Continuum Mechanics and Thermodynamics最新文献

The biharmonic operator naturally emerges in the modeling of various physical phenomena and plays, in particular, a prominent role in linear elasticity. Understanding its properties is, therefore, crucial in the consolidation of mechanical science. In his 1984 article, E.H. Mansfield introduced an alternative differential operator that can replace the Laplacian in the expression of the homogeneous biharmonic equation in (mathbb {R}^2). Despite its potential for further investigations and applications, this result has remained unnoticed so far. The present work generalizes and explores some of the properties of Mansfield’s operator through analytical derivations and symbolic computations. With the proposed generalization of the operator in (mathbb {R}^n), its essential properties are established, and its expressions in five new coordinate systems are derived. The unique status of the Laplacian and Mansfield’s operator in expressing biharmonic equations in (mathbb {R}^2) is assessed. The interrelation between the kernels of the biharmonic, harmonic, and Mansfield’s operators is demonstrated. Eventually, the structure of the biharmonic terms of the classical Michell solution is clarified through the introduction of the operator. In addition to popularizing and generalizing the original work of E.H. Mansfield, the results and methodologies described in this article can initiate further research, such as the study and extensions of the generalized operator, the reexamination of classical solutions to elasticity problems, and the definition of new polyharmonic operators.

The micromechanical models of the isotropic conductive and elastic polycrystalline solid featuring a general imperfect interface have been developed. In terms of conductivity, the imperfect grain bonding conditions assume a jump of temperature and normal heat flux across the interface. In terms of elasticity, they allow for a jump in displacement and normal traction vectors. The self-consistent homogenization scheme for a polycrystalline solid is formulated in terms of the induced dipole moments and property distribution tensors. The explicit formulas for the effective conductivity and elastic moduli of polycrystalline material are derived from the multipole expansion solution to the model problems under the zero dipole moment condition of the imperfectly bonded inhomogeneity in the effective medium. As a practical application of the developed model, the conductivity and elastic stiffness of the intergranular boundaries of polycrystalline cubic boron nitride are evaluated using the laboratory test data on its macroscopic thermal conductivity and elastic moduli. The interface parameters are found to be strongly dependent on the sintering temperature and hence can serve as an indicator of the consolidation degree of a sintered polycrystalline solid.

Modelling crystallization under stretch is a key topic for fatigue design of rubber-like antivibration parts. Nevertheless, most of the academic studies consider unfilled natural rubber while the industrial materials are fully formulated compounds filled with carbon blacks and exhibit a highly dissipative visco-elastic behavior. This behavior is very useful for antivibration systems but complicates the characterization and modelling of the phase change, as the addition of fillers and additives brings in numerous additional dissipation sources and intricates the time effects on the thermomechanical response and on crystallization. In this study, we use wellresolved WAXD synchrotron measurements to perform in situ measurements under various mechanical protocols. The objective is to characterize the evolution of the triplet {strain, stress, crystallinity index}, and their derivatives, for various time and mechanical solicitations. First, classic load/unload tension tests over a range of strain rates leading to non-equilibrium cases are achieved, to serve as a reference database on the compound studied. Then, a multi-relaxation cyclic test combining static and monotonic steps is applied in order to describe the crystallization state and kinetics around a relaxed state (sometimes called “equilibrium hysteresis”). The results provide a precious database to identify or challenge the existing thermodynamic models, for conditions seldom met in the literature: fully formulated material and various mechanical loading time histories.

Motivated by the abundance of fractals within the natural world, the present study deals with a novel extension of classical thermoelasticity to fractal materials by formulating a generalized vector calculus in non-integer dimensional spaces. Fractal media, characterized by complex geometries and anomalous transport behavior, are modeled as continua with non-integer mass dimensions. We develop a comprehensive thermoviscoelastic diffusion framework that incorporates nonlocal elasticity, fractional-order derivatives, and a three-phase-lag heat conduction model. The physical system considered is an infinitely extended Kelvin–Voigt viscoelastic medium with a cylindrical cavity subjected to time-dependent thermal and chemical boundary conditions. The governing equations are transformed into the Laplace domain and solved analytically, with time-domain solutions obtained via Zakian numerical inversion. Numerical simulations reveal the significant influence of fractal geometry, nonlocal effects, and fractal operators on temperature distribution, stress evolution, and diffusion behavior. These findings offer valuable insights for the design of advanced materials and nanoscale systems where classical models fall short.

In the framework of Extended Irreversible Thermodynamics and in agreement with the second law of thermodynamics, in this paper we derive a heat-transport equation which generalizes the classical Fourier law and contains non-local and non-linear contributions related to the heat flux. In order to delve mainly into the possible influence of the non-linear effects on the heat transfer at nanoscale, only meant as the direct consequence of the non-linear terms appearing in the heat-transport equation, we focus on the longitudinal propagation of heat flux in a thin layer in steady states. The analysis is performed by also employing enhanced boundary conditions which allow to account for the phonon-boundary scattering. A comparison with the results arising from another non-local and non-linear proposal of heat-transport equation is performed as well.

This paper presents a meshless numerical approach to the heat conduction analysis of multi-directional functionally graded materials (FGMs) with non-uniform boundary conditions (BCs) and heat sources. A novel three dimensional (3D) Chebyshev spectral approximation scheme is proposed to discretize and approximate the spatially varying material properties and boundary conditions. Without meshing, a simple and unified discrete governing equation is derived for 2D and 3D transient heat conduction problems of multi-directional FGMs. A series of numerical experiments are performed to validate the convergence and accuracy of the method, considering various FGMs, non-uniform BCs and heat sources. The results converge rapidly and agree well with analytical solutions and finite element simulation results. The advantage of this method is that it can handle non-homogeneous material with non-uniform BCs and heat loads without meshing, and is independent of the laws of material gradients and variation of BCs and heat sources, and has a consistent form in 2D and 3D problems.

This study investigates the influence of non-local thermal stresses on the mechanical behavior of fiber-reinforced poro-viscoelastic structures, utilizing the Lord-Shulman (L–S) model to incorporate thermal relaxation effects. The research employs normal mode analysis to derive and analyze the governing equations and boundary conditions, which integrate the Lord-Shulman heat conduction law and constitutive relations for viscoelastic porous materials. The study explores the effects of key parameters, such as relaxation time, porosity, and viscoelastic properties, on heat propagation and the thermo-mechanical response of the structures. Analytical solutions for the governing equations are obtained through normal mode analysis, yielding explicit expressions for critical physical quantities, including stress components, displacements, and temperature distributions within the material domain. These analytical results are then numerically evaluated for a specific material, with the findings presented graphically to illustrate the influence of the studied parameters. Additionally, the predictions of the Lord-Shulman (L–S) theory are compared with those of the classical heat conduction (C–T) theory, both in the absence and presence of viscous parameters, highlighting the differences and implications of the two approaches. This work provides valuable insights into the thermo-mechanical behavior of fiber-reinforced poro-viscoelastic structures under non-local thermal stresses.

We discuss the correspondence between discrete and continuum models of one-dimensional (1D) elastic structures. We propose positive definiteness as a consistency condition for continuum models. As a result, generalized continuum models may be required, as simple (Cauchy-type) continuum models may be insufficient. This theory is illustrated by a few physical examples.

Microstructural anisotropy plays a crucial role in determining the qualitative and quantitative mechanical properties of polymer foams. Moreover, their behavior at high strain rates remains an open question due to the limited studies on the topic, which poses a limit in the design of impact absorbing foam-made components (e.g., helmets). In this work we study the influence of microstructural anisotropy on the dynamic behavior of transversely isotropic closed cells polystyrene foams. We first produce isodensity foams with different degrees of microstructural anisotropy by using the gas foaming technique. We characterize the microstructure of the cells of the produced polystyrene foams by analyzing 2D images of the microstructure and computing the statistics of the shape indicators of the cells. We then characterize their dynamical behavior on split Hopkinson pressure bars at an average strain rate of 835 s^-1. Dynamic results are compared with quasi-static ones and confirm similar findings to the literature. We report a different strain rate sensitivity for the different foams. In particular, a higher sensitivity for foams when cells are aligned with the loading direction. Therefore, the strain rate sensitivity of the foam can be associated with the microstructural distribution of cell and the strain rate sensitivity of the base material.

Using the Stroh octet formalism, we study the problem of an elliptical incompressible liquid inclusion embedded in an infinite piezoelectric matrix under a uniform remote electromechanical loading. The liquid inside the inclusion can be electrically conducting, electrically insulating or isotropic dielectric. Closed-form solutions to the problem are derived. Furthermore, real-form expressions for the internal uniform electroelastic field of hydrostatic stresses, strains, rigid body rotation, electric fields and electric displacements within the liquid inclusion are obtained in terms of the generalized Barnett-Lothe tensors and the fundamental piezoelectricity matrix for the anisotropic piezoelectric matrix.