Calculation of load limits by the method of Norton- Hoff-Friaâ, behavior NORTON_HOFF

Transcription

1 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 1/12 Calculation of load liits by the ethod of Norton- Hoff-Friaâ, behavior NORTON_HOFF Suary: The liiting analysis akes it possible to deterine the acceptable loadings of a structure, of geoetry fixes given, ade up of a aterial having a criterion of resistance. One considers the case of loadings ade up of the su of a continued load and of another paraeterized by the load factor, which one seeks the bearable extree value. After a recall of the theoretical forulation, one presents the regularized kineatic approach applied to the criterion of resistance of Von Mises (ethod of Norton-Hoff-Friaâ) and put in work in Code_Aster. One will be able to refer to [bib4] for the various possible ethods of regularization suggested in the literature. One exposes then the calculation of the solutions of this nonlinear proble and postprocessing providing an estiate of the liiting load (a value by excess in all the cases, and when there is no peranent loading, a value by ).

2 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 2/12 Contents Contents 1 Theoretical forulation of the liiting analysis Definition of the liiting load Calculation of the load liits by a kineatic approach Regularization of the kineatic approach by the ethod of Norton-Hoff-Friaâ Digital aspects of the calculation of the liiting load Relation of behavior of Norton-Hoff Piloting Postprocessing of the calculation of the liiting load Features and checking Bibliography Description of the versions of the docuent... 12

3 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 3/12 1 Theoretical forulation of the liiting analysis 1.1 Definition of the liiting load One considers a solid occupying a field liited subjected to surface loadings F F 0 on the edge f and of the loadings of volue f f 0 on. The loading is distinguished F, f, paraeterized by positive reality, and the peranent loading F 0, f 0. The hoogeneous conditions of Dirichlet which are applied to the copleentary edge u of (an iposed displaceent or an initial unelastic deforation therics, plastic does not have an effect on the field of the working loads). One can find in [bib5] several other useful properties. The aterial constitutive of the solid has a criterion of resistance expressed by a scalar function of the constraints, negative for working stresses. The criterion used for a aterial of the perfect elastoplastic type with threshold of von Mises and selected here is: g =J y = D y 3 2. D. D y = y is the diverter of the tensor of the constraints, is the threshold of resistance in siple traction (like an elastic liit), possibly variable according to the zones of the solid considered. i being principal constraints of. Being given this criterion of resistance one seeks to calculate the value liits, called liiting load li, for which the structure can support the loadings li F F 0 and li f f 0. Strictly speaking, the value li indicate the liit of the bearable loadings, but for aterials obeying the Principle of Maxiu Plastic Work, this value is the liit of the supported loadings. 1.2 Calculation of the load liits by a kineatic approach In design the collapse two approaches are possible: static approach (in variables of constraints) and kineatic approach (in variables speeds). These approaches provide terinals of the liiting load: undervaluing for the approach static and raising for the kineatic approach. When both provide the sae result, the liiting load obtained is exact. The kineatic approach is that used in Code_Aster using finite eleents in displaceents. For the loading given (F,f), one defines the space the speeds kineatically acceptable and standardized by: V a 1 = { v adissible, v=0 sur u, L v = f. v d f F. v ds =1 } This standardisation forces the work of the loading (F,f) with being unit. Power of the peranent loading F 0,f 0 is noted: L 0 v. Fro the criterion of resistance in constraints g, one defines: the whole of working stresses by: G x ={ x, g x 0 } ( G x is convex for the criterion g )

4 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 4/12 the indicating function: G x ={ 0, si x G x, si x G x the function of support: = Sup IR 6 [. G ] Sup in can be reached only if is selected in G x, such as: = D Id (what ensures // D ). The optiu corresponds to g =0 = 2. y 3 D. D -1 / 2 G D 2 0 D D D 1 0 Figure 1.2-a: Optiu and graph of the fonctio N in 1D Fro where the function of support: v = y. 2 3 ε v.ε v Sup.div v = R v Sup.div v IR IR It is observed that the function aterial point, is not differentiable in 0., which is interpreted like the density of power dissipable at the One does not treat to date in Code_Aster possible internal surfaces of discontinuity within the solid [feeding-bottle 4]. The kineatic approach is defined using the convex functional calculus S e v 1 hoogeneous of degree one, for v V a defined on the whole field: S e v = v d L 0 v, positively This functional calculus is the integral on the field of the function of support the convex one G x, calculated in v and is interpreted like the axiu resistant power in the field speeds v (the contribution of resistance of interface on surfaces of discontinuity is supposed to be worthless). The function of support is positively hoogeneous of degree 1, and thus the functional calculus S e v also by consequence. With the criterion of Von Mises the functional calculus of power S e v is: S e v = [. y 2 3 v. vsup q.div v q IR ] d L v 0 éq where it is noted that only the fields v belonging to C={ v V a 1,div v=0 dans } provide finished values. Fields v ust thus check the condition known as of incopressibility div v=tr v =0. This is why it is necessary to use the quasi-incopressible eleents for a calculation of liiting load with the criterion of Von Mises [R ]. The liiting load li given by the kineatic approach is:

5 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 5/12 S li = Inf S e v = Inf e v 1 v V a v V a L v =Sup 0 L v 0 Inf v V a 1 S e v L v 1 With the optiu one obtains a solution u and the liiting load li (not unicity of u but unicity of li ). Thus, any loading L 0 v L v with 0 li is bearable. Beyond li, the proble of balance does not have a solution. Note: Note: There exist situations where, even if L 0 v is not bearable only, the cobination L 0 v L v, for 1 2, becoes it on a certain interval, and not only for two loadings colinéaires. The liiting load calculated for a two-diensional proble, in plane deforations, is necessarily higher than that obtained for this proble odelled in plane constraints. This result thus provides one raising. If one wishes to deal with a proble in plane constraints, it is necessary then to ake the kineatic approach on a three-diensional odeling. 1.3 Regularization of the kineatic approach by the ethod of Norton- Hoff-Friaâ The nueric work ipleentation of the kineatic approach requires the iniization of the notdifferentiable functional calculus S e v. Many techniques of regularization exist [bib4]. The ethod of Norton-Hoff-Friâa is used here [bib2], [bib7]. It rests on precursory work of Casciaro in It consists in replacing the function of support by the function of support regularized and differentiable NH. It is adjustable by a paraeter of regularization ( 1 2 ), of which the liiting value 1 conduit with convergence towards the function of support : NH = k 1 - éq The scalar k in [éq 1.3-1] is hoogeneous with a constraint. One notes the space acceptable speeds adapted to the proble of viscous flow for the law of Norton-Hoff of order : V a 1 ={v L,et v L, v=0sur u, L v=1} One defines on this space the regularized functional calculus S e v : The proble of iniization S e v = k 1- v d L 0 v Inf S v V a 1[ e v ] is well posed thanks to the properties of spaces L and adits a single solution u, for which the value reached by Inf is precisely. One notes the space of the incopressible fields of V a 1 by: V 1 a ={ v V 1 a tel que div v=0 } It is shown whereas this proble can be also written in the for of the research of pointsaddles, u Lagrangian following:

6 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 6/12 with: Max IR [ Inf v V 1 a A =k { A. v. v d L 0 v L v 1 } ] 2 / 2. 3 / 2. y = y éq It is noticed that A is increasing with (if E y, which is the case in practice) and hoogeneous with a constraint, and reains liited when 1. If one chooses k= y, then A = y 2 / 2 3 odulus is chosen E= y. and one finds the incopressible elastic proble when =2, if a Young It is thus noted that this potential [éq 1.3-2] defines a law of behavior giving the tensor of the constraints u by the relation of behavior of Norton-Hoff, to see [ 2.1]. One builds a decreasing continuation thus of and the liiting load li is the liit of this continuation when 1 (either n + ): li =li 1 Inf [ 1 v V a For the deonstration one will refer to [bib4] and [bib7]. Note: S e v ] =li S 1 e u éq If the intensity of the loading is aplified L L (whereas one does not consider a peranent loading L 0 =0 ), the solutions depend on the factor β according to the following relations: u = -1 u 1 ; D u = 1- D u 1 ; S e u = - S e u 1. With convergence for 1, the conclusion given by the solution u β is well the sae one as that given by u 1, since li = li 1 /.

7 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 7/12 2 Digital aspects of the calculation of the liiting load 2.1 Relation of behavior of Norton-Hoff The tensor of the constraints u check the relation of behavior of Norton-Hoff. The diverter of the constraints associated at the speed of deforation is: with tr u =0, and: A =k D u =A. D u. D u - 2. D u D u = A n. D u. D u 1 -n. D u / 2 y and n= 1 1. éq This behavior is integrated in the sae way that the elastoplastic incréentaux behaviors of Von Mises [R ]. Let us notice however that, in a point of integration, the calculation of the tensor of the constraints according to the tensor of the deforations is explicit, no iterative diagra being used. Moreover, no internal variable is necessary to the integration of this behavior. In Code_Aster, the calculation of the liiting load being independent of the oduli of elasticity, one chooses k= y, fro where A = y 2 3 when =2, for a Young odulus E= y. / 2. One finds the incopressible elastic proble thus Moreover, the continuation of the scalars is directly deduced fro the list of oents (fictitious) of calculation by: =110 1-t, so that when the oent increases, tends towards 1, and the behavior approaches a behavior perfect rigid-plastic, to see into unidiensional the curves [fig. 2.1-a]. In practice, one chooses the continuation of the values of [. 2.1-a]. Fig. 2.1-a. Stress-strain curve for various values of the oent t. t t a. Continuation of the values of the oent t, and values of corresponding.

8 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 8/12 The tangent operator, used in the option FULL_MECA ethod of Newton, is written thanks to [éq 2.1-1]: d D = A D - 2 Id Id 2 D 2 D D d D éq u with D, D vectors of the strains and deviatoric stresses writings in vectorial notations of WALPOLE-COWIN: 2.2 Piloting D = D 11, D 22, D 33, 2 D 12, 2 23 D = D 11, D 22, D 33, 2 D 12, 2 23 D D, 2 31 D D, 2 31 The proble is written in variational for in the following way on the space of the incopressible fields. For given, therefore at one oent t given, knowing the solution at the previous oent (noted u, ), to find, u IR V a such as: { L v = f.v d F.v ds =1 f u u. v d =L 0 v L v v V 0 éq L 0 is the peranent loading and L the loading controlled by the paraeter, cf [ 1.1], V 0 is a space of functions discretized on the basis of incopressible finite eleent, and is thus defined by a vector U degrees of freedo. This proble adits a single solution for all 1 2 (see [bib4]). For =2 the proble is of standard incopressible linear elasticity. The proble discretized at the oent t (thus for a value of, cf [. 2.1-a]) can be written (by oitting the boundary conditions to siplify): { F int U ; U ;..=F ext 0 F ext L U U =1 The search for ensuring the condition L U =1 is ensured by an algorith of piloting [R ]. Briefly, the principle is the following: by linearization of the equations relating to the interior forces, one obtains, for the iteration n algorith of Newton, cf [R ]: [ F int U Un ] K T [ U ]=[ F 0 ext F int U n ] [ F ext ] R cst R pilo éq One can now express the corrections of displaceents U and of ultipliers of Lagrange according to with the help of the resolution of this linear syste: [ U ]=[ U cst ] [ U pilo ] où [ U cst ]=K T 1 R cst et [ U pilo ]=K T 1 R pilo éq 2.2-3

9 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 9/12 One can substitute the correction of displaceent U according to its expression [éq 2.2-2] in the equation of control of the piloting of the syste L U =1 ; it results a scalar equation fro it in : L U U n U cst U pilo =1 that is to say f. U U n U cst U pilo d f F. U U n U cst U pilo ds=1 éq what in discretized for returns to: what leads to: e F ext. U U n U cst U pilo =1 1 F ext. U U n U cst e λ= e F ext. U pilo éq Postprocessing of the calculation of the liiting load Having obtained the solution, u, for each oent, therefore each given, it reains to use the continuation of to build the approxiation of the liiting load. For that one exploits the properties [éq 1.3-2], [éq 1.3-3], the fact that A is increasing and the property resulting fro iniization [éq 1.3-2] (see [bib7]). Of these two last, with 1 r s, one deduces that for u r and u s respective solutions (also checking the condition of incopressibility and standardisation) of [éq 1.3-2] for =r and s : A r u r. u r r / 2 d A s u s. u s s/ 2 d Associated with the property [éq 1.3-3], one draws for 1 r s, while noting r = Ar d : r A r u r. u r d r A r u r. u r r / 2 d 1r 1-1 s s A s u s. u s s/ 2 d 1 s éq Indeed, this property results fro the inclusion of functional spaces L r L s, for 1 r s, that is to say also, where hx play the part of a variable easureent (conditioned by the liit of resistance y ): 1 r r h x f x r 1 d r 1 s s h x f x s 1 d s The ters are noted continuation below, which one calculates in practice by postprocessing using u (external power being unit: = 1 1 A u.u /2 d 1 L 0 u éq 2.3-2

10 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 10/12 This continuation is thus decreasing for 1 and it is shown [bib7] that it converges towards, which li a good control allows. As one can undervalue (knowing that A1 of [éq 2.3-1]: y 2 3 ) the first ter li 2 y 3 u. u d L u 0 one thus calculates for each value of (thus of the oent t) the load liits by excess also converging towards li : li = y 2 3 u.u d L u 0 éq One judges quality of the approxiation of the liiting load li by coparison of the various values of who converge towards li by excess (in 1 ). These ters are calculated by digital integration at the points of Gauss of the finite eleents. Another interpretation of the interest which this continuation brings lies in the fact that it directly exploits the expression of the function of support of convex of resistance, i.e. the power dissipable in the odes of potential ruin, applied to the incopressible and standardized solutions calculated u. If the peranent loading is null: L 0 =0, one can easily exploit the stress field (alost statically acceptable) calculated with the solution u and to obtain a value by estiate of the liiting load, which would be necessarily a true lower liit if balance were checked exactly (see [bib4]). The continuation is thus calculated, which does not have on the other hand properties of onotony: = A. u. u d.sup x 3 D u 2. D u éq y This axiization (of the function called gauge of convex of resistance) is not calculated that at the points of Gauss of the finite eleents. Also the value obtained, for each, lower than [bib4], can be regarded only as one indication. On the other hand, always if the continued loads are worthless, it allows, with the value by excess, to provide a fraing of the load liits discretized proble. 3 Features and checking To carry out a calculation in Code_Aster in liiting analysis with the ethod of regularization of Norton-Hoff-Friaâ with the criterion of resistance of Von Mises, it is necessary: to define the odel 2D (plan or axis) or 3D with the quasi-incopressible finite eleents, odelings 3D_INCO_UPG, D_PLAN_INCO_UPG, or AXIS_INCO_UPG ;

11 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 11/12 to ensure the condition of incopressibility: GONF=0 in AFFE_CHAR_MECA ; to define only the characteristic of aterial s y, the liiting load being independent of E and, to define the peranent loading and that which is paraeterized by λ ; to define the discretization in tie, (in practice enters t in =1 and t ax =2 with 5 ); to carry out a non-linear calculation with the relation of behavior NORTON_HOFF with the order STAT_NON_LINE [U ], and piloting ANA_LIM. One can use linear research in practice to iprove convergence, and the subdivision of the step of tie, post-to treat calculation to obtain the load liits with the order POST_ELEM [U ]. The use of these orders is detailed in the docuent [U ]. With regard to postprocessing, the operator POST_ELEM product then a table which gives for each oent of calculation, i.e. for increasingly weak regularizations, 2 paraeters: the paraeter CHAR_LIMI_SUP an upper liit of the liiting load contains, by integration on each finite eleent and a su on L unit of the eleents of odel: = 2 y 3 u.u d L u 0 and, in the absence of constant loading, (CHAR_CSTE = ' NON'), the paraeter CHAR_LIMI_ESTIMEE contains an estiate D a lower liit λ correspondent with: = A. u. u d.sup x 3 D u 2. D u -1 y If a constant loading is present, (to infor then iperatively CHAR_CSTE = YES ), the paraeter PUIS_CHAR_CSTE represent the power of the constant loading in the field speed solution of the proble. Several tests of checking are available, in particular test SSNV124 [V ]. On this very siple proble, an analytical calculation akes it possible to obtain the exact liiting load in the direction of the loading, as well as the estiates produced by the ethod of regularization. For ore details one will refer to [bib4] and [bib5]. In addition copleentary validations were carried out within the fraework of coparative studies, like the benchark European LISA [bib8, bib10]: on calculations of liiting loads in 2D, 2D axis and 3D, the regularized kineatic ethod presented here akes it possible to gain a factor fro 6 to 10 over tie calculation copared to an increental elastoplastic calculation, and akes it possible to obtain a fraing of the liiting load, contrary to the ethods of the other participants. 4 Bibliography 1) ANGLES J., VOLDOIRE F., Modeling and calculation of the load liits of a fissured coponent, CR-MN , Sept ) FRIAA A., Law of Norton-Hoff generalized in plasticity and viscoplasticity, Doctorate, ) FRIAA A., FREMOND Mr., the ethods statics and kineatics in design the collapse and liiting analysis, Newspaper of Mechanics theoretical and applied, vol. 11, NO5, , ) VOLDOIRE F., Design the collapse and analyze liit of the structures, notes EDF HI- 74/93/082.

12 Titre : Calcul de charge liite par la éthode de Norton-H[...] Date : 25/02/2014 Page : 12/12 5) VOLDOIRE F., Analyzes liit of the fissured structures and criteria of resistance, note EDF/DER HI-74/95/026. 6) MICHEL-PONNELLE S., LORENTZ E. Finite eleents treating the quasi-incopressibility, docuent R D. 7) VOLDOIRE F., Put in work of the ethod of regularization of Norton-Hoff-Friaâ for the liiting analysis of the structures, notes EDF/DER HI-74/97/026. 8) LAHOUSSE A., VOLDOIRE F., Calculation of liiting load and benchark of the European project Brite EuRa LISA Notes EDF/DER HI-74/98/026/A. 9) [R ] S. MICHEL-PONNELLE, E. LORENTZ, Finite eleents treating the quasiincopressibility, ) VOLDOIRE F., Liit analysis by the Norton-Hoff-Friaâ regularising ethod. In Mr. Heitzer, Mr. Staat, LISA project carryforward 2001, publication of John von Neuann Institute for Coputing (2003). 5 Description of the versions of the docuent Author (S) Aster Organization (S) 5 F.VOLDOIRE EDF-R&D/AMA 8,4 E.LORENTZ, S.MICHEL-PONNELLE EDF-R&D/AMA Description of the odifications Initial text Modification of the anageent of the exhibitor of the law of Norton-Hoff