Thermomechanical Problems Research Articles

Tolerance modelling of free vibrations of medium thickness functionally graded plates

In many works different averaging methods based on the known Hencky-Bolle-type plate assumptions are applied to describe medium thickness functionally graded plates with a microstructure. The effect of the microstructure size plays a role in various thermomechanical problems of similar plates, cf. Brillouin [1], Baron [2], Jędrysiak [3], Cheng et al. [4]. Unfortunately, governing equations of most of averaged models do not include the effect of the microstructure size on the overall behaviour of these plates. This lack of them is corrected by using the tolerance modelling approach, cf. Woźniak and Wierzbicki [5], Woźniak et al. [6,7]. This method leads to model differential equations with slowly-varying, smooth coefficients. Hence, the new non-asymptotic model of microstructured medium thickness functionally graded plates is proposed. An example formulas of free vibration frequencies will be obtained – the fundamental lower frequency of macrovibrations and the first higher frequency of microvibrations related to the plate microstructure. A certain analysis of them will be shown for various distribution functions of material properties of functionally graded plates.

Mechanical Behavior and Damage Constitutive Model of Granite Under Coupling of Temperature and Dynamic Loading

Dynamic compression tests of Huashan granite were conducted using an improved split–Hopkinson pressure bar. The effects of the treatment temperature and strain rate on the mechanical behaviors (for example, the stress–strain curve, dynamic strength, elastic modulus, energy absorption, and failure mode) of the granite samples were explored. In addition, a statistical damage constitutive model for the rock was developed based on a Weibull distribution, and the influencing factors of the model parameters were analyzed. The results show that the enhancement effect of the strain rate on dynamic compressive strength under high temperatures still exists. However, the strain rate has no significant effect on the elastic modulus. The influences of the treatment temperature on the dynamic strength and elastic modulus are complex. There is a positive linear correlation between the energy absorbed by the sample and the incident energy. As the strain rate or incident energy increases, the failure modes of heat-treated samples change from axial splitting to pulverization. Under the same dynamic loading, an increase in the temperature can exacerbate the fragmentation degree of the sample. The proposed statistical damage constitutive model can accurately describe the effects of the treatment temperature and strain rate on the stress–strain responses of rock, and its parameters have definite physical meanings. Thus, the model is a very good tool for the analysis of thermo-mechanical coupling problems involved in deep rock mass engineering.

Thermodynamically consistent multiscale formulation of a thermo-mechanical problem with phase transformations

Fil: Said Schicchi, Diego. Leibniz Institut fur Werkstofforientierte Technologien ; Alemania

Performance Assessment of Variational Integrators for Thermomechanical Problems

Abstract Structure-preserving integrators are in the focus of ongoing research because of their distinguished features of robustness and long time stability. In particular, their formulation for coupled problems that include dissipative mechanisms is still an active topic. Conservative formulations, such as the thermo-elastic case without heat conduction, fit well into a variational framework and have been solved with variational integrators, whereas the inclusions of viscosity and heat conduction are still under investigation. To encompass viscous forces and the classical heat transfer (Fourier’s law), an extension of Hamilton’s principle is required. In this contribution we derive variational integrators for thermo-viscoelastic systems with classical heat transfer. Their results are compared for two discrete model problems vs. energy-entropy-momentum methods.

Second-order two-scale computational method for damped dynamic thermo-mechanical problems of quasi-periodic composite materials

In this paper, a novel second-order two-scale (SOTS) analysis method and related numerical algorithm are developed for damped dynamic thermo-mechanical problems of quasi-periodic composite materials. The formal SOTS solutions for these problems are constructed by the multiscale asymptotic analysis. Then we theoretically explain the importance of the SOTS solutions by the error analysis in the pointwise sense. Furthermore, the convergence result with an explicit rate for the SOTS solutions is obtained in the integral sense. In addition, a SOTS numerical algorithm is proposed to solve these problems effectively. Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed. • The SOTS solutions are constructed by the multiscale asymptotic analysis. • The importance of developing the SOTS solutions is theoretically explained. • The convergence result with an explicit rate for the SOTS solutions is obtained. • A SOTS numerical algorithm is proposed to solve these problems effectively.

Isogeometric analysis of coupled thermo-elastodynamic problems under cyclic thermal shock

The isogeometric analysis (IGA) method was extended for the solution of the coupled thermoelastodynamic equations. The dimensionless formulation was accepted in discretization of the uncoupled and coupled thermoelasticity equations and the Generalized Newmark method was used in the time integration procedure. First, the performance of the proposed method was verified against a two-dimensional benchmark example subjected to constant thermal shock with available exact analytical solutions. Then a two-dimensional half-space benchmark example under thermal shock was solved. Finally, cyclic thermal shock (CTS) loading was applied on the half-space problem. The results dedicated that IGA can be used as a suitable approach in the analysis of the general thermomechanical problems.

On renormalized solutions for thermomechanical problems in perfect plasticity with damping forces

We consider the quasi-static evolution of the thermo-plasticity model in which the evolution equation law for the inelastic strain is given by the Prandtl–Reuss flow rule. The thermal part of the Cauchy stress tensor is not linearized in the neighbourhood of a reference temperature. This nonlinear thermal part is imposed to add a damping term to the balance of the momentum, which can be interpreted as external forces acting on the material. In general, the dissipation term occurring in the heat equation is an integrable function only and the standard methods can not be applied. Combining truncation techniques and Boccardo-Gallouët approach with monotone methods, we prove an existence of renormalized solutions.

Transient Thermal Response of a Multilayered Geomaterial Subjected to a Heat Source

The analysis of layered geomaterials thermo-mechanical coupling response is important in design and analysis of a nuclear waste repository. Conventional analytical approaches usually have some intrinsic faults of ill-conditioned matrices for thick layers or accumulative numerical errors for a great quantity of layers for containing positive exponential functions. This paper presents a stable computational approach with no positive exponential functions to study the transient thermal responses of a layered geomaterial subjected to a heat source. Based on the governing equations of thermo-mechanical coupling problems, the analytical layer elements for a finite layer and a half–space are derived by utilizing Laplace and Hankel transforms. The analytical layer elements are then assembled into the total stiffness matrix and solved in the transformed domain. Finally, the Laplace–Hankel transform inversion is employed to obtain an actual solution in the physical domain. Numerical computations indicate that the difference of geomaterial mass properties between layers shows obvious effects on thermal response of the layered system, and these results can be served as a benchmark solution for future analyses of stratified material.

CFD–CHT calculation method using Buckingham Pi-Theorem for complex fluid–solid heat transfer problems with scattering boundary conditions

A three-dimensional CFD–CHT simulation method is presented and validated with a turbocharged single cylinder SI engine. Various ignition time and lambda strategies as well as variations of boost pressure are investigated with regard to cycle averaged component temperatures. This complements existing published works which experimentally studied crank angle resolved heat fluxes or temperature swings rather than averaged temperatures. Cyclical fluctuations in the pressure curves were measured and processed statistically using probability density functions for the heat transfer coefficient and the cylinder gas temperature. The corresponding joint probability density function considers their strong correlation. The interpretation as random variables enables a time-scale separation with a low-pass filter function. The thermomechanical problem of heat transfer is addressed with simplified models according to Woschni, Eichelberg, and Hohenberg. Previous investigations primarily focused on their predictive quality of instantaneous in-cylinder heat fluxes. In this paper, their effect on cycle averaged component temperatures is investigated and the corresponding different sensitivities to specific engine settings are presented and compared with measurements. It is shown that, by choosing the right model, the suggested simulation approach is an alternative to prevailing experimental methods in temperature analysis: all thermodynamic variations examined are in good agreement with theoretical predictions.

Serrated chip formation mechanism analysis using a modified model based on the material defect theory in machining Ti-6Al-4 V alloy

The serrated chip is a remarkable characteristic in the machining of the Ti-6Al-4 V alloy, but its formation mechanism is not completely understood. Finite element modeling is an effective method for studying the complex thermo-mechanical problems that occur during the machining process. The constitutive model plays a key role in the simulation, and Johnson-Cook (JC) model is the most commonly used model. However, the JC model neglects the coupled effects of the strain, strain rate, and temperature, and the material is assumed to be homogeneous. Thus, the JC model is applicable for studying macroscale machining. In fact, cutting is a complex thermo-mechanical process, and there are material defects in the Ti-6Al-4 V alloy. First, a modified JC model is proposed based on the THAN law, which is derived from material defects considering the coupled effects of the strain, strain rate, temperature, and strain softening. Then, the serrated chip morphology and cutting force predicted by the JC-THAN model are compared with experimental data. Finally, the serrated chip formation mechanism from macroscale machining to microscale machining is analyzed considering the material defects based on the JC-THAN model, and the dominated formation mechanism at each scale is found.

Semi-analytical solution of three-dimensional steady state thermoelastic contact problem of multilayered material under friction heating

Multilayer coatings offer a possibility to design the surface according to the different requirements, which attracts intense attentions from engineering and research community. The coupled thermo-mechanical contact problem of a multilayered material is of great interest. This paper firstly derives the frequency response functions (FRF) of thermoelastic fields through thermoelastic governing equations. The unknown coefficients in the FRFs are assembled in a linear system of matrix equations according to the thermal and mechanical loadings on surface and continuity condition of heat flux, temperature, displacement and stresses at each interface; then the coefficients are solved and expressed recursively. Based on the closed-form solution of FRFs, a fast semi-analytical method (SAM) is developed to solve the three-dimensional thermoelastic contact problem involved in arbitrary multilayered materials. There are no limits on the number or the thickness of layers, and material parameters can be varied arbitrarily. The present model is verified by literature and FEM and shows a high robustness and efficiency. Thermoelastic contact of multilayered materials with different coating designs under friction heating is further studied and the thermal effect is explored.

Thermo-mechanical strong coupling analysis on braking device of pipe belt conveyor

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.

A coupled thermo-mechanical bond-based peridynamics for simulating thermal cracking in rocks

A coupled thermo-mechanical bond-based peridynamical (TM-BB-PD) method is developed to simulate thermal cracking processes in rocks. The coupled thermo-mechanical model consists of two parts. In the first part, temperature distribution of the system is modeled based on the heat conduction equation. In the second part, the mechanical deformation caused by temperature change is calculated to investigate thermal fracture problems. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. Two benchmark examples, i.e., steady-state heat conduction and transient heat conduction with deformation problem, are performed to illustrate the correctness and accuracy of the proposed coupled numerical method in dealing with thermo-mechanical problems. Moreover, two kinds of numerical convergence for peridynamics, i.e., m- and $$\delta $$ -convergences, are tested. The thermal cracking behaviors in rocks are also investigated using the proposed coupled numerical method. The present numerical results are in good agreement with the previous numerical and experimental data. Effects of PD material point distributions and nonlocal ratios on thermal cracking patterns are also studied. It can be found from the numerical results that thermal crack growth paths do not increases with changes of PD material point spacing when the nonlocal ratio is larger than 4. The present numerical results also indicate that thermal crack growth paths are slightly affected by the arrangements of PD material points. Moreover, influences of thermal expansion coefficients and inhomogeneous properties on thermal cracking patterns are investigated, and the corresponding thermal fracture mechanism is analyzed in simulations. Finally, a LdB granite specimen with a borehole in the heated experiment is taken as an application example to examine applicability and usefulness of the proposed numerical method. Numerical results are in good agreement with the previous experimental and numerical results. Meanwhile, it can be found from the numerical results that the coupled TM-BB-PD has the capacity to capture phenomena of temperature jumps across cracks, which cannot be captured in the previous numerical simulations.

Application of an enriched FEM technique in thermo-mechanical contact problems

In this paper, an enriched FEM technique is employed for thermo-mechanical contact problem based on the extended finite element method. A fully coupled thermo-mechanical contact formulation is presented in the framework of X-FEM technique that takes into account the deformable continuum mechanics and the transient heat transfer analysis. The Coulomb frictional law is applied for the mechanical contact problem and a pressure dependent thermal contact model is employed through an explicit formulation in the weak form of X-FEM method. The equilibrium equations are discretized by the Newmark time splitting method and the final set of non-linear equations are solved based on the Newton–Raphson method using a staggered algorithm. Finally, in order to illustrate the capability of the proposed computational model several numerical examples are solved and the results are compared with those reported in literature.

Space\u2013time POD based computational vademecums for parametric studies: application to thermo-mechanical problems

Standard numerical simulations for optimization or inverse identification of welding processes remain costly and difficult due to their multi-parametric aspect and inherent complexity. The aim of this paper is to propose a non-intrusive strategy for building computational vademecums dedicated to real-time simulations of nonlinear thermo-mechanical problems. There is in essence, a set of precomputed space–time parametric solutions (snapshots), selected by an appropriate approach in the parameter space and stored in memory as quasi-optimal reduced bases (RBs) provided by the proper orthogonal decomposition method. Once the RBs are obtained, the computational vademecums can be used online and provide real-time space–time transient nonlinear thermo-mechanical solutions for any desired parameter value. The contributions of the paper consist in a space–time RBs interpolation approach with the Grassmann manifolds method, and a localized multigrid selection method that allows an automatic selection of snapshots in the parameter areas of interest for a given level of accuracy. As application, the welding simulation is considered with a transient non-linear thermo-mechanical model using the finite element method. It is shown that the moving frame allows an optimal design of the RBs. A good efficiency of the proposed approach is demonstrated. Computational vademecums can be used for optimization or inverse identification problems of welding.

Thermodynamics of viscoelastic rate-type fluids with stress diffusion

We propose thermodynamically consistent models for viscoelastic fluids with a stress diffusion term. In particular, we derive variants of compressible/incompressible Maxwell/Oldroyd-B models with a stress diffusion term in the evolution equation for the extra stress tensor. It is shown that the stress diffusion term can be interpreted either as a consequence of a nonlocal energy storage mechanism or as a consequence of a nonlocal entropy production mechanism, while different interpretations of the stress diffusion mechanism lead to different evolution equations for the temperature. The benefits of the knowledge of the thermodynamical background of the derived models are documented in the study of nonlinear stability of equilibrium rest states. The derived models open up the possibility to study fully coupled thermomechanical problems involving viscoelastic rate-type fluids with stress diffusion.

A Reduced Model of a Thermo-Elastic Nonlinear Circular Plate

Nonlinear vibrations of a circular plate subjected to mechanical and thermal loadings are presented in the paper. A model of the plate is based on the extended Mindlin theory, taking into account nonlinear geometrical terms and acting heat uniformly distributed along the plate span. The dynamics of a coupled thermo-mechanical problem is reduced from a set of partial differential equations to ordinary differential equations. Considering oscillations around the first natural frequency just one mode reduction is proposed. The analysis shows that elevated temperature shifts the resonance curve and new post-buckling oscillations arise. Depending on initial conditions for the post-buckling state various scenarios of bifurcations take place and transient irregular oscillations may occur. The proposed one degree of freedom model shows a good agreement with response of the model based on three or five-modes reduction.

Development of generalized interpolation material point method for simulating fully coupled thermomechanical failure evolution

Based on the conservation laws of mass, momentum and energy, the generalized interpolation material point (GIMP) method is further developed in this paper for fully coupled thermomechanical problems, with applications to model-based simulation of failure evolution. The fully coupled thermomechanical GIMP method (CTGIMP) considers the effects of both the temperature on deformation and the deformation on temperature. The discrete governing equations are formulated for both thermomechanical motion and heat conduction. A staggered solution scheme is designed to solve the coupled governing equations with explicit time integration. Representative examples with analytical solutions are presented to verify the proposed CTGIMP for coupled thermomechanical analyses. The CTGIMP is then used for model-based simulation of vibration-induced thermoplasticity, and of failure evolution in a snowy slope under the heat convection condition to demonstrate the potential of the proposed procedure in engineering applications.

Model order reduction for thermomechanical problems including radiation

AbstractThermal field problems including heat exchange by radiation lead to nonlinear system equations with a high number of inputs and outputs as radiation heat fluxes correspond to the fourth power of the temperature and thermal loads are distributed over the whole surface. In an alternative approach presented here, radiation is defined as a part of the load vector. Thus, the system matrices are constant. Furthermore, loads changing synchronously during operation are grouped into one column of the input matrix and load vector snapshots are used to consider the radiation heat fluxes. Hence, the Krylov Subspace Method can be applied to significantly reduce the system dimension and the computation times allowing transient thermal parameter studies. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermomechanical Extended Layerwise Method for laminated composite plates with multiple delaminations and transverse cracks

A Thermomechanical Extended Layerwise Method (TELM) is developed for the laminated composite plates with multiple delaminations and transverse cracks. The discontinuity of displacements and temperature induced by multiple delaminations is simulated by strong discontinuous function while the discontinuity of strain and temperature gradient between dissimilar layers is modeled by a weak discontinuous function. Transverse cracks are modeled using classical Extended Finite Element Method (XFEM). The coupled thermomechanical variational principle is employed to derive the Euler equations and the discrete forms. Since the displacement and temperature fields are solved simultaneously, a fully coupled time integration method is developed based on the Newmark integration algorithm and Crank-Nicolson scheme. The strain energy release rate for multiple delamination fronts and stress intensity factors for transverse cracks are calculated by the Virtual Crack Closure Technique and the Interaction Integral Method, respectively. The proposed method is applied to the steady-state thermomechanical problems, elastic and thermomechanical dynamic problems for isotropic and composite plates with transverse cracks and multiple delaminations.