Supplementary Material—Selected details of the numerical
method and its implementation
The combination of boundary conditions imposed on the faces of the reduced unit cell was outlined in the body of
the paper. The reduced cell is shown in Figs. 2 and 3. Fig. 3 represents the cell for a normal band with  u = 0,
while, for  u  0, the front and back faces the unit cell take the form of a parallelogram as shown in Fig. 2. The
designations of the surfaces and edges to which the boundary conditions are applied are the same in both cases.
Based on the explanations given earlier, we illustrate the specification of the boundary conditions for the reduced
cell with respect to the initial configuration for only several of the boundary surfaces. The others are prescribed in a
similar manner. For Surface-Top, the conditions are

u x
u x

tan   , x     L  x
tan   , x    x .
u1  x10 , L20  x10 tan  u  , x30  1 x10  1 ,
2
10
, L20  x10
3
10
, L20  x10
u
30
2
30
3
20
10

tan  u    2 ,
u
30
For Edge-Top-Left/Edge-Top-Right

u L
u  L

tan   , x     L  L tan      ,
tan   , x   u  L , L  L tan   , x   0.
u1  L10 , L20  L10 tan  u  , x30  1 L10  1 ,
2
3
10
, L20  L10
10
, L20  L10
u
u
30
2
20
10
2
u
u
30
3
10
20
10
30
For Edge-Bottom-Middle, u1  0,0, x30   0 and u2  0,0, x30   0 . For Surface-Back, u3  x10 , x20 ,0  0 ; while for
Surface-Front, u3  x10 , x20 , L30   L30 3 . Note that in these equations, xi0 refer to the coordinates of the surface nodes,
and Li0 to the halve edge lengths of the unit cell (see Fig. 2) in the undeformed configuration.
The additional displacement rates of the top surface of the cell, 1 and  2 , emerge by enforcing continuity
of tractions across the (top and bottom) interfaces between the outer blocks and the cell: the expressions for the force
increments exerted by the blocks on the top surface of the unit cell, F1 and F2 , are given in (14). The following
method is used to impose this force continuity with the code ABAQUS [50]. A dummy node M, which is not
attached to the unit cell, is created with displacement rates u1M and u2M , and 1 and  2 are coupled (i.e., forced to
be the same) to the corresponding displacement rates of this dummy node, i.e. 1  u1M and  2  u2M . In order to
apply F1 and F2 , two additional nodes are created, N1 and N2, which are connected to node M via two springs
(CONN2D2 elements of the ABAQUS element library), see Fig. 3. The extra nodes can be regarded as being
attached to the upper block such that the incremental point forces applied to the top surface of the unit cell through
these springs read:
F1   AT N11 sin    c  u1N1  u1M  , F2  AT N 22 cos    c  u2N2  u2M  ,
(A-1)
where c is the stiffness of the springs. To solve the equations in (A-1) containing multi-point constraints, an
ABAQUS user subroutine is written (see MPC subroutine in [50]), by using which ABAQUS iteratively determines
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the values of u1N1 , u1M   1  , u2N2 and u2M    2  corresponding to the prescribed values of F1 and F2 . It is worth
emphasizing that only displacements boundary conditions are applied to the unit cell as shown in the equations
above; F1 and F2 are used by the ABAQUS user subroutine to simply determine the values of 1 and  2 .
It has been verified that the convergence of the results has no dependence on the stiffness c of the springs.
However, as discussed in detail in [54] the smaller the spring constant c the more rapidly the iterations converge in
ABAQUS: in this study c = 6 10-26 EL0 is used, where E is the Young’s modulus of the base material of the cell
(see Section 3), and L0 is the halve edge length of the cubic unit cell employed in this study.
A FORTRAN code has been written to run the ABAQUS calculations with a flow chart of the code given
in Fig. A-1. After evaluating the quantities needed for imposing the boundary conditions, the code writes an
ABAQUS input file and runs the ABAQUS job. Calculations are performed in several successive steps using restart
files. At each step, the FORTRAN code updates the boundary conditions, prepares an ABAQUS restart-input file,
runs the ABAQUS job, and joins the results-file (.odb file in ABAQUS terminology) of the new calculation with the
main results-file which includes the results of all the previous steps. Each new step corresponds to a succeeding
calculation with updated boundary conditions. ABAQUS further divides each step into a number of increments to
find a converged solution. In Fig. A-1, G is an integer that is used to define the step size: the change in the
predominant axial strain in the outer blocks in one step is related to G through E22  G  105 . The initial value of
the parameter G, G0 (see Fig. A-1), can be taken as large as desired (G0 = 2000 is used in this study). If ABAQUS is
not able to find 1 and  2 iteratively for the assigned G value, i.e. if the step size is too large, a new value of G is
assigned: Gnew = int (k  G), where the function “int” gives the integer part of the number and k < 1 (k = 2/3 is used
in this study).
The post-processing of the main results-file is performed with an ABAQUS-Python script (see [50]). The
evolution of the volume-averaged stresses and strains in the unit cell, the forces acting on the surfaces of the unit
cell, the orientation angle of the band are all calculated by using this script. The volume-averaged stress components
are calculated by looping over all elements:
e
 I k k

  ij v 
 ,
ij  e 1  k 1 C
V
P
where P is the total number of elements, I is the number of integration points in an element (I = 8 for C3D8, which
is the element used in this study, see [50]),  ij and v are, respectively, the local Cauchy stress and volume values at
the corresponding integration point, and V C is the total volume of the unit cell at the current configuration.
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Fig. A-1. Flow chart for the numerical scheme.
INPUT PARAMETERS
CALCULATE THE
BOUNDARY
CONDITIONS
PERFORM THE FE
CALCULATION
ITERATIONS FOR
DO NOT CONVERGE
ITERATIONS FOR
CONVERGE
FIRST LOCALIZATION:
SWITCH THE BOUNDARY
CONDITIONS
END OF CALCULATION
References
54. Tekoğlu C. 2014 Representative volume element calculations under constant stress triaxiality, Lode parameter,
and shear ratio. Int. J. Solids Struct. In press. (doi: 10.1016/j.ijsolstr.2014.09.001)
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