The implicit corotational method and its use in the derivation of nonlinear structural models for beams and plates

Giovanni Garcea, Antonio Madeo, Raffaele Casciaro
2012 Journal of Mechanics of Materials and Structures  
What we call the implicit corotational method is proposed as a tool to obtain geometrically exact nonlinear models for structural elements, such as beams or shells, undergoing finite rotations and small strains, starting from the basic solutions for the three-dimensional Cauchy continuum used in the corresponding linear modelings. The idea is to use a local corotational description to decompose the deformation gradient in a stretch part followed by a finite rigid rotation. Referring to this
more » ... ferring to this corotational frame we can derive, from the linear stress tensor and the deformation gradient provided by linear elasticity, an accurate approximation for the nonlinear stress and strain tensors which implicitly assure frame invariance. The stress and strain fields recovered in this way as functions of generalized stress and strain resultants are then used within a mixed variational formulation allowing us to recover an objective nonlinear modeling directly suitable for FEM implementations through a black-box process which maintains the full details of the linear solutions, such as shear warping and other subtle effects. The method is applied to the construction of three-dimensional beam and plate nonlinear models starting from the Saint-Venant rod and Kirchhoff and Mindlin-Reissner plate linear theories, respectively. This paper has been developed within the national joint research project "Performance-based modeling and analysis of nonlinear structures," supported by the Italian Ministry of University Scientific and Technology Research (MIUR). We would like to thank all the participants in the project for their comments and suggestions. Keywords: geometrically exact beam and shell theories, corotational description, postbuckling analysis. 509 510 GIOVANNI GARCEA, ANTONIO MADEO AND RAFFAELE CASCIARO The great majority of beam and shell models are based on geometric exact theories such as those developed in [Cosserat and Cosserat Auricchio et al. 2008]. Models so generated are geometrically exact, that is, exactly frameindependent, but, being based on simplified assumptions in the constitutive laws relating strain and stress resultants, are generally unable to describe important details already present in the corresponding linear models. This is evident, for example, in the classical Antman-Simo nonlinear beam model where the assumed simplified constitutive law lacks the shear/torsional coupling manifested by the three-dimensional Saint-Venant linear solution [de Saint-Venant 1855] and more subtle nonlinear couplings associated with the section warping, such as the axial-torsional second-order coupling recognized in [Wagner 1936]. Models derived as Ritz-Galerkin approximations by introducing a three-dimensional displacement field in a variational principle allow for more detailed modeling, at least in principle. This approach was followed, for instance, in [Pai and Nayfeh 1994; Pai et al. 1998; Petrov and Géradin 1998a; 1998b; Nayfeh and Pai 2004; Pi et al. 2005; Kim et al. 2005;]. However, extending to finite kinematics the three-dimensional displacement provided by the linear theory, like the nonlinear beam model of [Petrov and Géradin 1998a; 1998b], appears somewhat overcomplex and also requires ad hoc simplifications in order to eliminate spurious locking. Models obtained by the use of problem-dependent engineering nonlinear strain measures, like the beam and shell models of Nayfeh and Pai (see [Nayfeh and Pai 2004]), are only aimed at an essential simplified modeling. On the other hand the availability of linear structural models for fibred continua derived, using a small displacements hypothesis, from three-dimensional Cauchy equations through appropriate assumptions on the statics and kinematics of the body is notable. Its use, as a basis to generate a corresponding nonlinear model, is then attractive due to the possibility of recovering all the effort spent in developing linear theories in a simplified context. The aim of this paper is to exploit this possibility through the use, in the continuum description, of the corotational approach initially proposed in a FEM context in [Wemper 1969; Belytschko and Glaum 1979; Rankin 1986; Nour-Omid and Rankin 1991] and used to construct a nonlinear finite element starting from a linear one. In this paper we show that, by transferring this idea from the element to the continuum, we can derive a standard methodology to obtain a frame-indifferent nonlinear modeling which maintains all the details of the embedded linear solution. We call the proposed method the implicit corotational method (ICM). The main idea is to associate a corotational frame (observer) to each point of the three-dimensional continuum so allowing the motion in the neighbor of the point to be split in a pure stretch followed by a pure rotation, according to the decomposition theorem [Malvern 1969; Bonet and Wood 1997]. It will be shown that, using the small strain hypothesis and rotation algebra, the linear stress and linear strain solution fields, when viewed in this corotational frame, can provide accurate approximations for the Biot nonlinear stress and strain tensor fields. Once the corotational rotation is appropriately defined, the local statics and kinematics of the model are recovered from the linear solution as a function of the stress/displacement resultants. Stress and strain fields are then introduced within a mixed variational principle in order to obtain the constitutive laws directly in terms of stress/strain resultants. This completes the ICM definition of the nonlinear model. The thus-obtained nonlinear model retains all the details of the three-dimensional linear solution, including torsion/shear warping, while its objectivity is ensured implicitly. Furthermore, the use of the mixed formulation and the greater accuracy with which ICM recovers the stress field allow an accurate THE IMPLICIT COROTATIONAL METHOD 511 512 GIOVANNI GARCEA, ANTONIO MADEO AND RAFFAELE CASCIARO The present article describes a coupled experimental/computational study of damage development in confined ceramic tiles impacted by spherical metal projectiles. The principal objective is to calibrate the material parameters in the Deshpande-Evans constitutive model for an armor alumina and assess its utility in predicting trends in damage development with impact velocity. The nature of the damage at the impact site is probed through optical and scanning electron microscopy of cross-sections through the impact site as well as surface profile measurements. Once calibrated, the model is used in finite element simulations and shown to predict reasonably accurately the variation in the size of the comminuted zone beneath the impact site with incident projectile velocity. The numerical simulations also provide new insights into the spatial and temporal evolution of subsurface damage and deformation processes as well as the role of metal face sheets in the these processes. mathematical sciences publishers msp JOURNAL
doi:10.2140/jomms.2012.7.509 fatcat:lqle7qihjzgf7eyddayxs3hiyu