Code_Aster. Elastoplastic integration of the behavior models of Summarized

Transcription

1 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 1/27 Elastoplastic integration of the behavior models of Summarized Von Mises : This document describes quantities calculated by the operator STAT_NON_LINE necessary to the implementation of the quasi static nonlinear algorithm describes in [R ] in the case as of elastoplastic behaviors. These quantities are calculated by the same subroutines in operator DYNA_NON_LINE in the case of a dynamic stress [R ]. This description is presented according to the various keywords which make it possible to the user to choose the desired behavior model. The behavior models treated here are: the behavior of Von Mises with isotropic hardening (linear or not linear) the behavior of Von Mises with linear kinematic hardening (models of Prager) the integration method used is based on a direct implicit formulation. From the initial state, or from the time of preceding computation, one calculates the stress field resulting from an increment of strain. The tangent operator is also calculated.

2 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 2/27 Contents 1 Introduction Behavior models described in this document Goal of the intégration general Notations and assumptions on the déformations Partition of the strains (small strains) Réactualisation Conditions initiales Relation of Von Mises with hardening isotrope Form of the behavior models Relation VMIS_ISOT_LINE Relation VMIS_ISOT_TRAC Relation VMIS_ISOT_PUIS Relation VMIS_JOHN_COOK tangent Operator. Option RIGI_MECA_TANG Computation of the stresses and the variables internes tangent Operator. Option FULL_MECA Local variables of behaviors VMIS_ISOT_LINE, VMIS_ISOT_PUIS, VMIS_ISOT_TRAC and VMIS_JOHN_COOK Relation of Von Mises with kinematic hardening linéaire Statement of the behavior model, cases général Form of the behavior model in 1D tangent Operator. Option RIGI_MECA_TANG Computation of the stresses and variable internes tangent Operator. Option FULL_MECA Local variables of model VMIS_CINE_LINE Bibliographie24... Appendix 1 Relation VMIS_ISOT_TRAC: complements on the intégration25... isotropic Appendix 2 Hardening in stresses planes26...

3 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 3/27 1 Introduction 1.1 Behavior models described into this document In operator STAT_NON_LINE [U ] (or DYNA_NON_LINE [U ]), two types of behaviors can be treated: the incremental behavior: key word factor BEHAVIOR, the behavior in nonlinear elasticity: key word factor BEHAVIOR. For each behavior one can choose: the behavior model: key word RELATION, mode of computation of the strains: key word DEFORMATION. For more details, to consult the document [U ] user's manual, the behaviors described here not raising but of the key word factor BEHAVIOR. The relations treated in this document are: VMIS_ISOT_LINE: VMIS_ISOT_TRAC: VMIS_ISOT_PUIS: VMIS_JOHN_COOK: VMIS_CINE_LINE: Von Mises with linear isotropic hardening, Von Mises with isotropic hardening given by a curve of tension, Von Mises with isotropic hardening given by an analytical curve, Von Mises with isotropic hardening of Johnson-Cook, Von Mises with linear kinematic hardening. 1.2 Goal of integration to solve the nonlinear total problem posed on the structure, the document [R ] described the algorithm used in Code_Aster for the nonlinear static (operator STAT_NON_LINE) and the document [R ] described the method used for the nonlinear dynamics (operator DYNA_NON_LINE). These two algorithms lean on the computation of local quantities (in each point of integration of each finite element) which result from the integration of the behavior models. A each iteration n of the method of Newton [R ] one must calculate the nodal forces R u n i =Q T n n i (options RAPH_MECA and FULL_MECA) the stresses i being calculated in each point of integration of each element starting from displacements u i n must build also the tangent operator to compute: K i n (option FULL_MECA). via the behavior model. One Before the first iteration, for the phase of prediction, one calculates K i 1 (option RIGI_MECA_TANG). The computation of K i 1, which is necessary to the phase of initialization [R ] corresponds to the computation of the tangent operator deduced from the problem of velocity. n This operator is not identical to that which is used to compute: K i by option FULL_MECA, during iterations of Newton. Indeed, this last operator is tangent with the discretized problem in an implicit way. One describes here for behavior models VMIS_ISOT_LINE, VMIS_ISOT_TRAC, VMIS_ISOT_PUIS, VMIS_JOHN_COOK and VMIS_CINE_LINE, the computation of the tangent matrix of the phase of prediction K i 1, then the computation of the stress field from an increment of strain, the computation of the nodal forces R and tangent matrix K i n.

4 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 4/27 2 General notations and assumptions on the strains All the quantities evaluated at previous time are subscripted par. the quantities evaluated at time tt are not subscripted. The increments are indicated par. One has as follows: Q=Qtt =QtQ=Q Q. For the computation of derivatives, one will note: Q derived from Q ratio with tensor ~ time from the stresses. operator déviatoire: i j = 1 3 k k i j. eq A,, E,, K equivalent value of Von Mises: eq= 3 2 i j i j increment of strain. elasticity tensor. moduli of the isotropic elasticity, respectively: coefficients of Lamé, Young modulus, Poisson's ratio and modulus of compressibility. 3 K =32 modulate compressibility t T average thermal coefficient of thermal expansion. time. temperature. positive part. To compute: the tangent operators, one will adopt the convention of writing of the symmetric tensors of order 2 in the form of vectors with 6 components. Thus, for a tensor a : a= t [ a xx a yy a zz 2 a xy 2 a xz 2 a yz ] One introduces the hydrostatic vector 1 and the matrix of deviatoric projection P : 1=t [ ] P=Id

5 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 5/ Partition of the strains (small strains) One writes for any time: with t= e t th t p t e t=a 1 T t t th t=t t T t T ref Id or in a more general way: th T = T T T def T ref T ref T def = T T T ref et th T ref = 0 A depends on time t via the temperature. The thermal coefficient of thermal expansion T t is an average coefficient of thermal expansion which can depend on the temperature T. The temperature T ref is the reference temperature, i.e. that for which thermal thermal expansion is supposed null if the average coefficient of thermal expansion is not known compared to T ref, one can use a temperature of definition of the average coefficient of thermal expansion T def (definite by the key word TEMP_DEF_ALPHA of DEFI_MATERIAU) different from the reference temperature [R ]. What leads to: t=a. 1 T t t th t p t This choice is made by preoccupation with a coherence with elasticity: it is necessary to be able to find the same solution in elasticity (operator MECA_STATIQUE) and elastoplasticity (operator STAT_NON_LINE) when the characteristics of the material remain elastic. This choice leads to the discretization: with: and = p A 1 th A 1 =A 1 t t A 1 t th =t tt T ref t T T ref Id

6 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 6/ Reactualization In STAT_NON_LINE, under the key word factor BEHAVIOR, several modes of computation of the strains are possible: PETIT SIMO_MIEHE [R ] (which carries out computation in large deformations for an isotropic hardening) GDEF_HYPO_ELAS [R ] which carries out the computation in large deformations, but with an hypo-elastic formulation, and which is usable for an unspecified hardening) GROT_GDEP [R ] (which carries out computation in large displacements and large rotations, but in small strains) PETIT_REAC (which is a substitute with computation in large deformations, valid for small increments of load, and for small rotations [bib2]). This last possibility consists in reactualizing the geometry before you calculate : One writes x= x 0 u i 1 u i n, the computation of the gradients of ui n is thus made with the geometry x instead of the initial geometry x Initial conditions They are taken into account via, p, u. In the event of poursuite or resumption of a preceding computation, there is directly the initial state, p, u on the basis of, p, u preceding computation at specified time. 3 Relation from Von Mises with isotropic hardening 3.1 Form of the behavior models These relations are obtained by key keys VMIS_ISOT_LINE, VMIS_ISOT_TRAC and VMIS_ISOT_PUIS. One describes here these relations into small strain (DEFORMATION=' PETIT'): { p = 3 2 ṗ. = A 1 th eq eq R p 0 ṗ=0 si eq R p 0 ṗ 0 si eq R p =0 p : p : th : vitesse de déformation plastique, déformation plastique cumulée, déformation d'origine thermique : th = T T ref Id The function of hardening R p is deduced from a monotonous simple traction test and isotherm

7 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 7/27 the user can choose a linear hardening (relation VMIS_ISOT_LINE) or a given curve of tension either points by points (relation VMIS_ISOT_TRAC), or by an analytical statement (relation VMIS_ISOT_PUIS). Constitutive law VMIS_JOHN_COOK differs from the preceding ones in the meaning where the function of hardening depends on the velocity of the cumulated plastic strain and the temperature. One describes here these relations into small strain (DEFORMATION=' PETIT'): { p = 3 2 ṗ. = A 1 th eq eq R p, ṗ,t 0 { ṗ=0 si eq R p, ṗ,t 0 ṗ 0 si eq R p, ṗ,t =0 p : p : ṗ : th : vitesse de déformation plastique, déformation plastique cumulée, vitesse de déformation plastique cumulée, déformation d'origine thermique : th = T T ref Id The function of hardening R p, p,t is deduced from a series of traction tests simple monotonous at different strainrate and different temperature Relation VMIS_ISOT_LINE the data of the characteristics of materials are those provided under the key word factor ECRO_LINE or ECRO_LINE_FO of operator DEFI_MATERIAU [U ]. /ECRO_LINE =_F (D_SIGM_EPSI = E T, SY = y ) /ECRO_LINE_FO =_F (D_SIGM_EPSI = E T, SY = y ) ECRO_LINE_FO corresponds if E T and y depends on the temperature and is then calculated for the temperature of the current Gauss point. The Young modulus E and the Poisson's ratio are those provided under the key keys factors ELAS or ELAS_FO. In this case curve of tension is the following one: I.e.:

8 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 8/27 Note: L=E L sil y E L = y E T L y E si y L E y is the elastic limit (the choice of y falls to the user: it can correspond at the end of linearity of real curve of tension, either to a lawful or conventional elastic limit. At all events, one uses here the single value defined under ECRO_LINE ). When the criterion is reached one a: eq R p=0. To identify R( p), one uses the properties of the uniaxial stress state: L 0 0 = eq = L donc p= P L = L and the criterion is written: L L R p=0 E thus L y =E T L y E =E T L E p y E L y 1 E T E =E p T is = E. E T L y E E T p from where the linear function of hardening: R p= E T E E E T p y Relation VMIS_ISOT_TRAC the data of the material are those provided under the key word factor TENSION =_F (SIGM = F), of operator DEFI_MATERIAU. F is a function with one or two variables representing curves of tension simple. The first variable is obligatorily the strain, the second if it exists is the temperature (parameter of a three-dimensions function). For each temperature, curve of tension must be such as: the X-coordinates (strains) are strictly increasing, the slope between 2 successive points is lower than the elastic slope between 0 and the first point of the curve. To interpolate temperature compared to the, Code_Aster transforms initially all the curves = f given by the user into curves σ( p)=r( p) in the following way: if E is the elastic slope between 0 and the first point of the curve, the point, becomes the point p, with p= / E. That is to say temperature considered, if there exists k such as [ k, k1 ] where k the index of the curves of tension contained in the three-dimensions function indicates, one point by point builds the curve R p, while interpolating from R p, k and R p, k 1 for all the values of p meeting of the values of the X-coordinates of the curves k and k+1 (if these two curves are prolonged linearly or by a constant function):

9 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 9/27 If n k and n k+1 are the numbers of points of the curves k and k+1, the number of points n of the curve R( p,θ) is worth in the general case n k +n k+1 1 (case where all the non-zero X-coordinates are distinct). If is apart from the intervals of definition of curves of tension, one extrapolates in accordance with the prolongations specified by the user in DEFI_NAPPE [U ] and according to the preceding principle. Note: To avoid generating important errors of approximation or even obtaining by extrapolation of bad curves of tension, it is not to better use linear prolongation in DEFI_NAPPE. If the prolongation of the shortest curve IS EXCLUDED, one stops the interpolation at this place and the prolongation of the interpolated curve IS ALSO EXCLUDED. One thus obtains in all the cases a linear function of hardening per pieces:

10 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 10/27 R p, q =s i s i1 s i p i1 p i p p i for p [ p i, p i+1 ] i1 n, with p 1 =0 the Young modulus corresponding to the temperature is calculated in the following way: E=E k q q k E q k1 q k1 E k k where, for i=k or i=k+1, E i is the elastic slope enters 0 and the first point of the curve = f corresponding to the temperature i. It is this Young's modulus which is used in the integration of the behavior model. The elastic limit with the temperature is worth: y =R 0,= 1 The user must also give the Poisson's ratio and a fictitious modulus Young (who serves only to compute: the elastic stiffness matrix if key word NEWTON=_F (MATRICE=' ELASTIQUE') is present in STAT_NON_LINE) by the keywords: /ELAS =_F (NU =, E = E ) /ELAS_FO =_F (NU =, E = E ) Relation VMIS_ISOT_PUIS the data of the material are those provided under the key word factor ECRO_PUIS or ECRO_PUIS_FO of operator DEFI_MATERIAU [U ]. ECRO_PUIS=_F (SY= y, A_PUIS =a, N_PUIS =n) the curve of hardening is deduced from the uniaxial curve connecting the strains to the stresses, whose statement is: = E a y y E y n for y what gives the curve of hardening: 1 R p= y y E n a y p The curve representative of such a function takes the following form, for various values of N:

11 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 11/ Relation VMIS_JOHN_COOK the data of the material are those provided under the key word factor ECRO_COOK or ECRO_COOK_FO of operator DEFI_MATERIAU [U ]. ECRO_COOK =_F (A=A, B=B, C=C, N_PUIS=n, M_PUIS=m, EPSP0=epsp0, TROOM=troom, TMELT=tmelt,) the curve of hardening is written in the following way: R p, p,t = AB p n 1C ln p ṗ 0 or in a more concise way: R p, p,t = AB p n 1C p 1 T m { with p = p si p ṗ ṗ 0 0 and T = 1 si p ṗ 0 { T 1 T T room T melt T room m T room T melt T room si T T room 0 si T T room 3.2 tangent Operator. Option RIGI_MECA_TANG the goal of this paragraph is to calculate the tangent operator K i 1 (computation option RIGI_MECA_TANG called with the first iteration of a new increment of load) starting from the results known at previous time t i 1. For that, if the tensor of the stresses with t i 1 is on the border of the field of elasticity, the condition is written: ḟ =0

12 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 12/27 who must be checked (for the continuous problem in time) jointly with the condition: with: f =0 f, p= eq R p So on the other hand the tensor of the stresses with t i 1 is inside the field f 0, then the tangent operator is the operator of elasticity. The quantities intervening in this statement are calculated at previous time t i 1, which are the only known ones at the time of the phase of prediction. One thus obtains: because of being f With Note: ḟ = f f p ṗ= f f p ṗ= f 2 2 P f p ṗ = f 2 2 P f p ṗ, deviative. = = t i- 1 = = t i 1, p = p = p t i-1 and p =p = p t i- 1 One does not take account in this statement of the variation of the elastic coefficients with the temperature. It is an approximation, without important consequence, since this operator is only used to initialize the iterations of Newton. On the other hand, the dependence of the tangent operator compared to the thermal strains is well taken into account on the level of the total algorithm [R ]. 3 One has then: ṗ 3 eq 2 eq R' p ṗ= 0 what leads to: ṗ= 3 eq. 3R ' p thus { p = {9 2. 3R ' p, si f, p= 2 eq R p = 0 eq 0, si eq R p0 ij = K kk ij 2 ij ij p Note:: The information eq R p = 0 is stored in the form of a local variable which is worth 1 in this case and 0 if eq R p.

13 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 13/27 The tangent operator binds the vector of virtual strains to a stress vector virtual. The tangent stiffness matrix is written for an elastic behavior: and for a plastic behavior: = K 1 12P = K 1 12 P C p s s with s the vector of the deviatoric stresses associated with and: s T = 11, 22, 33, 2 12, C p = eq 2 3R ' { = 1 si eq R p = 0 0 sinon, 2 31 defined by: In the case of the first increment of loading, therefore if the state at previous time corresponds in a nonconstrained initial state, the tangent operator is identical to the operator of elasticity. 3.3 Computation of the stresses and the local variables the decomposition of the strains makes it possible to write: = p A 1 th Maybe, by taking the spherical and deviatoric parts: = p 2 because th = 0. tr = tr 3K tr th tr p = 0. By direct implicit discretization of the behavior models for isotropic hardening, one obtains then: 2 = p 2 eq 2 tr = 3K 3K tr 3K tr 3K tr th eq R p p 0 p=0si eq R p p p 0si eq =R p p One defines, to simplify the notations, the tensor e such as:

14 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 14/27 Two cases arise: eq R p p e = and tr e = tr. in this case: p= 0soit = = e eq =R p p in this case: p 0 thus: e eq R p One from of deduced the algorithm from resolution: so e eq R p then p=0soit = = e e R p so eq then it is necessary to solve:, therefore: e eq R p e = p eq thus while factorizing and by taking the equivalent value of Von Mises: = e p eq 2 eq eq that is to say: because: eq = eq =R p p e eq =R p p 3 p It is a scalar equation in p, linear or not according to R p. p is calculated in the following way: if hardening is linear (relation VMIS_ISOT_LINE), one obtains directly: p= e eq y R ' p R '3 R '= E E T E E T with if hardening is given by a curve of tension closely connected per pieces, (relation VMIS_ISOT_TRAC), one benefits from the linearity per pieces to determine exactly p (see Annexe25); in the case of a hardening defined by a model in power (relation VMIS_ISOT_PUIS), p is R p e p 3 p eq solution of the nonlinear equation: =0. This equation is solved by an iterative method (algorithm of the secant type). In the vicinity of the origin, one linearizes R p, because the derivative R ' = E an E a y p1 n 1 is infinite in p=0. Thus if p p 0, one replaces R p by R lin p = y p p 0 R p 0 y, which avoids the search for a solution numerically almost null. In practice, one chooses p 0 =10 10.

15 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 15/27 Once p determined, one calculates by: and = e eq 3 p. e e eq tr =tr e. Options RAPH_MECA and FULL_MECA carry out both the preceding computation, which clarifies the computation of R u i n. It is noticed that actually, R u i n =Q T i n according to u i n, but i 1 et u i n. Note: Typical case of the plane stresses. where i n is calculated not The model of Von Mises with isotropic hardening (VMIS_ISOT_LINE, VMIS_ISOT_PUIS or VMIS_ISOT_TRAC) is also available in plane stresses, i.e. for modelizations C_PLAN, DKT, COQUE_3D, COQUE_AXIS, COQUE_D_PLAN, COQUE_C_PLAN, PIPE, TUYAU_6M. In this case, the system to be solved comprises an additional equation. This computation is detailed in appendix Tangent operator. Option FULL_MECA option FULL_MECA makes it possible to calculate the tangent matrix K i n with each iteration. The tangent operator who is used for building it is calculated directly on the preceding discretized system (one notes to simplify: =, p= p p ) and one writes the statements only in the isothermal case. If the tensor of the stresses is on the border of the field, f =0 then one has, by differentiating the form of the model of normality in = : 2 p =2 = p p 3 : d p. eq eq 2 3 eq where p,, represent infinitesimal increases around the solution in the incremental elastoplastic problem obtained previously. Like: 3 2 :d eq =R ' p dp by carrying out the tensor product of the first equation by one a: 2 : : =2 p. eq while eliminating p from the two last equations:

16 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 16/27 2 : : = 1 3 R ' p So on the other hand if the tensor of the stresses is inside the field f 0, then the tangent operator is the operator of elasticity.. While expressing p and : in the first equation, one obtains: with: 2 = 3 p eq C p. :, C p = R' p p eq eq 1 R ' p 3 The positive part : makes it possible to gather in only one equation the two conditions: either f 0, which implies p=0 or f =0 One obtains then: = 2 a C p a : while posing: a=1 3 p R p p The tangent operator binds the vector of virtual strains to a stress vector virtual. The tangent stiffness matrix is written for an elastic behavior: = K 1 12P and for a plastic behavior: = K a P C p a s s with s the vector of the deviatoric stresses associated with and: s T = 11, 22, 33, 2 12,2 23, 2 31 defined by:

17 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 17/27 1 si conduit à une plastification ={ 0 sinon et. 0 It is noted that the tangent operator with the system resulting from the implicit discretization differs from the tangent operator to the problem of velocity (RIGI_MECA_TANG). One finds it while making: p=0 in the statements of C p and a. 3.5 Local variables of behaviors VMIS_ISOT_LINE, VMIS_ISOT_PUIS, VMIS_ISOT_TRAC and VMIS_JOHN_COOK behavior models VMIS_ISOT_LINE, VMIS_ISOT_PUIS and VMIS_ISOT_TRAC produce two local variables: p cumulated equivalent plastic strain, and indicator of plasticity at time considered (useful for the computation of the tangent operator). VMIS_JOHN_COOK uses two local variables besides the two preceding ones: p plastic strainrate equivalent cumulated to time less, and t the increment of time step at time less.

18 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 18/27 4 Relation from Von Mises with linear kinematic hardening 4.1 Form of the behavior model, general case This relation is obtained by key word VMIS_CINE_LINE of the key word factor BEHAVIOR. She is written (always in small strains): { p = 3 2 ṗ X X eq = 3 2 ṗ X=C p { ṗ=0 X X eq = A 1, th = T T ref Id X eq y 0 si X eq y 0 ṗ 0 si s X eq y =0 th éq y is the elastic limit (the choice of y falls to the user: it can correspond at the end of linearity of real curve of tension, either to a lawful or conventional elastic limit At all events, one uses here the single value defined under ECRO_LINE). C is the coefficient of hardening deduced from the data by a simple traction test. In this case (uniaxial stress tensor, strain tensor plastics isochoric and orthotropic): X L 0 0 L 0 0 = X= X L X L 2 X eq = L 3 2 X L and the X L =C L p =C L L E

19 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 19/27 material characteristics are those provided under the key word factor ECRO_LINE or ECRO_LINE_FO of operator DEFI_MATERIAU : /ECRO_LINE =_F (D_SIGM_EPSI = E T, SY = y ) /ECRO_LINE_FO =_F (D_SIGM_EPSI = E T, SY = y ) ECRO_LINE_FO corresponds if E T and y depends on the temperature and is then calculated for the temperature of the current Gauss point. The Young modulus E and the Poisson's ratio are those provided under the key keys factors ELAS or ELAS_FO. For L y E = E y L y T L E, but one also has: { L 3 2 X = L y X L =C L L E from where, while eliminating X L and while identifying: C= Form of the behavior model in 1D E E T E E T. For reasons of performances the relation is also written in 1D for a use with of the finite elements of standard beam multifibre. The preceding equations are identical, the quantities L, X L and L are scalars. The material characteristics are those provided under the key word factor ECRO_LINE of operator DEFI_MATERIAU [U ] : /ECRO_LINE = _F ( D_SIGM_EPSI= E T [ R éel] ) SY= y [ R éel] SIGM_ELS= sigmels [R éel] EPSI_ELU= epsielu [ R éel] the operands SIGM_ELS and ESPI_ELU make it possible to define the limits which correspond to the limiting states of service and ultimate, classically used during study in civil engineering. SIGM_ELS = sgels Definition of the ultimate stress of service. EPSI_ELU = epelu Definition of the ultimate limiting strain.

20 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 20/27 These limits are compulsory when one uses the behavior ECRO_CINE _1D (cf [U ] nonlinear Behaviors, [U ] DEFI_MATER_GC). In the other cases they are not taken into account. The modelization supported is 1D, the numbers of local variables is of 6. V1 : Criterion ELS: CRITELS. This variable gives information compared to the absolute limit of service. This variable represents the absolute value of the stress divided by the ultimate stress in the ELS of the material. If this variable is in [0,1] the material respects the ELS. V2 : Criterion ELU: CRITELU. This variable gives information compared to the ultimate absolute limit. This variable represents the absolute value of the total deflection divided by the limiting strain with the ELU of the material. If this variable is in [0,1] the material respects the ELU. V3 : Kinematic hardening: XCINXX. In 1D only a scalar is necessary. V4 : Plastic indicator: INDIPLAS. Indicate if the material exceeded the elastic criterion. V5 : nonrecoverable dissipation: DISSIP. During seismic computations it can be useful for the user to know nonrecoverable dissipated energy. The variable DISSIP represents the nonrecoverable office plurality of energy. The nonrecoverable increment of energy is written in the form: Eg= 1 2 E V6 : thermodynamic dissipation: DISSTHER. The thermodynamic increment of dissipation is written in the form: Eg= y ṗ. 4.3 Tangent operator. Option RIGI_MECA_TANG the goal of this paragraph is to calculate the tangent operator K i 1 (computation option RIGI_MECA_TANG called with the first iteration of a new increment of load) starting from the results known at previous time t i 1. For that, if the tensor of the stresses with t i 1 is on the border of the field of elasticity, the condition is written: ḟ =0 who must be checked (for the continuous problem in time) jointly with the condition: with f =0 f = f,x = X eq y So on the other hand the tensor of the stresses with t i 1 is inside the field f 0, then the tangent operator is the operator of elasticity. One poses: dev = X et ={ 1 si X eq y =0 0 sinon The problem of velocity is written in this case: variable interne

21 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 21/27 { ={ p y 2 X. X C 2 0 si s X eq y 0 ij = K kk ij 2 ij ij p si X y =0 The tangent operator binds the vector of virtual strains to a stress vector virtual. The tangent stiffness matrix is written for an elastic behavior: and for a plastic behavior: = K 1 12 P = K 1 12 P C p s s with s the vector of the deviatoric stresses associated with dev defined by: and: s T = dev 11, dev 22, dev 33, 2 dev 12, 2 dev 23, 2 31 dev C p = y 2 C In the case of the first increment of loading, therefore if the state at previous time corresponds in a nonconstrained initial state, the tangent operator is identical to the operator of elasticity. 4.4 Computation of the stresses and local variables the direct implicit discretization of the continuous relations results in solving: {2 p =2 2 2 =3 2 X= C C X C p X eq y p=0 si X eq y p 0 sinon 2 p X y tr = 3 K 3 K tr 3 K tr 3 K tr th One still poses: The first equation is also written: e = C C X.

22 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 22/27 while cutting off X= C C = p X y X C p has each term, one obtains: C C X = X 3 2 or, by means of the flow model: e = X C p y X 2 p C P y One still obtains a scalar equation by p taking some the equivalent values of Von Mises: what gives directly: e eq = y 3 2 C p 2 p= e eq y 3 2 C 2 And is obtained by: = p By noticing that: p = 3 2 p X = 3 y 2 p e because: e eq one thus has: X y = e e eq = C The local variables X are calculated by: Note: Typical case of the plane stresses. s e eq y +. e e eq X= C C X C p = C C X 3 2 C p e The direct taking into account of the assumption of the plane stresses in the integration of the model of Von Mises with linear kinematic hardening was not made in Code_Aster. To take into account this assumption, i.e. to use an elastoplastic behavior of Von Mises with a linear kinematic hardening (model of Prager) with modelizations C_PLAN, DKT, COQUE_3D, COQUE_AXIS, COQUE_D_PLAN, COQUE_C_PLAN, PIPE, TUYAU_6M, one can: that is to say to use the method of condensation static (due to R. of Borst [R ]) which makes it possible to obtain a plane state of stresses with convergence of the total iterations of the algorithm of Newton; that is to say to use behavior VMIS_ECMI_LINE (cf [R ]). e eq

23 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 23/ Tangent operator. Option FULL_MECA option FULL_MECA makes it possible to calculate the tangent matrix K i n with each iteration. The tangent operator who is used for building it is calculated directly on the preceding discretized system (one notes to simplify: =, p= p p ) and one writes the statements only in the isothermal case. One poses dev 1 si p0et X. 0 = X and ={ 0 sinon the tangent operator binds the vector of virtual strains tangent stiffness matrix is written: =K 1 12 a 2 P C p s s to the stress vector virtual. Then the with s the stress vector associated with dev by: and: C p = y 2 s T = dev 11, dev 22, dev 33, 2 dev 12, 2 dev 23,2 31 dev 1 2C a 1 with a = C p 2 y and a 2 =a C p y 4.6 Local variables of model VMIS_CINE_LINE the local variables are 7: the tensor X stored on 6 components, the scalar variable.

24 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 24/27 5 Bibliography 1) P. MIALON, Elements of analysis and numerical resolution of the relations of elastoplasticity. EDF - Bulletin of the Management of the Studies and Searches - Series C - N , p ) E.LORENTZ, J.M.PROIX, I.VAUTIER, F.VOLDOIRE, F.WAECKEL Initiation with the thermo - plasticity in the Code_Aster, EDF/DER/HI - 74/96/013 Description of the versions of the document Aster Author ( Notes Organization of the modifications 5 J.M.Proix, E.Lorentz S) (S) Description 5, P.Mialon EDF-R&D initial Text J.M.Proix EDF-R&D/AMA 8.5 Correction page 10, cf R drives REX J.M.Proix EDF-R&D/AMA Modification page 7 of the drafting on the way of computation (p) (cf cardindexes 15001) S. Fayolle EDF-R&D/AMA Addition of VMIS_JO HN_COO K Relation VMIS_ISOT_TRAC : complements

25 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 25/27 Annexe 1 on integration the implicit discretization of the behavior model results in solving an equation in (see 3.3): One solves p the equation exactly13 while e eq 3 p R p p=0 drawing left the linearity per pieces. One examines initially if the solution could be apart from the limits of the points of discretization of the curve, i.e., if is a possible R p solution. For that p p n : if, then one is in the following situation e eq 3 p p n n 0 : if the prolongation on the right is linear then: formulate with: if the prolongation Δp= σ e eq H n 1 α n 1 +3μ : α n 1 = σ n σ n 1 p n p n 1 H n 1 =σ n 1 +α n 1 ( p p n 1 ) if not, the solution is to be sought Δp= σ e eq σ n 3μ in the interval p such as: and then the solution is [ p i, p i+1 ] : with: σ i +1 >σ e eq +3μ( p p i+1 ) formulate σ i σ e eq +3μ( p p i ) isotropic Hardening Δp= σ e eq H i α i +3μ et +Δp [ p p i, p i+1 ] in plane stresses α i = σ i +1 σ i ; H p i +1 p i =σ i +α i ( p p i ) pour i=1 à n 1 i

26 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 26/27 Annexe 2 In this case, the system to be solved comprises an equation moreover:. The following system then is obtained 33 =0 : With this assumption, is not entirely { 3 2 = 2 2 p eq tr =3 K tr eq R p p 0 p=0 si eq R p p p 0 si eq =R p p 33 =0 known: cannot be only calculated from 33. Note: In the case as of modelizations u i n other than C_PLAN, therefore for example for the modelizations of shells (DKT, COQUE_3D), the assumptions on the transverse terms of shears and are defined by these modelizations (in general, the behavior related to the transverse shears linear, elastic and is uncoupled from the equations above). These terms thus do not enter on account here. One poses with entirely known from = q y and q n of elasticity, therefore is unknown u i q 33 = y system 0 0 = q q 22 et y. Compared to the preceding, there is an additional unknown. If thus i.e. If not, y the technique eq R p p alors p=0 2 =, of resolution y=0. consists in expressing according to. One then obtains y a nonlinear scalar p equation in. One poses:. In the same way that p for L integration e = q except plane stresses, one obtains:. But this statement utilizes e 2 y = 1 3 p R p p

27 Titre : Intégration des relations de comportement élasto-p[...] Date : 31/10/2014 Page : 27/27 an additional unknown: In particular: however and Like: y One obtains an equation 33 2 y 33 = p R p p flexible tr =3 K tr q 3 K y 3K and e = 33 : with: That is to say: by noticing y that p : and while clarifying y 33 = 2 3 y 3K tr 3 K th e 33 = tr 3 =0 tr y= 1 3 p R p p tr e 3 K y 3 y 4 3 K 1 tr e = 3K 3 K tr 3K tr q 3 K th 3 p R p p = e 33 tr e 3 p 1 3 R p p, one obtains: to defer in, K the equation in (identical y= e 33 = e 33 tr e 3 =0 tr e p E p2 1 R p p to the preceding cases): p where expresses itself according to since e 33 e 2 y eq 3 p R p p =0 y : The scalar equation in p thus obtained y = y is always nonlinear p. This equation is solved by a research method of zeros of functions, based on an algorithm of secant. Once the known solution one calculates then. p y