This document presents the hypoelastic formulations of large deformations used in Code_Aster for the constitutive laws of the type von Mises.

Transcription

1 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : /3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 Models of large deformations GDEF_LOG and GDEF_HYPO_ELAS Summarized: his document presents the hypoelastic formulations of large deformations used in Code_Aster for the constitutive laws of the type von Mises. GDEF_LOG : this model, due to C.Miehe, N.Appel and M.Lambrecht [3] is a model of large deformations based to a logarithmic curve measure, with a particular stress tensor in duality. It is valid whatever the behavior in small strains and has the advantage of providing a symmetric tangent matrix. No kinematical modification of the local variables is necessary. GDEF_HYPO_ELAS : this model is resulting from an approach due to Simo and Hughes []. It is based on the notion of a turned configuration objectifies in which the derivatives are carried out. his generic approach makes it possible to treat the constitutive laws with hardenings isotropic and kinematical, with or without viscosity, with formulation of a hypoelastic type. hese two models allow an integration incrémentalement objectifies constitutive laws like SIMO_MIEHE the model. However, like all the hypoelastic models, these constitutive laws in any rigor are limited to the weak elastic strain. One and the illustrates in this document the capacities of these models advantages compared to the approximation PEI_REAC.

2 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : /3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 Contents Introduction3... Algorithm GDEF_HYPO_ELAS3.... Elements of cinématique Constitutive laws... hypo-élastoplastiques5 3. objective Integration: algorithm GDEF_HYPO_ELAS Algorithm of integration Choices of the skew-symmetric tensor and intégration Advantages and limitations of the méthode Algorithm GDEF_LOG...4. Description summary...4. Algorithm... 5 Validity of the large models déformations Identification of the paramètres Comparison with PEI_REAC Approximation of the large deformations by PEI_REAC Comparison on a exemple Bibliographie Description of the versions of the document Appendix : tangent operator for GDEF_HYPO_ELAS Principle of the computation of the tangent matrix: variational writing and general statement...9. Computation of the tangent matrix for both formalismes... erm...3 erm...3 erm erm erm erm Assessment Appendix : Computation of the strains logarithmiques Notations: Statement of the stresses in Lagrangian configuration Statement of the tangent operator in Lagrangian configuration effective Computation of the strains logarithmiques3...

3 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 3/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 Introduction Most constitutive laws of Code_Aster are usable under the assumption of the small disturbances (HP), which makes it possible to confuse the geometrical configurations initial and current. However, when the strains become important (one in general fixes the limit at 5%), this assumption is not checked any more. he notions of derivatives particulate and partial are then different, and of this fact the constitutive laws formulated incrémentalement lose their objective character (independence of the mechanical state according to the observer); a tedious consequence is the possible evolution of the stresses for a rigid body motion, opposite with physics. In order to find objectivity, essential thus to guarantee a good reliability of result, various strategies large deformations are possible. he object of this document is to present the formalisms set up in Code_Aster to treat the constitutive laws with hardenings isotropic and kinematical and criterion of Von Mises. One presents firstly the formalism hypoelastic of Simo-Hughes [] usable in Code_Aster via key word DEFORMAION=GDEF_HYPO_ELAS of operator COMP_INCR. One result presents then one allowing to check the objectivity of this integration and to check that the formalism PEI_REAC is put at fault. In a second part, one presents formalism GDEF_LOG, due to C.Miehe, N.Appel and M.Lambrecht [3] which is a model of large deformations based to a logarithmic curve measure, with a particular stress tensor in duality. It is valid some is the behavior in small strains and present the advantage of providing a symmetric tangent matrix. No kinematical modification of the local variables is necessary. Algorithm GDEF_HYPO_ELAS. Elements of kinematics the kinematical elements in the continuous case can be found for example in [3]. One will be interested here in the case directly discretized in time allowing to define the quantities used in the formalism presented in this document. One considers a closed initial continuous field 0 R 3, whose each point is located by its coordinates X 0, undergoing a strain field making it pass in the configuration : : 0 R 3 One will note x the coordinates of this point in the current configuration. he deformed shape evolving in the course of time, one actually defines, by the means of the temporal discretization, a family of field n corresponding each one to one time t n of the history of evolution of the field. In the case of the formalism large deformations treated here, it is necessary to introduce 4 configurations for the field and its evolution (cf Figure ): initial 0 configuration of reference (i.e. for which the strains are null), configuration n at the beginning of time step running t n+ =t n t, configuration n + at the end of this time step, and a configuration medium of time step n+, formalism being integrated with a rule of point medium. Eq

4 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 4/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 F n Ω n Ω 0 F n + Ω n + ~ f f n + n + f n + F n + Ω n + Figure : Configurations necessary and gradients of transformation From these configurations, one defines the fields of displacements and the gradients of transformation to pass from the one to the other. he quantities making pass from the initial configuration to a given configuration are noted in capital letter U,F and the quantities connecting two configurations deformed between them are noted into tiny u, f. able recapitulates the various quantities and their statements. Configuration starting Configuration of arrival Gradient Displacement of transformation 0 n U n F n =I d grad 0 U n 0 n + U n + F n + =I d grad 0 U n + 0 n + F = n + F F n n+ n n + f n+ = I d + grad n u - = F. F n + n = f n I d n n + u f n+ = I d + grad n u = F n +. F n - = F n + n + f n+ = f n +. f - n - = F n +. F n able : summary of displacements and gradients of transformation

5 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 5/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 From these gradients of transformation, it is possible to define strain rates L as well as tensor rates of rotation and strain rate d : { L. = Ḟ. F. d. = L L Eq... = L L.. with. indicating the configuration n, n or n. he Eulerian strain tensor enters the configurations n and n results from these definitions: e n + = [ I f f d n+ n + ] Eq 3 what makes it possible to write d n + = t eulerian strain) f n + e n + f n + (strain rate is then quite related to the the last quantity to be introduced for our algorithm is the incremental displacement gradient, relating to the configuration n + and defined by: h = grad n u f n+ n + Eq 4 his last makes it possible to determine the rate of rotation in the same configuration by the relation: = n + t [h h ] n + n + Eq5 From these elements of kinematics, it is possible to define hypoelastic constitutive laws whose integration is objective in large deformations. he following paragraph presents this kind of formulation of the constitutive laws. 3 Hypo-elastoplastic constitutive laws In this section, the phenomenologic class of model of plasticity (here independent of time) with hypoelasticity is considered. It constitutes an ad hoc extension of the writing of the models in small strains, which allows certain generics and represents an advantage in the context of a computer code: one will see in the chapter according to whether it is possible to carry out its numerical integration in a way equivalent to that of the small strains. his class of models is to be opposed to the class hyper elastic, based on the thermodynamic approach of the continuums. In this context, a free energy, being able for example to be regarded as a function of the temperature and the strain of Green-Lagrange, is defined; the evolutions of the stresses and possibly of the local variables result from this. One can quote for example the case of the model hyper elastic of Signorini (cf [4]) in elasticity and the Simo-Miehe formalism in hyperelastoplasticity (cf [3]). An hypo-elastoplastic constitutive law is generally built in five stages. (I) Following the example of the additive decomposition of the small strains, strain rate d is first of all broken up into an elastic part and a plastic part: d=d e d p Eq 6

6 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 6/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 (II) a derivative of the stress of Kirchhoff =det F is then determined by an incremental relation function of the elastic rate of strain: =C :[d d p ] Eq7 with x a derivative objectifies to define and C the elasticity tensor. (III) One builds a field of reversibility convex defining acceptable space of the stresses from a function f : E ={,q, S X R m+ f,q,0} Eq 8 with S the space of the stresses and q all m the local variables representing the kinematic hardening of the material and the variable scalars (including isotropic hardening). (iv) the laws of evolution of these local variables follow a principle of normality (one considers only the associated constitutive laws here): d p = f,q, Eq9 q = g,q, with 0 the plastic multiplier, f,q, defining the direction of yielding and g, q, the evolution of the other local variables. (v) the writing of the conditions of load/discharge, classically represented by Kuhn-ucker and the condition of coherence: { 0 f,q, 0 Eq 0 f,q,= 0 his class of model is thus characterized by a strong analogy with the formalisms small strains, with an incremental writing of the stresses which is not without raising some difficulties of numerical integration: indeed, in order to prevent the evolutions of stresses by a rigid body motion, it is advisable to have an objective integration of the equation (Eq7). he object of the following chapter is to describe the objective algorithm of integration used in Code_Aster, which had in Simo-Hughes. 3. Objective integration: algorithm GDEF_HYPO_ELAS the preceding chapter defined the class of material hypo-elastic, of which one of the characteristics is to calculate the stresses incrémentalement. It is advisable to have an integration objectifies of it in order to during secure against an evolution of the stresses a rigid body motion. For that, two large types of approaches exist: approaches by definition of an objective derivative (of Jaumann type or Binds), and the approaches by definition of an objective turned reference frame in which the derivatives are calculated. hese approaches are in fact equivalent (see []) for the continuous problem, but the integration of objective derivatives often pose problem (cf [8]). he algorithm describes here is based on a turned reference frame, which had in Simo-Hughes []. he idea is to use a description cancelling the effects of rotations in temporal derivatives. his kind of approach is formalized thus. hat is to say a skew-symmetric tensor of order unspecified. One can build the equation of evolution of a clean orthogonal tensor X,t SO 3 from following form:

7 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 7/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 = t = 0 = Eq With given, the solutions of (Eq) generate a sub-group of the orthogonal special group (subgroup defined by a parameter), representing a turned local reference frame. One can then define the stress of Kirchhoff and strain rate in this reference frame: : = D = d the temporal derivative of the stress of Kirchhoff turned is then: = [ ] := Eq Eq3 what reveals an objective derivative (from where equivalence enters the approaches). Choice of depends objective derivative obtained. Constitutive law of the hypo-elastoplastic type defined in paragraph 3 is rewritten in this reference frame: i D := D ; D= D e D p ii = a:[ D D p ] iii E = {,Q, S X R m F,Q,0} Eq4 iv D p F,Q, = ; Q= g,q, v 0; F,q,0 ; F,q, =0; Ḟ, q, his system of equations is integrated numerically by an algorithm of point medium describes hereafter. 3. Algorithm of integration (I) the first stage consists in discretizing temporally the equations (Eq4). A rule of classical point medium is first of all used for derivative of the stress of Kirchhoff in the turned reference (Eq4 (II)) : n+ n = t n + = t a n + = a n+ = a : n + : n+ n+ [ f : D n + [t d n+ n+ e n + f ] n + ] n + n + Eq5 and, finally, while posing e = n+ f n+ e n + f n+ : n + = n a n+ : n+ e n + n+ Eq6 the equation (Eq6) makes it possible to see that the algorithm is objective (invariance of the stress by rotation of rigid body): f n + SO 3 e n + =0 n+ = n Eq7 (II) his integration of the stress is then coupled with an algorithm of radial return completely similar to that used in small strains. he complete algorithm which rises for a criterion of the type von Mises with

8 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 8/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 isotropic and/or kinematical hardening is described in Box hereafter. In practice, in the case of treated plasticity, the elastic matrix is constant, except introducing a command variable like the temperature; in this case however, it is the elastic matrix at the time t n which is actually used. One thus finds the same type of diagram as in [R5.03.0] or [R5.03.6]. Note: he direct integration of the equation (Eq4) (II) by a derivative of the type Jaumann J =a:[d D p ]= is necessarily explicit on the term of transport of derivative, generating secondary stresses which tend towards 0 with the step of load. It is this kind of difficulty which is circumvented by the use of an objective local coordinate system. (I) Knowing the fields of displacement, compute gradients of transformation: f n+ = I d + grad n u f n+ = I d + grad u, f n n+ = f n +. f n (II) Increment of strain in the turned reference frame: e n + = f n + [ I d f n+ f n + ] n+ f n + (III) relative Rotation tensors: r n + = n + n, r = n + n + n + (iv) Elastic Prediction: pred =r n + n r n + a :[r n + n + q pred n + =r n + q n r n+ pred n + =dev[ pred n+ q n + e n + pred ] r ] n + f pred n+ = pred n + 3 K Y n (v) plastic Correction if necessary: SO f pred pred n + 0 HEN. n + =. n + IF NO = f pred n+ / H K 3 pred n + = pred n + n + q n + =q n + pred n + pred 3 H pred n+ n+ = n 3 pred n+

9 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 9/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 FIN IF able 3. : Box - algorithm of resolution (III) the last point relates to the choice of the skew-symmetric tensor and thus of the computation of the tensors rotations which rise from the equation (Eq). 3.3 Choice of the skew-symmetric tensor and integration the choice which is made in Code_Aster is to use the tensor rate of rotation ; this choice led to an algorithm equivalent to an objective derivative of Jaumann (but with an integration incrémentalement objectifies). he stage (III) of Box contains consequently the stages described hereafter in Box. he last stage of the Box, which constitutes the integration of the equation (Eq), is related to the rule of the point medium in SO 3. One will be able to refer to [] for a demonstration of this relation. (I) (II) (III) incremental Displacement gradient h = grad n u f n + n + ensor rate of rotation = n + t [h h n + n + ensor rotation of the following increments =exp[ t n + ] n + n ] n + =exp[t ] n n+ able 3.3 : Box - Integration of the tensors rotation Moreover, the computation of the exponential term in Boîte (III) is carried out in practice by the sin formula of Eulerian Rodrigues: exp=..., cf [9] 3.4 Advantages and limitations of the method the algorithm describes above has advantages and limitations which are pointed out here. A comparative study of the various approaches in large deformations can be found in [5]. Among the advantages let us quote the following: Generics of the algorithm: it makes it possible to treat a large number of constitutive laws without particular modifications; thus, the following models of Code_Aster are currently available, corresponding to hardenings isotropic and kinematical, like with viscoplasticity: VMIS_ISO_LINE, VMIS_ISO_RAC, VMIS_ISO_PUIS, VMIS_CINE_LINE, VMIS_CIN_MEMO, VMIS_ECMI_LINE, VMIS_ECMI_RAC, VMIS_CIN_CHAB, VMIS_CIN_CHAB, LEMAIRE, VENDOCHAB, VISC_CIN_CHAB, VISC_CIN_CHAB, VISC_CIN_MEMO Similarity with the small strains: this generics is related to the similarity between the algorithm of radial return presented and that classically used in small strains; only the stresses and the variable kinematics of entry and output must be affected by a relative rotation of r

10 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 0/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 guaranteed Objectivity: as underlined by Eq7, the algorithm makes it possible to be secured against any evolution of stresses by a rigid body motion. It presents some disadvantages all the same: It is first of all restricted with the isotropic and relatively weak elastic strain (hypoelasticity) ; at the more theoretical level, the hypoelastic models can lead to a non-zero dissipation during a closed cycle of elastic strain, which is not physical ; because of these disadvantages, one will prefer as much as possible formalism SIMO_MIEHE for the models with isotropic hardening. Only the constitutive laws of the type VMIS are treated: one does not treat cases of criterion of resca or Hill for example; moreover, for the models with kinematic hardening, it is necessary to transport the kinematical tensor at previous time in the objective reference frame at current time; he computing times are relatively important; this is due to an expensive algorithm and an asymmetric tangent matrix which is not optimal (see additional), which increases the nombre of iterations of Newton with convergence. It is noted that the number D iterations with convergence is notably more important than that observed by means of the formalism veryelastic of Simo Miehe. he analytical study of the possibility of writing a tangent matrix for these models is described in appendix.

11 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : /3 Responsable : Renaud BARGELLINI Clé : R Révision : Algorithm GDEF_LOG his algorithm, due to C.Miehe; N.Appel and M.Lambrecht [3] have the same selected advantages as those until now but they based on an energy formulation and the stiffness matrix is provided in [3]. 4. Summary description he model is based on a logarithmic strain defined by: E= log [ F. F ]. he definition of this statement is provided in appendix. he stress is defined in space logarithmic curve like dual of E, so that the density of power mechanical is expressed : Ė. It is not a classical stress tensor, but one can deduce the usual tensors from them: = : P, with P a tensor of projection (of order 4 in 3D) and the tensor of the Piola- Kirchhoff stresses of first species. he tensors of Cauchy and Kirchhoff will be written in a usual way J == F Indeed the mechanical power being written : Ė= : Ḟ, one obtains: : Ė= : E F : Ḟ = : Ḟ = : P :Ḟ what defines the tensor of projection P = E F. One can also calculate the second tensor of Piola-Kirchhoff S according to : : Ė=S : =S : Ċ, with = C I d = F F I d the tensor of the strains of Green- Lagrange. hen : Ė= : E C :Ċ =S : E Ċ thus S = : P with P = C If selected physics is particular, the model allows however to keep the additive decomposition of the elastic strain and plastics classic in HP with E p = logf p. F p. Such a choice is always licit. hat simply amounts adopting a definition for the elastic strain. However, this one proves to be coherent with a multiplicative decomposition, in the absence of rotation (coaxial situation). Moreover, the plastic incompressibility is assured because: tr E p =log J p. he elastic strain energy e of the model also takes the same shape as that of the small strains, but by adopting the notions of stress and strain specific to this formalism: e = E E p E = :C :. his formulation has certain advantages: as for GDEF_HYPO_ELAS, the kinematical dimension of the model is confined upstream and downstream from the integration of the behavior; this was one of the principal elements for the choice of the formalism; all the models of behavior available in small strains are a priori available, with condition of course that has a physical meaning ( the hypoelastic large deformations are well adapted to the metal behaviors, and not to the behaviors of the concrete). so the model HP admits an energy statement, it will be the same for the model large deformations: the tangent matrix is thus symmetric, which was not the case of GDEF_HYPO_ELAS ; the only difficulty seems a priori concentrated in the definition of the logarithmic strain, but the article [3] provides a calculation algorithm distinguishing the difficult cases (multiple clean entities); the model, according to the examples presented by the authors, results very close to those obtained by a classical formalism give to multiplicative decomposition;

12 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : /3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 the model can be wide with the cases of anisotropy (initial or induced), which can be an important source of motivation (in particular for the behavior of alloys containing Zirconium). Moreover, the article [3] provides a form of the tangent matrix in the configuration using (called nominal); however, as it is based on a writing starting from the Piola-Kirchhoff stresses of first species, asymmetric, which classical and is never carried out in Code_Aster, one prefers here to compute: to use the second tensor of Piola-Kirchhoff the internal forces and the matrix tangent on the initial configuration, while referring to [5] for example. 4. Algorithm Preprocessing : E n + = log [ F n +.F n + ] = log [C n + ] calculated by spectral decomposition: If i are the eigenvalues of C n+ and the N i associated eigenvectors, then selected strain measurement is written E n = log i N i N i. i =,3 his measurement makes it possible to obtain an additive decomposition: E=E e E p, with E p = logf p. F p and E e = log F e.f e Moreover one can also write, between time n and time n : E n = E n E One calls then the constitutive law HP It provides the tensor of the stresses, defined by n = E ; E n, n, n where n all the local variables at time and the n forced n at time It n represents is thus necessary to recompute n according to the stresses of Cauchy n stored at time n (forced which are written: n =S n : P n. However that requires the transformation of n in S n and the computation of P n which can be expensive. One thus chooses to store the tensors as local variables. Postprocessing: he tensor of the Piola-Kirchhoff stresses of nd species is obtained by : S n + = n + : P n +, with P n+ = C E n + ; this quantity is calculated via an algorithm presented in [3] and quoted in appendix. he tensor of the stresses of Cauchy is obtained par. J ==F. S.F =F. : P. F One obtains the tangent modulus, in configuration known as Lagrangian, by derivative of ep S n + = n + : P n + : Ṡ n + =C n + E ep n + = n E n : Ċ n + with C ep n + =P n + ep : E n+ : P n+ n + : L n + where the tangent operator resulting from the constitutive law represents and L n is the tensor of a nature 6 (in 3D) defined by L n + =4 C C E n+ One can then calculate on this configuration the internal forces and the tangent matrix, on the initial configuration, as in [4].

13 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 3/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 he formalism is tested in particular on the case test of tension-rotation for elastoplastic models of Von Mises with isotropic or kinematical hardening (cf tests SSND06, SSND07). As the figures show it below, objectivity is preserved (forced invariant by rotation in hardenings isotropic and kinematical) and the good accuracy (results identical to SIMO_MIEHE into isotropic).

14 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 4/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 of von Mises (MPa) InstantContrainte s of Von Mises (MPa) InstantFigu re 4: Responses for the model GDEF_LOG in tension - rotationscontraintes

15 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 5/3 Responsable : Renaud BARGELLINI Clé : R Révision : Validity of the models of large deformations 5. Identification of the parameters It also should be specified that the formalism here presented and studied does not extend the validity of the constitutive laws in the field as of large deformations, it nothing but does propose an objective derivative of it. o clarify this matter, it is possible to consider the case of an elastoplastic model with linear isotropic hardening. his kind of model of plasticity is valid physically in small strains; its use is extended to the large deformations, but its physical validity can be precisely called in question. Moreover, one identification made on tests in small strains must be potentially reconsidered; on figure, one presents a curve of tension modelled by a linear isotropic hardening: the tangent modulus must inevitably be defined compared to the beach of strain considered. In small strains, it seems more judicious to use E ; if the strains are more important, it seems more judicious to use E as value of the slope of hardening. But it is felt well that it is the physical validity of the model itself which should be reconsideration. σ E E Figure : Identification in great or small strains 6 Comparison with PEI_REAC 6. Approximation of the large deformations by PEI_REAC the principle of the formulation PEI_REAC simply consists in reactualizing the geometry of the problem during iterations of Newton (and not at the end of each time step). his means that all the quantities intervening in the equations of the problem are evaluated on the present configuration. Anything else is not modified compared to the case small disturbances. 6.. Kinematical description kinematical description is the same one as that of the small disturbances. his means that increment of strain is calculated by: Δε= Δu i Δu i ε able i being reactualized configuration. he total deflection is then the sum of each one of these increments of linearized strain, calculated on different configurations. It is thus delicate to give him a

16 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 6/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 physical meaning and it is better to use it like an indicator of the reached level of strain. he assumption of additive decomposition of the strains is applied. 6.. Elastoplastic behavior model In the statement of the relation elastic stress-strains, one saw (eq.7) the need for using an objective derivative: =C :[ p ]. With PEI_REAC one replaces objective derivative by simple derivative in time: it is thus not objective. Consequently, the use of PEI_REAC is thus not appropriate to large rotations but it is it with the large deformations, under certain conditions [0]: very small increments, very small rotations (what implies a quasi-radial loading) elastic strain small in front of plastic strains, isotropic behavior Balance and stamps tangent In term of finite elements, the resolution by PEI_REAC implies with each step of load the resolution of the same nonlinear system as in small strains []: L int u i,t i B. λ i = L ext t i B.u i = u d t i able 6..3 With the difference close the internal forces are formally calculated by: L int u i,t i =Q u i : σ able where the operator Q depends on displacements. In this frame, the computation of the tangent matrix carries out to: K n i = Lint u σ =Qu: u n i,t i u Q u u n, ti i u : σ ui n, ti able the first term is the contribution of the behavior, similar to what was presented in small transformations, to the difference which this contribution is evaluated here in present configuration. he second term is the contribution of the geometry which is not present in small transformations. In the frame of the resolution PEI_REAC, this term is not present in the computation of the tangent matrix. One thus has: K i n =Q u: σ u u i n, t i able the absence of the geometrical contribution in the tangent matrix can sometimes make convergence difficult. 6. Comparison on an example the formalism PEI_REAC (cf [6]) is based on an actualization of the geometry for the computation of the increment of strain before integrating the behavior of way identical to the small strains. his allows

17 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 7/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 a simple processing of the large deformations, but in a very approximate way, not objectifies and being able to generate great errors. o be convinced some, let us consider the alternate tension-rotation of a cube; for more details on the test, one will refer to [7] for example. Figure 3: Example of tension rotation of a cube During the phases of rotation, the stress must remain constant: a rigid body motion does not generate stresses (in static all at least and without viscosity). he figure below PEI_REAC presents the response obtained with a behavior VMIS_ISO_LINE for the strains and GDEF_HYPO_ELAS. he type of strain PEI_REAC is put at fault whereas GDEF_HYPO_ELAS is valid (and provides a response identical to SIMO_MIEHE). Figure 4: Von Mises stress in tension-rotation

18 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 8/3 Responsable : Renaud BARGELLINI Clé : R Révision : Bibliography Simo J.C., Hughes.J.R., Computational Inelasticity, Mechanics and Materials, Interdisciplinary Applied Mathematics vol. 7, Springer-Verlag Edition, New York, 998. Handbook use of U4.5. Aster, Behavior Nonlinear Handbook 3 of reference R5.03. Aster, Modelization élasto- (visco) - plastic with isotropic hardening in large deformations, Handbook 4 of reference R Aster, Constitutive law hyper elastic. Presqueincompressible material. Note 5 intern EDF HI-74/98/006/0, Introduction into the Code_Aster of an elastoplastic model of behavior in large deformations with isotropic hardening, V. Cano E. Lorentz, 998. Handbook 6 use U4.5. Aster, Behaviors Nonlinear. Handbook 7 validation Aster ssna06, alternate ension-rotation of an element. Isotropic plastic model. Great 8 transformations, plane stresses, plane strains generalized, Leblond, 989. Handbook 9 of references R Aster, static and dynamic Modelization of the beams in large rotations. Report 0 EDF/DER-MMN 536/07: Plastic large deformations. Modelization in Aster by PEI_REAC. E.Lorentz, June 4th, 996 Manual of references R Aster, Algorithm of SA_NON_LINE CR-AMA Model of large deformations compatible with most behaviors. Ratio of internship of Julie Coloigner. R. Bargellini, J. - M. Proix. Miehe 3 C., Call N., Lambrecht Mr.: Additive Anisotropic plasticity in the logarithm strain space: modular kinematic formulation and implementation based one incremental minimization principles for standard materials., Methods Computer in Applied Mechanics and Engineering, 9, pp , 00. Handbook 4 of reference R Aster, nonlinear elastic behavior model in large displacements. CR-AMA Improvement of the tangent matrix of the formalism of large deformations GDEF_HYPO_ELAS. R. Bargellini, J. - M. Proix. 6 Computational methods for plasticity Wiley 008, of EA of Souza Neto, D. Peric, DRJ. Owen Description 8 of the versions of the document Aster (S) Organization Author (S) Description R.BARG, J.M.PROIX R & D /AMA ELLINI J.COLOIGNER Univ. Rennes initial ext, J.M.PROIX R & D /AMA Addition of the description of the modifications R.BARGELLINI of the algorithm using derivative of Dregs, of the computation of the tangent operator, and algorithm GDEF_LOG able 8 Appendix : tangent 8

19 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 9/3 Responsable : Renaud BARGELLINI Clé : R Révision : operator for GDEF_HYPO_ELAS In the reference [ ], two algorithms are proposed to solve the equations of elastoplasticity with a hypo formulation - elastic: that presented in this document (GDEF_HYPO_ELAS), and a simpler algorithm, using an approximation of derivative of Dregs, which we describe here. GDEF_HYPO_ELAS (derivative of Jaumann) SIMO_HUGHES_ (derivative of Dregs) Knowing the fields of displacements, one calculates the following quantities:, Definition of f n+ = I d + grad n u rotation tensors f n+ = I d + grad n u= I d f n f n+ = f n+. f n incremental gradient of displacement Rates of rotation and - h n / =grad n u f n/ = I d f n/ clean rotation tensor orthogonal 3/ensor =t n / = [ h h n/ n / - ]= f n / f n/ and of rotation, Nothing Definition r n + =exp[] n n + / =exp[/] n r n + = n + n strains r n +/ = n + n+ / :, Definition of the strains e n + = [ I d f n+ f n + ] e n+ = f n + e n+ f n + :, Given for the integration e n + = [ I d f n+ f n + ] of behavior GDEF_HYPO_ELAS (derivative { n=rn + n rn + n =[r e n+ n + x n =r n + q n r n+ r n + contrainte à l'instant n ] déformations variables cinématiques of Jaumann) able 9: Box e n+ = f n + e n+ f n +

20 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 0/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 wo 9 of resolution Some is the value of, hese two algorithms are incrémentalement objective: let us consider a pure rotation: then Principle of the computation e n + = [ I f f d n + n+ ]=0 of n+ =r n n r n 9. the tangent matrix: variational writing and general statement his paragraph is generic for all the formalisms large deformations and constitutive laws. It is inspired in particular by [5] and makes it possible to define the tangent matrix, whose computation in the case as of formalisms described before will be then detailed. he frame of the analysis is obviously that of the finished transformations. he purpose is to solve the definite mechanical problem by the equilibrium. his comes down to determine field of displacement acceptable which cancels the functional calculus U of the virtual wors, which can be written G U,V, in the initial configuration: with the space of acceptable 0 V ad V V 0 ad, G U, V = 0 P : grad 0 V b V d 0 0 t. V d 0 =0 displacements, and the voluminal forces b t and surfaces and the first tensor of P the Piola-Kirchhoff stresses. By comparison, its equivalent in small strains is: In particular, the notion V V 0 ad, G U, V = : grad V b V d t. V d of configuration of field is in small strains without object, since all are equivalent. he dependence of comes from it from the dependence G U from in through the gradient P U of transformation. In order to solve F= I d grad 0 U the problem 0, one linearizes it compared to G U,V =0 V V ad the unknown around an arbitrary state U definite par. the linearized problem U * then consists in determining the field which cancels the linearized U functional calculus of the virtual wors, i.e.: with directional derivative 0 trouver U tel que V V ad, L U,V = G U *, V DG U *,V [ U ]=0 DG U *,V [ U ] of at the point in the direction G U,V. One U * introduces to solve U the problem an infinitesimal displacement around; the gradient U of transformation U * is written then like a function of such as, and the directional derivative F =F U * U of is defined by: By G U,V observing DG U *,V [ U ] = d [ d 0 P F : grad 0 V bv d 0 0 t.v d 0] =0 = d [ d 0 P F : grad 0 V d 0] =0 the composed derivative rules simply, one can express this directional derivative as follows: with the material tangent DG U *,V [ U ]= 0 A : grad 0 U : grad 0 V d 0 matrix A= P F F U *. his then makes it possible to define the linearized equation of the virtual wors in the hardware configuration: to find such as, For 0 reasons U of V V ad simplicity 0 A : grad 0 U : grad 0 V d 0 = 0 P : grad 0 V b V d 0 0 t. V d 0

21 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : /3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464, it is preferable to remain in initial configuration for the definition of the integrals (in particular for the definition of the volume forces and surface) and to calculate the gradients in the present configuration; in fact besides the choice had been retained for the integration of the formalism of Simo-Miehe (see [6]). One thus makes use of the relation (with the gradient compared to the grad 0 a= grad n a.f grad n present configuration) to write directional derivative as follows: Moreover, the use DG U *,V [ U ]= 0 A : [ grad n U. F ] : [ grad n V. F ] d 0 of the first Piola-Kirchhoff stress, asymmetric, is problematic for the definition of the internal forces and the tangent matrixes; one thus makes use of his definition to express, in indicielle P =J F - = F - notation A : because for a tensor of A ijkl = P ij = F ip jp = ip F kl F kl F kl order. From where: with X the tangent X jp = X X jk kl matrix X lp F jp ip F - jk F lp DG U *,V [ U ] = 0 A ijkl [ grad n U kr F rl ] [ grad n V is F sj ] d 0 = [ 0 ] [ grad n U kr F rl ] [ grad n V is F sj] d 0 of ip F F jp ip F jk F lp kl = [ ip F 0 F jp F rl F sj ip F jk F lp F rl F kl sj] [ grad n U kr ] [ grad n V is ] d 0 = [ is F 0 F rl ir kl ks] [ grad U n kr] [ grad n V is ] d 0 = 0 C : grad n U : grad n V d 0 C iskr = is F kl F rl ir ks the problem. his tangent matrix is made up of two distinct terms. Second is called geometrical stiffness, and first is related on the formalism of large deformations used, and possibly to the constitutive law: it is him whom it is necessary here to calculate. It will be noted that this statement is in agreement with [6]; from an integration point of view of code, that makes it possible not to have to modify this one too basically insofar as it is possible to use the routines dedicated to Simo-Miehe. It is pointed out that in small strains, the tangent matrix is written simply. Computation of the tangent C HPP iskr = is matrix kr 9. for the two formalisms As announced in introduction, this paragraph is voluntarily very detailed so that an attentive reader can correct the possible errors of reasoning and/or computation. he preceding paragraph arrives at the conclusion which the term with calculating is. If one discretizes temporally F. F the configurations as presented of Figure, he is written indifferently: It is thus necessary mainly. F F n + =. f n + f n+ n + to evaluate the computation of derivative of the stress of Kirchhoff. he formalisms presented

22 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : /3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 and considered here use the elastoplastic constitutive laws written in small strains. In this frame, one always has, where represent all n+ =g n, n, x n ; n n the scalar local variables of isotropic hardening and the variable tensorial x n of kinematic hardening; the statement of the tangent behavior is then sought in the form: For the two algorithms n + f n+ = n+ n n f n + n + n n f n+ n+ x n x n f n + considered here, it is thus necessary to evaluate each term: : provided by the constitutive law. n+ n called in small strains. : specific to each. n f n+ algorithm, it contains however a general term: : not currently calculated e n + f n+ n + 3. by the constitutive laws. In elasticity, it is worth the identity. In plasticity, its exact statement n is specific to each constitutive law. : specific to each 4. n f n+ algorithm. And in the case of a kinematic hardening : : there still specific n+ 5. x n to each constitutive law: specific to each 6. x n f n+ algorithm. wo aspects are to be put forward now. First of all the feeling which the computation of the tangent matrix will be long, with large numbers of terms; from a numerical point of view, this is likely to have a consequent cost. Moreover, the presence of two narrower terms with the constitutive law called by the formalisms: there one loses in advance the generic character of the formalisms, which was one of their strong point for computation him even; moreover, it is understood why the authors themselves advance that the notion of consistent tangent modulus is not available for the class of formalisms considered here (see [], Remark 8.3., pp 9) the continuation presents successively the computation of each term. erm his term is directly n + n resulting from the constitutive law in small strains; no computation is thus necessary. erm his term, as specified n f n+ front, depends on the formalism considered. It will be seen however that it contains general terms. For algorithm SIMO 9.. _HUGHES_ For this algorithm, one

23 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 3/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 A. From where wo derivatives n =e n + = f n+ e n + f n+ are thus n f n+ = e n+ = f n f n + e e f n+ f n + f n + n+ f n + f n + f n+ n + e n + f n + f n+ with calculating here, of which one is necessary twice; we will clarify them., because By recapitulating e n + ij = - f f n + kl f n + n+ f n + = [ f n+ = [ f n+ li f n+ li f n+ f n+ f n+ ij = [ f f n+ ip n+ pj ] f n + kl f n + kl = ik pl f n + mj f n + f n + mj f n + = ik f n + lj f n + ip f n + = [ ik f n + ip f n + these two f n+ = I d f n + mk f n+ lj f n + = ik pl f n + lj f n + mi ] f n+ mi f n + mk ] mk f pj f n + n + ip f pj f n + n+ pj f n + rs ip f n+ rs f n + kl pk ] f n + pr f n + sj rk sl lj pj f n + kl results, one can thus write the first term (the derivative of the strain being calculated in a dedicated routine, one leaves it in the statements without clarifying it): For algorithm GDEF_HYPO_ELAS n ij f n + kl = f n + f n + kl = f n + pi f n + kl = f n + pi e n + pq ip e n + pq f n + qj e n + pq f n + qj f n + pi e n + pq f n + kl f n + qj f n+ pi e n + pq f n + qj f n + kl f n + kl f n + qj [ tk f n + tr f n + rk ]e n + st [ f n + li f n + sj f n + si f n + lj ] 9.. For this algorithm, one always takes and the strain is written = : From where Compared to = n [ r e r n+ n + n + ] SIMO_HUGHES n qp = r n +/ qa e f n + kl f n + n +/ ab r n + / pb e n + / ab r r kl f n + n + / qa r n +/ pb r n +/ qa e n +/ n + / pb ab kl f n + kl _, this term is longer because it utilizes the derivatives of the rotation tensors, with and. One thus r n + / = n + n+ / calculates n + =exp[] n : n + / =exp[/] n o finish the computation of

24 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 4/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 r n +/ qa f n + kl = n + qc f n + n + / kl = exp qd f n kl = exp qd mn n + qc n + / = [ exp qd mn n + qc n +/ = [ exp qd mn n + qc n +/ n+/ ca ca n + qc f n + kl n dc n/ ca n n/ qc mn f n +/ rs f n +/ rs f n + n dc n + / kl exp/ cm n +/ mw na / rs mn f n +/ rk sl n dc n +/ rs cm n +/ na exp/ mw / rs mn f n +/ n dc n +/ kl cm n +/ ca n / mn n/ mn f n kl ca na exp/ mw / rs ca / rs f n + / de f n +/ de f n + kl ca / rs f n + / dk el n de wn] / rs f n + / n kl wn] n wn this term, it misses two more derivatives. therefore, the derivative of - = f n+ / f n + / exponential mn = mn f n+ / tu = f n + / rs f n+ / tu f n+ / nt mu mt nu f n +/ tr f n+ / su rs = f n +/ mr f n +/ sn f n + / nr f n + / sm tensorial. his one is presented in Appendix; it will be retained especially here that it is iterative and asks one enough a large number of computations. his term is thus of a complex writing and inevitably of a cost high enough for this algorithm. erm 3 his term, commun run with the two n+ n formalisms, prevents the tangent matrix from being of single exact statement for any behavior. Indeed, it would be necessary that each routine of integration of the behaviors provides it, which is not currently the case. he algorithms presented and studied thus lose one their major attractions which was their generics. Nevertheless, one can calculate his statement for example in the case of the model of Von Mises with isotropic and kinematical hardening linear. he criterion is form: with the solution f n+, x n +, p n + = dev n+ x n + 3 y K p n+ 0 is obtained x n + =C n + by: with and his kind of n+ = pred n + n= n a: n + n algorithm n= dev pred n + x n dev pred n + x n dev pred n + x n 3 K p y n = is = 3 3 K C p

25 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 5/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 completely similar to the method used in small strains (radial return), which makes it possible not to modify structure of the code and was selection criteria of these algorithms. In this case, One sees intervening three n + n = n + pred n + pred n + = n + n I =I pred d d n n = I d n dev pred dev pred n + distinct terms: the derivative of the deviator : the derivative of the plastic dev ij kl = pred dev n n + pred dev pred n + n + pred n + dev pred n + n + ij tr kl 3 ij = ik jl il jk 3 ij kl multiplier: the derivative of the norm pred n + = n + dev kl 3 K C n + pred dev kl pred dev = 3 K C n kl on the surface of load: With final, by noting n ij pred n = dev kl n pred dev ik jl jk il n ij n kl the first tensor identity I d 4s of a nature 4 defined by:, one obtains for this term I d 4sijkl= ik jl jk il : his statement is unfortunately not n + ij = I 4s d ijkl n kl n ij n mn 3 K C pred n I 4s d ijmn n ij n mn d I 4s mnkl 3 mn kl valid for nonlinear hardenings erm 4 his term is again n f n+ specific to each algorithm considered here. For the algorithm Simo 9.. _Hughes_ In this algorithm, the stress of Cauchy is written From where: For algorithm n = f n + n f n+ GDEF_HYPO_ELAS n = f f n + ip n pq n+ jq = f n + ijkl f n+ ik pl n pq f n + jq f n + ip n pq jk ql kl 9.. In this algorithm, = ik n lq f n + jq f n + ip n pl jk = n lq ik f n + jq f n + iq jk

26 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 6/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 the stress of Cauchy is thus written: And like, and by means of n =r n+ n r n + n qp = r r n+ qa n ab n + pb f n kl f n+ kl = r n+ qa f n + kl n ab r n + pb r n + qa n ab r n + pb f n + kl r n+ qa = n+ qx n ax =exp [] qb n bx n ax preceding results, With the final one: he derivative r n+ qa = exp [ ] qb mn f n +/ rs f n + kl mn f n + / rs f n + n bx n ax kl exp[] qc = mn f n+ / nk f n+ / lm f n + / mk f n + / ln n cx n ax of exponential n qp = f n + kl n ab f n + / nk f n + / lm f n + / mk f n + / ln n cx r n + pb exp[] qc mn tensorial is given in Appendix. erm 5 his term, present exp[] n ax r n + pc qa n mn bx only n + x n n + in the case of a kinematic hardening, is of the same type as the term, in the sense that it is specific to n each constitutive law with kinematic hardening and thus prevents the writing of a generic tangent matrix for the formalisms. Let us give its statement for example in the case of the model of Von Mises to linear kinematic hardening. One has then: with and From where which requires n+ = n a: n n n= pred pred pred =dev n a: n x n n + n pred the computation = x n pred dev x n x n dev x n of four terms: the derivative of the deviator, the same form as for isotropic hardening: the derivative of the predictor x dev n ij = ik jl il jk 3 ij kl x n kl : the derivative of the multiplier pred x n dev = I d pred 3 K p y n, from where the derivative = of the norm 3 K C = pred kl n kl 3 K C

27 itre : Modèles de grandes déformations GDEF_LOG et GDEF_H[...] Date : /0/03 Page : 7/3 Responsable : Renaud BARGELLINI Clé : R Révision : 0464 on the surface of load: By combining the various n ij pred = dev kl relations, one obtains finally: Following the example of the term, n + ij x n kl = n ij n mn 3 K C pred I 4s d ijmn n ij n mn I d pred ik jl jk il n ij n kl 4s mnkl 3 mn kl the statement given n + n here is valid only for linear kinematic hardening. erm 6 his last term exists x n f n+ obviously only for kinematic hardening. It depends moreover of the selected formalism. For SIMO_HUGHES_ In this case, 9.. one has, from where x n :. It is noticed x n = f n + q n f n + that this = f q f n + ip n pq n+ jq f n + ijkl f n+ kl n ij statement is same form as for. By analogy, one thus f n + kl obtains: For GDEF_HYPO_ELAS In x n ij =x f n + n lq ik f n+ jq f n+ iq jk kl 9.. this case one has, from where :, which has x n =r n + q n r n+ the same x n = r q r n + n n + f n+ f n+ form again as: One thus obtains by analogy n f n+ : Assessment wo important points x n qp = f n + kl q n ab f n +/ r exp[] qc n + pb mn nk f n +/ lm f n +/ mk f n+ / ln n cx exp[] n ax r n + pc qa n mn bx are to be retained mainly: the statements are heavy, with many terms with calculating, which forecasts important computing times, because of certain terms dependant on the constitutive law used, it is not possible to give a generic statement for the tangent matrixes. Consequently, unless programming for each constitutive law the computation which corresponds to him, the tangent matrix cannot be exact. One will find in the ref.