Effect of Resonant Aggregate Parameters in Metaconcrete Thin Plates on Flexural Bandgaps: Numerical Simulations

Carvalho, André Luiz Louzeiro; Gomes, Cássio Bruno Florêncio; Santos, José Maria Campos dos; Miranda Jr., Edson Jansen Pedrosa de

doi:10.1590/1980-5373-MR-2022-0547

André Luiz Louzeiro Carvalho

Instituto Federal do Maranhão (IFMA), Programa de Pós-Graduação em Engenharia de Materiais (PPGEM), Avenida Getúlio Vargas, 4, 65030-005, São Luís, MA, Brasil.

Cássio Bruno Florêncio Gomes

Instituto Federal do Maranhão (IFMA), Programa de Pós-Graduação em Engenharia de Materiais (PPGEM), Avenida Getúlio Vargas, 4, 65030-005, São Luís, MA, Brasil.

José Maria Campos dos Santos

Universidade Estadual de Campinas (UNICAMP), Faculdade de Engenharia Mecânica (FEM), Departamento de Mecânica Computacional (DMC), Rua Mendeleyev, 200, 13083-970, Campinas, SP, Brasil.

Edson Jansen Pedrosa de Miranda Jr.^**e-mail: edson.jansen@ifma.edu.br

Instituto Federal do Maranhão (IFMA), Programa de Pós-Graduação em Engenharia de Materiais (PPGEM), Avenida Getúlio Vargas, 4, 65030-005, São Luís, MA, Brasil.

Instituto Federal do Maranhão (IFMA), Extensão Itaqui Bacanga (EIB), Departamento de Ensino (DE), Rua Afonso Pena, 174, 65010-030, São Luís, MA, Brasil.

Instituto Tecnológico Vale (ITV), Rua Professor Paulo Magalhães Gomes, Bauxita, 35400-000, Ouro Preto, MG, Brasil.

http://orcid.org/0000-0003-1100-9169

*e-mail: edson.jansen@ifma.edu.br

Material	Mass density (kg/m³)	Young’s modulus (GPa)	Poisson’s ratio
C30 Concrete³⁷37 Mehta PK, Monteiro PJM. Concreto: Microestrutura, propriedades e materiais. 2ª ed. São Paulo: IBRACON; 2014. [Portuguese].	2,500	30	0.20
Aluminum³⁸38 Qian D, Shi Z. Using PWE/FE method to calculate the band structures of the semi-infinite beam-like PCs: Periodic in z-direction and finite in x–y plane. Phys Lett A. 2017;381(17):1516-24.	2,730	77.6	0.35
A36 Steel¹⁹19 Briccola D, Cadoni E. Split Hopkinson bar tests on metaconcrete: Modeling and numerical simulations. EPJ Web Conf. 2021;250:1-9.	7,800	210	0.30
Copper³⁹39 Callister WD, Rethwisch DG. Ciência e engenharia de materiais: Uma introdução. 8ª ed. Rio de Janeiro: LTC; 2012. [Portuguese].	8,950	117	0.34
Lead¹⁴14 Kettenbeil C, Ravichandran G. Experimental investigation of the dynamic behavior of metaconcrete. Int J Impact Eng. 2018;111:199-207.	11,300	16	0.44
Silicone¹²12 Mitchell SJ, Pandolfi A, Ortiz M. Metaconcrete: Designed aggregates to enhance dynamic performance. J Mech Phys Solids. 2014;65(1):69-81.	1,100	0.001	0.47
Natural rubber¹²12 Mitchell SJ, Pandolfi A, Ortiz M. Metaconcrete: Designed aggregates to enhance dynamic performance. J Mech Phys Solids. 2014;65(1):69-81.	900	0.01	0.49
Low density polyethyene (LDPE)¹⁹19 Briccola D, Cadoni E. Split Hopkinson bar tests on metaconcrete: Modeling and numerical simulations. EPJ Web Conf. 2021;250:1-9.	1,100	0.100	0.45
Nylon¹⁴14 Kettenbeil C, Ravichandran G. Experimental investigation of the dynamic behavior of metaconcrete. Int J Impact Eng. 2018;111:199-207.	1,150	1	0.40
Granite⁴⁰40 Bauer LAF. Materiais de construção. 6ª ed. Rio de Janeiro: LTC; 2019. [Portuguese].	2,700	34	0.28

ID	Core material	Coating material	Coating thickness (mm)	Core diameter (mm)	Resonator diameter (mm)
D1	Steel	Silicone	1	8	10
D2				11	13
D3				14	16
D4				17	19

ID	Core material	Coating material	Core diameter (mm)	Coating thickness (mm)	Resonator diameter (mm)
T1	Steel	Silicone	9	0.5	10
T2				2	13
T3				3.5	16
T4				5	19

ID	Core material	Coating material	Coating thickness (mm)	Core diameter (mm)
AS	Aluminum	Silicone	1	17
SS	Steel
CS	Copper
LS	Lead

ID	Coating material	Core material	Coating thickness (mm)	Core diameter (mm)
SS	Silicone	Steel	2	9
SR	Natural rubber
SP	LPDE
SN	Nylon

ID	Core material	Coating material	Cover material	Shell material
ASAS	Aluminum	Silicone	Aluminum	Silicone
SSSS	Steel		Steel
CSCS	Copper		Copper
LSLS	Lead		Lead

Core diameter (mm)	Coating thickness (mm)	Cover thickness (mm)	Shell thickness (mm)	Total diameter (mm)
8	1	3.5	1	19

\begin{matrix} – α \frac{\partial^{2} u_{z}}{\partial t^{2}} = \frac{\partial^{2}}{\partial x^{2}} (D \frac{\partial^{2} u_{z}}{\partial x^{2}} + β \frac{\partial^{2} u_{z}}{\partial y^{2}}) \\ + 2 \frac{\partial^{2}}{\partial x \partial y} (γ \frac{\partial^{2} u_{z}}{\partial x \partial y}) + \frac{\partial^{2}}{\partial y^{2}} (D \frac{\partial^{2} u_{z}}{\partial y^{2}} + β \frac{\partial^{2} u_{z}}{\partial x^{2}}), \end{matrix}

\begin{matrix} \sum_{G_{1} =- \infty}^{+ \infty} [\begin{matrix} - ω^{2} α (G_{3} - G_{1}) + {[\frac{1}{D (G_{3} - G_{1})}]}^{-1} {(k + G_{1})}_{x}^{2} {(k + G_{3})}_{x}^{2} \\ + {[\frac{1}{β (G_{3} - G_{1})}]}^{-1} {(k + G_{1})}_{y}^{2} (_{k}^{+} \end{matrix} \\ +2 {[\frac{1}{γ (G_{3} - G_{1})}]}^{-1} {(k + G_{1})}_{x} {(k + G_{1})}_{y} {(k + G_{3})}_{x} {(k + G_{3})}_{y} \\ + {[\frac{1}{D (G_{3} - G_{1})}]}^{-1} {(k + G_{1})}_{y}^{2} {(k + G_{3})}_{y}^{2} \\ + {[\frac{1}{β (G_{3} - G_{1})}]}^{-1} {(k + G_{1})}_{x}^{2} {(k + G_{3})}_{y}^{2}] u_{k} (G_{1}) = 0, \end{matrix}

[1] *e-mail: edson.jansen@ifma.edu.br