Stability domain and optimal design of a metamaterial made-up of a beam lattice with diagonal cables

. The elastic stability of a plane metamaterial made up of a mesh of orthogonal rods coupled with diagonal cables is studied. The stability domain and the optimal values of the stiffness of the cables are determined.


Introduction
In this work explicit solutions are obtained for the stability domain of a bidimensional periodic metamaterial made up of a mesh of beams and cables.Fig. 1 shows the metamaterial and the chosen periodic cell (RVE).This metamaterial was already studied in [1] in the simpler case of absence of cables by assigning a priori the wave lengths of critical modes.
In [2], by introducing suitable simplifying assumptions, the wave lengths of critical modes were exactly determined by means of Floquet-Bloch theory.The main idea of the paper was to couple periodic patterns of rods with periodic patterns of extremely flexible cables to enhance the stability performance of the obtained metamaterial.Here, we extend our preceding results by considering the more general case in which vertical and horizontal rods have different stiffness.

Mechanical assumptions
The studied metamaterial is shown in Figure 1.In the following analysis we will neglect the axial and shear deformations of rods and denote by J and J µ the inertia moments of horizontal and vertical rods, respectively.Without loss of generality, we assume 1 µ < .Further, we assume 0, 0, where c J and c A are the moment of inertia and the cross-section areas of diagonal cables and min{ , } is the minimum cross section area of the rods.We distinguish (see Figure 1) two types of joints: type 1, connecting four rods and four cables, and type 0, connecting four cables.As a consequence of (1), equilibrium of a type 0 node is identically satisfied and will not be considered in the following.
The chosen RVE is composed of four rods (green in Figure 1) with length l , four internal diagonal cables of length 2 l (red in Figure 1) and four external cables of length 2l , whose stiffness is halved since these are shared with the neighboring RVEs.

Equilibrium in precritical states
In case of null shear macrostress ( 0) T = , as a consequence of (1), the axial compressive forces in the rods are , , x y x y P l P l where x Σ and y Σ are the traction macrostresses in the horizontal and vertical direction, respectively.In case of pure shear, ( 0 ) cables in compression immediately buckle (dotted lines in Fig. 2), while cables in tension exert the axial forces T N (continuous lines in Fig. 2 ) and the axial compressive forces T P occur in rods.These are given by 2 , , where ρ and e k are a distribution factor and a dimensionless measure of elastic cables stiffness (see [2] for details): Here, as usual in Eulerian stability analysis, the mechanical response is assumed linear in precritical states.Then the distribution factor is determined with reference to the unstressed initial state in which the stiffness of cables reduces only to its elastic part e k .Under the assumption (1) it is possible to superimpose the two cases above analyzed, to obtain ( ) ( ) , .
x y x T x y T y P P P T l P P P T l

Floquet-Bloch analysis
The critical displacement vector field is expressed as (see Fig. 2) where x is the node position vector, which is written as (7) The critical displacement vector is determined according to Floquet-Bloch's theorem where ω is a dimensionless wave vector, with , , ω ω π π ∈ − .Due to the axial inextensibility of rods, only the following cases are feasible: The degrees of freedom of the RVE are chosen as and its Hermitian stiffness matrix k is determined as where b k and c k are the stiffness matrices of the rods and cables, respectively.k depends on both the wave numbers 1 2 ( , ) ω ω and the applied macrostress through the parameters x q and y q : , , 2 2 which can assume either positive real values (compressed rods) or imaginary values (rods in traction).We impose that in stable or critical states the second variation of the total energy is positive semidefinite 1 ( , , ) 0. 2 The adopted procedure is detailed in [2].In case a) the stability condition (12) reduces to cot cot 0 x x y y q q q q µ + ≥ (13) and the critical state condition is always attained for 1 2 ω ω π = = .Figure 3 depicts the corresponding critical mode.Notice that, under the assumption (1), the stability condition (13) is not influenced by the stiffness of cables, which is negligible with respect to those of rods.In case b1), from the stability condition (12) we obtain ( ) q q q q q q q k k q q q q q q q q q q q q µ µ µ µ µ µ where is a dimensionless measure total stiffness of cables, which is sum of an elastic part e k , already given in ( 4), and a geometric part In Figure 3 it is shown the critical mode occurring when Similarly, in case b2), from the stability condition (12) we find ( ) x y y x x y x q q q q q q q k k q q q q q q q q q q q q µ µ µ The critical mode III, occurring in case b2) when h k k = is similar to the mode II rotated by π/2.In absence of cables, the stability conditions in cases b1) and b2) reduce to the conditions x y y q q q q q q q q q q µ µ µ

Stability domain
From the above analysis we conclude that the stability domain of the material is determined by the conditions  ( ) In order to obtain a bidimensional representation, we define the dimensionless macrostresses , such to determine the critical mode I, in which the boundaries are the continuous lines with dot marks; the case of absence of cables, in which the boundaries are the inner continuous lines (without dots), deduced by (17).Notice that, in view of (19), to each point of these stability domains correspond infinite macrostresses.
is represented.This is the minimum value ensuring the stability of the considered macrostress.The minimum value of k on the boundary of the stability domain is attained at the point A≡(-1, -1) and is equal to 2 π .Among the critical macrostresses corresponding to this point, we have equiaxial compression and pure shear.The maximum value of optimal stiffness is attained at the points (-4, ∞) and (∞, -4) and is equal to 2 8π .
Figure 6 shows the stability boundaries in the three dimensional space ( ) , ,

Figure 1 :
Figure 1: The metamaterial subjected to a generic macrostress

Figure 2 :
Figure 2: A square mesh of rods and cables a) no translations of nodes occur:

Figure 3 :
Figure 3: Critical modes I (case a) and II (case b1) boundaries of the stability domains in the plane   shown for different values of µ.Two limit cases are considered: the case in which the cables have a stiffness max( , )

Figure 4 :
Figure 4: Stability frontiers for different values of µ In Figure 5, in the case 1 µ = , the optimal dimensionless stiffness of cables in order to preserve the scale, all macrostress components have been divided by the same quantity Ex Σ .

Figure 5 :
Figure 5: Stability domain and optimal values k for 1 µ =