ2(d)) and for the associated growth-rate θ(d) (the dominant mode). Plastic deformation concentrates preferably along zero rate extension lines for non positive velocity gradient ratio α=D22/D11α=D22/D11 (D denoting deformation rate tensor), and along lines parallel to minor principal stress direction for biaxial stretching (α > 0). For sufficiently viscous materials, inertia plays a negligible role (maximum plastic strain-rate considered in this paper equals 20 s−1, and thickness does not exceed 2 cm), the wavelength associated with the dominant mode is much greater than thickness, and Dudzinski and Molinari's model gives the associated growth-rate very accurately. This growth-rate is a root of a polynomial equation, that we re-establish starting from the equations of our 3D model. For non viscous materials, inertia is no longer negligible for non positive values of α.