Thermomechanical Problems Research Articles

Thermal stresses in bimetallic plate under induction heating by nonstationary electromagnetic field

A thermomechanical problem is formulated for a bimetallic plate subjected to induction heating by a nonstationary electromagnetic field. As the object of the study, a plate with layers made of stainless steel and copper is selected. The stress intensities in the constituent layers of the plate are numerically analyzed on its external surfaces and at the contact interface of the constituent layers. The analysis is conducted based on the amplitude-frequency characteristics and the duration of the nonstationary electromagnetic action. The research is carried out for two cases of electromagnetic carrier wave frequencies including those equal to the first resonance frequency and those extending outside the resonance frequency range of the electromagnetic field.

Enhanced physics-informed neural networks without labeled data for weakly and fully coupled transient thermomechanical analysis

In this paper, a series of enhanced physics-informed neural networks (PINN) models without labeled data is proposed to solve the weakly and fully coupled thermomechanical problems. In these models, to better predict the thermal and mechanical responses, PINNs consisting of different deep neural networks (DNN) representing temperature, displacement, and stress are specifically constructed. Furthermore, to elevate the accuracy and avoid possible training failure, several advanced algorithms are developed to ensure the effectiveness of imposing boundary conditions, refining sampling distributions, and enhancing training strategy. A notable aspect of the enhanced PINNs is their independence from expensive, labeled data, relying solely on the temporal and spatial information embedded within the sampling points. The effectiveness and accuracy of the enhanced PINNs are validated through extensive numerical examples, including heat conduction and both weakly and fully coupled thermomechanical problems. The comparation between original PINN and enhanced PINN illustrates the necessity of involving these enhanced methods. The results demonstrate the significant potential of PINN methodologies in engineering areas involving complex thermomechanical processes.

Topology optimization for minimizing the mean compliance under thermo-mechanical loads using element-free Galerkin method

This paper presents a numerical model for addressing thermo-mechanical coupling problems in topology optimization, utilizing the element-free Galerkin (EFG) method. A multi-parameter density-based topology optimization framework is introduced to interpolate the thermal stress coefficient, encompassing thermal conductivity, thermal expansion coefficient, and elastic modulus. Two numerical examples are provided to investigate the model's characteristics and associated considerations, including the impact of EFG node distribution, quantity, calculation points layout, design variables, and filtering techniques on optimized results. Numerical findings indicates that the present model allows for adaptable adjustments in nodes number and distribution, calculation points layout, choice of design variables, and application of various filtering techniques while maintaining consistent background grids. Although these adjustments may affect convergence rates and final objective values, they can promise satisfactory optimization structures. Additionally, the study highlights the critical influence of filtering radius on optimization outcomes and the objective value, recommending a value of approximately 16∼20 calculation points.

Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

AbstractImmersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods.

Finite element analysis of the thermal and thermo-mechanical coupling problems in the dry friction clutches using functionally graded material

Abstract The friction clutch is an important machine part in the torque transmission system, where it is responsible for gently transmitting the power from the engine to the gearbox and damping the tensional vibration. At the beginning of the engagement between the clutch’s elements, a large amount of heat is released due to frictional sliding between the elements of the clutch system, generating very high temperatures. In this work, the two- and three-dimensional finite element models were developed from scratch to explore the clutch disk’s temperature and contact pressure behaviors using an alternative frictional material such as functionally graded material (FGM). The two assumptions for the thermal loads were considered: uniform wear and uniform pressure assumptions. The thermal–structural problem for the new frictional material was solved numerically and then presented as the temperature distributions and the contact pressure at any instant of sliding time. Also, a comparison was made between the clutch system behaviors when using FGM with different frictional materials. The results showed that the FGM has superior results to the other available frictional materials.

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

A new hybrid local radial basis function collocation method for 2.5D thermo-mechanical modelling of continuous casting of steel

This study presents a new strong-form meshless method to solve the thermo-mechanical problem of the solidification process in the continuous casting of steel. A two-dimensional slice that travels in the casting direction is modelled in the Lagrangian system. The newly developed mechanical model is one-way coupled to the thermal model, where the heat flux due to the mould, sprays, rolls and radiation are imposed to solve heat transfer in the strand. The resulting temperature and metallostatic pressure govern the Norton-Hoff visco-plastic model used for computing shrinkage of the solid shell and induced residual stresses. The results are used to estimate critical areas susceptible to hot-tearing formation. The mechanical model uses a generalised plane strain assumption that includes linear strains perpendicular to the slice and enables the computation of the straightening of the strand. The thermo-mechanical model is spatially discretised with a local radial basis function collocation method (LRBFCM). The mechanical part includes a new hybrid method that combines LRBFCM with finite differences for increased stability. The presented work shows how the developed model is used to assess the impact of casting velocity on the solid shell shrinkage and the probability of hot-tearing occurrence in the continuous casting of square billets.

Prediction of thermal contact resistance for reusable heat-pipe cooled thermal protection system based on an inverse thermo-mechanical coupling method

Thermal contact resistance (TCR) is a critical characteristic on increasing or decreasing thermal energy transmission efficiency between two bodies in thermal management systems. However, it is difficult to accurately determine the thermal contact resistance in actual engineering structures, due to the complicated influence factors, such as pressure, temperature, and physical properties. In the work, a new method is presented to estimate the thermal contact resistance for a three-dimensional (3D) reusable heat-pipe cooled thermal protection system based on boundary measurements, by solving transient inverse thermo-mechanical coupling problems. Moreover, the thermal contact resistance varies with interface pressure, temperature, and spatial position, which is more practical and challenging. The thermo-mechanical coupling analysis is conducted by the finite element method, and the inversion is carried out by the gradient-based algorithm. First, the present method is validated by identifying the constant TCR, based on the available experimental data. Then, the thermal contact resistance with the functional form in the 3D reusable thermal protection system is accurately estimated. Finally, the convergence stability and robustness of the proposed method are evaluated, by considering the effects of initial guess value and measurement error, respectively. The accurate determination of thermal contact resistance of the reusable thermal protection system effectively avoids the overweight of aircraft and improves its utilization. The present work provides a novel approach for the determination of thermal contact resistance in actual engineering applications.

A thermodynamically consistent phase field model for brittle fracture in graded coatings under thermo-mechanical loading

This paper presents a thermodynamically consistent phase field formulation designed for investigating quasi-static brittle fracture under thermo-mechanical loading conditions in functionally graded materials. The governing equations for linear momentum and crack evolution are derived by minimizing the free energy functional, which includes elastic strain energy and surface energy, with respect to displacement field and the phase field variable, respectively. A thermodynamically consistent formulation is employed to establish the governing equation for heat conduction. The proposed formulation is converted into a set of finite element equations using the Ritz method and implemented in COMSOL Multiphysics, a commercial finite element software which allows for writing governing equations in their strong form. A segregated solver is utilized to solve iteratively the coupled problem. The proposed model is verified against existing mechanical and thermo-mechanical phase field models, demonstrating good agreement with the literature. To explore the model’s capabilities, a graded layer composed of Zirconia and a Titanium alloy with a surface crack undergoing thermo-mechanical loading is considered as a case study for this investigation. Additionally, the model incorporates a thermal conductivity degradation proposed. A parametric study is performed to investigate the effect of varying parameters like volume fraction, thermal loading rate, and the thermal conductivity degradation on the load–displacement curve of the graded layer. These findings contribute to the understanding of thermo-mechanical behavior of surface graded layers and serve as a foundation for addressing more complex thermo-mechanical problems.

Thermomechanical problem of functionally graded spherical shells based on homogenization schemes: Data-driven volume fraction optimization with material uncertainties

Mechanical analysis and optimal design for functionally graded materials (FGMs) structures are significant. The present paper analyzes the parameter uncertainties problem for the thermomechanical response of the FGMs spherical shells with volume fraction power distribution. Here, the interval random uncertainty model is recommended to describe the uncertainties of each component of material parameters in FGMs, and a data-driven artificial neural network (ANN) fast calculation for FGMs spherical shells thermomechanical problems with uncertainties analysis is established and trained. The thermomechanical coupling response analysis and volume fraction optimization analysis of the FGMs spherical shell with material uncertainties are realized for zirconia-titanium FGMs, with the maximum normalized Von Mises stress as the structural safety index and the ANN as the fast calculation method. In addition, the Von Mises stress of the FGMs spherical shell and a ZrO2-Ti alloy double-layer spherical shell under different loads is compared. The results show that material parameter uncertainties, especially those in Young's modulus, thermal expansion coefficient, and uniaxial yield limit significantly affect structural safety. The structural safety index λ calculated based on the volume fraction optimization design of FGMs spherical shell is significantly lower than that of the ZrO2-Ti alloy double-layer. The ANN method greatly improves the computational efficiency for uncertain problems and maintains high accuracy, compared with the statistical analysis after many calculations by randomly selecting material parameters. This study provides a path for the safety design of FGMs structures with uncertain parameters.

A thermo-mechanical numerical method for modelling heating due to frictional sliding

Heating due to frictional sliding is an important phenomenon in tribological applications. The present study develops a numerical method based on the finite elements (FE) for modelling frictional sliding induced heating. More specifically, the method is designed for applications where the behavior of the tool part of the frictional contact couple is not critical so that it can be idealized as a rigid body. The contact between the tool and the target, assuming linear elastic material, is modelled with the penalty method. The FE-discretized balance of linear momentum is time discretized with the Newmark scheme, and the FE-discretized heat equation is time discretized with the backward Euler scheme. The global coupled thermo-mechanical problem is solved with a globally iterative staggered approach. The frictional contact model is verified against analytical solution of a rotating blade pressed against a plane. Finally, a validation simulation of a pin-on-disc tribology test is carried out.

Thermo-mechanical coupled peridynamics simulation of concrete failure under fire scenarios

A coupled thermo-mechanical bond-based peridynamical (PD) method is developed to simulate thermal cracking processes in concrete. Based on the bond-based PD heat conduction and motion equations, the PD differential operator (PDDO) is used to express the classical computational fluid dynamics basic equations in a non-local integral form, establishing a PD model for the thermo-mechanical coupling problems. In this model, concrete is considered a heterogeneous material composed of aggregates and mortar, with material parameters assumed to be temperature-dependent. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. The model is applied to simulate the transient heat transfer problem of a homogeneous plate. The results show good agreement with the analytical solution, providing evidence for the correctness and accuracy of the proposed coupled numerical method. The damage behavior of the borehole heated concrete plate is analyzed, followed by the simulation of the damage to the concrete plate under a fire scenario. The numerical results accurately predict the temperature distribution within the heterogeneous concrete and the resulting material damage. This study provides new theoretical basis for the fire resistance design and high-temperature performance improvement of concrete materials and structures.

Plasticity-Induced Heating: Revisiting the Energy-Based Variational Model.

Temperature evolution during plastic deformation is of great importance for the design of manufacturing processes, as well as for the analysis and prediction of tool wear. However, the results from experimental- and numerical-type research are still often contradictory. In this paper, we analyze methods for estimating plasticity-induced heating directly from displacement fields that can be recorded during experiments or extracted from simulation results. In terms of computational methodology, the thermodynamically motivated energy-based variational formulation of the coupled thermo-mechanical boundary-value problem is adapted to the problem at hand. Since an analysis of this variational formulation exhibits challenges and distinct inconsistencies with respect to the problem at hand, an alternative approach is proposed. This alternative approach is essentially a purely thermal finite element simulation, and it is conducted using a heat source term that is empirically based on the fraction of irreversible deformation work converted to heat. Our approach estimates plasticity-induced heating based on the strain and strain rate data derived from displacement fields. We therefore incorporate thermo-visco-plastic constitutive behavior (Johnson-Cook) with a thermodynamically motivated model that specifies the fraction of plastic work converted to heat (the Taylor-Quinney coefficient).

МЕТОДЫ МОДЕЛИРОВАНИЯ ТЕМПЕРАТУРЫ И ПЕРЕМЕЩЕНИЯ МАТЕРИАЛА ПРИ СВАРКЕ ТРЕНИЕМ С ПЕРЕМЕШИВАНИЕМ

Numerical modeling of the friction stir welding (FSW) process is challenging due to numerous complexities. This is a fully coupled thermomechanical problem that involves large plastic deformation, heat flow, nonlinear contact, and friction. Accurate modeling of the FSW process will provide a better understanding of the material flow and temperature distribution during the process, which is important for optimizing process parameters and reducing the likelihood of weld defects during FSW. Various approaches to modeling the temperature and material distribution in the FSW process are considered, such as the Euler or Lagrange approach, smoother particle hydrodynamics (SPH), coupled Euler and Lagrange methods (CEL), and arbitrary Lagrange and Euler methods (ALE). In this work, temperature and material distributions during the FSW process of a dissimilar aluminum-copper joint were simulated by finite element analysis using CEL and ALE approaches. In addition, it was shown that the temperature distribution along the weld line is asymmetrical: the maximum temperature on the advancing side is higher than on the retreating side.

Variational formulation of thermomechanical problems

This article proposes that a 4D space-time continuum is used for building variational thermomechanical continuum models. In order to identify physical constants in reversible processes, physically justified hypotheses were formulated. They are the hypotheses of complementary shear stress, classical dependence of momentum on velocity, and heat flow potentiality (generalized Maxwell–Cattaneo law). The Duhamel–Neumann law was assumed to be classical. In the considered model, the generalized Maxwell–Cattaneo and Duhamel–Neumann laws were not introduced phenomenologically. They were derived from the compatibility equations by excluding thermal potential from the constitutive equations for temperature, heat flow, and pressure. Dissipation channels were considered as the simplest non-integrable variational forms, which are linear in the variations of arguments. As a result, a variational principle that generalizes L.I. Sedov’s principle was developed. It is a consequence of the virtual work principle and termed as the difference between the variation of the Lagrangian of reversible thermomechanical processes and the algebraic sum of dissipation channels. It was proved that for the classical thermomechanical processes, with second-order differential equations, there can only exist six dissipation channels. Two of them determine dissipation in an uncoupled system – in the equations of motion and heat balance. The remaining four channels define coupling effects in coupled problems of dissipative thermomechanics.

The Impact of Various Heating Rates on Real-Time Degree of Hybrid Fire Simulation

Global fire performance of structures in fire is proven to be more advantageous in many cases of engineering practice than the prescriptive fire resistance based on isolated structural member testing. Hybrid fire simulation (HFS) is a novel well-suited method trending in recent years for analysis of global performance of structures in fire. In the principles of this method, the part of a structure which has unknown behavior or is uncertain to be numerically modeled (subjected to fire) would be physically tested, while the rest of the structure is numerically simulated. HFS method enables capturing the beneficial interaction mechanisms evolving between fire-exposed structural members and the adjacent cooler substructure. Due to the continuous temperature increase in a fire test and the existing thermal inertia as well as the rate- and temperature-dependent material behavior of structures exposed to fire, a real-time performance in hybrid fire simulation counts as a necessity. This challenge is more critical for hybrid fire simulations with higher applied heating rates relevant to structural fire engineering. Within scope of this paper, (a) a robust and rigorous approach for real-time HFS is presented; (b) a series of proof-of-concept studies of different hybrid fire simulations with various applied heating rates are carried out for a thermomechanical benchmark problem; (c) the important results of four representative hybrid fire simulations with relevant heating rates to structural fire engineering are discussed; (d) the importance of an appropriate calculation method for stiffness update of the fire-exposed structural member over HFS procedure is highlighted, and e) the precision and accuracy of the applied HFS approach with respect to interface error and real-time degree are evidenced.

Regularized coupling multiscale method for thermomechanical coupled problems

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM.

A deep material network approach for predicting the thermomechanical response of composites

Recent progress combining micromechanics and machine learning holds promise for accurately and rapidly predicting the mechanical behavior of complex composite materials. This study extends the Deep Material Network (DMN), a binary-tree network trained on linear direct-numerical-simulation data, to predict the thermo-elasto-viscoplastic behavior of composites. This extension incorporates the thermal expansion properties homogenization into the network’s building blocks and expands the online formulation to consider thermal boundary conditions. By comparing various implementations of these building blocks, we demonstrate the network’s extrapolation to thermomechanical problems. We then show how this extended DMN can be applied to uncertainty quantification and inverse design problems.

Multi–materials topology optimization using deep neural network for coupled thermo–mechanical problems

In this paper, a density-based topology optimization method using neural networks is used to design domains composed of multi-materials under combined thermo–mechanical loading. The neural network produces the topology density field by minimizing a loss function defined for the problem. Length scale control is conducted using a Fourier space projection. JAX, a high-performance automatic differentiation (AD) Python library is used to build an end-to-end differentiable neural network. The model can handle the sensitivity analysis automatically using the backpropagation process (automatic differentiation of loss function) and the need for solving adjoint equations manual sensitivity analysis is removed. The optimization problem is solved by minimizing the loss function of the network and other optimization algorithms such as the Method of Moving Asymptotes (MMA) are not required. The performance of the method is demonstrated through several examples and compared with those obtained from the popular Solid Isotropic Material with Penalization (SIMP) method. The network is able to handle high-resolution re-sampling, resulting in a more refined and smooth variation of optimal topologies.

Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation, the numerical manifold method (NMM) adopts a double covering system, which can easily deal with discontinuous deformation problems and has a high calculation accuracy. Aiming at the thermo-mechanical (TM) coupling problem of fractured rock masses, this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field, deduces related system equations, and proposes a penalty function method to deal with boundary conditions. Numerical examples are employed to confirm the effectiveness and high accuracy of this method. By the thermal stress analysis of a thick-walled cylinder (TWC), the simulation of cracking in the TWC under heating and cooling conditions, and the simulation of thermal cracking of the Swedish Äspö Pillar Stability Experiment (APSE) rock column, the thermal stress, and TM coupling are obtained. The numerical simulation results are in good agreement with the test data and other numerical results, thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.