Of Contemporary Multiplicative Plasicity Models

Strojnícky časopis – Journal of MECHANICAL ENGINEERING, VOL 69 (2019), NO 2, 15 - 26 A STUDY ON THE ‘COMPATIBLITY ASSUMPTION’ OF CONTEMPORARY MULTIPLICATIVE PLASICITY MODELS ÉCSI Ladislav1, JERÁBEK Róbert1, ÉLESZTŐS Pavel1 1Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Applied Mechanics and Mechatronics, Nám. Slobody 17,812 31 Bratislava, Slovakia, e-mail: [email protected] Abstract: Contemporary multiplicative plasticity models are now generally accepted as “proper material models” for modelling plastic behaviour of deformable bodies within the framework of finite-strain elastoplasticity. The models are based on the assumptions that the intermediate configuration of the body is stress-free or locally unstressed, for which no plastic deformation exists that meets the conditions of compatibility. The assumption; however, has never really been questioned nor justified, but was rather taken as an axiom and therefore considered to be generally true. In this study, we take a critical look at the assumption from both, physical and mathematical points of view, in order to investigate whether contemporary multiplicative plasticity models are indeed continuum based and if there are alternatives to them. KEYWORDS: Multiplicative plasticity models, finite-strain elastoplasticity, continuum theory, compatibility 1 Introduction Contemporary multiplicative plasticity models are now generally accepted as “proper material models” for modelling plastic behaviour of deformable bodies within the framework of finite-strain elastoplasticity [1 - 3]. The models are based on the theory of single-crystal plasticity, developed by Asaro and his associates [4 - 6], which describes the micromechanics of irreversible plastic deformations in the material. In order to model the plastic flow in the material, the models use a multiplicative split of the deformation gradient into a Lagrangian elastic part and a Lagrangian plastic part, where in addition is assumed that the intermediate configuration of the body is stress-free [1] or locally unstressed [2], for which no plastic deformation exists that meets the conditions of compatibly [2]. Therefore, the models treat the kinematics of deformation differently between the initial and current configurations of the body, where the motion, the displacement field and the deformation gradient, respectively are considered in accordance with the theory of nonlinear continuum mechanics [7 - 9, 23, 24], but not between the configurations where one is an intermediate configuration of the body. Here the motion and the displacement field are disregarded and as a result, the deformation gradient loses its physical meaning [10 - 12]. Moreover, the models offer a solution to problems with a particular order of deformations only, namely when the body first undergoes plastic deformations and then elastic deformations at its every constituent. Virgin materials just do not deform in this way, as they cannot undergo plastic deformations until they have undergone elastic deformations so that the models fail to imitate even the true physics of the deformation process. From the physical point of view, the assumption of a stress-free/locally unstressed intermediate configuration also sounds strange, especially if one considers the fact that this state ought to exist when the body is mechanically loaded, and therefore is stressed in both, the initial and current configurations of the body. Moreover, to prove that the formulation of the models is thermodynamically consistent also poses a problem and according to the authors, it will not be possible until the elastic and plastic displacement fields are somehow related to the elastic and plastic parts of the deformation gradients. DOI: 10.2478/scjme-2019-0015, Print ISSN 0039-2472, On-line ISSN 2450-5471 2019 SjF STU Bratislava The aim of this paper is to take a critical look at the ‘compatibly assumption’ of contemporary multiplicative plasticity models, in order to investigate how much the assumption is justified from both, physical and mathematical point of view, so that one could decide whether the models are continuum based and if there are alternatives to them. 2 Compatibility conditions and their effect on the solution of the respective problems describing the behaviour of a deformable body Compatibility conditions play an inevitable role in making sure that a deformable body, which was continuous in its initial configuration before deformation, remains continuous after deformation and that the continuity applies to the displacement field too during the solution. In fact, meeting the compatibility conditions is a necessary requirement to make sure that a strain or deformation tensor characterizing the deformation at a material point of the body is integrable, or more exactly that a line integral of the above tensors along any path between two material points of the body, determined by their position vectors x0 and x respectively, is independent of the integration path [13, 14]. In small-strain elasticity, the compatibility conditions are described by the following equations [13, 14] small small ( ε) = ikr jls ij, kl == e r e s θ 0, (1) where 1 2u 2u smallε = small = e e r +j == e e θi e e . ijk rj, i k r ijk k r i j (2) 2 xj x i x r x i x j Here small ε=1/ 2 u + u T is a small-strain tensor, θ= e =1/ 2 u e is the ( ) i i ijk k, j i axial vector of infinitesimal rotations, ω=1/ 2 − u uT = =− e e e e ( ) ij i j ijk k i j is a skew-symmetric infinitesimal rotation tensor and ijk is the Levi-Civita or permutation symbol. Eqn. (1) then indirectly ensures the existence of the following path-independent integral, on condition that the body is simply-connected [13, 14] xx θ x= θ x + θ dd x = θ x + small ε x, ( ) ( 00) ( ) ( ) (3) xx00 where the path independence follows from Eq. (1) and the Stokes’ theorem small small (ε) =d x n ( ε) = ds 0, (4) cs where c is a closed curve, a perimeter of a simply-connected surface s, determined by arbitrary two not intersecting pats between the two material points, whose position vectors are and respectively. Here n is the unit outward normal vector of the surface s . Then in the same manner one can calculate the displacement field as [14], xx u x= u x + u dd x = u x +small ε + ω x, ( ) ( 00) ( ) ( ) (5) xx00 where considering the Stokes’ theorem again 16 2019 SjF STU Bratislava Volume 69, No. 2, (2019) small small ( ε+ ω) d x = n ( ε + ω) ds = 0, (6) cs and the fact that in small-strain elasticity ωθ = − and small εθ = , the right-hand-side of Eq. (6) becomes zero [14], which ensures that the integral Eq. (5) becomes independent of the integration path. Eq. (1) represents six constraint equations, which are called the conditions of compatibility of Barré de Saint-Venant [15]. The conditions can alternatively be expressed by a vanishing, fourth-order Saint-Venant tensor W =Wijkl e i e j e k e l in an n-dimensional space [16] Wijkl= ij,,,, kl + kl ij − ik jl − jl ik = 0. (7) However, it should be noted that in a three-dimensional (3D) Euclidian space the independent components of the Saint-Venant tensor W coincide with the independent components of the incompatibility tensor R = ( small ε) defined by Eq. (1) [16], so the two vanishing tensors actually define identical compatibility conditions in the 3D Euclidian space. The determination of the compatibility conditions when the deformation of the body is characterized by a finite-strain or deformation tensor is much more difficult. The difficulty stems from the fact that in limiting state when infinitesimal deformations take place in the body, each finite-strain tensor reduces to the small-strain tensor. Several attempts have been made to determine the compatibility conditions using finite-strain formulation combined with Riemann geometry, as in [17-19]. The most important results of these are the facts that in all cases the sufficient conditions of compatibility are either somehow related to the vanishing Riemann curvature tensor Eq. (8) (such as Eq. (7) which is closely related to Eq. (8), or Eq. (1) which is just a Bianchi identity of the Riemann curvature tensor [16]) or are defined in terms of a Christoffel symbol of second kind k = u k// x p 2 x p u j u i [18] for ij ( ) ( ) i =1,2,3, where xi are Cartesian coordinates and ui are curvilinear coordinates [20] n n n m n m n Rijk= ik,, j − ij k + ik mj − ij mk = 0. (8) It should; however, be noted that in Cartesian coordinates, in which the majority of 3D problems of nonlinear continuum mechanics are formulated, both, the Christoffel symbol of k n second kind ij and the Riemann curvature tensor Rijk are zero tensors and as a result, the compatibility conditions above are explicitly satisfied [20]. Then, the sufficient conditions of compatibility become significantly simpler and can be defined by a vanishing curl of the deformation gradient tensor [14] Frk x ΦX( ,t) =0Fijk = e i e r 0( 0 x) = 0 = 0 = 0, (9) X j XX where 0 ( •) = ( •) / X is the nabla operator expressed in terms of material coordinates, F= F( X,t) = 0 x is the deformation gradient, x = ΦX( ,t)is the position vector of a spatial point in the current configuration of the body defined in terms of a vector function called motion and X is the position vector of a material point. Because the curl of a gradient of a vector is zero, the compatibility conditions defined by Eq. (9) are always explicitly satisfied. The proof of the above assertion is simple [21]. One just needs to post-multiply 0 F by a constant vector c= const 0,use some identities and manipulate the expression into a curl of Volume 69, No.

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