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
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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. 2, (2019) 2019 SjF STU Bratislava 17 a gradient of a scalar variable = (X,,t) which is well-known to be zero, i.e.
0 0() = 0 0 (X0,t) = [22]. Then we have
T T (0F) = c 0( F = c) 0( 0 x) c = 0 0 ( x c) = (10) = 0 0 (X ,t) = 0 , 0 F = 0 . and since c is an arbitrary nonzero vector, Eq. (10) implies that the compatibility conditions, defined by Eq. (9) hold true for any deformation gradient tensor Fx=0 .Then Eq. (9) and the Stokes’ theorem (Fig. 1) Fd X = x d X = N F dS = N x dS = 0, 0 ( 0) 0 0( 0) 0 (11) 00 cc SS00 implies that F is integrable and that its integral is independent of the integration path 0 0 0 0 QP S0 on condition that the surface S0 is simply connected, where c = PQ + QP is a 0 closed curve, S0 is the surface enclosed by the c curve and N is the unit outward normal vector of the surface S0 .
Fig. 1 Proper kinematics of motion of elastoplastic deformation when the body first deforms plastically and then elastically
It also should be noted, that Eqs. (9)-(11) hold true in any surface S0 ,a cross-section of the body containing the points 00PQ, which was created by cutting the initial volume of the body into half by an imaginary cutting plane. Then the integral of the deformation gradient becomes
XXQQ x= x + F d X, or ΦXΦXFXX , t = , t + , t d . QPQP( ) ( ) ( ) (12) XXPP
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Moreover, because according to the theory of nonlinear continuum mechanics [9] x= x( X,t) =Φ( X , t) = Φ( X ,0) +0 u = X + 0 u( X , t) = X + 0 u , (13) the deformation gradient can equivalently be expressed as
00 F= F( X,,.tt) = 0 x = I + 0 u( X) = I + 0 u (14) Then the curl of the deformation gradient (Eq. (9)), and the fact that the curl of a gradient of 0 any vector is zero (see Eq. (10) when replacing Fx=0 by 0 uX( ,t) ), will result in the following, explicitly satisfied compatibility conditions for the Lagrangian displacement field 0 0 0 =+0F 0 I 0 u( X,,.tt) = 0 0 u( X) = 0( 0 u) = 0 (15) Eq. (15) then along with the Stokes’ theorem
00u X,,,t d X = N u X t dS = 0 0( ) 0 0( ) 0 (16) 0 c S0 ensures the integrability of the material gradient of the Lagrangian total displacement field 0 0 uX( ,t) either, whose path-independent integral then takes the form
XXQQ 0uu= 0 + 0 uX d, or 0 uX , t = 0 uX , t + 0 uXX , t d . QPQQPP00( ) ( ) ( ) (17) XXPP Then after substitution of Eq. (14) into Eq. (12) and carrying out the integration, while taking into account Eqs. (17) and (13) respectively, we have
XXXQQQ x =ΦX,,,t = ΦX t ++= I00 uXΦX d t ++= d X uX d QQPP( ) ( ) ( 00) ( ) XXXPPP 00 (18) =Φ( XPPQQQPP ,t) − X + X + u( X , t) − u( X , t) = 00 =XQQQQQ + u( X ,t) = X + u , which is just the same equation as Eq. (13) for any material point 00Q of the initial volume 0 of the body. The above results simply imply that meeting the compatibility conditions Eq. (9) and (15) results in the integrability of both, the deformation gradient
F= F( X,,t) = 0 x and the material gradient of the total Lagrangian displacement field 00 00u = u( X,t) respectively, whose path independent integrals along any path between any two material points 0PQ, 0 0 respectively, located in any imaginary plane of the body allow for the determination of the relative motion (Eq. (12)) and the relative displacement (Eq. (17)) between the two points. As a result, the most important consequence of the integrability is the existence of the motion x = ΦX( ,,t) the Lagrangian displacement field 0uX( ,t) and the well-known relationship from nonlinear continuum mechanics (Eq. (18)) that allows for the determination of the position vector of each material point 00Q of the body at any time t 0 during the whole deformation process.
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3 On the compatibility assumption of contemporary multiplicative plasticity models Contemporary multiplicative plasticity models are based on the assumption that the intermediate configuration of the body is either stress-free [1] or locally unstressed [2] and that therefore no plastic deformation exists that meets the conditions of compatibility. The deformation gradient in the models is then decomposed multiplicatively into a Lagrangian elastic part and a Lagrangian plastic part only FFXFFFXFX=( ,,,.t) =el pl = el( t) pl ( t) (19) Since the left-hand-side (l-h-s) of the deformation gradient is integrable, as we have shown it in the above, the right-hand-side (r-h-s) of Eq. (19) must be integrable too, otherwise, the total displacement field could not be calculated. This is; however, possible only if both, the elastic FFXel= el ( ,t) and plastic FFXpl= pl ( ,t) parts of the deformation gradient are also integrable. Moreover, the Eq, (14) form of the deformation gradient and the compatibility conditions Eqn. (15) too indirectly imply the integrability of the elastic and plastic parts of the deformation gradient, i.e. there must exist a Lagragian plastic displacement field 00upl= u pl ( X,t) and a Lagragian elastic displacement field 00uel= u el ( X,t) such that
0 0pl 0 el FFX=( ,,,,,t) ==+0 xI 0 uXI( t) =+ 0 uX( t) + 0 uX( t) and (20) 0 0el 0 pl =0F 0 0 u( X,,,,t) = 0 0 u( X t) + 0 u( X t) = 0 (21) or that 0u( X,,,,t) =+ 0 uel( X t) 0 u pl ( X t) otherwise the L-H-S and the R-H-S of Eqs. (19) and (14) cannot be the same. The most important consequence of these is that the multiplicative split (Eq. (19)) cannot be the final form of the deformation gradient. Indeed, if one considers the fact that the conditions of compatibility for the plastic part of the deformation gradient are explicitly satisfied over the initial volume 0 of the body, just in the same way as in the case of the deformation gradient (see Eq. (10) for the proof where one just needs to replace F with F pl bellow), i.e.
i pl− el pl pl i X ΦX( ,t) 0 pl 0F = 0( 0 X) = 0 = 0 = 0( 0 u) = 0, (22) XX
pl pl 0 pl then FFX= ( ,t) and 0 u are according to the Stokes’ theorem integrable and their 00 inegrals are independent of the inegration path PQ S in any imaginary (cutting) plane 0 i pl− el pl S0 (see Fig. 1). As a result, there must exist a plastic motion X = ΦX( ,,t) a plastic Lagrangian displacement field 00upl= u pl ( X,t) and a relationship between them such that,
XXQQ iXXFX= i + pl d, or pl−− elΦXΦXFXX pl , t = pl el pl , t + pl , t d , QPQP( ) ( ) ( ) (23) XXPP
XXQQ 0uupl= 0 pl + 0 uX pl d, or 0 uX pl , t = 0 uX pl , t + 0 uXX pl , t d , QPQQPP00( ) ( ) ( ) (24) XXPP i pl− el pl00 pl pl XQQQQQQQ=Φ( X,,.tt) = X + u( X) = X + u (25) Exactly, in the same manner, one can prove that the conditions of compatibility for the elastic part of the deformation gradient are explicitly satisfied over the intermediate volume
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i of the body (see Eq. (10) for the proof where one just needs to replace 0 with i and F with Fel bellow), i.e. pl− elΦX el i ,t elx ( ) i el (26) iF == i( i x) iii = i = i( i u) = 0, XX
el el i i el then FFX= ( ,t) and i u are according to the Stokes’ theorem integrable and their i integrals are independent of the integration path PQS i in any imaginary (cutting) plane i pl− el pl i Si (see Fig. 1). As a result, there must exist an elastic motion x = ΦX( ,,t) an elastic Eulerian displacement field iu pl= i u el( i X,t) and a relationship between them such that,
ii XXQQ x= x + Fel d i X, or pl−− elΦXΦXFXX el i , t = pl el el i , t + el i , t d i , QPQP( ) ( ) ( ) (27) ii XXPP ii XXQQ ielieluu= + ieli uXuXuX d, or ieli , t = ieli , t + ieli uXX , t d i , Q P i Q( Q) P( P) i ( ) (28) ii XXPP pl− el el i i i el i i i el xQQQQQQQ=Φ( X,,.tt) = X + u( X) = X + u (29) Then after substitution of Eq. (25) into Eq. (29) and considering the fact that the Eulerian i el i el i i elastic displacement field uQQQ= u( X ,t) , defined over the intermediate volume , 00el el 0 becomes a Lagrangian one uQQQ= u( X ,,t) defined over the initial volume , that is 00uel== u el X,,,,t i u el pl− elΦX pl t t then we have QQQQQ( ) ( )
x =pl−− elΦ el pl el ΦX pl,,,,.t t = XuX +0 pl t + 0 uX el t = Xuu + 0 pl + 0 el QQQQQQQQQQ( ) ( ) ( ) (30) Comparing Eq. (30) with Eq. (18) it is easy to see that the function expressing the overall motion x ==ΦXΦΦX,,,tpl−− el el pl el pl t t is actually a composite function and that QQQ( ) ( ) the total Lagrangian displacement field is just the sum of the Lagrangian elastic and the 0 0pl 0 el Lagrangian plastic displacement fields, i.e. uQQQQQQ( X,,,t) =+ u( X t) u( X t) at any material point 00Q of the body, as it was expected. Moreover, the respective gradients of all pieces of the motion and all displacement fields in the above meet the corresponding conditions of compatibility and therefore are integrable. As a result, the deformation gradient Eq. (19) takes it final Lagrangian form
pl−− el el pl el pl x ΦΦX( ,,tt) pl− elΦX pl ( ,t) x i X F= F( X,t) = I + 0 u = = = = 0 X pl− elΦXXXX pl( ,t) i (31) el pl0 el 0 pl 0 el 0 pl =FXFXI( ,t) ( , t) =+0 uX( , t) + 0 uXI( , t) =++ 0 u 0 u , where (see. Fig. 1)
i el i 00el el xX uX( ,t) u( X,,tt) u( X ) −1 FXIIIFXel( ,,tt) = = + = + = + pl ( ) (32) iXXXXX i i
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i X 0uXpl ( ,t) FXIpl ( ,,t) = = + (33) XX and where considering the fact hat iu el( i X,,tt) = 0 u el ( X ) and the mathematical identity
ii XXXXX −1 IFXFX= = =pl( ,tt) = pl ( ,) , (34) iXXXXX i i i
i el i el i el i the spatial gradient i u of the Eulerian displacement field u= u( X,t) can be expressed in the following Lagrangian form
i el i 00el el uX( ,t) u( X,,tt) X u( X ) −1 iu el = = = F pl ( X,.t) (35) i iiXXXX The most important implications of the above results are the facts, that • the plastic part of the deformation gradient FXpl ( ,t) is integrable over the initial volume 0 of the body, • the elastic part of the deformation gradient FXel( i ,t) is integrable over the intermediate volume i of the body, • and as a result, the deformation gradient FX( ,t) is integrable over the initial volume of the body, while the respective conditions of compatibility are explicitly satisfied in all the above cases. As a result, the assumption that there is no plastic deformation that meets the conditions of compatibility [1, 2], on which the whole theory of contemporary multiplicative plasticity models is based, is simply not justified from the mathematical point of view.
4 On the stress-free/locally unstressed configuration assumption of contemporary multiplicative plasticity models Contemporary multiplicative plasticity models are based on the assumption that the intermediate configuration of the body is either stress-free [1] or locally unstressed [2] and as a result, no plastic deformation exists that meets the conditions of compatibility. The assumption of a stress-free/locally unstressed intermediate configuration is; however, not compatible with the theory of nonlinear continuum mechanics, as it violates proper stress transformations resulting from the invariance of the internal mechanical power. The invariance of the internal mechanical power plays an inevitable role in making sure that the total and updated Lagrangian formulations are equivalent and therefore the analysis results, when used in the weak solution of the respective problem, are not affected by the particularities of the formulation, i.e. that the material model can be formulated in whatever stress space in terms of internal power conjugate stress measures and strain or deformation rates without influencing the analysis results. This is achieved by two postulates, namely that the force acting on the surface of an infinitesimal volume element and the rate of the change of the internal mechanical energy accumulated in the element in the body’s initial and current configurations be the same [8, 9], which was later on extended by Écsi and Élsztős and their associates to cover constitutive equations in rate forms [10-12]. Upon using the same procedure, the Cauchy’s stress theorem iTPN= i i , the Nanson’s formula for the
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pl pl −T infinitesimal surface element transformation dSFSi = J ( ) d 0 and the formula for the pl pl pl infinitesimal volume element transformation dVi = J dV0 , where J = det(F ) , adapted to the intermediate configuration of the body [9], one can arrive at a non-symmetric stress tensor iPPX= i( i ,t) acting over the intermediate volume i (in the intermediate configuration) of the body, and two other stress tensors iσ= i σ( i X,t) and iτ= i τ( i X,,t) similar to the Cauchy’s and Kirchhoff stress tensors respectively as follows (Fig. 2)
Fig. 2 The stress tensor acting over the intermediate volume i of the body during elastoplastic deformations when the body first deforms plastically and then elastically
i i i i dfT= dS0 = PN dS 0 = PST d 0 ==== dSi PN dS i PS d i TT PFFFSFpl el pl pl el i (36) pl i pl−T i( ) ( ) F τ el i =Jd PFSPF ( ) 0 =pl = pl = pl = σ , JJJ where
i i pl plT i τ τ= F S ( F) , σ = pl are symmetric stress measures. (37) J Then from the invariance of the internal mechanical power we have
pl T PF( ) −−11 dW=PFFFPFF:::, dV = pl dV = i pl dV (38) 0 pl ( ) ii ( ) 0 ii J or
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TT −1 dW==SESFFFSFFFF::: dVT = dV pl pl el pl = dV 0 ( ) 0 ( ) ( ) ( ) 0 0 0 0 (39) TT−−11sym sym =iτ : F el F F pl dV = i σ : F el F F pl dV , ( ) ( ) 0 ( ) ( ) i 0 i or simply
−1 dW=iPFF: pl dV = iτ : i d dV = i σ : i d dV , ( ) ii 0 ii0 (40) T −1 sym where id= F el F F pl . ( ) ( ) Eqs. (36) and (37) simply state that from the nonlinear continuum mechanics point of view the intermediate configuration cannot be either stress-free or locally unstressed unless the body is stress-free or locally unstressed in all configurations of the body. Moreover, one can find conjugate pairs of stress measures and strain or deformation rates (see Eq. (40)) defined over the intermediate volume i just in the same manner as over the initial 0 and current volumes in nonlinear continuum mechanics. As a result, the assumption of a stress-free or locally unstressed intermediate configuration is not justified physically from the nonlinear continuum theory point of view. However, it should be noted, that all the problems herein except for the stress-free/locally unstressed configuration with contemporary plasticity models can be eliminated by modifying the kinematics of elastoplastic deformations in accordance with the theory of nonlinear continuum mechanics, which then will result in the entirely new nonlinear continuum theory for finite-deformations of elastoplastic media already published by Écsi and Élesztős and their associates [10-12].
CONCLUSION In this paper, a study on the ‘compatibility assumption’ of contemporary multiplicative plasticity models was presented. The study shows that the compatibility conditions are explicitly satisfied in the cases of both, the deformation gradient and the respective gradient of the displacement field and their elastic and plastic parts. Therefore, all, the deformation gradient and its elastic and plastic parts and the respective gradients of the displacement fields are integrable. Moreover, it has also been shown that the overall motion is a composite function, consisting of an elastic part and a plastic part. Therefore the overall displacement field and its elastic and the plastic parts have both, Lagrangian and Eulerian forms in all configurations of the body. In addition to this, it was demonstrated that the intermediate configuration of the body cannot be stress-free or locally unstressed if the models are to be continuum based. Apart from this, internal mechanical power conjugate pairs of stress measures and strain or deformation rates exist over the intermediate volume of the body. As a result, the ‘compatibility assumption’, on which the whole theory of contemporary multiplicative plasticity models is based, is neither mathematically nor physically justified and the resulting material models are in reality not continuum based.
ACKNOWLEDGEMENTS The paper was supported by the grant from KEGA no. 017STU-4/2018 entitled "Theoretical and Practical Solution of Structures on Elastic Linear and Nonlinear Foundation".
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[18] Acharya A. “On compatibility conditions for the left Cauchy-Green deformation field in three dimensions”, Journal of Elasticity 56, pp. 95 – 105, 1999. [19] Amrouche C, Ciarlet PG, Grate L, Kesavan S. “On Saint Venant’s compatibility conditions and Poincaré’s lemma”, In: Mathematical Problems in Mechanics. C. R. Acad. Sci. Paris 342, Ser. I, pp. 887 – 891, 2006. [20] Schäfer HN, Schmidt JP. “Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers”, Heidelberg: Springer, 2014. ISBN- 978-3-662-43443-7. [21] Continuum mechanics/Curl of a gradient of a vector. On official website of Wikiversity, https://en.wikiversity.org/wiki/Continuum_mechanics/Curl_of_a_gradient_of_a_vector , accessed May 16 2019. [22] Schey HM. “Div, grad, curl and all that”, An informal text on vector calculus. 3nd ed. NY: Norton & Company, 1997. ISBN-0-393-96997-5. [23] Jančo, R. “Solution of the thermo-elastic-plastic problems with consistent integration of constitutive equation”, Strojnícky časopis – Journal of Mechanical Engineering 53 (4), pp. 197 – 214, 2002. [24] Halama, R., Markopoulus, A., Jančo, R, Bartecký, M. “Implementation of MAKOC cyclic plasticity model with memory“, Advanced in Engineering Software 113, pp. 34 – 46, 2017. DOI: 10.1016/j.advengsoft.2016.10.009
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