Three-dimensional modeling of strain localization in Cosserat continuum theory

Transcription

1 hree-dimensional modeling of strain localization in Cosserat continuum theory A.R. Khoei, S. Yadegari and M. Anahid Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 Center of Excellence in Structures and Earthquake Engineering, Deartment of Civil Engineering, Sharif University of echnology, P.O. Box , ehran, Iran Abstract. In this aer, a higher order continuum model is resented based on the Cosserat continuum theory in 3D numerical simulation of shear band localization. As the classical continuum models suffer from the athological mesh-deendence in strain softening models, the governing equations are regularized by adding the rotational degrees-of-freedom to conventional degrees-of-freedom. he fundamental relations in three-dimensional Cosserat continuum are resented and the internal length arameters are introduced in the elasto-lastic constitutive matrix to control the shear bandwidth. Finally, the efficiency of roosed model and comutational algorithm is demonstrated by a 3D stri in tensile. A comarison is erformed between the classical and Cosserat theories and the effect of internal length arameter is demonstrated. Clearly, a finite shear bandwidth is achieved and the load-dislacement curves are uniformly converged uon different mesh sizes. Keywords: Localization, Dislacement discontinuity, Cosserat theory, 3D FEM, Return maing algorithm. 1. INRODUCION Localization of deformation refers to the emergence of narrow regions in a structure where all further deformation tends to localize, in site of the fact that the external actions continue to follow a monotonic loading rogram. he remaining arts of the structure usually unload and behave in an almost rigid manner. Indeed such localization is almost certain to occur if strain softening or non-associated behavior exists, though it can be triggered even when ideal lasticity is assumed. he henomenon has a detrimental effect on the integrity of the structure and often acts as a direct recursor to structural failure. It is observed for a wide range of materials, including rocks, concrete, soils, metals, alloys and olymers, although the scale of localization henomena in the various materials may differ by some orders of magnitude: the band width is tyically less than a millimeter in metals and several meters for crystal faults in rocks. From a mechanical oint of view the driving forces behind localization henomena are material instabilities, that is, the constitutive relationshi violates the stability criterion that the inner roduct of the stress rate and the strain rate is ositive. Obviously, this inner roduct becomes negative when, in a uniaxial tension or comression test, the sloe of the homogenized axial stress - axial strain curve is negative. We call this henomenon 'strain softening'. By using the terminology 'homogenized' we refer to the fact that initial flaws and boundary conditions necessarily induce a nonhomogeneous stress state in a secimen during testing. In articular, during rogressive failure of the secimen these flaws and local stress concentrations will cause strongly inhomogeneous deformations of the secimen. he rocedure that is normally utilized to derive stress-strain 176 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

2 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 relations, namely dividing the force by the virgin load-carrying area and dividing the dislacement of the end of the secimen by the original length so as to obtain stress and strain resectively then no longer reflects what haens at a micro-level and loses hysical significance. In many hysical henomena, whose behavior is described by differential equations in a well osed manner, abrut discontinuities in the main solution variables may develo. Such henomena are frequently noted in solid mechanics where lastic failure or fracture can localize in surfaces on which discontinuities of stresses and dislacements occur. his henomenon was observed as early as 1972 in an early work describing a general aroach to finite element solution of lasticity roblems by Nayak and Zienkiewicz [1]. hey exerienced that even with a very coarse finite element discretization, quite narrow lastic zones occur under a unch when lastic softening is assumed. On the contrary, when the material is hardening the lasticity zone extends ractically throughout the whole domain. hough not noted at the time, it became clear some years later, that material softening of the general form assumed by Nayak and Zienkiewicz, will always lead to localization with an abrut discontinuity. Analysis of strain localization has been an imortant subject in the attemt to imrove the numerical simulation of structure failures. he resence of strain-softening in the constitutive laws brings great difficulties to classical (local) continuum theories [2 4]. Although it is ossible to include discontinuities in the analysis (by use of secial finite elements) this is comlex as the osition of such discontinuities has to be assumed a riori. Early efforts were devoted to obtaining failure surfaces and their associated safety factors. Later, finite element techniques allowed a more recise analysis of the stress and strain fields, using more sohisticated material models. However, the roblem is no longer mathematically well osed after the onset of localization in strain-softening materials, because local continuum allows for an infinitely small band width in shear or in front of a crack ti [5]. At the numerical level, these difficulties translate in mesh deendence of solutions. Various techniques have been imlemented to rovide a hysically accetable solution. Some imose restrictions on the constitutive moduli in the ost-localization regime, while others artificially restrict the size of finite element by comarison to the localization zone. he former is based on enriching the continuum with non-conventional constitutive relations in such way that an internal or characteristic length scale is introduced. Bifurcation analysis techniques based on the early work of homas [6] and Rice [7] were adoted by many researchers to determine the shear band localization (de Borst [8], Runneson et al. [9], Pijaudier- Cabot and Benallal [10] and Simo et al [11]). Non-local theories are the Cosserat continuum [12, 13], the higher gradient theories (riantafyllidis and Ainfantis [14]), and the integral theory or the gradient theory (de Borst and Mulhlaus [15] and Muhlhaus and Ainfantis [16]). he later is based on a mesh refinement technique using the normal, continuous, aroximations to all the variables (Pastor et al. [17], Belytschko and abbara [18], Zienkiewicz et al. [19], Lewis and Khoei [20] and Khoei and Lewis [21]). he aim of this research is to cature the localization henomena by restriction on the material roerty matrix in the ostlocalization regime based on the Cosserat continuum theory. International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

3 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 he Cosserat continuum theory, first roosed by Cosserat brothers in 1909 [22], has attracted many attentions in last two decades. he first imlementation of the theory into a finite element code was made by de Borst [23] and de Borst and his coworkers [13, 15, 24]. his theory has two main characteristics. Firstly, the rotational degree-of-freedom is taken into account in addition to translation degrees-of-freedom. In fact, the introduction of rotational degreeof-freedom leads to the existence of moment stresses (moment er area) in addition to the stresses of classic continuum. Secondly, the internal length scales is introduced in the field of constitutive equation. his arameter, which lays the most imortant role in controlling shear bandwidth, relates coule stresses to micro-curvature. ejchman and Wu [25] roosed the Cosserat continuum model as the regularization aroach to analyze strain localization roblems. An alication of adative strategy was resented by Peric et al. [26] in numerical simulation of strain localization using Cosserat continua. Iordache and Willam [27] roosed the microolar Cosserat continua to examine the regularization roerties of discontinuous bifurcation roblems. he Cosserat theory was alied to the localization behavior of associated and nonassociated materials by Carmer et al. [28]. he Cosserat theory was imlemented not only to redict the henomena of localization, but also to simulate the variety of other roblems ranging from the mechanics of rocks to owder forming rocesses. A microstructure lastic continuum was develoed by Chambon et al. [29] based on the local second gradient theory in strain localization of geomaterials. he theory was roosed by Forest et al. [30] to study the localization atterns at a crack ti in generalized single crystal lasticity. Sulem and Cerrolaza [31] alied the Cosserat theory to study the scale effect in measuring of strength arameters of rocks in indentation test. A study of localized deformation attern in granular media was erformed by Nubel and Huang [32]. A ressure-deendent elasto-lastic Cosserat continuum was resented by Li and ang [33] in modeling strain localization using a consistent return maing algorithm. A finite strain elasto-lasticity Cosserat formulation was resented by Neff [34] based on the multilicative decomosition of deformation gradient, following the earlier work of Sievert et al. [35]. he Cosserat continuum theory was emloyed by Mori [36] to owder forming rocesses due to microscoic rotations of owder articles. he main objective of resent aer is to extend the 2D Cosserat theory resented by Khoei et al. [37, 38] for strain softening lasticity to three-dimensional modeling of shear band localization. A numerical solution is develoed based on the higher order continuum model in rediction of localization henomena. he fundamental relations in Cosserat continuum are resented for 3D solid roblems. he governing equations are regularized by adding the rotational degrees-of-freedom to conventional degrees-of-freedom and an internal length scale in the field of constitutive equation. Generally, due to ath deendency of the solution in nonlinear analysis, the loading is alied in an incremental manner with an iterative linearization, using the Newton-Rahson method in each increment. 2. COSSERA CONINUUM HEORY In Cosserat theory, the indeendent rotation vector (micro-rotation) is contributed 178 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

4 zz zz xz yz zy through the governing equations, which is distinct from the macro-rotation introduced by the gradient of dislacement vector. hese micro-rotations are included as a art of degree of freedom for a oint aside of dislacement vector. hus, the strain comonents in Cosserat continuum can be defined as ij ui, j eijkk (1) ij i, j x z where e ijk is the ermutation symbol. Due to contribution of micro-rotations, the Cosserat strain tensor is no longer symmetric. However, the substitution of micro-rotations with macro-rotations results in symmetry tensor which is not the subject of this study. In above relation, k ij is the torsion-curvature, called wryness tensor, and is conjugated to the coule stress tensor m ij. In Figure 1, two different tyes of coule-stresses are resented which can be interreted as torsion and bending coules acting on the surface. hese two coule stresses have different characteristic lengths which can be related to the curvature tensor through the elasticity tensor. In Cosserat theory, it is assumed that the y zx xz zx zy Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 xx yx xy yy xx zx yx xy Figure 1.he stress and coule-stress comonents in three dimensional sace yy interaction between two articles of the body occurs through the means of traction vector t i ds and the means of moment vector m i ds. he surface forces and coules can be then exressed as follows t n, m n i ij j i ij j (2) Alying the Green s theorem with resect to body forces and body coules, the equilibrium governing equations in Cosserat continuum can be written as [39] ij, j f i ui (3) e c I ij, j ikl kl i i where f i and c i denote the body force and body coule, resectively, and r and I indicate the mass density and rotational inertia. he second equilibrium equation resents the moment of momentum, and due to this equation the symmetry of stress tensor is no longer available, i.e. s ij Ks ji, that can be comlied with its conjugate in strain sace e ij hree-dimensional Cosserat Elasticity In order to obtain an aroriate elastic modulus for isotroic Cosserat medium, two International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

5 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 assumtions need to be taken into account. First, consider that the stress and coulestress comonents do not interact together so we will have two distinct equations, each of them dealing with either stresses or coulestresses. In this case, the governing equation for stress comonents is slightly different from the classical continuum because of lack symmetry in stress tensor, as mentioned earlier. hus, the fourth order tensor that relates the stress to strain tensor has three constants, one more than its relevant classical theory, in which this additional arameter is called G c. Secondly, in order to relate the coule-stresses to curvatures we assume no interaction between bending and torsion coule stresses in constitutive equation, thus the number of indeendent constants can be reduced to two arameters. Combining the stress and coule-stress vectors [ %, ] to a single generalized vector, with % and m denoting by % [,,,,,,,, ] xx yy zz xy yx yz zy xz zx [,,,,,,,, ] xx yy zz xy yx yz zy xz zx (4) And combining the strain and curvature vectors to a single generalized strain vector as [, % ], where % and k are % [,,,,,,,, ] xx yy zz xy yx yz zy xz zx [,,,,,,,, ] xx yy zz xy yx yz zy xz zx (5) Hence, the constitutive relation for an elastic Cosserat medium can be defined as s=d e e, where D e is the linear elastic oerator defined as D e diag[ D1, D2, D2, D2, D3] where (6) 2Gc1 2Gc2 2Gc2 1 D1 2Gc2 2Gc1 2 Gc 2, c1, c Gc2 2Gc2 2Gc 1 GGc GGc D2 GGc GG c D3 diag 2 Glt, 2 Glt, 2 Glt, 2 Glb, 2 Glb, 2 Glb, 2 Glb, 2 Glb, 2Gl b (7) where l denotes the internal length arameter for both torsion and bending coule-stresses, which indicates hysically the width of shear bands in localized region. l t can be determined by the torsion of a circular cylindrical rod, and the bending of a circular cylindrical rod can be used for determination of l b. aking the internal length arameters into the curvature vector (5) to make all comonents dimensionless, the new forms of m and k can be rewritten as [ / l, / l, / l, / l, / l, / l, / l, / l, / l ] xx t yy t zz t xy b yx b yz b zy b xz b zx b [ l, l, l, l, l, l, l, l, l ] xx t yy t zz t xy b yx b yz b zy b xz b zx b 2.2. hree-dimensional Cosserat Elastolasticity (8) In order to introduce the internal length scale into the set of constitutive equations, the classical model of ressure-indeendent J 2 -elasto-lasticity is generalized by introducing additional degrees of freedom within the Cosserat continuum. Hence, an extension of classical J 2 -flow theory in Cosserat continuum is defined as [24] 1 J2 as 1 ijsij a2sijsji a3 2 ijij l (9) where s ij is the deviatoric stress tensor, l is either l t or l b, and a 1,a 2 and a 3 are the material arameters. In the absence of coule-stresses, i.e. m ij =0 and s ij =s ji, equation (9) reduces to 180 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

6 J2 ( a1a2) s ij s ij (10) 3. COMPUAIONAL ALGORIHM Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 in which we need to enforce a 1 +a 2 =1/2 in order to obtain the classical exression for J 2. hus, the yield criteria can be resented in a similar manner to classical continuum as F J (11) where denotes the yield stress, which is a function of the effective lastic strain. Equation (9) can be rewritten in a matrix format as where 0.5 (3 2) ( ) 0 1 Τ J2 P 2 where P= diag [ P1, P2, P2, P2, P3] (12) (13) 2/3 1/3 1/3 P 1 = 1/3 2/3 1/3 1/3 1/3 2/3 a1 a2 P 2 = a2 a 1 P = diag 2 a, 2 a, 2 a, 2 a, 2 a, 2 a, 2 a, 2 a, 2a (14) As in classical lasticity, the lastic strain is related to yield criteria as follow & F 3 & & P 2 ( ) (15) where l is the lastic multilier. he effective lastic strain, defined in equation (15), can be evaluated by ( ) P 3 (16) he algorithm resented here is comletely similar to classical continuum algorithm when the von-mises yield criterion is adoted. In what follows, we first resent the return maing algorithm to obtain the lastic multilier Dl at each increment, the consistent tangent matrix is then extracted to achieve the quadratic convergency rate in Newton-Rahson algorithm Return Maing Algorithm he trial stress at the beginning of the new ste can be calculated as e t 0 D (17) In the above equation it is assumed that the stress increment is related to strain increment through the elastic material roerty matrix. he new stress state at the end of the increment is the sum of the stress at the beginning of the ste and the stress increment, i.e. 0 (18) Substituting equations (15) and (17) into equation (18), we obtain the following exlicit exression 3G I+ P ( ) 1 (19) in which the only unknown arameter is Dl. If lasticity occurs (i.e., F( ), a t, 0 ) 0 correction for lastic flow must be alied. o this end the yield condition F(,, n n ) 0 where the subscrit n denotes the value of a quantity after correction for lastic flow, is develoed in a truncated aylor series around ( t, ) as [23] 0 t International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

7 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 F e F t 0 H D F(, ) (20) where H is the hardening arameter and can be defined similar to that of the classic continuum as H 0. Substituting jf/js from equation (15) into equation (20), we obtain an exlicit exression for the lastic multilier Dl as ( t, 0 ) 3G F H (21) With exression (21) for the magnitude of the lastic strain increment, the comutation for elasto-lasticity in Cosserat continuum is comleted in analogy to a conventional lasticity model. he lastic strain increment in each ste can be determined from equations (15) and (21) and then, the corrected value of stress can be obtained from equations (17) and (19). For numerical calculations the coefficients of P are chosen as a 1 =a 2 =1/4 and a 3 =1/2 [24] Consistent angent Modulus In order to achieve the full advantage of quadratic convergence rate in Newton s algorithm, the consistent tangent modulus needs to be introduced for Cosserat continuum model. he stress udate can be comuted in standard elasto-lasticity by an integral along a given ath from the initial state to the final state as follows De d 0 0 (22) where the elasto-lastic tangent modulus is defined as D e (23) Algorithmically, the stress udate is calculated as S(, ) i (24) where S is a nonlinear maing oerator deends on the numerical method of lastic strain integration within the increment De i defined as (25) he consistent (algorithmic) tangent modulus is defined as [40] D (26) in which it is in general non-symmetric and for finite, large stes differs significantly from D e. An exlicit definition of D e cons for associated J 2 flow lasticity can be obtained by differentiating from equation (19) as where i i j1 e cons 0 & D ) 0 0 d j i i S (, i ) e F & & 0, i 0, i ) e 3 D 2 P 1 e 1 P P P P = D 3/2 (27) (28) where D ) e is the so-called seudo-elastic stiffness oerator. Following the standard rocedure for derivation of D e, which emloys the consistency condition F(, ) 0, the consistent tangent modulus D e cons can be extracted as ) ) ) e e e e Dnn D D (29) cons D ) e H n D n where n=jf/js. From equation (28), it can be seen that when strain increment is very small (i.e., at limit DlY0), D e cons turns into D e. 182 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

8 Fixed on ydirection u 4.2 mm Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 z Fixed on xdirection y 4. NUMERICAL SIMULAION RESULS x In order to demonstrate the effective erformance of roosed comutational algorithm, the imlementation of Cosserat continuum model on the lastic flow in localization analysis is illustrated. he Cosserat finite element analysis is carried out for 20-noded hexahedral elements with 6 degrees-of-freedom at each node, including three rotational and three translational degrees-of-freedom. A stri in tensile with strain softening Cosserat lasticity is analyzed numerically, as shown in Figure 2. hree-dimensional numerical analysis is comared with its two-dimensional model to verify the correctness of roosed model and the caability of the algorithm in caturing shear band localization. he effect of Cosserat theory on regularization of the results is illustrated by erforming a comarison with classical analysis. he effect of internal length arameters in Cosserat continuum is shown on the shear band width and corresonding loaddislacement curves. Fixed on zdirection Figure 2.he geometry and boundary conditions of a 3D stri in tensile A stri with 60 mm width, 120 mm height and 22.5 mm deth is subjected to a uniform dislacement in x direction. he stri is restrained in x direction at the left hand side and y direction on the uer edge of fixed suort. All external nodes in x y lanes are restrained in z direction (Figure 2). he rotational degrees of freedom are free for all nodal oints. he 3D numerical simulation is erformed by using 20 noded-hexahedral elements. he material arameters chosen are as follows; E=4000 N/mm 2, v=0.49, G c =2000 N/mm 2, s Y =100 N/mm 2, and the lastic softening modulus H=-120 N/mm 2. he internal length arameter varies from 1mm to 3mm. A weak zone is assumed on the uer left edge of the late to trigger the localization mechanism. he material arameters of soft zone are similar to stri excet s Y =98 N/mm2 and v=0.30. In Figure 3, the deformed meshes of 3D (16x8x3) and 2D (16x8) models are shown for both the Cosserat and classical theories at u=4.2 mm. he 2D and 3D Cosserat analyses are erformed at different internal length International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

9 (a) Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 (b) (c) (d) Figure 3. he deformed meshes of 3D (16G8G3) and 2D (16G8) models for a stri in tensile using Cosserat theory at u=4.2mm ; a) l b =1mm, l t =0 b) l b =2mm, l t =0, c) l b =3mm, l t =0, d) he classical theory 184 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

10 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 (a) (b) (c) (d) Figure 4.he effective lastic strain contours of 3D (16x8x3) and 2D (16x8) models for a stri in tensile using Cosserat theory at u=4.2mm ; a) l b =1mm, l t =0 b) l b =2mm, l t =0, c) l b =3mm, l t =0 d) he classical theory International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

11 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 Force (KN) D-Cosserat heory (l= 1 mm) 2D-Cosserat heory (l= 2 mm) 2D-Cosserat heory (l= 3 mm) 3D-Cosserat heory (l= 1 mm) 3D-Cosserat heory (l= 2 mm) 3D-Cosserat heory (l= 3 mm) 3D Classical heory Dsilacement (mm) Figure 5.he load dislacement curves for a stri in tensile using 3D (16x8x3) and 2D (16x8) models; A comarison between the Cosserat and classical continuum analyses (a) (b) (c) (d) Figure 6.he deformed meshes of 3D (24x12x3) analyses for a stri in tensile using Cosserat theory at u=4.2mm ; a) l b =1mm, l t =0 b) l b =2mm, l t =0, c) l b =3mm, l t =0 d) he classical theory 186 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

12 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 (a) (c) Figure 7.he effective lastic strain contours of 3D (24x12x3) analyses for a stri in tensile using Cosserat theory at u=4.2mm a) l b =1mm, l t =0 b) l b =2mm, l t =0, c) l b =3mm, l t =0 d) he classical theory (b) (d) 3D-Cosserat heory ( l = 1 mm) 3D-Cosserat heory ( l = 2 mm) 3D-Cosserat heory ( l = 3 mm) 24x12x3 Classical heory Force (KN) Dislacement (mm) Figure 8.he load dislacement curves for a stri in tensile using 3D (24x12x3) model; A comarison between the Cosserat analysis with different internal lengths and classical theory International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

13 Cosserat heory (16x8x3 l = 1 mm) Cosserat heory (16x8x3 l = 2 mm) Cosserat heory (16x8x3 l = 3 mm) Cosserat heory (24x12x3 l = 1 mm) Cosserat heory (24x12x3 l = 2 mm) Cosserat heory (24x12x3 l = 3 mm) Classical heory (16x8x3) Classical heory (24x12x3) Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 arameters of l b =1,2,3 mm and l t =0. Obviously, the Cosserat analyses regularize the localization zone while the classical solution shows a sudden jum on the edge of localized region. In Figure 4, the effective lastic strain contours are resented for both 3D and 2D models, which can be comared with classical theory. he 3D effective lastic strain contours are almost identical with those obtained from 2D simulation results. he effect of internal length arameter in Cosserat theory can be clearly observed on the shear band width. As can be seen, by increasing the internal length, the shear band width becomes wider in localized zone. In this figure, a non-smooth shear band localization is evident in the effective lastic strain contour of classical theory. he variations of redicted reaction with dislacement are lotted in Figure 5 for 3D and 2D models using Cosserat theory. he 2D Cosserat results can be comared with those reorted by de Borst in reference [24]. As can be seen from this figure, with reduction of internal length arameter the load-dislacement curve converges to the classical solution, in which for zero value of internal length arameter the classical result Force (KN) Dislacement (mm) Figure 9.he load dislacement curves for a stri in tensile using 3D Cosserat analysis; A comarison between two different meshes has been achieved. In order to investigate the accuracy of numerical simulation results, a mesh of 24x12x3 hexahedral elements is emloyed for the roosed stri roblem. Figure 6 resents the deformed meshes at different internal length arameters in Cosserat theory and that obtained from classical theory. he mesh deendency is obvious in classical theory while the Cosserat theory reserves a smooth behavior in deformed shae of different internal lengths. For classical solution, the localized band width is much smaller than revious one obtained from coarse mesh (Figure 3) and the mesh distortion can be observed. In Figure 7, the effective lastic strain contours are shown for both classical and Cosserat theories. In Cosserat theory, the maximum values of effective lastic strain are similar to those obtained from coarse mesh in Figure 4, while for classical solution its maximum value is almost twice the coarse mesh. In Figure 8, the load dislacement curves are lotted for both techniques. As can be exected, by reducing the value of internal length arameter to zero the Cosserat theory converges to classical theory. In order to 188 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

14 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 demonstrate the mesh objectivity in Cosserat theory, the variations with dislacement of reaction force are resented in Figure 9 for the coarse and fine meshes. Remarkable agreement can be observed between two different meshes. 5. CONCLUSION In the resent aer, the Cosserat finite element analysis was resented for the history-deendent material of elastolasticity with secial reference to strain localization in 3D roblems. he governing equations were regularized by adding rotational degrees-of-freedom to the conventional degrees-of-freedom in Cosserat continuum theory. he fundamental equations of Cosserat elasto-lasticity was resented in three-dimensional framework, including: the kinematic relation, stressstrain relationshi, yield criterion, flow rule, hardening rule and consistent tangent matrix. he internal length arameters were introduced in the elasto-lastic constitutive matrix to control the shear bandwidth. he caability of comutational algorithm was demonstrated through the numerical analysis of a 3D stri in tensile. he Cosserat finite element analysis was carried out for 20- noded hexahedral elements with 6 degreesof-freedom at each node, including three rotational and three translational degrees-offreedom. A comarison was erformed between the classical and Cosserat theories and the effect of internal length arameter was demonstrated. he existence of mesh deendency and instability has been shown in classical continuum theory. Clearly, a finite shear bandwidth is achieved and the load-dislacement curves are uniformly converged uon different mesh sizes. [2] [3] [4] [5] [6] [7] [8] lastic stress analysis, Generalization for various constitutive relations including strain softening, Int J Num Meth Eng, 1972; 5: Bazant ZP, Belytschko B, Chang P. Continuum theory for strain softening. J Eng Mech Div, ASCE 1984; 110: Pietruszczak S, Stolle DE. Deformation of strain softening materials, Part I: Objectivity of finite element solutions based on conventional strain softening formulations. Comut Geotech, 1985; 1: Belytschko, Fish J, Englemann BE. A finite element with embedded localization zones. Comut Meth Al Mech Eng, 1988; 70: Benallal A, vergaard V. Nonlocal continuum effects on bifurcation in the lane strain tension comression test. J Mech Phys Solids, 1995; 43: homas Y. Plastic Flow and Fracture in Solids. Academic Press, New York, Rice JR. he localization of lastic deformation. in: W.. Koiter (Ed.), heoretical and Alied Mechanics, North-Holland, Amesterdam, 1977; De Borst R. Bifurcations in finite element models with a nonassociated flow law. Int J Numer Anal Meth Geomech, 1988; 12: REFERENCES [1] Nayak GC, Zienkiewicz, OC. Elasto- [9] Runneson K, Ottossen NS, Peric D. Discontinuous bifurcation of elasto- International Journal of Civil Engineerng. Vol. 4, No. 3, Setember

15 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 [10] [11] [12] [13] [14] [15] [16] [17] lastic solution at lane stress and lane strain. Int J Plasticity, 1991; 7: Pijaudier-Cabot G, Benallal A. Strain localization and bifurcation in a nonlocal continuum. Int J Solids Struct, 1993; 30: Simo JC, Oliver J, Armero F. An analysis of strong discontinuities induced by strain-softening in rateindeendent inelastic solids. Comut Mech, 1993; 12: Fleck NA, Hutchinson JW. A henomenological theory of strain gradient lasticity. J Mech Phys Solids, 1993; 41: De Borst R, Sluys LJ, Muhlhaus HB, Pamin J. Fundamental issues in finite element analysis of localization of deformation. Eng Comut, 1993; 10: riantafyllidis N, Ainfantis E. A gradient aroach to localization of deformation. I: Hyerelastic materials. J Elasticity, 1986; 16: De Borst R, Muhlhaus HB. Gradientdeendent lasticity: formulation and algorithmic asects. Int J Num Meth Eng, 1992; 35: Muhlhaus HB, Ainfantis E. A variational rincile for gradient lasticity. Int J Solids Struct, 1991; 28: Pastor M, Peraire J, Zienkiewicz OC. Adative remeshing for shear band localization roblem. Arch Al Mech, 1991; 61: [18] [19] [20] [21] [22] [23] [24] [25] [26] Belytschko, abbara M. H-adative finite element methods for dynamic roblems with emhasis on localization. Int J Num Meth Eng, 1993; 36: Zienkiewicz OC, Huang M, Pastor M. Localization roblems in lasticity using finite elements with adative remeshing. Int J Num Anal Meth Geomech, 1995; 19: Lewis RW, Khoei AR. Numerical analysis of strain localization in metal owder forming rocesses. Int J Num Meth Eng, 2001; 52: Khoei AR, Lewis RW. H-adative finite element analysis for localization henomena with reference to metal owder forming. Finite Elem Anal Des, 2002; 38: Cosserat E, Cosserat F. heorie des Cors Deformable. Hermann, Paris, De Borst R. Simulation of strain localization: a rearaisal of Cosserat continuum. Eng Comut 1991; 8: De Borst R. A generalization of J2-flow theory for olar continua. Comut Meth Al Mech Eng, 1993; 103: ejchman J, Wu W. Numerical study on atterning of shear bands in Cosserat continuum. Acta Mech, 1993; 99: Peric D, Yu J, Owen DRJ. On error estimates and adativity in elastolastic solids: alications to the 190 International Journal of Civil Engineerng. Vol. 4, No. 3, Setember 2006

16 Downloaded from ijce.iust.ac.ir at 1:42 IRS on Saturday October 13th 2018 [27] [28] [29] [30] [31] [32] [33] numerical simulation of strain localization in classical and Cosserat continua. Int J Num Meth Eng, 1994; 37: Iordache MM, Willam K. Localized failure analysis in elastolastic Cosserat continua. Comut Meth Al Mech Eng, 1998; 151: Cramer H, Findeiss R, Steinl G, Wunderlich W. An aroach to the adative finite element analysis in associated and non-associated lasticity considering localization henomena. Comut Meth Al Mech Eng, 1999; 176: Chambon R, Caillerie D, Matsuchima. Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies. Int J Solids Struct, 2001; 38: Forest S, Boubidi P, Sievert R. Strain localization atterns at a crack ti in generalized single crystal lasticity. Scrita Mater. 2001; 44: Sulem J, Cerrolaza M. Finite element analysis of the indentation test on rocks with microstructure. Comut Geotech, 2002; 29: Nubel K, Huang W. A study of localized deformation attern in granular media. Comut Meth Al Mech Eng, 2004; 193: Li X, ang H. A consistent return maing algorithm for ressuredeendent elastolastic Cosserat [34] [35] [36] [37] [38] [39] [40] continua and modeling of strain localization. Comut Struct, 2005; 83: Neff P. A finite-strain elastic lastic Cosserat theory for olycrystals with grain rotations. Int J Eng Science, 2006; 44: Sievert R, Forest S, rostel R. Finite deformation Cosserat-tye modeling of dissiative solids and its alication to crystal lasticity. J de Physique, 1998; 8: Mori K. Finite element simulation of owder forming and sintering. Comut Meth Al Mech Eng, 2006; 195: Khoei AR, abarraie AR, Gharehbaghi SA. H-adative mesh refinement for shear band localization in elastolasticity Cosserat continuum. Comm Nonl Science Numer Simul, 2005; 10: Khoei AR, Gharehbaghi SA, abarraie AR, Riahi A. Error estimation, adativity and data transfer in enriched lasticity continua to analysis of shear band localization. Al Math Model, 2006; in ress. Malvern L. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Simo JC, aylor RL. Consistent tangent oerator for rate indeendent elasto-lasticity. Comut Meth Al Mech Eng, 1985; 48: International Journal of Civil Engineerng. Vol. 4, No. 3, Setember