Supplementary Information
In this section, a detailed study is presented on how the on-axis properties change according to the
values of the geometric parameters li as well as  i where (i=1,2). Starting with the transition in the
Poisson’s ratios  i 3 and  3i  i  1, 2  from positive to negative values as 1 or  2 decrease and pass
through the value of zero, we note that whilst the on-axis Poisson’s ratios in the Ox1  Ox3 and
Ox2  Ox3 planes for loading in Ox3 direction change from positive to negative (or vice-versa) as 
changes from positive to negative values in a continuous and smooth manner, this is not the case for
the Poisson’s ratios in the same planes when loading in the Ox1 or Ox2 direction where the change
in sign of the Poisson’s ratio occurs in a discontinuous asymptotic manner.
This behaviour may be physically explained by the fact that when the structure is pulled along the
aforementioned directions, the system locks when the angle becomes zero, in analogy to what
happens in the 2D model (Evans et al. 1994). This means that further pulling in that direction will not
result in any further deformation of the structure, hence it would not be possible to change the angle
 i from a negative value to a positive value simply by pulling in the Oxi direction and similarly to
change the system from a positive value of  i to a negative value. This behaviour is also
accompanied by a tendency of the Young’s moduli Ei to approach infinity as the angles  i approach
zero, indicating locking of the structure. From a mathematical point of view, this asymptotic
behaviour can be explained by means of the equations (22-27) obtained in the analytical model for Ei
and  i 3 for  i  1, 2  which simplify to:
Ei 
X i2
d [ i ]
1

4
k
h
2
i 
V li sin 2 i
d i
(A1)

1
 i3 
d  3 
i
i 
d i

X i cos i
X 3 sin i
 X
 X
i
cot i

(A2)
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Both of these expressions contain the term d  i in the denominator which contains the term sin i  .
i 
Due to the presence of the sin i  term in the denominator of the equations for  13 and  23 , the
Poisson’s ratio varies asymptotically with the angle as i  0 . (The term sin (  i ) is positive when
0  i  
2
, negative when  
2
 i  0 and zero when i  0 .) This means that as the angle
approaches 0° (from the left) the Poisson’s ratio tends to -∞, whilst as the angle approaches 0° (from
the right) the Poisson’s ratio tends to +∞. Also, this presence of the term sin i  in the denominator
of  i 3 suggests that as the angle  i approaches ±90°, the Poisson’s ratio goes to zero. As noted
above, this behaviour is also accompanied by the tendency for Ei   as i  0 which can be
explained by the presence of the term sin
2
i  in the denominator of
positive apart when i  0 in which case sin
2
i   0 , i.e. i
Ei (the term sin 2 i  is always
goes to zero, independently from
which side, Ei tends to infinity).
In contrast to all this, the profile of the variation of the Poisson’s ratios in these same Oxi  Ox3
planes  i  1, 2  for loading in the Ox3 direction with  i , is very different since in such cases the
change in the sign of the Poisson’s ratio occurs gradually and continuously. This may be explained by
the fact that in this case, the change in the sign of the Poisson’s ratio occurs due to the term
d i  becoming zero, which term is found in the numerator of the expression for  3i and not found in
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the expression for E3 . Physically this means that one can pull the structure from a negative angle to a
positive angle in a one continuous motion without encountering any locking positions. A closer look
at these expressions will however suggest that for loading in the Ox3 direction, these systems would
still lock upon tensile loading, something which occurs when 1   2  90o . The case when
2
1   2  90o would make the denominator of E3 and that of  31 and  32 become equal to zero.
o
Here it should be noted that if only one of these angles reaches 90 , then the system would not be
locked since deformation can continue through an increase in the angle in the other plane, something
which obviously may not be observed in 2D systems. The same would happen when
1   2  90o but this time, locking would occur due to compressive loading.
If we now look at how the different parameters l1 , l2 and l3 affect the magnitude of the Young’s
moduli and the Poisson’s ratios in the Ox1  Ox3 and Ox3  Ox2 plane, we note that the parameter l3 ,
i.e. the length of the vertical rib, has an effect such that increasing its value will decrease the value of
the Young’s moduli E1 and E2 but increases the value of E3 . This latter effect may, prima facie,
appear as surprising as it means that a decrease in density will result in a stiffening of the structure,
but this may be easily explained by the fact that irrespective of the value of l3 , the extension of the
unit cell dimension in the Ox3 dimension as a result of a stress applied in Ox3 direction is
independent of the value of l3 . However, the actual dimension of the un-deformed unit cell will
increase with an increase in the value of l3 , hence the resulting strain from the same amount of stress
in the Ox3 direction will be larger for systems having small values of l3 compared to systems having
large values of l3 . On the other hand, the decrease in the magnitude of the Young’s moduli E1 and
E2 on increasing l3 is due to the fact that for a given amount of stress in the Ox1 or Ox2 direction,
an increase in l3 will result in a larger cross-sectional area of the unit cell in the orthogonal direction
to where the stress is applied with the result that forces acting on each of the ribs cause the hinging
deformation to be larger in magnitude, i.e. for a given stress in the Ox1 or Ox2 direction, the extent
of hinging increases as l3 increases, and thus E1 and E2 decrease. Such trends in the moduli may
also be observed in the 2D model and may be explained using similar arguments.
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In the case of the Poisson’s ratios, an increase of the parameter l3 will result in a decrease in the
magnitude of  13 and  23 and an increase in the magnitude of  31 and  32 . These dependencies of
the Poisson’s ratios  13 ,  23 ,  31 and  32 on l3 may be explained using similar arguments as
previously presented. Obviously, there is no effect on the magnitude of the Poisson’s ratios  12 and
 21 since these have a constant value of zero which is independent of the geometric parameters used.
The relationship between the Young’s moduli and Poisson’s ratios and the values of l1 and l2 is
slightly more complex since these parameters will also have an effect on the magnitude of the
moment acting on the hinge for a given force acting on the ribs. To explain these effects, one may
look at the simplified systems where l1  l2  l and 1   2   , in which case the expressions for the
Young’s moduli and Poisson’s ratios (equations (22-30)) simplify to  i  1, 2  :
Ei 


2k h
1
 2 2

l3  2l sin    l sin   
 i3 
l cos  
Xi
cot   
cot  
X3
l3  2l sin  
E3  2kh
 3i 
l3  2l sin  
l 4 cos3  
X 3 tan   l3  2l sin   tan  

Xi
2
l cos  
2
(A3)
(A4)
First of all, it may be noted that the simplified expressions suggest that in the case when
l1  l2  l and 1   2   , the on-axis Poisson’s ratio in the Ox1  Ox3 and Ox2  Ox3 planes are
such that as in the 2D case,  3i1   i 3 . However, in the 3D case  3i1  2 i 3 whilst in the 2D case
 xy1   yx (i.e. there is a factor of two which results from the 3D geometry).
Furthermore, these simplified equations clearly highlight the fact that an increase in the value of l
results in a decrease in the value of the Young’s moduli in all three directions. This can be explained
by the fact that an increase in the value of l will result in a larger moment being applied, something
which is caused both by an increase in the magnitude of the force acting perpendicularly to the rib
(there is an increase in the surface area onto which the stress is applied) and also due to the increased
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leverage effect that results when l increases. In the case of the Poisson’s ratio, for physically realisable
structures where l3  0 , an increase in l will result in a change of the aspect ratio which will affect its
Poisson’s ratio, as was the case for the 2D model. In fact, an increase in l will result in a bigger
relative increase in X i  i  1, 2  than in X 3 with the result that as l increases,  i 3 increases whilst
 3i decreases. In other words, as l increases, for  13 and  23 the ratio between the numerator and the
denominator of the equation increases resulting in an increase of both Poisson’s ratios whereas for the
Poisson’s ratios  31 and  32 , the reverse applies; an increase in l results in a decrease in the Poisson’s
ratio since on increasing l the ratio between the numerator and the denominator of the equation
decreases.
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