Analysis Of Singularly Perturbed Thermoelastic Coupling Model

It is significant for thermoelastic coupling model excited by laser in engineering.The distribution of temperature field is necessary for the study of thermoelastic coupling model.We need to adjust the traditional Fourier heat conduction law due to the laser excitation time is too short(usually femtosecond).Therefore,it is necessary to establish the distribution of temperature field by using the non-Fourier heat conduction law.It is popular for the numerical solutions of temperature field model to use numerical analysis and computer simulation in the past,and it is hard to obtain its analytical solutions.So far,there are few studies for the asymptotic solution of temperature field model and the position determination of heat conductivity coefficient by adopting singularly perturbed analysis method.In this paper,the temperature field model of monolayer material is constructed by ap-plying non-Fourier heat conduction law,that is,a class of singularly perturbed hyperbolic equations with small parameters in unbounded domain.The expansion of the problem is obtained by using singularly perturbed method.The existence and uniqueness of the internal and external solutions are obtained by estimating the maximum modulus of the in-ternal and external solutions and the maximum modulus estimates of the time derivatives,and the formal asymptotic expansion of the solutions is obtained.The uniform validity of asymptotic solution is obtained by covariance estimation,and the distribution of temper-ature field in unbounded domain is obtained.Secondly,a nonlinear singularly perturbed hyperbolic equation with discontinuous coefficients is obtained by considering the one-dimensional temperature field in which the heat conduction coefficient jumps due to the sharp change of temperature.The expansion of the problem is obtained by using singular-ly perturbed method.The existence and uniqueness of the internal and external solutions are obtained by estimating the maximum modulus of the internal and external solutions and the maximum modulus of the time derivative,and the position relation of the jump of the heat conduction coefficient is determined.The joint method is used to connect the two sides of the joint where the heat conduction coefficient jumps,and the asymptotic expan-sion of the solution is obtained.Secondly,the uniform validity of asymptotic solution is obtained by covariance estimation.Extend one-dimensional problems to In this paper,the temperature field model for the jump of heat conduction coefficient is discussed.A class of nonlinear singularly perturbed hyperbolic equations with discontinuous coefficients are obtained.The expansion of the problem is obtained by using the singularly perturbed two-parameter expansion method.The existence and uniqueness of the internal and external solutions are obtained by giving the maximum modulus estimation,and the position re-lation of the jump of the heat conduction coefficient is determined by Fourier transform,and the joint between the two sides of the jump position of the heat conduction coefficient is connected by the slit method.The asymptotic expansion of the solution is obtained.Secondly,the uniform validity of the asymptotic solution is obtained by covariance esti-mation,and the distribution of the complete temperature field is obtained.Through odd The relation between the non Fourier temperature field and the Fourier temperature field is given,and the concrete state of the non Fourier temperature field is described.The main contents are as follows:1.The singularly perturbed solution for the temperature field distribution of non-Fourier.Thanks to the non-Fourier heat conduction law,we construct a temperature field model in single layer material,which is a class of singularly perturbed hyperbolic parabol-ic equations with small parameters in some unbounded domain.1.Firstly,the asymptotic expansion of this class of singularly perturbed hyperbolic equations is performed by us-ing the singularly perturbed method.The internal and external solutions of the problem are obtained,and the corresponding formal asymptotic solutions are constructed.The existence and uniqueness of internal and external solutions are obtained by estimating the solution and the existence and uniqueness theorem of classical solution.Secondly,the singularly perturbed theorem In this paper,the initial layer correction of this class of singularly perturbed hyperbolic equations is carried out,and the estimate of the deriva-tive of time is obtained.The remainder estimate is obtained and the uniform validity of the asymptotic solution is obtained,and the distribution of the temperature field in the unbounded domain is obtained.In addition,the relation and difference between the non Fourier temperature field distribution and the Fourier temperature field distribution are given,and the concrete state of the non Fourier temperature field is described.2.A class of singularly perturbed two-parameter solutions of non-Fourier tempera-ture field with jump heat conduction coefficient.The temperature field model,a class of singularly perturbed hyperbolic equations with small parameters on a bounded domain,is constructed by using the non-Fourier heat conduction law.A nonlinear singularly per-turbed hyperbolic equation with discontinuous coefficients is obtained.The asymptotic solution of the problem is obtained by using the singularly perturbed two-parameter ex-pansion method.Secondly,the position expression of heat conduction coefficient jump is determined by the method of separating variables,and the two sides of the heat con-duction coefficient jump position are jointed by the split-joint method.Then the asymp-totic expansion of the form of solution is obtained.Secondly,the uniform validity of the asymptotic solution is obtained by covariance estimation,and the distribution of the complete temperature field is obtained.3.The singularly perturbed solution of the non-Fourier temperature field with three dimensional jump in heat conduction coefficient.The temperature field model,a class of singularly perturbed hyperbolic equations with small parameters in unbounded domain,is constructed by using the non-Fourier heat conduction law.A nonlinear singularly per-turbed hyperbolic equation with discontinuous coefficients is obtained.The asymptotic solution of the problem is obtained by using the singularly perturbed two-parameter ex-pansion method.Firstly,the expansion of the problem is obtained by using the singularly perturbed method.The existence of the internal and external solutions is obtained by es-timating the solution and the existence and uniqueness theorem of the classical solution.Only Sex.Secondly,by using the singularly perturbed theory,the initial layer correction of this class of singularly perturbed hyperbolic equations is carried out,and the estimate of the derivative of time is obtained.By means of Fourier transform,the position expression of heat conduction coefficient jump is determined,and the joint of the two sides of the heat conduction coefficient jump position is connected by the slit method,and the asymp-totic expansion of the solution is obtained.Finally,the uniform validity of the asymptotic solution is obtained by covariance estimation,and the distribution of the temperature field with discontinuous heat conduction coefficient is obtained.4.Vibration of elastomer when temperature field and stress field are coupled.A thermoelastic coupling model is constructed by combining the non-Fourier temperature field with a class of equations.Because the temperature field changes rapidly,the ther-mal conductivity jumps and the thermoelastic coupling model jumps.According to the singular perturbation theory,the asymptotic solution of the problem is obtained.Second-ly,the transient displacements of discontinuous sound fields at t=0(initial layer)and t=t*(jumping layer)are solved by traveling wave method.The properties of the asymp-totic solutions of the initial layer and jumping layer are studied.At t=0,the thermal stress field appears as a boundary layer with exponential attenuation.Then,considering t*<t<T,y<?(s)and t*<t<T,y>?(s),the solutions of the thermal stress field are different due to the different thermal conductivity of the temperature field.We obtain more accurate higher-order approximate solutions by using the two-parameter expansion method of asymptotic expansion with ? at the left end and ?? at the right end.5.The one-dimensional thermoelastic coupling is extended to the three-dimensional thermoelastic coupling,which is carried out in the unbounded domain.According to the singular perturbation analysis method,the singular perturbation asymptotic expansion is carried out on both sides of the jump of the thermal conductivity,and the formal asymptot-ic solution is obtained.The transient displacement of discontinuous sound field is solved by Fourier transform.The properties of the asymptotic solutions of the initial layer and the jumping layer are studied.The higher order approximation solutions are obtained by using the two-parameter expansion method.The left end is approximated by the order of?and the right end is approximated by the order of ?.In the course of the study,we have synthetically applied the knowledge of ordinary differential equations,partial differential equations,mathematical and physical equations,nonlinear acoustics,mathematical analysis,singular perturbation theory and so on,which not only enriched the study of non-Fourier temperature field model,but also discussed the deep thermoelastic coupling model.