The main objective of this paper is to introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures. The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to describe the deformation and mechanical behaviors of smart nanostructures subjected to thermal, and piezoelectric loadings. Because of difficulty of experimental research problems associated with the proposed theory. Therefore, we propose a new boundary element method (BEM) formulation and algorithm for the solution of such problems, which involve temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation. The computational performance of the proposed methodology has been demonstrated by using the generalized modified shift-splitting (GMSS) iteration method to solve the linear systems resulting from the BEM discretization. GMSS advantages are investigated and compared with other iterative methods. The numerical results are depicted graphically to show the size-dependent effects of thermopiezoelectricity, thermoelasticity, piezoelectricity, and elasticity theories of nanostructures. The numerical results also show the effects of the size-dependent and piezoelectric on the displacement components. The validity, efficiency and accuracy of the proposed BEM formulation and algorithm have been demonstrated. The findings of the current study contribute to the further development of technological and industrial applications of smart nanostructures.

Nanoscience is that science through which atoms can be moved and manipulated in order to obtain the properties we need in a specific field of life, as for nanotechnology, it is concerned with manufacturing devices that can be used to study the properties of nanomaterials [

In the present paper, we introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures to describe the mechanical behaviors of deformed nanostructures subjected to various types of mechanical, thermal, and piezoelectric loadings. Also, we develop a new boundary element formulation for solving the deformation problems associated with the proposed theory. The numerical results illustrate the size-dependent effects on the thermo-piezoelectric, thermoelastic, piezoelectric, and elastic smart nanostructures. The numerical results also show the effects of the length scale parameter and piezoelectric coefficient on the displacement components, and confirm the validity, efficiency and accuracy of the proposed BEM formulation and algorithm.

Consider a size-dependent thermopiezoelectric nanostructure occupies the cylindrical region

where

In the two-dimensional plane, all quantities are independent of

The rotation component is

where

The electric field components are

The strain tensor and the mean curvature vector

where

The true couple-stress vector

where the true couple-stress vector

The force-stress tensor can be decomposed into the following two parts

where

The electric field and mechanical deformation can induce polarization

where

The governing equations of size-dependent thermopiezoelectric problems in smart nanostructures subjected to various types of mechanical, thermal and piezoelectric loadings can be expressed as

The entropy balance equation

where

The force equilibrium equation

where

The moment equilibrium equation

The Gauss’s law for electric field can be expressed as

where

Substitution of

The constitutive relations of size-dependent thermopiezoelectric nanostructures can be written as:

The heat flux vector equation

The symmetric force-stress equation

where

the couple-stress equation

the electric displacement equation

Also, the force-traction vector

where

The Lamé elastic constants

where

The electric permittivity of the material can be defined as

where

The material length scale parameter used in couple stress theories can be written as

where

Now, the total force-stress tensor

Hence, the governing

where

Now, the normal heat flux

The considered boundary conditions may specify either temperature change

Displacements

Rotation

and electric potential

where

Now, we can write the boundary integral equations for temperature, displacements, rotation, and potential as follows

where the superscripts

The integral

Now, it is convenient to rewrite

where the generalized displacements

This leads to the following linear algebraic equations system

where

which can be written as

where

To illustrate the numerical calculations computed by the proposed methodology, we consider the thermopiezoelectric nanoplate with free boundary conditions on the sides, as shown in

The solid line represents the Case A that corresponds to the size-dependent thermo-piezoelectric plates

The efficiency of our proposed methodology has been demonstrated through the use of the GMSS iteration method [

Method | Iter. | CPU time | Rr | Err. | |
---|---|---|---|---|---|

GMSS | 20 | 0.0115 | 1.94e−07 | 1.46e−09 | |

SSOR | 50 | 0.0559 | 5.47e−07 | 1.69e−07 | |

PGSS | 60 | 0.0725 | 6.99e−07 | 2.48e−06 | |

GMSS | 30 | 0.0534 | 0.17e−06 | 2.03e−08 | |

SSOR | 80 | 0.2235 | 1.69e−05 | 4.49e−06 | |

PGSS | 100 | 0.3759 | 1.13e−04 | 0.55e−05 | |

GMSS | 50 | 0.1754 | 2.19e−05 | 1.78e−07 | |

SSOR | 250 | 0.7936 | 1.78e−04 | 3.59e−05 | |

PGSS | 270 | 0.8947 | 1.19e−03 | 4.56e−04 |

BEM | FEM | Analytical | ||||
---|---|---|---|---|---|---|

0.01 | 1.67878122 | 0.17597343 | 1.67878017 | 0.17597329 | 1.67878120 | 0.17597340 |

0.1 | 0.27564102 | 0.01015923 | 0.27564089 | 0.01015898 | 0.27564099 | 0.01015919 |

1.0 | 0.04096853 | 0.00281463 | 0.04096849 | 0.00281456 | 0.04096851 | 0.00281462 |

—A new theory called size-dependent thermopiezoelectricity for smart nanostructures is introduced.

—Because of the benefits of the BEM such as dealing with more complicated shapes of nanostructures and not requiring the discretization of the internal domain, also, it has low CPU time and memory. Therefore, it is versatile and efficient method for modeling of size-dependent thermopiezoelectric problems in smart nanostructures.

—A new BEM formulation is developed for solving the problems associated with the proposed theory, which involves temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation.

—The BEM is accelerated by using the GMSS, which reduces the total CPU time and number of iterations.

—The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to explain the differences between size-dependent thermopiezoelectricity, size-dependent thermoelasticity, size-dependent piezoelectricity and size-dependent elasticity theories of nanostructures.

—Numerical findings are presented graphically to show the effects of the size-dependent and piezoelectric on the displacement components.

—The computational performance of the proposed methodology has been demonstrated.

—The validity and accuracy of the proposed BEM technique have been demonstrated.

—From the proposed model that has been carried out using BEM formulation, it is possible to conclude that our proposed technique is more convenient, cost-effective, highly accurate, and has superiority over FDM or FEM.

—The proposed technique can be applied to study a wide variety of size-dependent problems in smart nanostructures subjected to mechanical, thermal and piezoelectric loadings.

—It can be concluded that our study has a wide variety of applications in numerous fields, such as electronics, chemistry, physics, biology, material science, optics, photonics, industry, military, and even medicine.

—Current numerical results for the proposed theory and its related problems, may provide interesting information for nanophysicists, nanochemists, nanobiologists, nanotechnology engineers, and nanoscience mathematicians as well as for computer scientists specializing in nanotechnology.