ABSTRACT
We performed a series of element tests by the Granular Element Method (GEM). The algorithm of GEM is based on the stiffness relaxation method. The incremental relationship between force and relative displacement at each contact point is given by linear springs in normal and tangential directions. Contact forces obey the Coulomb friction law and no-tension rule. Peripheral particles, placed on the surface of specimen, translate without rotations as follows:
where X P and xP are the coordinates of the particle centers in initial and current states and T is the deformation gradient. As we need to give no rigid rotation to the specimen, we assume that the deformation gradient is symmetric. Then the strain, which is positive for compression, is defined by
where I is the unit tensor. The number of independent variables required for the boundary control is six, irrespective of the number of the peripheral particles. The resultant contact force applied from inner region to a peripheral particle P is
where f PC is a contact force at contact point C and C denotes the summation over the contact points. Then the Cauchy stress is given by
where V is the current volume of the specimen and P denotes the summation over peripheral particles. The data used in the simulation tests are listed in Table 1, and the numerical specimen is shown in Figure 1. The specimen was initially packed with isotropic stress: σ0 = 200 kPa. Then it was loaded along five loading directions on the π plane as shown in Figure 2. The starting points of stress-probe tests were the intersections of these loading paths and a circle whose radius is s = 60 kpa. A stress-probe is a set of incremental loading t|dσ | and reversal loading −t|dσ |. Then the recoverable and irrecoverable parts give the elastic and plastic strains respectively. To specify probe directions t= dσ/|dσ | at a stress point, we define three base vectors associated with each loading direction as shown in Figure 3.The first base vector n is the unit normal of the cone shown in the figure. The second base vector is the direction of current stress:
The third base vector is perpendicular to these vectors:
A plane of stress space in which a set of 72 stressprobes were given is called the probe plane. As shown in Figure 3, all the probe planes include the first base vector n, and the direction of probe plane is referred by the angle α from the direction of the second base vector l. The probe plane is called the reference plane
when α= 0◦; the orthogonal plane when α= 90◦. The magnitude of each stress-probe |dσ | was 1 kPa.
