Correction: A nonlocal analysis of thermal strain in elastic medium with two-parameter spatial nonlocal heat conduction model without energy dissipation

A nonlocal analysis of thermal strain in elastic medium with two-parameter spatial nonlocal heat conduction model without energy dissipation

Experimental results showed that the effective thermal conductivity of silicon nanowire is smaller than the bulk thermal conductivity, while that of carbon nanotube (CNT) is usually much larger than its bulk counterpart. In order to resolve this paradox, a nonlocal heat conduction model for one-dimensional materials is proposed. This nonlocal model indicates that the different heat conduction boundary conditions of silicon nanowire and CNT lead to the different behaviour of their thermal conductivities in comparison with their bulk counterparts. Furthermore, the nonlocal effect of heat flux on the surfaces of the CNT makes the thermal conductivity of the single-wall CNT more than seven orders of magnitude higher than its bulk thermal conductivity. The thermal conductivities of the single-wall and multi-wall CNTs obtained by using the nonlocal model show an agreement with the experimental ones.

The elastic medium can usually reduce the vibration of supporting structure, and the impact of the interaction on the vibration characteristics of the structure is similar to the characteristic of nonlinear energy sink. At present, the dynamic research of beam on elastic foundation considering soil motion has been paid more and more attention. According to the modified Winkler model, the finite-depth elastic medium can be considered to the nonlinear energy sink mass, and the vibration energy dissipation capacity and parameter optimization of the elastic medium supporting finite-length beam under half sine pulse are studied. The Galerkin truncation is applied to the discretization of the governing equations. The numerical solution of the beam coupling system with simple support on the elastic medium is obtained by applying the fourth-order Runge–Kutta method. Based on this, the input energy ratio of the elastic medium dissipation is investigated. Furthermore, through the analysis and optimization of targeted energy transfer and dissipation, the dissipation effect of the finite range elastic medium on the vibration energy of its supporting beam is revealed, and the optimal parameter range of the elastic medium is proved. The results show that after adjusting the elastic medium parameters by technical means, the nonlinear energy sink can absorb most of the vibration energy of the beam quickly and effectively, and the optimal energy dissipation ratio can reach 95.16[Formula: see text]. The quantitative evaluation of the energy dissipation in elastic medium within soil–structure interaction effect is realized.

Purpose The purpose of this study is to investigate the effects of dual-phase lag thermoelastic coupling on the propagation of disturbances in a thermosensitive solid sphere. By considering non-local thermal transfer and a non-local thermoelastic heat equation with memory-dependent derivatives, the study aims to provide a deeper understanding of microstructural interactions and thermal variations. This research seeks to offer valuable insights for the development of advanced materials with improved thermal and mechanical properties. Design/methodology/approach The study employs a novel non-local fractional heat conduction model that incorporates memory effects and is influenced by a magnetic field. It examines the microstructural interactions and thermal variations within a thermosensitive solid sphere. The model uses a non-local thermoelastic heat equation with memory-dependent derivatives to determine quasi-static non-local thermoelastic stress and displacement. Numerical solutions for various physical fields are derived using a numerical Laplace inversion technique, and the results are presented graphically. Findings The study finds that the proposed non-local fractional heat conduction model effectively captures the memory effects and the influence of a magnetic field on the thermal and mechanical behavior of the solid sphere. The numerical solutions reveal significant insights into the quasi-static non-local thermoelastic stress and displacement within the sphere. The graphical results demonstrate the impact of instantaneous time and other fractional thermoelastic models on the medium’s physical properties, highlighting the importance of considering non-local effects in thermoelastic analyses. Research limitations/implications The research is limited by the assumptions made in the model, such as the specific form of the memory-dependent derivatives and the idealized conditions of the solid sphere. These limitations may affect the generalizability of the findings to other materials and geometries. Future research could explore different forms of memory effects and extend the model to more complex geometries and boundary conditions. The implications of this study suggest that incorporating non-local effects and memory-dependent derivatives can significantly enhance the accuracy of thermoelastic analyses in advanced materials. Practical implications The findings of this study have practical implications for the design and development of advanced materials with enhanced thermal and mechanical properties. By understanding the effects of dual-phase lag thermoelastic coupling and non-local thermal transfer, engineers and material scientists can develop materials that better withstand thermal stresses and deformations. This research could lead to the creation of more resilient materials for use in high-temperature environments, such as those in aerospace, automotive and energy sectors. Social implications The development of advanced materials with improved thermal and mechanical properties has significant social implications. These materials can lead to safer and more efficient technologies, reducing the risk of failure in critical applications such as transportation and energy infrastructure. Additionally, the insights gained from this research can contribute to the development of sustainable materials that minimize environmental impact and enhance the longevity of products, ultimately benefiting society as a whole. Originality/value This study is original in its approach to incorporating memory effects and non-local thermal transfer into a fractional heat conduction model influenced by a magnetic field. The novel methodology provides a deeper understanding of the microstructural interactions and thermal variations within a thermosensitive solid sphere. The value of this research lies in its potential to inform the design and development of advanced materials with superior thermal and mechanical properties, offering significant benefits for various high-performance applications.

Statics and dynamics of nanorods embedded in an elastic medium: Nonlocal elasticity and lattice formulations

Non-local continuum theory helps to analyze the influence of all the points of the body at a material point. Involvement of non-local factor, i.e., size effect in heat conduction theory enhances the microscopic effects at a macroscopic level. The present work is concerned with the generalized thermoelasticity theory based on the recently introduced non-local heat conduction model with dual-phase-lag effects by Tzou and Guo. We formulate the generalized governing equations for this non-local heat conduction model and investigate a one-dimensional elastic half-space problem. Danilovskaya’s problem is taken, i.e., we assume that thermal shock is applied at the traction free boundary of the half-space. Laplace transformation is used to solve the problem and numerical method is applied to solve the problem by finding Laplace inversion through the Stehfest method. Various graphs are plotted to analyze the effects of different parameters and to mark the variation of this non-local model with previously established models.

We consider a space-time nonlocal heat conduction model with balance laws in the form of integral equations (so-called strong nonlocality). The model identifies two internal parameters---the time \ensuremath{\tau} and the space h scales of nonlocality. In going from the strong nonlocal model to its approximations of various accuracy in the form of partial differential equations, which correspond to weak nonlocality, we introduce two limiting relations between \ensuremath{\tau} and h as \ensuremath{\tau},h\ensuremath{\rightarrow}0. In the diffusion limit, which preserves the thermal diffusivity a=${\mathit{h}}^{2}$/\ensuremath{\tau}=const as \ensuremath{\tau},h\ensuremath{\rightarrow}0, the strong nonlocal model gives a hierarchy of parabolic equations with an infinite speed of heat waves. In the wave limit, which preserves the ratio v=h/\ensuremath{\tau}=const as \ensuremath{\tau},h\ensuremath{\rightarrow}0, a hierarchy of hyperbolic equations has been obtained. The hyperbolic equations imply a finite speed of heat waves. These results suggest that for diffusion (low-k) and propagative (high-k) regimes distinct models are responsible for the space-time evolution of the temperature and heat flux. The connection with phonon hydrodynamic theory and applications to other problems are discussed.

Nonlocal heat conduction in suspended graphene

Functionally graded (FG) sandwich structures stand as one of the most representative composite structures owing to the thermal resistance and the energy absorption property in non-unfiorm thermal environment. Additionally, accurately estimating thermoelastic damping (TED) is of great importance for the design of high-performance micro/nano-resonators. Nevertheless, the classical TED models fail on the micro/nano-scale structures due to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. The nonlocal heat conduction model and modified coupled stress theory are responsible for the size-dependent effects. To address this issue, present study aims to conduct the size-dependent TED model of FG sandwich microbeam resonators for TED analysis by incorporating the nonlocal dual-phase-lag (NDPL) heat conduction model and the modified coupled stress theory (MCST). It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are formulated, and then, the analytical solution is solved by complex frequency method. Exact and closed-form expressions for TED can be obtained through the complex frequency method. The results are validated, and the parameter effects of the nonlocal thermal parameter, the material length-scale parameter, the power-law index and the vibration modes on the TED are analyzed. The results show that the energy dissipation can be reduced by the size-dependent effects resulting in improving the quality factor of microstructures. It is expected that these results may provide a theoretical basis for predicating TED in the design of FG sandwich micro/nano-resonators with high quality factor in extreme heat transfer environment.

A non-classical local gradient theory of nonferromagnetic thermoelastic dielectrics is presented, incorporating both the local mass-displacement process and the heat-flux gradient effect. The process of local mass displacement is related to the changes in material microstructure. The nonlocal heat conduction law is also addressed in the model. Thus, the generalized relationship between the higher-grade heat and entropy fluxes is adopted. The gradient-type constitutive relations and governing equations are derived using the fundamental principles of continuum mechanics, non-equilibrium thermodynamics, and electrodynamics. Due to the contribution of higher-grade flux, the nonlocal law of heat conduction is obtained. The constitutive relations for isotropic materials with the corresponding additional material constants are derived. To illustrate the local gradient theory and to show the electro-thermo-mechanical coupling effect in isotropic materials, a straightforward problem is analytically solved for a layered non-piezoelectric structure under non-uniform temperature distribution. The analytical results reveal that the thermal polarization effect can also be pronounced in isotropic materials. To illustrate the model considering the effect of nonlocal heat conduction, the propagation of spherical thermoelastic harmonic waves in a homogeneous and isotropic elastic medium with non-classical heat conduction law is studied.

It is well-acknowledged by the scientific community that Eringen’s nonlocal integral theory is not applicable to nanostructures of engineering interest due to conflict between equilibrium and constitutive requirements. In this paper, a well-posed two-phase nonlocal integral elasticity with the bi-Helmholtz kernel is developed to study the size-dependent buckling response of Bernoulli-Euler beams under non-uniform temperatures. The governing equation is derived by invoking the variational principle of virtual work, and the temperature effect is equivalent to the thermal load along the axial direction, which is determined by nonlocal heat conduction. The two-phase nonlocal integral constitutive equation is transformed into a differential one equipped with four constitutive boundary conditions, and then exact solutions for the buckling loads of the beam with various boundary edges are obtained. Numerical results are validated by comparing them with those from local elasticity. Moreover, the effects of parameters related to the two-phase nonlocal elastic model and the nonlocal heat conductive model are investigated.

Weakly nonlocal heat conduction in rigid solids

This study investigates the size-dependent buckling behavior of Timoshenko nanobeams resting on Winkler-Pasternak foundations in a non-uniform thermal environment. A non-uniform temperature distribution is established through nonlocal heat conduction. Subsequently, the equivalent thermal load due to the obtained temperature distribution and boundary constraints is derived using the governing equations of the axial thermal deformation of the beams based on the stress-driven nonlocal elastic model. To obtain the critical buckling load of the nanobeams, the quadrature element method is used to numerically resolve the eigenvalue problem. In the numerical simulation section, we have presented a series of examples to analyze the effects of length-to-height ratios, nonlocal scale parameters, and Winkler-Pasternak foundation parameters on the buckling loads of the nanobeams under various boundary conditions. Moreover, we examine the effect of comprehensively considering both the elastic and thermal nonlocality on the thermal loads and finally mechanical buckling loads.

Lots of generalized heat conduction models have been developed in recent decades, such as local thermal nonequilibrium model, phase lagging model, and nonlocal heat conduction model. But no attempt was made to prove which model is better (or worse) than others, or whether there is a certain relationship between these different models. With this inspiration, we establish the nonlocal bioheat transfer equations with lagging time, and the two and three-temperature bioheat transfer equations with considering all the carrier's heat conduction effect are also constructed. Comparing the two (or three)-temperature equation model with the nonlocal bioheat transfer models with lagging time, one may obtain: the lagging time of temperature gradient τtand the nonlocal characteristic length λq in the space derivative items of heat flux have the same effect on heat transfer; when the heat transport occur among N energy carriers with considering the conduction effects of all carries, the heat transfer processes are dependent upon the high-order effect of τqN-1, τtN-1 and λt(2N-1) in nonlocal dual phase lag bioheat transfer model. This phenomenon is very important for biological and medical systems where numerous carriers may exist on the cellular level.

Investigation both actions of elastic foundation parameters and small scale effect on axisymmetric bending of annular single-layered graphene sheet resting on an elastic medium