Issue 52

A. Ayadi et alii, Frattura ed Integrità Strutturale, 52 (2020) 148-162; DOI: 10.3221/IGF-ESIS.52.13 149 and growing body of literature has investigated the nonlinear elastoplastic behavior of structures as it appears in a wide range of applications and industrial processes. For instance, crashworthy structures show a complex combination of structural nonlinearities in which plastic deformation is expected to contribute to the energy absorption [3]. Another example is drop test of some mobile devices such as mobile phones, personal digital assistants and laptops which end up generally with permanent deformation [4]. Due to complexity of geometry, boundary and loading conditions, there are limited practical problems in plasticity where analytical solution is available. Therefore, numerical analysis, and in particular, the finite element method has emerged as a viable alternative in addressing this kind of problems. Within this framework, accuracy and efficiency of finite elements have gained fresh prominence. To avoid the problem of singularity in the stiffness matrix and also to improve the accuracy of numerical results, there has been renewed interests in finite element with in-plane rotational degrees of freedom called also drilling rotations [5]. Since the first successful element with drilling variables proposed by Allman [6], a considerable amount of literature has been published on this subject [7–13]. Working along similar lines, many three-dimensional solid element formulations with rotational DOFs have been also proposed [14–24]. Ayad et al.[24], developed two eight-node hexahedral elements SFR8 and SFR8I based on the so-called Space Fiber Rotation concept (SFR). This approach, firstly introduced by Ayad [25], considers 3D virtual rotations of a nodal space fiber within the finite element that improves the displacement vector approximation. Similarly, many authors implemented this concept in different area of research [16, 17, 23–25, 27]. Despite the abundance of literature on the finite elements with rotational DOFs, only a few have been proposed to account for material and geometric nonlinear problems. These includes the works of Gruttmann et al. [28], Ibrahimbegovic et al. [29–31], Rebiai et al. [13] and Zouari et al. [32]. In the work of Zouari et al. [32], this approach has been successfully adopted to develop two four-node membrane elements, named PFR4 and PFR4I, to analyze linear and nonlinear geometric plane problems. This paper presents a new eight-node membrane finite element PFR8 in order to analyze elastoplastic nonlinear problems. The element’s formulation is based on a planar adaptation of the SFR concept which results in a simpler and more concise formulation than the previous developed membrane elements with rotational DOFs. The remaining part of the paper proceeds as follows. In section 2, the formulation behind the proposed quadrilateral element is detailed. In section 3, the model is framed within the nonlinear analysis through the small strain elastoplastic constitutive model. Finally, and before highlighting the concluding remarks, numerical results relative to a series of benchmarks are presented. B ASIC CONSTITUTIVE EQUATIONS OF CONTINUUM ELASTOPLASTICITY fter initial yielding the material behavior will be partly elastic and partly plastic. The phenomenon is characterized by an irreversible deformation where the total strain can be expressed:     e p ij ij ij      (1) where   e ij  and   p ij  are the elastic and plastic components of the total strain respectively. The elastoplastic behavior is governed by a scalar function called yield function of the form:   , 0 ij     (2) where ij  denotes the indicial form of the stress tensor and  represents hardening parameters. As shown in Fig. 1, the yield condition can be visualized as a surface in the stress space where the size of the surface depends on the value of the parameter  . In this paper, the elastoplastic model based on the Von-Mises associated yield criterion is adopted. Thus, the above yield condition can be expressed as:   , 0 p e y         (3) where e  is the Von-Mises effective stress and y  is the yield stress. A