A general and efficient multistart algorithm for the detection of loss of ellipticity in elastoplastic structures

Abstract : The present paper proposes a new efficient and robust algorithm for evaluating the loss of ellipticity criterion. While commonly used in two-dimensional models for thin metal sheet forming processes, it is rarely evaluated in three-dimensional structures due to the computational cost. The proposed algorithm is based on a Newton-Raphson scheme and a multisampling optimization method based on a discretization method of the half unit sphere. First the new process is compared to the existing methods in the literature and then it is applied to a structural problem, namely tubes in torsion. The evolution of the loss of ellipticity in these structures is analyzed leading to conclusions about the failure of the structure. Meanwhile, the stability of the discretized problem is analyzed in order to better understand the loss of regularity of the finite element method problem. These results are then used to predict the failure of an experimentally tested torsion sample. K E Y W O R D S elastoplasticity, finite deformation, loss of ellipticty, numerical method, strain localization 1 INTRODUCTION Even though strain localization is one of the most critical phenomena leading to the failure of elasto-plastic structures, its emergence is still not fully understood. Indeed, the term "localization" itself is interpreted differently depending on the context. In a loose sense, localization means the development of high strains in a narrow region of the body, like in a shear band with through thickness necking in a thin plate in tension. A more precise definition is the emergence of strain rate discontinuities through surfaces usually associated with loss of ellipticity, and we use this latter definition in the present work. In some situations, both definitions may even coincide, for instance, a one-dimensional bar displaying a softening behavior experiences simultaneously necking and loss of ellipticity, 1,2 or in thin plates modeled under plane stress conditions. 3 However, these definitions do not coincide for three-dimensional (3D) models in general. When localization starts in complex structures and whether it leads to catastrophic failure is still an open problem in many situations , especially when certifying industrial components. The example of tubes loaded in torsion, 1 shows that even an apparently simple structure can lead to difficulties in defining localization and the localization's influence on the safety of the global structure. When dealing with localization, a first distinction needs to be made between types of loss of ellipticity. Strong ellipticity corresponds to definite positiveness of the eigenvalues of the symmetrized acoustic tensor whereas ellipticity refers to the absence of vanishing eigenvalue of the general acoustic tensor. 4,5 Int J Numer Methods Eng. 2019;1-25. wileyonlinelibrary.com/journal/nme
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Moubine Al Kotob, Christelle Combescure, Matthieu Mazière, Tonya Rose, Samuel Forest. A general and efficient multistart algorithm for the detection of loss of ellipticity in elastoplastic structures. International Journal for Numerical Methods in Engineering, Wiley, 2019, ⟨10.1002/nme.6247⟩. ⟨hal-02358732⟩

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