Tunable shear stiffness in a metamaterial sheet

Abstract

In this paper a metamaterial sheet constituted by a periodic pattern of square tensegrity cells (T-bar) subjected to a uniform equibiaxial pressure is studied. In particular, a minimal mass lattice is analytical determined by imposing stability and material failure conditions. Interestingly, the shear stiffness of this optimized lattice is very small in comparison to the other moduli and exhibits a linear dependence on the applied load. This behavior suggests the interesting possibility of tailoring new materials with force tunable shear modulus.

Appendices

Appendix 1

Here we give the explicit form of the relation

$$\beta = \bar{\beta }+ N_{[1]}.$$

(40)

In view of (18), (19) and (20) the incremental compressive axial force $N_{[1]}$ associated to the biaxial compression force $fl=1$ can be determined as

$$\begin{aligned} \begin{array}{l} N_{[1]}=\dfrac{4 k_e}{k_{ss1}}=\dfrac{k_e}{k_e +2 k_{ec}}=\dfrac{1}{\dfrac{\alpha (\beta -1) \sqrt{f}}{\xi \epsilon _y \sqrt{\beta l}}+1}. \end{array} \end{aligned}$$

(41)

It is should be noted that Eq. (40), based on the superposition principle, can be applied to the present non-linear problem since no geometrical stiffness terms are present in (41). By using (21) and (22), we further find

$$N_{[1]}=\dfrac{\beta \lambda }{\alpha (\beta -1) + \beta \lambda }.$$

(42)

Now, for sake of simplicity, we consider that in all the numerical examples here presented we always have $(\beta -1)< 0.004$. Thus we limit our analysis to the case $(\beta -1)\ll 1$ for which we have

$$N_{[1]}=1-\dfrac{\alpha (\beta -1)}{\lambda }+O\left( (\beta -1)^2\right).$$

(43)

Then (40) can be written as

$$\bar{\beta }= \left( \beta -1\right) +\dfrac{\alpha (\beta -1)}{\lambda }+O\left( (\beta -1)^2\right)$$

(44)

Finally, we find the allowable range for the parameter $\alpha$. This is determined by the condition

$$\bar{\beta }\le \alpha (\beta -1)$$

(45)

that, in view of (44), leads to

$$\begin{aligned} \left\{ \begin{array}{ll} \lambda >&{}1\\ \alpha \ge &{} \dfrac{\lambda }{\lambda -1}. \end{array}\right. \end{aligned}$$

(46)

Appendix 2

In Fig. 7 we plot the optimal length $l_0$ for different values of the load f and limit strain $\varepsilon _y$. In the figure, by considering a steel based metamaterial, we assume $E=207$ GPa. Moreover we consider a prestress parameter $\beta =1.02$ and circular sections, so that $\xi ^2=4/\pi$.

Fig. 7

Optimal truss dimension