Non-Equilibrium Heating of a Solid Surface by a Short-Pulse Laser: A Closed-Form Solution Including Thermo-Mechanical Coupling

Laser shortpulse heating triggers non-equilibrium energy transport in the surface region of the metallic substrate. In this case, volumetric entropy generation is governed by the non-equilibrium energy transport due to coupling of electron and lattice subsystems as well as thermomechanical coupling in the lattice system. In the present study, non-equilibrium energy transport inside the metallic substrate is modelled using an electron kinetic theory approach. Volumetric entropy generation inside the substrate material during non-equilibrium energy transport is formulated. The effect of thermomechanical coupling on the energy transport is included in the analysis. Temperature and volumetric entropy profiles are computed for silver. It is found that an electron temperature well in excess of lattice site temperatures occurs in the surface vicinity of the substrate material. Volumetric entropy generation due to electron-lattice coupling dominates the other sources of entropy generation. Thermomechanical coupling has no significant effect on the volumetric entropy generation due to a small thermal displacement of the irradiated surface, which is in the order of 10-10 m at the centre of the irradiated spot.

Laser short-pulse heating of metallic surfaces is involved with nonequilibrium energy transport in the region irradiated by a laser beam. In this case, the Fourier heating model fails to predict correct temperature rise in this region. Moreover, for completeness of analysis, the thermomechanical coupling needs to be incorporated in the governing equations. In the present study, electron kinetic theory approach is introduced to model the heating process and thermomechanical coupling is formulated and accommodated in the energy transport equation. Temperature and stress fields are computed numerically for silver. It is found that electron temperature well in excess of lattice site temperature occurs in the surface vicinity of the substrate material. Although lattice site temperature rise is low (~170°C), stress levels as high 3 2 10 8 Pa are computed in the region heated by a laser beam. The thermal expansion of the surface at the irradiated spot center reaches 0.5 nm after 4 ns of the heating period.

In this paper, laser short-pulse heating of gold-chromium two-layerassembly is considered. Since, the heating period is in the order ofpicoseconds, nonequilibrium energy transport is considered when modellingthe heating process. An electron kinetic theory is employed to derive thegoverning equations of energy transport while elasto-plastic analysis iscarried out when modelling the thermomechanical response of the substratematerials. In the analysis, thermomechanical coupling is considered and itis accommodated in the energy transport equation. A numerical method using afinite difference scheme is used to discretize the governing equations. Itis found that temperature attains considerably high values across thegold-chromium interface, which in turn results in excessive stress levels inthis region.

Laser short-pulse heating of solid surfaces results in non-equilibrium energy transport in the region irradiated by the laser beam. Owing to the large temperature gradients in the lattice subsystem, high stress levels develop in the surface region of the substrate material. In the present study, temperature and stress fields in the substrate material are presented for the case of the laser short-pulse heating of gold. Electron kinetic theory and a two-equation heating model are introduced to account for non-equilibrium energy transport during the laser heating pulse. Laser pulses exponentially decaying with time are accommodated in the simulations. It is found that lattice site temperature gradients attain high values inspite of the low magnitude of the lattice site temperature. This, in turn, results in high stress levels in the surface region of the substrate material. Thermal stress is compressive owing to high thermal strain development and low displacement of the surface.

A closed-form solution for temperature and stress fields is presented for short-pulse laser heating of solid substrate. Thermomechanical coupling is incorporated in the governing equation of energy. The temperature, thermal stress, and displacement are nondimensionalized with the appropriate parameters. The Lie point symmetry method is adopted to solve the governing equation of energy. It is found that temperature decay is sharp in the region next to the surface where laser heating occurs. Thermal stress is negative at the surface and it decays sharply with increasing distance below the surface, which is more pronounced in the early heating period.

Anisotropic functionally graded nano-beam models and closed-form solutions in plane gradient elasticity

Elastic displacement of surface due to laser picosecond pulse heating of gold

Laser short pulse heating: Influence of pulse intensity on temperature and stress fields

Entropy analysis in solids during laser shortpulse heating gives insight into the nonequilibrium energy transport process taking place in electron and lattice subsystems. In this study, laser picosecond heating of an Au-Cr two-layer assembly is considered. Entropy production due to nonequilibrium energy transport in the region irradiated by a laser beam is formulated. Thermomechanical coupling is accommodated in the analysis. The governing equations of energy transport and displacement field are solved numerically. The gold layer (0.3 w m) is situated at the top of the chromium base material. It is found that the lattice site temperature increases sharply in the chromium layer next to the Au-Cr interface. The coupling of electron lattice subsystems results in higher volumetric entropy production than that produced in lattice and electron subsystems.

SUMMARY In this study we investigate the dynamics of the region that includes Greece, the Aegean Sea and Asia Minor. In a least-squares inversion we solve for a continuous strain rate field, and corresponding velocity field, that satisfies 872 GPS data. The estimate of the geodetic strain rate field provides constraints for our dynamic analysis. Next, we separately solve the depth integrated 3-D force balance equations for depth-integrated deviatoric stresses within the lithosphere, in which body force input comes from differences in vertically integrated vertical stress, or differences in gravitational potential energy per unit area (GPE). These GPE estimates calibrate the absolute magnitudes of deviatoric stresses that are acting within the lithosphere. Further, we investigate the sensitivity of our stress field solutions by using two different crustal structure models: one from compiled crustal structure estimates obtained primarily from relatively recent seismic observations and the other from the Crust 2.0 model. In an iterative least-squares inversion we then solve for stress field boundary conditions that, when added to the contribution of deviatoric stresses associated with GPE differences, provides a best fit to the directions of principal axes and relative magnitudes of the principal axes of the rates of strain obtained in the kinematic analysis. Robust features that arise from the boundary condition solution are NNE forcing along the southern boundary east of about 33° E (0.5–1.2 × 1012 N m−1), with a rapid anticlockwise rotation of forces to the west of this, along with an outward pulling force (∼0.4 × 1012 N m−1) directed SSW along the entire Hellenic Arc segment. This force system along the Hellenic Arc can be interpreted as a result of slab rollback. The total depth integrated 3-D deviatoric stresses in the final dynamic solution provides an excellent match to the deformation indicators throughout the region, with vertically integrated stress magnitudes of order 0.5–2.5 × 1012 N m−1. We use constraints from derived stress magnitudes, together with GPS-defined scalar values of strain rate magnitude, to define bulk effective viscosities of the lithosphere. Depth-averaged effective viscosities for the entire lithosphere are high within the Black Sea, of order 0.7–3 ×1023 Pa-s, relative to surrounding continental lithosphere. North Anatolian shear zone, northern Aegean Sea and Gulf of Corinth are characterized by low depth averaged viscosities of order 1–5 ×1021 Pa-s. Deviatoric stresses from GPE differences and boundary condition effects combine in surprising ways in some regions, resulting in near total stress cancellation in areas such as the southern Aegean Sea and portions of the central Anatolian block. GPE differences combine with boundary condition effects along the eastern segment of the North Anatolian Fault (NAF) in a way that is compatible with the hypothesis that motion on the NAF was facilitated by slab detachment beneath East Anatolia and dynamic uplift of East Anatolian Plateau. In general, GPE differences play a nearly equal role as boundary condition influences in their contribution to the total deviatoric stress field. The low depth integrated deviatoric stress magnitudes throughout the region suggest that zones of active deformation are facilitated by dramatic weakening mechanisms throughout the lithosphere.

In this study, the formula to calculate the impact factor of coupling (ξ) was proposed, considering the thermo-mechanical coupling problem of friction pairs for braking device during braking. First, a correlation between independent physical field parameters was provided by analyzing the interaction parameters of stress and temperature field. Next, the thermal analysis and thermo-mechanical strong coupling analysis were completed for friction pairs using heat transfer module and structural mechanics module of COMSOL multiphysics software on the basis of the actual operating situation of a pipe belt conveyor. The distribution and variation characteristics of the temperature and stress fields were studied with comparative analysis for braking friction pairs during braking. The mathematical expression on the impact factor of coupling (ξ) depending on braking time (t) was established. Finally, the model with thermo-mechanical strong coupling analysis was verified and compared with the thermal analysis model in terms of the measured surface temperature during braking. Simulation results revealed that the impact factor of coupling (ξ) decreased with braking time (t). However, the change rate of function initially decreased and then increased with time. An inflection point existed at t = 1.5396 s in the function ξ of t. In comparison with the small fluctuation of the simulations with thermal analysis model, the simulations with thermo-mechanical strong coupling analysis model were in good agreement with the trend of experimental results, and all the maximum relative errors were less than 4 %. Thus, the simulation with thermo-mechanical strong coupling analysis was more reliable than thermal analysis. This work could provide valuable insights in solving complex multiphysics coupling analysis.

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that the numbers of symmetries and Lie brackets are reduced significantly as compared to the integer order for all dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. The Lie algebras for the integer and fractional order equations are mentioned using the subsequent computations of Lie brackets and by inspection. Interestingly, it is observed that for the one-dimensional fractional heat equation, the Lie algebra obtained by inspection of symmetries is similar to the result obtained by computation of Lie brackets, which is . The Lie algebra using the symmetries of the two-dimensional heat equation is observed to be , whereas using the Lie brackets the algebra is deduced to be . Hence, it can be concluded that the Lie algebra obtained from the nonzero Lie brackets can be conflated to the algebra which is obtained by inspection. Further, the subsequent Lie algebras are mentioned for the three and four-dimensional integer and fractional equations and the conservation laws are explicitly stated.

Solutions of heat or diffusion equations with the boundary conditions which is a dynamic random field are discussed. This kind of method can be used to obtain the description of heat equations or diffusion equations based on observed physical reality, ie ordinary differential equations, representing heat or diffusion propagation, with a boundary condition that satisfies stochastic differential equations. The heat or diffusion equations obtained from the method are the compared to the heat equation or the stochastic diffusion. The comparison is emphasized on the existence and properties of Green functions.

Thermomechanical coupling and second sound in superfluid flows

In this paper, the governing differential equation of a beam with axial force is studied using the Lie symmetry method. Considering the inhomogeneous beam and non-uniform axial load, the governing equation is a fourth-order linear partial differential equation with variable coefficients with no closed-form solution. We search for a favourable coordinate system where the governing equation has a simpler-form or a closed-form solution. A favourable coordinate transformation is found using the Lie transformation group method. The system of determining equations for the governing equation of a beam with non-uniform axial load is derived and then solved to find a favourable coordinate system dependent on the spatially variable stiffness, mass, and axial force. The class of non-uniform axially loaded beams which have a closed-form solution is determined. The fixed-free boundary condition is imposed to find the invariant closed-form solution. A comparison between the analytical solution derived by the Lie symmetry method and the numerical solution is presented. Lie symmetry analysis yields hitherto undiscovered closed-form solutions for non-uniform axially loaded beams.