Large strain nite element analysis of a local second gradient model: application to localization

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 00; 54:499 5 (DOI: 0.00/nme.433) Large strain nite element analysis of a local second gradient model: application to localization Takashi Matsushima, Rene Chambon ; ; and Denis Calerie Institute of Engineering Mechanics and Systems; University of Tsukuba; --; Tennodai; Tsukuba City; Ibaraki; ; Japan Laboratoire 3S Grenoble Universite Joseph Fourier; Institut National Polytechnique; C.N.R.S. U.M.R. 55; B.P. 53X; 3804 Grenoble Cedex; France SUMMARY A large strain nite element formulation based on a local second gradient plasticity model is presented. The corresponding constitutive equations were developed as a direct extension of microstructured (SIAM J. Appl. Math. 973; 5(3): ; Arch. Rational Mech. Anal. 964; 6:5 78) or micromorphic (In Mechanics of Generalized Continua, IUTAM Symposium, Kroner (ed.), Springer: Berlin, 968; 8 35; J. Math. Mech. 966; 5(6): ) continua in which a mathematical constraint between the micro kinematics description and the usual macrodeformation gradient eld has been introduced. This constraint is enforced in a weak sense by the use of Lagrange multipliers in order to avoid diculties with the C continuity, for the nite element method. Corresponding nite elements are then constructed involving the Lagrange multipliers eld. A geometrically non-linear -D nite element code is developed within a framework of an incremental method. For every step, a full Newton Raphson involving a numerical consistent tangent stiness operator for the complete model (i.e., the second gradient terms as well as the classical ones) is done and some numerical tests allow to validate the method and to discuss the inuence of the geometrical non-linearity. Copyright? 00 John Wey & Sons, Ltd. KEY WORDS: localization; large strain; second gradient plasticity model; Lagrange multipliers; F.E.M.. INTRODUCTION Strain localization is an important phenomenon which appears almost always when a structure is close to rupture. This phenomenon is present in every kind of material, including metals, polymers, geomaterials. Even if it is possible to predict the occurrence of localization with Correspondence to: Rene Chambon, Laboratoire 3S, Grenoble Universite Joseph Fourier, Institut National Polytechnique, C.N.R.S. U.M.R. 55, B.P. 53X, 3804 Grenoble Cedex, France E-ma: rene.chambon@hmg.inpg.fr Contract=grant sponsor: Japan Society for the Promotion of Science for Young Scientists Received 3 November 000 Copyright? 00 John Wey & Sons, Ltd. Revised 3 July 00

2 500 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE usual models [], the post-localization regime cannot be modelled with classical non-viscous constitutive equations [], and computations suer from a strong mesh dependency (size and orientation). In this case enhanced models are necessary. Since the pioneering work of Aifantis [3], many enhanced constitutive equations have been proposed to model localized zones. They can be classied into three dierent famies: the non-local models advocated by Bazant and his co-workers [; 4 6] mainly used for concrete, the gradient plasticity theories [3; 7 3] which have been applied for various materials, and the Cosserat theory [4; 5] usually applied to geomaterials. In fact this recent attention paid to such media may be traced back to some seminal works mainly made for modelling the behaviour of crystalline solids, which have been done essentially during the 960s, except the pioneering work of Cosserat [6]. It is worth mentioning various theories such as couple-stress theory [7], microstructured continuum theory [8; 9], strain (or second) gradient theory [0] micromorphic theory [], multipolar theory [], micropolar (or generalized Cosserat) theory [3], non-local theory [4; 5], etc. The more recent studies mentioned in the beginning of this introduction have been elaborated to remedy the pathological behaviour of classical models in localized computations and then present computational applications of the studied enhanced models. However, most of them do not deal with geometrical non-linearity. Few papers present works taking into account large strains for such models. It is worth mentioning the papers of Xikui Li [6] and Steinmann [7; 8] about large strain extensions of gradient plasticity or Cosserat theories. The purpose of this paper is to develop a large strain nite element formulation suitable for the analysis of localization phenomena. Since localization occurs close to rupture, geometric non-linear eects can be expected. Therefore, it should be incorporated into the post-localization analysis. It is also important to clearly put forward what is neglected when the changes of geometry are not taken into account. The work detaed in this paper has been done within the microstructured materials framework which can be traced back to Mindlin [8; 0] and Germain [9; 9]. The proposed theory has some attractive advantages [30]. Dierent from the gradient plasticity models without higher order stresses, it is a local theory with higher order stresses, in which the stresses depend only on the local kinematic history. Moreover, it is a theory with consistently dened dual quantities which allow a clear denition of energy. Besides, it can be easy adapted to every kind of classical constitutive equation (elasto-plastic, damage or hypoplastic) though this paper presents only an elasto-plastic version. Some analytical solutions of -D boundary value problems has been exhibited in the restricted case of the so-called small strain assumption [3]. The method proposed (i.e., the use of Lagrange multipliers to enforce the relation between the strain and the strain gradient) has already been successfully used for -D problems [3]. It is worth mentioning that the constitutive equation used, shares some simarities with the one developed independently by Fleck and co-workers [33; 34]. The rst section is devoted to the basic equations of local second gradient models. Balance equations are written in the virtual work form. The Lagrange multiplier eld is introduced to weaken the relation between the deformation gradient and its space derivatives, and to avoid the use of C continuous elements in the subsequent nite element analysis. The basic initial boundary value problem to be solved is detaed in the second section, and the time step problem is introduced. The third section is devoted to the iterative method used to solve the time step problem. Then the theory is applied to a D problem in the fourth section. The used nite elements share some simarities with some of the ones advocated by Shu and Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

3 LARGE STRAIN FINITE ELEMENT ANALYSIS 50 co-workers [35] for second gradient linear elasticity. The constitutive equations are presented in the fth section. It is a straightforward extension of a classical elasto-plastic softening Von Mises model, which implicitly involves an internal length. Some remarks about various aspects are explained in the sixth section, the iterative algorithm used is detaed. Some D numerical tests, performed with the presented D code, are compared with the D known solutions [3; 3] in the seventh part. A biaxial test exhibiting shear bands is computed to verify the eciency of the proposed nite element method in the eighth part. Mesh dependency, inuence of the internal length, dierence between large and small strain and convergence are specically studied. Finally some comments are given in the conclusion. Let us nally give the principles of our notations. A component is denoted by the name of the tensor (or vector) accompanied with tensorial indices. All tensorial indices are in lower position as there is no need in the following for a distinction between covariant and contravariant components. Other indices have other meanings. The summation convention with respect to repeated tensorial indices has been adopted.. BASIC EQUATIONS.. Variational principle Let us begin with the following weak form of the balance equations written for the strain gradient theory viewed as a particular case of the microstructured continuum theory [8; 9; 30]: ( ) t + u i? k t d t P e? = 0 () where, superscripts t and? denote, respectively, quantities at a given time t and virtual quantities, t is the classical Cauchy stress tensor, k t is the corresponding double stress tensor, u? i is a kinematically admissible virtual displacement eld, xi t are the current coordinates of the points of the studied body and P e? is the external virtual work generated by the corresponding external forces. Assuming for simplicity that there is no body double force and that the boundary of t is regular which means that it is possible to dene an external normal in every point of this boundary, external virtual work reads as: P? e = t t f t i u? i d t + t (p t i u? i + P t i Du? i ) d t () where, fi t is the body force per unit mass, t is the mass density, pi t is the external (classical) forces per unit area, and Pi t an additional external (double) force per unit area, all applied on a part t of the boundary t. D denotes the normal derivative of any quantity q; (Dq =(@q=@x k )n k ). The constitutive equations are assumed here to give the values of t and t as functions of the local kinematic history up to the time t generalizing the classical denition of constitutive equations [36]. It has to be emphasized here that this is a very big dierence with the second gradient models used up to now which are non-local ones, except the models developed by Fleck et al. [33; 34] and Chambon et al. [30 3]. Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

4 50 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE.. Strong form of the balance equations As usual, starting Equation (), using the divergence theorem and integrations by part allow us to get the strong form of the balance equation for every internal point of the k t + t fi t = 0 (3) Simarly the links between p t i ; P t i, and the local values (on the boundaries) of the stress and the double stress can be deduced [8 0]: t n t n t kn t D t k Dt k Dx t k n t Dt k Dx t n t k + Dnt l Dx t l kn t n t k t Dnt Dxk t k t = pi t (4) and kn t n t k t = Pi t (5) where Dq=Dx denotes the tangential derivatives of any quantity q. n k k.3. Equations with Lagrange multipliers Unlike classical modelling, the previous way of dealing with second gradient models implies the use of C functions for the displacement eld as the second derivatives of the displacement are involved in the variational principle written in Equation (). This generates some diculties when Finite Elements are used to solve boundary value problems. In order to avoid such complexities it is usual to weaken the constraint between a function and its derivatives and to introduce a corresponding eld of Lagrange multipliers related to a weak form of the constraint. So let us introduce, as new independent variables, the derivatives of the displacement eld ui t with respect to the current conguration, denoted as v t in the following. As usual, in this paper the corresponding virtual quantity is denoted as v.? According to this new viewpoint, Equation () is transformed as follows: ( ) ( ) i? t +k t d t t? t v? d t P e? = 0 (6) t where t denotes the eld of Lagrange multipliers. The Lagrange multipliers can be interpreted as the so-called microstress in the framework of the microstructured materials [30; 3]. Moreover, the elds u i and v have to meet the following weak form constraint: ( t i v t d t = 0 (7) t It is now natural to write the external virtual power as P e? = t fi t u i? d t + (pi t u i? + Pi t v ikn? k) t d t (8) t t Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

5 LARGE STRAIN FINITE ELEMENT ANALYSIS 503 Simarly it is now natural to assume that the constitutive equations give t and t as functions of the history of u t i and v t..4. Time discretization: the time step problem The problem of the evolution of a body assumed to obey a local second gradient constitutive equation can now be formulated, with the purpose of buding a C 0 nite element model. We are looking for two elds ui t and v t such as that for every time t ]0;T] Equations (6) (8) are met for every kinematically admissible virtual elds u i? and v? and for every virtual eld?. Kinematically admissible here means that u? i and v? vanish on the part of the boundary where ui t and v t are assumed to be known and are suciently smooth throughout the domain. The constitutive equations which give t and t as functions on the history of ui t and v t are assumed to be met as well. The values of the functions fi t ; pi t and Pi t are assumed to be known (at least on some part of the boundary for the last two ones). As usual it is necessary to discretize the time in order to be able to solve the problem numerically. In fact the problem is not solved for every time t but for a sequence of time steps t n+ = t n+ t n, such that T = =n+ = t. In the next section we wl consider the following time step problem. The solution of the general problem is known up to the time t t, and we are looking for the values of ui t and v t at time t which meet Equations (6) (8) at time t, the values of fi t ; pi t and Pi t being given. In order to solve this problem we wl construct an iterative procedure based on a Newton Raphson method in the next section. 3. EQUATIONS FOR THE ITERATIVE PROCEDURE 3.. General formulation Let us deta here the central point of the iterative procedure corresponding to a given time step. Let us assume that we know an approximation of the solution at time t. This means that we know a conguration called. Note that the index of time is now the number of the iteration (for a given time step) whereas in the previous section it is the number of the time step). We are now looking for a better (unknown) approximation called +. First of all for every kinematically admissible eld, it is possible to dene residuals denoted as R t? n and Q t? n with the following equation: i l l i l + v?? t n i ] d P t? n e = R t? n (9) v ) d = Q t? n (0) Let us note that R t? n is a linear form with respect to u i? and v?, and that Qt? n is a linear form with respect to?. Then simar equations are written in the unknown conguration + assuming that this new conguration is the solution, which means assuming that the corresponding residuals vanish. Writing both sets of equations in conguration and subtracting the ones corresponding to conguration from the ones corresponding to Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

6 504 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE conguration + yields: [ i + l + i l (? + l + t n+ i k det F det F k + ) ) ( l + v? ( + det i where det F is the Jacobian of F = + i = Let us denote: ) + k l + k det F l ] det F ) d ( P t? n+ e P t? n e )= R t? n (v + ) () det F v ) ]d = Q t? n () and obeys the following equation: d + = det F d (3) du = x + x dv d = v + = + v (4) d ikl = + ikl ikl d = + Using a Taylor expansion, assuming that du, dv, d, d ikl, and d are small on the same order and discarding terms of degree greater than one yield after some algebra, the following linearized equations: i l =(P t? n+ ik l v? ] d d ikl 0 + m m + ik 0 l d d P t? n e ) R t? n (5)? t n i v t nm m i k dv ) d = Q? t n (6) As seen in Equation (8), the term P t? n+ e P t? n e in Equation (5) depends on fi t, pi t and Pi t. Usually fi t are position independent. For simplicity it is assumed in the sequel that Pi t =0. So if pi t are dead loading forces which means that they are position independent too, then: P t? n+ e P t? n e = 0 (7) Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

7 LARGE STRAIN FINITE ELEMENT ANALYSIS 505 This is assumed in the following. Adding the corresponding terms in the case of following forces (for pi t ) such as pressure originated ones is straightforward and can be done as in the standard (without gradient eects) case. Let us emphasize at the end that Equations (5) and (6) have been obtained through proper linearization of Equations (9) and (0). This appears clearly when observing Equations (5) and (6). Terms (amongst others) such as or ikl appearing in the left-hand side of Equation (5) are clearly due to the assumption that congurations and + are dierent. For the sake of simplicity, only the two-dimensional problem wl be considered in the following. So now the indices i; ;::: {; }. This can be used for truly two-dimensional problems or in the case of plane strain problems. Analysis of plane stress problems requires appropriate developments [37], as in this case, gradients with respect to the third direction have to be taken into consideration. 3.. Two-dimensional formulation The detaed form of the above equations for two-dimensional formulation is derived here. Let us rst introduce the two matrices [C (4 4)] and [D (8 8)] dened as d d = C ] (4 4) d d d d. d = [ D (8 8) Let us emphasize that it is assumed that these matrices are gained through a consistent linearization [38] of the stress point algorithm employed for the integration of the constitutive equation between t t and t (see Section 6.). Let us dene the following 0-dimensional @dv (9) t n [du @dv dv dv dv dv d d d (0) Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

8 506 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE [U? ] T is dened simarly for the corresponding virtual quantities. The corresponding matrix form of Equations (5) and (6) is then given by: [U? ] T [E ][du ] d = R t? n Q t? n () where [E ]isa0 0 matrix dened as follows: E (4 4) 0 (4 8) 0 (4 4) I (4 4) E (8 4) D (8 8) 0 (8 4) 0 (8 4) [E ]= E3 (4 4) 0 (4 8) 0 (4 4) I (4 4) E4 (4 4) 0 (4 8) I (4 4) 0 (4 4) () with [ E t n (4 4) ] = [ C t n [ E t n ] (8 4) = (4 4)] [ E3 t n ] 0 0 (4 4) = 0 0 (3) (4) (5) 0 0 Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

9 LARGE STRAIN FINITE ELEMENT ANALYSIS 507 [ E4 t n ] (4 4) @u 0 v I (4 4) is as usual the 4 4 v v v v (6) 4. FINITE ELEMENT METHOD 4.. Shape functions Equation () is now discretized using the nite element method. The quadrateral elements used are detaed hereafter. They have eight nodes for u i, four nodes for v and one node for as described in the left-hand side of Figure. The parent element is dened in space with ( ) and ( ) (see the right-hand side of Figure ). Finally all the variables are described with respect to these parent coordinates. The usual quadratic serendipity shape function denoted as are used for u i. The linear shape function denoted as are used for v, whereas is assumed constant: i (; )=[ (; ) (; ) ::: 8 (; )] u u i (= ;= ) u i (0; ). u i ( ; 0) (7) Figure. Quadrateral element and parent element used in this study: (a) quadrateral element; and (b) parent element. Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

10 508 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE (; )=[ (; ) (; ) 3 (; ) 4 (; )] v v (= ;= ) v (; ) v (; ) v ( ; ) (8) (; )= (=0;=0) (9) It is our choice here for the sake of simplicity to assume constant Lagrange multipliers on each element. Other choices can be made [35] and have to be veried in the case of the problems studies here. The mapping from (x ;x )to(; ), is described in terms of the same shape functions as for the displacement u i (; ) (isoparametric element). 4.. Element stiness matrix The transformation matrices, [T ] and [B], which connect [du ] with the nodal variables are obtained through the common nite element procedure: [du ]=[T ][du (; ) ] (30) [du (; ) ]=[B][dUnode ] (3) where, t n [du (; ) @ dv dv d d ] (3) [du node ]T = [ du ( ; ) du ( ; 0) du (0; ) du ( ; 0) du (0; ) du ( ; ) du ( ; ) dv ( ; ) du ( ; ) dv ( ; ) dv ( ; ) dv ( ; ) dv ( ; ) dv ( ; ) d (0; 0) d (0; 0) d (0; 0) d (0; 0) du (0; ) du (0; ) dv ( ; ) dv ( ; ) du (; ) du (; ) dv (; ) dv (; ) dv (; ) du (; ) du (; ) dv (; ) dv (; ) dv (; ) dv (; ) dv (; ) ] du (; 0) du (; 0) (33) Finally the element stiness matrix [k ] is obtained from Equation () as follows: elem [U? ] T [E ][du ] d =[U? node] T [B] T [T ] T [E ][T ][B] det J d d[du node ] [U? node] T [k ][du node ] (34) Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

11 LARGE STRAIN FINITE ELEMENT ANALYSIS 509 where det It is worth mentioning here that in the following applications, the integration procedure uses Gauss method. In order to avoid possible locking in the plastic range, the elements are underintegrated, which means here that only four Gauss points are used to compute the element stiness matrix. Obviously the same method holds for the computation of the residual terms detaed in the following Element residual terms The residual terms R t? n + Q t? n in Equation () can be computed for one element as follows, dening the elementary out of balance forces [f ( R t? n where, [ ]= Q t? n )elem = P t? n e [U? HE ]: [ ::: ::: ::: The external virtual power, P t? n e surface force term. (35) [B] T [T ] T [ ] det J d d [U node]? T [f HE ] v v ] (37), consists of two terms; the body force term and the boundary 4.4. Global stiness matrix Adding Equations (34) and (36) for each element and eliminating the virtual displacement vector, [U node? ], the global nite element equations are nally obtained: [K ][U node ]= [FHE ] (38) where [K ] is the global stiness matrix obtained by assembling the element stiness matrices [k ] (see Equation (34), [F HE ] is the global out-of-balance force vector obtained by assembling the elementary out-of-balance forces [f HE ] (see Equation (36)), and [Unode] is the global vector of the correction of the nodal degree of freedom. They are the unknown of the nth iteration of the time step from t t to t. After solving this auxiary linear system, the current conguration is actualized and a new iteration is started. 5. TWO-DIMENSIONAL ELASTO-PLASTIC CONSTITUTIVE RELATION In order to check our nite element model, the following constitutive equation is used. It is an elasto-plastic model belonging to the famy described by Chambon et al. [30]. It can be seen as a classical elasto-plastic model with a second gradient term. However, it is not a Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

12 50 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE fully large strain elasto-plastic model where for instance a multiplicative decomposition of the displacement gradient is used. It is our choice to begin with a simple model like this one. The main reason is that such a model is a straightforward generalization of the one-dimensional model studied by Chambon et al. [3] for which small strain analytical solutions and large strain numerical solutions are avaable [3]. As far as the rst gradient part is concerned, the model implemented in this study is based on the Prandtl Reuss elasto-plastic model as follows: =3Kė (39) O s = { G ( 6e lim ) G ( G G G s kl kl s s ) ( e lim ) where, O s is the Jaumann rate of the deviatoric Cauchy stress tensor, are deviatoric strain rates, is the mean stress rate and ė is the mean strain rate. K; G and G are the bulk modulus, the shear moduli before peak and after peak, respectively. is the second invariant of the Green Lagrange deformation tensor, e lim is a parameter of the model. For the second gradient part, we start from the more general isotropic linear relation derived by Mindlin [8; 30], involving six parameters corresponding to ve independent coecients. We used a particular case depending on only one parameter as it can be seen in the following equation. O O O O O O O O D D= D= 0 0 D= D= 0 D= 0 0 D= 0 D= D= 0 D= 0 0 D= D 0 D= D= 0 = 0 D= D= 0 D D= 0 0 D= 0 D= D= 0 D= 0 0 D= 0 D= D= 0 0 D= D= D where v is the material derivative of v, and O k is the Jaumann double stress derivative dened by Equation (4) O k = k + lk! li + imk! m + p! pk (4) where! li is the spin v v v v v v v (40) (4) Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

13 LARGE STRAIN FINITE ELEMENT ANALYSIS 5 6. SOME REMARKS ON THE TIME STEP ALGORITHM 6.. Kinematic assumption and consistent tangent stiness matrix During a given time step, the velocity of each material point is assumed to be constant: this allows to compute the velocity gradient used in the classical part of the constitutive equation. Simarly the second gradient rate ( v ) is also assumed to be constant, which allows to compute the stress rate and the double stress rate using the previous assumption. In some cases sub time increments have been used with the same assumption over the time step. This assumption has been made for simplicity and as a rst step. It is not the straightforward generalization of the one done for the one-dimensional problem [3]. However, it can be seen in the following that the present numerical computation gives simar results as the previous ones in the particular case of one-dimensional problems [3]. This assumption allows us to compute the stress and the double stress p at the end of a given time step. We now have to compute the consistent tangent stiness matrices [C (4 4)] (see Equation (8)) and [D (8 8)] (see Equation (9)), which are clearly dierent from the constitutive matrices obtained with Equations (39) (4). Matrices [C (4 4)] and [D (8 8)] are proper linearizations of the computation of and p with respect to a (small) change of the nal state. A close form computation of such matrices is not straightforward. So we use a forward nite dierence approximation of the consistent tangent stiness matrices. Such a procedure is time consuming as computations of stress and double stress have to be done, respectively, 4 and 8 times, however it is convenient to work rst in this manner. 6.. Small strain assumption It is quite meaningful to compare the results obtained in the large strain computation framework with those corresponding to the small strain one. The following small changes enable us to deduce the needed small strain computations from the general ones described above. At rst, all the variables should be dened with respect to the initial conguration in small strain computation. Therefore, x i = x + i = xi 0 is applied to each previous equation. This change results in the simpler form of matrix [E ]: C (4 4) 0 (4 8) 0 (4 4) I (4 4) 0 (8 4) D [E (8 8) 0 (8 4) 0 (8 4) ] ss = 0 (4 4) 0 (4 8) 0 (4 4) I (4 4) I (4 4) 0 (4 8) I (4 4) 0 (4 4) The transformation matrix, [T ], is also changed into [T ] ss in the same manner, and accordingly, the element stiness matrix, [k ], and the element out-of-balance force vector, [f HE ], are computed from [E ] ss and [T ] ss. As far as the constitutive equation is concerned, in the stress point algorithm, Jaumann corrections are not considered in small strain. In fact in the so-called small strain assumption, it is postulated that there is no dierence between the dierent congurations. So the equation can be obtained only by replacing Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5 (43)

14 5 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE Table I. The nite step algorithm for one loading step.. Initial conguration: stress t t, double stress t t, coordinates x t t. Assumption on the nal conguration Initialization of [U node ] update coordinates: x for n = 3. Beginning of the iteration n: 4. For each element: For each integration point: compute the strain rate, the rotation rate and the second gradient rate (6.) compute n and n using the constitutive equations (6.) update the stress and the double stress = t t + n, = t t + n, compute the compliance matrices by means of perturbation methods [C (4 4) ] and [D (8 8)] (3., 6.) Compute the element stiness matrix [k ] Compute the element out of balance forces [f HE ] 5. Compute the global stiness matrix [K ] 6. Compute the global out of balance forces [F HE ] 7. Compute [U node ] by solving [K ][U node ]= [FHE ] 8. Check the accuracy of the computed solution if convergence: go to 9 if no convergence: update the new assumed nal conguration, n = n +,goto3 9. End of the step current values (stress, double stress, deformation gradients...) in Equations (6) and (7) by the corresponding rates. Results seen in Equation (43) are an lustration of the well-known [39] (at least for classical continuum media) link between large strain assumption and the presence of initial stress in the incremental balance equations Algorithm of a time step computation In order to clarify the various aspects detaed above, the algorithm is detaed in Table I. 7. ONE-DIMENSIONAL PROBLEMS In order to validate the implemented -D code, one-dimensional compressions are computed and the results are compared with analytical solutions known in the case of small strain [3] as well as with numerical solutions given by one-dimensional code [3]. 7.. Boundary conditions and material properties Figure shows the boundary condition used in the -D code. In order to avoid a twodimensional eect, the condition, u = 0, is applied in the upper and lower boundaries along Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

15 LARGE STRAIN FINITE ELEMENT ANALYSIS 53 Figure. Boundary condition in -D compression with -D code. Figure 3. Constitutive relation in -D case: (a) rst grade term; and (b) second grade term. the bar. The right end of the bar is xed (u i = 0) and the external compressive force is applied to the left end. The additional external forces P i are assumed to be zero at both ends. Material properties in the -D code, which correspond to those in -D code shown in Figure 3, are chosen as follows: for the standard rst gradient term, u = u 3 =0; u = u (x ) is applied to the two-dimensional linear isotropic material, and the following relations are obtained: K G = A (for strain hardening part) (44) K G = A (for strain softening part) (45) For simplicity, we assume the condition, G = 0. We obtain K = A = 7:5 MPa and G = 3 4 (A K)=6:875 MPa. These parameters are completely unrealistic but they have been chosen only to verify the validity of our two-dimensional code. In order to t to the second gradient part D =0:08 MPa m has been chosen. Clearly the second gradient one-dimensional model used implicitly denes two internal lengths, one (namely D=A ) corresponding to the unloading regime of the rst gradient part of the model, and the other (namely D=A ) corresponding to the softening loading regime. As soon as the peak point is reached, the problems solved exhibits a loss of uniqueness. Dierent solutions are found by changing the initialization of [U node ] (item of Table I). This is a generalization for the second gradient models of the method proposed by Chambon et al. [40] for usual rst gradient continuum models. Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

16 54 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE Figure 4. Relation between global deformation and external force in -D compression test under small strain assumption. Figure 5. Deformation patterns for solutions, and 3 in -D compression test under small strain assumption. 7.. Small strain computation Figure 4 shows the global force displacement curve. It is a comparison between the results obtained with the present two-dimensional code and the analytical solutions [3]. The number of elements, N elem is 50. According to the analytical study avaable only under the small strain assumption [3], there are four possible solutions after the yield point (bifurcation point) with the material properties and the boundary conditions chosen above. Solution is the homogeneous one, and the others are heterogeneous (localized) solutions. These solutions are patches of partial solutions involving either the unloading part of the model (in this partial solution the strain is an exponential function of the coordinate) or the softening loading part (and in this case the partial solution is a sine function of the coordinate). The two internal lengths mentioned above govern these solutions. Detas of these solutions have been given by Chambon et al. [3]. All the results computed with -D code (shown with open circles in the gure) are in good agreement with those calculated by analytical solutions [3]. Figures 5 and 6 show the distributions of v along the bar for the four possible solutions. In Figure 5 the dierences between the numerical solutions and analytical ones cannot be distinguished, whe some dierence is seen in Figure 6. The eect of the number of elements is more important in solution 4, which exhibits a very large slope in its force displacement curve as shown in Figure 4. In the large strain case (see Section 7.3), the slope is less large and the comparison between -D and -D computations is better. The solution with larger number of element, N elem = 500 is closer to the analytical solution. In the case of solution 4 it is clear that this convergence is not monotonic, however, we do not systematically study the convergence of the solutions. Further studies are needed concerning this point Large strain computation It is well known that geometric non-linearity increases the diculties of the computations. Therefore the time step has to be adusted according to the problem to be solved. Figure 7 shows the global stress strain relationships in large strain computations both with the -D code and the -D code described in Matsushima et al. [3]. All the curves given Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

17 LARGE STRAIN FINITE ELEMENT ANALYSIS 55 Figure 6. Deformation patterns for solution 4 in -D compression test under small strain assumption. Figure 7. Relation between global deformation and external force in -D compression test under large strain assumption. Figure 8. Deformation patterns for solutions, 3, and 4 in -D compression test under large strain assumption. Figure 9. Mesh size independence for solution 3 in -D compression test under large strain assumption. by the -D code are in good agreement with those obtained by the -D code, which proves that these two codes work well. The eect of the large strain assumption is clearly seen in their paths. In solutions 3 and 4, the calculations have been stopped when an element reaches the zero-stress state ( = 0). The solution corresponding to solution in small strain can be obtained neither by -D code nor by -D code due to the eect of large strain; the large strain assumption aects the deformation patterns, and there is a threshold strain over which solution does not exist. Figure 8 shows the deformation patterns in the possible solutions. In large strain computation the nodal variable v t corresponds to the displacement t =@x t. It is then chosen to plot for both computations the variable, vik t (@xt k =@x0 ), as a function of the reference coordinates. The accordance between the results by -D code and those by -D code is quite satisfactory. Figure 9 shows the mesh size independence of solution 3, which exhibits a clear distinction between the present model and that based on classical continuum. Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

18 56 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE 8. TWO-DIMENSIONAL PROBLEMS 8.. Boundary conditions and material properties A bi-axial test is computed in this section as an example of -D problem. Figure 0 shows the initial conguration of the specimen. It is 0:5 m wide and m high. The forces per unit area at both sides of the specimen are set to zero. The top is assumed to be a smooth rigid plate remaining horizontal. Through the plate a compressive force, F a, is applied. The vertical displacement of this top plate is denoted by u a. The bottom plate is also smooth, rigid and also remains horizontal. The central point of the bottom plate is xed to avoid rigid body displacement. Its horizontal displacement as well as its vertical one was prescribed to be zero. The external additional double forces per unit area P i are assumed to be zero all along the boundaries. The material obeys the constitutive equation detaed in Section 5. The bulk modulus K is assumed to be constant. The elastic shear modulus avaable for unloading is assumed to be constant, whe an exponential function is assumed as follows for the shear modulus after the yield point so that the material could reach its residual state smoothly: ( ) G G = G exp ( e lim ) (46) G e lim res where G is the value of the shear modulus ust after yielding. The second order part of the constitutive equation is described in Section 5. In this study the ratio, G = G is constant and D varies depending on the computation in order to allow for dierent values of the implicit internal lengths (see Section 7.). The other parameters are chosen as follows: G =50MPa; G = MPa; e lim =0:0; K =97:3856 MPa (47) Figure shows the second stress invariant s as a function of the second strain invariant. Figure 0. Initial conguration and boundary condition for bi-axial test. Figure. Designed s relationship. Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

19 LARGE STRAIN FINITE ELEMENT ANALYSIS 57 Figure. Two-dimensional mesh dependency study (Global stress strain curve). Figure 3. Two-dimensional mesh dependency study (Deformation pattern): (a) 0 by 0 mesh; (b) 30 by 5 mesh; and (c) 40 by 0 mesh. In order to obtain a shear band deformation mode, a 5% reduction of yield strength is assumed (e lim =0:95e lim in fact) at the bottom left-hand side element. 8.. Results and discussion First, a mesh dependency study is performed. A set of large strain computations are conducted for four dierent meshes (0 5; 0 0; 30 5; 40 0) involving elements having the same aspect ratio. Figure shows the global force as a function of the global strain. The convergence of the computation towards a solution not depending on the mesh is clearly exhibited in this gure. Figure 3 shows the deformation pattern of the ner three meshes for Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

20 58 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE Figure 4. Eect of parameter D on the global stress strain curve. Figure 5. Eect of parameter D (0 by 0 mesh): (a) D = :0e 4; (b) D =:0e ; and (c) D =:0e. the same global strain. It has to be emphasized rst that, contrary to many gures showing such patterns, the displacements are not magnied. From Figure 3 it can be checked that the observed width of the band is intrinsic and independent of the mesh size. Second a study of the inuence of the (implicit) internal length is done. Figure 4 shows large strain computations of the bi-axial test for three dierent values of D which implicitly means (see Section 7.) three dierent internal lengths, but for the same mesh. The smaller the internal length, the larger is the apparent softening eect for the global force strain curve. Figure 5 shows the deformation pattern corresponding to the same global apparent strain and for the three dierent values of D (three dierent internal lengths). It is quite clear that D governs the shear band width, and that this width, conversely, aects the global stress strain relationship. Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

21 LARGE STRAIN FINITE ELEMENT ANALYSIS 59 Figure 6. Dierence between large and small strain calculations (30 by 5 mesh): (a) large strain calculation; and (b) small strain calculation. Figure 7. Convergence through N R iteration in bi-axial test under large strain assumption. Figure 6 shows a comparison between a small strain computation and a large strain one for the same mesh and the same values of D and of the global apparent strain. Let us emphasize once more that the displacements of the nodes are not magnied. Clearly, small strain assumption results are dierent from the large strain ones. The stiness due to geometrical non-linear eects is necessary to obtain realistic deformation patterns. On the contrary the apparent band width seems to be unaected by the small strain assumption. Figure 7 is an example of convergence. It is quite satisfactory with respect to the accuracy attained at the end of a time step. However, the method used in the work presented in this paper (i.e., a full Newton method using consistent tangent stiness matrices) implies a quadratic Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

22 50 T. MATSUSHIMA, R. CHAMBON AND D. CAILLERIE convergence. A careful study of this convergence reveals that the stiness matrices used exhibit maor dierences between terms coming from the rst gradient part of the model and the ones coming from the second gradient part. After the Gaussian elimination, the slight dierences between the stiness matrix for a given iteration and the same matrix for the following iteration seems to vanish due to round o errors, which yields a linear convergence diagram. 9. CONCLUSIONS Two-dimensional large strain second gradient nite element models based on the strain gradient theory developed by Chambon et al. [30] was formulated and implemented. A simple -D deformation problem was simulated in order to validate the model. A problem involving a shear band has been simulated and the results are quite satisfactory. This work is a rst step towards ecient large strain computation of second gradient models. Some points need further studies. Coupling the proposed theory and computation codes with more classical methods such as remeshing for instance has to be done. Improvements of stress point algorithms, better solving of the auxiary linear systems can be tackled in the future. Use of the presented work with other rst gradient model is another realistic goal for the future. Simarly it should be interesting to develop true large strain second gradient elasto-plastic constitutive equations which means a theory involving a complete thermomechanical in the large strain framework. ACKNOWLEDGEMENT T. M. is grateful for nancial support for his work at Laboratoire 3S, Grenoble, France (August 997 August 998) from Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. REFERENCES. Rice J. The localization of plastic deformation. In International Congress of Theoretical and Applied Mechanics. Koiter WD (ed.), North Holland: Amsterdam, Paudier-Cabot G, Bazant ZP. Nonlocal damage theory. Journal of Engineering Mechanics, ASCE 987; 3(0): Aifantis EC. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, ASME 984; 06: Bazant ZP. Mechanics of distributed cracking. Applied Mechanical Review 986; 39(5): Bazant ZP. Imbricate continuum and its variational derivation. Journal of Engineering Mechanics, ASCE 984; 0(): Bazant ZP, Lin F-B. Non-local yield limit degradation. International Journal for Numerical Methods in Engineering 984; 6: Zbib HM, Aifantis EC. A gradient-dependent ow theory of plasticity: application to metal and so instabities. Applied Mechanical Review 989; 4: Zbib HM, Aifantis E. On the localisation and post localisation of plastic deformation. part I. On the initiation of shear bands. Res Mechanica 988; 3: Zbib HM, Aifantis E. On the localisation and post localisation of plastic deformation. part II. On the evolution and thickness of shear bands. Res Mechanica 988; 3: de Borst R, Muhlhaus H-B. Gradient-dependent plasticity: formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering 99; 35: Pamin J. Gradient dependent plasticity in numerical simulation of localization phenomena. Dissertation, Delft University of Technology, Delft, Vardoulakis I, Aifantis E. A gradient ow theory of plasticity for granular materials. Acta Mechanica 99; 87: Zervos A, Papanastasiou P, Vardoulakis I. A nite element displacement formulation for gradient elastoplasticity. International Journal for Numerical Methods in Engineering 00; 50: Copyright? 00 John Wey & Sons, Ltd. Int. J. Numer. Meth. Engng 00; 54:499 5

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