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traditionally , the quasi - static deformation of soils has been described by using constitutive laws .
they are empirical relations between the stress and the strain involving a certain number of material parameters , which , in the simplest models , can be measured in experimental tests @xcite .
however , the more sophisticated models involve so many parameters that their direct experimental meaning and their identification becomes impossible .    in the last years
the numerical simulations have been used as an alternative to study the behavior of soils .
usually , disks or spheres are used in order to capture the granularity of the materials @xcite .
the simplicity of their geometry allows to reduce the computer time of calculations .
however , they do not take into account the diversity of shapes of the grains in realistic materials .
a more detailed description is presented here by using randomly generated convex polygons .
the interaction between the polygons can be handled by letting the polygons interpenetrate each other and calculating the force as a function of their overlap @xcite .
this approach has been successfully applied to model different processes , like fragmentation @xcite , damage @xcite , strain localization and earthquakes @xcite .
the contribution of this work is to determine the constitutive relation of this discrete model material in the regime of quasi - static deformations .
the results show that simple mechanical laws at the grain level are able to reproduce the complex macroscopic behavior of the deformation of soils .
the details of the particle model are presented in sec .
[ model ] .
in addition to the normal contact force mentioned above , the tangential contact force law is implemented by a coulomb friction criterion , and the boundary conditions are modeled by the introduction of a flexible membrane that allows to fix the stress value . the calculation of the constitutive relations is presented in sec .
[ c - r ] .
we discuss the results in the framework of the classical theory of elasto - plasticity .
the summary and the perspectives of this work are presented in sec .
[ conclusion ] .
the polygons representing the particles of this model are generated using a simple version of the voronoi tessellation : first , we choose a random point in each cell of a regular square lattice , then each polygon is constructed assigning to each point that part of the plane that is nearer to it than to any other point .
each polygon is subjected to interparticle contact forces and boundary forces as we explain below .    when two polygons overlap , two contact points appear form the intersection of their edges .
the contact line is defined by the segment connecting these two intersection points .
the contact force is calculated as    @xmath1    here @xmath2 and @xmath3 denotes the normal and tangential unitary vectors with respect to the contact line , and @xmath4 and @xmath5 are the stiffness in the respective directions .
the overlapping length @xmath6 is the ratio between the overlap area of the polygons and the length of their contact line .
@xmath7 defines the elastic tangential displacement of the contact , that is given by the time integral starting at the begin of the contact    @xmath8    where @xmath9 is the heaviside function and @xmath10 denotes the tangential component of the relative velocity @xmath11 at the contact .
@xmath11 depends on the linear velocity @xmath12 and angular velocity @xmath13 of the particles in contact according to    @xmath14    the branch vector @xmath15 connects the center of mass of particle @xmath16 with the point of application of the contact force .
this point is taken as the center of mass of the overlapping polygon .
( [ t - d ] ) defines a limit of elasticity in the contact force . when the contact force satisfies @xmath17 the contact slips , giving rise to a plastic deformation .
the external forces are applied on the boundary through a flexible membrane which surrounds the sample .
such a membrane is calculated using an iterative algorithm , that selects the segments of the external contour whose bending angle is smaller than a threshold angle @xmath18 @xcite . on each selected segment @xmath19 , we apply an external force of the form    @xmath20    here @xmath21 and @xmath22 are the unit vectors of the cartesian coordinate system . @xmath23 and @xmath24 are the components of the stress we want to apply on the sample , as it is presented in sec .
[ c - r ] .
the contact forces and the boundary forces are inserted in newton s equations of motion which is solved numerically using a predictor - corrector algorithm . in order to enhance the stability of the numerical method and to allow for rapid relaxation ,
some viscous forces are included both in the contacts and in the boundaries :    @xmath25    the contribution of these forces is almost negligible in the quasi - static regime where velocities are small .
they are included only to reduce the acoustic waves emitted when the system goes from one equilibrium state to the other .
@xmath26 is the effective mass of the particles in contact , and @xmath27 is the mass of the particle @xmath16 in contact with the membrane .
there are three characteristic times in the simulation : the relaxation time @xmath28 , the loading time @xmath29 and the characteristic period of oscillation @xmath30 . here
@xmath31 is the mean mass of the polygons .
this leads to a minimum set of dimensionless parameters , whose selected values are shown in the table .    [
cols="^,^,^",options="header " , ]
the characterization of the macroscopic state of a granular material in static equilibrium is usually given by the cauchy stress tensor .
the derivation of this tensor over a representative volume @xcite leads to    @xmath32    the sub - scripts @xmath16 and @xmath33 in eq .
( [ cauchy ] ) denote the components of vectors and tensors . here
@xmath34 is the point of application of the boundary force @xmath35 .
this force is defined in eq .
( [ fbound ] ) .
@xmath36 is the area enclosed by the boundary .
the sum goes over all the boundary forces of the sample . inserting eq .
( [ fbound ] ) in eq .
( [ cauchy ] ) leads to    @xmath37.\ ] ]    these sums can be converted into integrals over closed loops .
then , the calculation of such integrals leads to    @xmath38 . \label{stress1}\ ] ]    thus , the stress eigensystem coincides with the cartesian coordinate system used .
we can reduce the notation introducing the _ pressure _ @xmath39 and the _ shear stress _
@xmath40 in the components of the _ stress vector _
@xmath41    = \frac{1}{2 } \left [ \begin{array}{c }
\sigma_1+\sigma_3 \\                                        \sigma_1-\sigma_3   \end{array } \right ] .
\label{stv}\ ] ]    in the same way , the incremental strain tensor can be calculated as the average of the displacement gradient over the area of the sample .
it has been shown @xcite that this average can be transformed into a sum over boundary segments of the sample    @xmath42    here @xmath43 is the @xmath44 counterclockwise rotation of the boundary segment @xmath45 .
the displacement of the segment @xmath46 is calculated from the linear displacement @xmath47 and the angular rotation @xmath48 of the polygon , according to    @xmath49    the vector @xmath50 connects the center of the segment with the center of mass of the polygon to which it belongs .
the eigenvalues @xmath51 , @xmath52 of the symmetric part of @xmath53 define the _ volumetric _ and _ shear _ component of the strain as the components of the _ incremental strain vector _ :    @xmath54    = \left [ \begin{array}{c } d\epsilon_1+d\epsilon_3 \\                                         d\epsilon_1-d\epsilon_3   \end{array } \right ] .
\label{strain}\ ] ]    from the * macro - mechanic * point of view , each state of the sample is related to a single point in the stress space , and the quasi - static evolution of the system is represented by the movement of this point in the stress space .
the resulting deformation during the transition from stress state @xmath55 to @xmath56 is given by the incremental strain @xmath57 .
in advance , let us separate the incremental stress in its _
elastic _ ( recoverable ) and _ plastic _ ( irrecoverable ) components :    @xmath58    following the procedure proposed by bardet @xcite both components can be obtained as it is shown in fig .
initially , the sample is in the stress state @xmath55 .
loading the sample from @xmath55 to @xmath56 the strain increment @xmath57 is obtained .
then the sample is unloaded , back to the original @xmath55 , and one finds a remaining strain @xmath59 , that corresponds to the plastic component of the incremental strain . for small stress increments
the unloaded stress - strain path is almost elastic .
thus , the difference @xmath60 can be taken as the elastic component of the strain .
this procedure is implemented on one sample choosing different stress directions .
[ de1 ] shows the load - unload stress paths and the corresponding strain response when an initial stress state with @xmath61 is chosen , corresponding to @xmath62 and @xmath63 , a larger stress in vertical direction .
the end of the load paths in the stress space maps into a strain envelope response @xmath64 in the strain space .
likewise , the end of the unload paths map into a plastic envelope response @xmath65 . the _ yield direction _ @xmath66 can be found from this response , as the direction in the stress space where the plastic response is maximal .
this is close to @xmath67 in this case .
the _ flow direction _
@xmath68 is given by the direction of the maximal plastic response in the strain space , which is close to @xmath69 in this example .
we found that these directions do not agree , corresponding to a _ non - associated flow rule _ as it is observed in experiments on realistic soils @xcite .
if one approximates the plastic envelope response by its projection in the flow direction , the elastic response can be written as the simple form    @xmath70    the so - called _ hardening modulus _
@xmath71 is the ratio between the modulus of the maximal plastic strain and the modulus of the incremental stress . the function @xmath72 is defined as zero if @xmath73 ; otherwise it is valued to @xmath74 .
the unit vectors @xmath75 and @xmath76 define the flow direction and the yield direction , respectively .
the vector @xmath77 defines the direction and magnitude of applied load .
2    both elastic and plastic envelope responses are calculated from different stress values .
the results are shown in fig .
[ elpl ] . the elastic response , calculated from eq .
( [ elastoplastic ] ) , has a centered ellipse as envelope response for all the cases .
since the direction of this response does not always correspond to the direction of the volumetric strain increment , the general form for the elastic stiffness must be written as    @xmath78d\tilde{\sigma}. \label{elastic}\ ] ]    here @xmath79 and @xmath80 are the classical parameters of the elasticity , i.e. the young modulus and the poisson ratio .
they are not material parameters because they depend on the stress state we take .
moreover , an additional variable @xmath81 must be included in this relation , taking into account the anisotropy of the elastic response .
a limit of isotropy is found around @xmath82 ( see fig .  [ elpl ] ) .
below of this line the parameters of elasticity are almost constants with @xmath83 . above this limit
the stiffness decreases as a result of open contacts , giving rise to an anisotropic elastic response .    in a previous work @xcite , the elasto - plastic quantities resulting form eq .
( [ plastic ] ) and eq .
( [ elastic ] ) have been evaluated as a function of the stress state . since the mechanical response of soils depends not only on the initial stress state but also on the way how this state is reached @xcite , these results are only valid in the case of monotonic load . in the classical theory of elasto - plasticity ,
the dependence of the strain response on the history of the deformation is described by the evolution of the so - called _
yield surface_. this surface encloses a hypothetical region in the stress space where only elastic deformations are possible @xcite .    we attempt to detect the yield surface by using a standard procedure proposed in experiments with sand @xcite .
[ yield ] shows this procedure : initially the sample is subject to isotropic pressure .
then the sample is loaded in the axial direction until it reaches a the yield - stress state with pressure @xmath39 and shear stress @xmath40 . since plastic deformation
is found in this stress value , the point @xmath84 can be considered as a classical yield point .
then , the classical theory assumes the existence of a yield surface containing this point . in order to explore the yield surface ,
the sample is unloaded in the axial direction until it reaches the stress point with pressure @xmath85 and shear stress @xmath86 inside the elastic regime .
then the yield surface is constructed by taking different directions in the stress space for re - loading . in each direction , the new yield point must be detected by a sharp change in the slope in the stress - strain curve , indicating plastic deformations .
[ de ] shows the strain response taking different load directions in the same sample . if the direction of the reload path is the same as that one of the original load ( @xmath87 ) , we observe a sharp decrease of stiffness when the load point reaches the initial yield point , which corresponds to the origin in fig .
however , if we take a different direction of re - loading , we find that the decrease of the stiffness with the loading becomes smooth .
since there is no straightforward way to identify those points where the yielding begins , the yield function , as it was introduced by drucker & prager @xcite in order to describe a sharp transition between the elastic and plastic regions is not consistent with our results .
the incremental elasto - plastic response of a voronoi tessellated sample of polygons has been examined in the framework of the classical theory .
the resulting constitutive relation leads to non - linear , anisotropic elasticity , where the classical parameters of elasticity , the young modulus and the poisson ratio , are not material constants .
the plastic response reflects the non - associated features of realistic soils . here
the classical analysis of drucker & prager is not applicable , because it is not possible to determine an elastic regime .
future work will be oriented to a micro - mechanically based description of these elasto - plastic features .
since the mechanical response of the granular sample is represented as a collective response of all the contacts , it is expected that the macro - mechanical response can be completely characterized by the inclusion of some field - variables , which contain the information about the micro - structural arrangements between the grains .
some statistical variables like the fabric tensor have been included as internal variables @xcite .
this description , however , does not seem to offer a complete characterization of the constitutive response .
more salient aspects , such as the yielding of the contacts , and the fluctuations of the stress inside the granular material , might offer a more complete set of internal variables for the description of the macroscopic state .
this is , in our point of view , an important challenge in the future .
we thank f. darve , p. vermeer , f. kun , and j. strm for helpful discussions and acknowledge the support of the _ deutsche forschungsgemeinschaft _ within the research group _
modellierung kohsiver reibungsmaterialen_.
we investigate the constitutive response of two - dimensional packed samples of polygons using molecular dynamics simulation . 
 the incremental elasto - plastic response is examined in the pre - failure regime . besides the young modulus and the poisson ratio , an additional parameter must be included , which takes into account the anisotropy of the elastic response . 
 the plastic deformations are described by the introduction of the yield and the flow directions . 
 these directions do not agree , which reproduces the non - associated feature of realistic soils . in order to detect the yield surface , 
 different loading - unloading - reloading tests were performed . during the reload path , it is found that the yielding develops continuously with the amplitude of loading , which does not allow to identify a purely elastic regime .    
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