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Considering only the static deformation of a unit cell, [12] obtained the equivalent elastic moduli of the 2. The derivation of frequency-dependent elastic moduli The elements of this matrix are frequency dependent complex quantities because 6 is a function of w and

Figure 8 Considering only the static deformation of a unit cell, [12] obtained the equivalent elastic moduli of the 2. The derivation of frequency-dependent elastic moduli The elements of this matrix are frequency dependent complex quantities because 6 is a function of w and

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FIG. 1: Bottom-up approach (involving an hierarchy of anal- ysis with beam element, unit cell and lattice structure) for analysing the frequency-dependent elastic moduli of lattice materials (a) Typical representation of a hexagonal cellular structure in a dynamic environment (such as the honeycomb as part of a host structure experiencing wave propagation, vibrating structural component etc.). The curved arrows are symbolically used to indicate propagation of wave (b) One hexagonal unit cell under dynamic environment (c) A dy- namic beam element for the damped bending vibration with two nodes and four degrees of freedom) An insightful mechanics-based bottom-up framework is developed for probing the frequency dependence of lattice material microstructures. Under a vibrating condition, effective elastic modul of such microstructured materials can become negative for certain frequency values, leading to a1 unusual mechanical behaviour with a multitude of potential applications. We have derived the fun damental theoretical limits for the minimum frequency, beyond which the negative effective modul of the materials could be obtained. An efficient dynamic stiffness matrix based approach is devel oped to obtain the closed-form limits, which can exactly capture the sub-wavelength scale dynamics The limits turn out to be a fundamental property of the lattice materials and depend on certait material and geometric parameters of the lattice in a unique manner. An explicit characterization o the theoretical limits of negative elastic moduli along with adequate physical insights would accel erate the process of its potential exploitation in various engineered materials and structural system: under dynamic regime across the length-scales.
FIG. 1: Bottom-up approach (involving an hierarchy of anal- ysis with beam element, unit cell and lattice structure) for analysing the frequency-dependent elastic moduli of lattice materials (a) Typical representation of a hexagonal cellular structure in a dynamic environment (such as the honeycomb as part of a host structure experiencing wave propagation, vibrating structural component etc.). The curved arrows are symbolically used to indicate propagation of wave (b) One hexagonal unit cell under dynamic environment (c) A dy- namic beam element for the damped bending vibration with two nodes and four degrees of freedom) An insightful mechanics-based bottom-up framework is developed for probing the frequency dependence of lattice material microstructures. Under a vibrating condition, effective elastic modul of such microstructured materials can become negative for certain frequency values, leading to a1 unusual mechanical behaviour with a multitude of potential applications. We have derived the fun damental theoretical limits for the minimum frequency, beyond which the negative effective modul of the materials could be obtained. An efficient dynamic stiffness matrix based approach is devel oped to obtain the closed-form limits, which can exactly capture the sub-wavelength scale dynamics The limits turn out to be a fundamental property of the lattice materials and depend on certait material and geometric parameters of the lattice in a unique manner. An explicit characterization o the theoretical limits of negative elastic moduli along with adequate physical insights would accel erate the process of its potential exploitation in various engineered materials and structural system: under dynamic regime across the length-scales.
The frequency dependent elastic stiffness matrix of a beam element is obtained based on an efficient dynamic stiffmess method [39, 40], which is a high fidelity ap- proach at low to high frequencies compared to the con- ventional “static” finite element method. For character- izing the frequency-dependent elastic moduli, the con- ventional “static” finite element method could require very fine discretization for higher frequencies that may be practically impossible to achieve in a complex lat- tice metamaterial. The displacement field within the ele- ments can be expressed by complex frequency dependent shape functions in dynamic stiffness method, leading to a radically significant computational efficiency at higher requencies. The major advantages of this method and derivation of the frequency-dependent elastic moduli of the hexagonal lattices is provided as supplementary ma- terial. Expressions of the frequency-dependent Young’s moduli and shear modulus [41]can be obtained based on the concepts of structural mechanics using the elements of [D(w)] matrix as (refer to the supplementary material or derivation) For detailed description regarding the elements of dy- namic stiffness matrix [D(w)] involved in the above ex- pressions, refer to the supplementary material. It can be noted in the above expressions that the elements of [D(w)] matrix are functions of the frequency-dependent 1) gy tbe ae, 2
The frequency dependent elastic stiffness matrix of a beam element is obtained based on an efficient dynamic stiffmess method [39, 40], which is a high fidelity ap- proach at low to high frequencies compared to the con- ventional “static” finite element method. For character- izing the frequency-dependent elastic moduli, the con- ventional “static” finite element method could require very fine discretization for higher frequencies that may be practically impossible to achieve in a complex lat- tice metamaterial. The displacement field within the ele- ments can be expressed by complex frequency dependent shape functions in dynamic stiffness method, leading to a radically significant computational efficiency at higher requencies. The major advantages of this method and derivation of the frequency-dependent elastic moduli of the hexagonal lattices is provided as supplementary ma- terial. Expressions of the frequency-dependent Young’s moduli and shear modulus [41]can be obtained based on the concepts of structural mechanics using the elements of [D(w)] matrix as (refer to the supplementary material or derivation) For detailed description regarding the elements of dy- namic stiffness matrix [D(w)] involved in the above ex- pressions, refer to the supplementary material. It can be noted in the above expressions that the elements of [D(w)] matrix are functions of the frequency-dependent 1) gy tbe ae, 2
Since the discovery of the Young’s modulus over three centuries ago, it has been generally recognised as a pos- itive quantity. This can be mathematically explained in the light of equation 7. Since m #4 0, this implies that wh, p, > 0 for any lattice with finite-length beams. A static analysis normally used to obtain the classical Young’s modulus can be viewed as a dynamic analysis with w = 0. Therefore, according to equation 7 it is not possible to observe a negative Young’s modulus as W,,p, > 9. Only when a dynamic equilibrium is con- sidered, our results show that for cellular metamaterials the Young’s moduli can be negative, apparently contra- dicting notions established for centuries. It should be noted that a similar observation has been made in the context of acoustics metamaterials with sub-wavelength scale oscillators (see the review paper [30] for more dis- cussions). The result derived through equation 7 is the first explicit analysis towards establishing the existence of negative Young’s modulus in the context of dynamics of elastic cellular metamaterials. FIG. 2: The real and imaginary parts and the amplitude of the normalised value of EF; and E2 as a function of frequency. Here the first frequency value when the Young’s moduli be- come negative is marked by ‘+’.
Since the discovery of the Young’s modulus over three centuries ago, it has been generally recognised as a pos- itive quantity. This can be mathematically explained in the light of equation 7. Since m #4 0, this implies that wh, p, > 0 for any lattice with finite-length beams. A static analysis normally used to obtain the classical Young’s modulus can be viewed as a dynamic analysis with w = 0. Therefore, according to equation 7 it is not possible to observe a negative Young’s modulus as W,,p, > 9. Only when a dynamic equilibrium is con- sidered, our results show that for cellular metamaterials the Young’s moduli can be negative, apparently contra- dicting notions established for centuries. It should be noted that a similar observation has been made in the context of acoustics metamaterials with sub-wavelength scale oscillators (see the review paper [30] for more dis- cussions). The result derived through equation 7 is the first explicit analysis towards establishing the existence of negative Young’s modulus in the context of dynamics of elastic cellular metamaterials. FIG. 2: The real and imaginary parts and the amplitude of the normalised value of EF; and E2 as a function of frequency. Here the first frequency value when the Young’s moduli be- come negative is marked by ‘+’.
Here, w@,, represents the fundamental inflection fre- quency for shear modulus, where the shear modulus changes sign. Unlike the case of Young’s moduli, t dependent closed-form expression 3 for s he frequency- hear modulus shows a compound effect of multiple elements of the [D(w)] matrix. It is possible to obtain the a tight bound for the frequency, beyond w expression of hich the shear modulus becomes negative. Expanding the closed-form expression of Gj2 in a Taylor series, the following fun- damental inequality regarding the frequency for negative value of Giz can be derived (refer to the s material for detailed derivation) upplementary
Here, w@,, represents the fundamental inflection fre- quency for shear modulus, where the shear modulus changes sign. Unlike the case of Young’s moduli, t dependent closed-form expression 3 for s he frequency- hear modulus shows a compound effect of multiple elements of the [D(w)] matrix. It is possible to obtain the a tight bound for the frequency, beyond w expression of hich the shear modulus becomes negative. Expanding the closed-form expression of Gj2 in a Taylor series, the following fun- damental inequality regarding the frequency for negative value of Giz can be derived (refer to the s material for detailed derivation) upplementary
FIG. 3: The real and imaginary parts and the amplitude of the normalised value of Giz as a function of frequency for two different values of h/l. Bounds of w@,,, the frequency at the which the Giz becomes negative, are calculated from equation 8. Here the upper and lower bounds of frequency where Giz becomes negative for the first time are shown by ‘x’ and ‘+’ respectively.
FIG. 3: The real and imaginary parts and the amplitude of the normalised value of Giz as a function of frequency for two different values of h/l. Bounds of w@,,, the frequency at the which the Giz becomes negative, are calculated from equation 8. Here the upper and lower bounds of frequency where Giz becomes negative for the first time are shown by ‘x’ and ‘+’ respectively.
Table 1: The relationship between the boundary conditions and the shape functions for the bending vibration of beams. where v € C* is the vector of constants to be determined from the boundary conditions. For steady-state dynamic response, the displacement field within the element can be expressed as
Table 1: The relationship between the boundary conditions and the shape functions for the bending vibration of beams. where v € C* is the vector of constants to be determined from the boundary conditions. For steady-state dynamic response, the displacement field within the element can be expressed as
Eq. (14) for the above four sets of boundary conditions, one obtains
Eq. (14) for the above four sets of boundary conditions, one obtains
The stiffness and mass matrices can be obtained following the conventional variational formulation [8].
The stiffness and mass matrices can be obtained following the conventional variational formulation [8].
“igure 1: Deformed shapes and free body diagrams under the application of direct stresses and shear stress. The undeformed shapes of the hexagonal cell are indicated using blue colour for each of the loading conditions.
“igure 1: Deformed shapes and free body diagrams under the application of direct stresses and shear stress. The undeformed shapes of the hexagonal cell are indicated using blue colour for each of the loading conditions.
Xeplacing the expressions for the stiffness components from equation (24) and (30), the shear modulus c: Thus the rotation of joint B can be expressed as The strain components in direction - 1 and direction - 2 under the application of stress 0; are given by
Xeplacing the expressions for the stiffness components from equation (24) and (30), the shear modulus c: Thus the rotation of joint B can be expressed as The strain components in direction - 1 and direction - 2 under the application of stress 0; are given by
Figure 2: (a) Frequency-dependent responses of a pinned-pinned beam under the application of a unit moment at the right edge (b) Frequency dependent Young’s modulus EF; (with h/l = 1) of hexagonal lattices with 6 = 30° and ¢,, = 0.05 and ¢x = 0.002 (obtained using the analytical expressions and finite element method). A validation of the finite element (denoted by FE) model for static case (i.e. w — 0) is shown in the inset along with a convergence study for the number of unit cells. The validation is presented with respect to the static analytical expression provided by [12] (denoted by GA). a single beam element first. In figure 2(a) the responses of a pinned-pinned beam under the application of
Figure 2: (a) Frequency-dependent responses of a pinned-pinned beam under the application of a unit moment at the right edge (b) Frequency dependent Young’s modulus EF; (with h/l = 1) of hexagonal lattices with 6 = 30° and ¢,, = 0.05 and ¢x = 0.002 (obtained using the analytical expressions and finite element method). A validation of the finite element (denoted by FE) model for static case (i.e. w — 0) is shown in the inset along with a convergence study for the number of unit cells. The validation is presented with respect to the static analytical expression provided by [12] (denoted by GA). a single beam element first. In figure 2(a) the responses of a pinned-pinned beam under the application of
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