Coupled heat conduction and multiphase change problem accounting for thermal contact resistance

International audienceIn this paper, heat conduction coupled with multiphase changes are considered in a cylindrical multilayer composite accounting for thermal contact resistance depending on contact pressures and roughness parameters. A numerical simulation is proposed using both analytical developments and numerical computations. The presented modeling strategy relies on an algorithm that alternates between heat conduction accounting for volumetric heat sources and a multi-phase change model based on non-isothermal Avrami's equation using the isokinetic assumption. Applications to coiling process (winding of a steel strip on itself) are considered. Indeed, phase changes determine the microstructure of the final material and are responsible for residual stresses that create flatness defects. A Finite Element modeling is used for validating the presented solution and numerical results are presented and discussed

Inverse three-dimensional method for fast evaluation of temperature and heat flux fields during rolling process

International audienceMonitoring and controlling flatness during the rolling process becomes critical for ensuring the product quality. Flatness defects are due to highly three-dimensional phenomena. Indeed, strips with different widths are rolled during the same campaign and cooling systems are heterogeneous along the axial direction to modify the thermal expansion of the roll. Therefore this paper presents a fully three dimensional inverse analytical method to determine the temperature field and heat fluxes (especially at the surface of the roll) by interpreting measurements of temperature done with several thermocouples fully embedded in the roll body and aligned along the axial direction. Since the method is dedicated to on-line interpretation and designed as a tool for adapting the rolling parameters during the rolling process, iterative methods are not studied to avoid long computation times, which justify the development of an analytical solution of the problem. The computation time displayed by Scilab 5.3 with a quadcore 2.8 GHz is around 0.07 s/cycle. The 3D unsteady heat equation is solved analytically in the roll, managing only one assumption so that restrictions of the measurement system (i.e., successive times) are taken into account. The solution is validated by comparing the outputs (surface temperature) and a prescribed temperature field (corresponding to hot rolling conditions). A satisfying 1.1% error is obtained. The accuracy is therefore promising. Furthermore noise sensitivity is evaluated by adding random values to the inputs (temperature computed at a depth of 0.5 mm under the surface) and the accuracy has not bee compromised (1.8%). Therefore good noise robustness is demonstrated

A new sensor for the evaluation of contact stress by inverse analysis during steel strip rolling

International audienceKnowledge of the contact stress between roll and strip becomes a critical factor in modern, high-speed rolling mills. In this paper, an inverse analytical method is developed to determine the contact stress in the roll gap by measuring the stress tensor with fibre optics at only one point inside the roll. Unlike many inverse methods, no matrix inversion is needed because the very small contact length would lead to ill-conditioned matrices. Iterative methods are also not studied because short computation times are desired. This approach uses the theory of elasticity on the assumption that the problem is isothermal and planar and relies on the expansion of holomorphic functions into a power series. On the other hand, the computation time is studied to rapidly optimise the industrial parameters during the rolling process. Hot, cold and temper-rolling simulations are given to demonstrate the accuracy of the method and the feasibility of this new kind of sensor, taking into account the restrictions (e.g., frequency of acquisition) of the local measurement system

Curvature of an elasto-plastic strip at finite strains : application to fast simulation of coils

International audienceThis work is part of the framework of a fast modeling of winding aiming at improving knowledge of residual stress evolution in steel strips and therefore their flatness during the coiling process. An exact analytical solution of an elasto-plastic strip with isotropic hardening at finite strains under an imposedtransformation of curvature is developed. Issues related to flow rules for non-differentiable yield functions (Tresca) have been broached and a unique solution is obtained. The equivalence for this transformation, between von Mises and Tresca yield functions is demonstrated. This solution contributes to an efficient model by terms of computation times that aims at simulating coiling by taking into account inelastic deformations and enabling parametric studies in order to improve the process

Non-linear simulation of coiling accounting for roughness of contacts and multiplicative elastic-plastic behavior

International audienceIn this paper numerical simulations of coiling (winding of a steel strip on itself) and uncoiling are developed. Initial residual stress field is taken into account as well as roughness of contacts and elastic-plastic behavior at finite strains, considering the Tresca yield function and isotropic hardening. The main output is the residual stress field due to plastic deformations during the process. This enables to quantify additional flatness defects. The presented coiling simulation relies on a modeling strategy that consists in dividing each time step into two sub-steps. Each sub-step can be solved semi-analytically and numerical optimizations enable to obtain a general solution. Thus reasonable computation times are reached and parametric studies can be performed in order to develop coiling strategies considering the process parameters. Comparisons with previous models from the literature are presented. Moreover the comparison with a Finite Element simulation presents the same order of magnitude, however it shows that direct computations using classical FE codes are difficult to perform in terms of computation times and stability if an explicit integration scheme is chosen. Numerical results are also given in order to determine the effect of some parameters such as roughness, yield stress, applied force, strip crown or mandrel's radius

Curvature of an elasto-plastic strip at finite strains: application to fast simulation of coils

This work is part of the framework of a fast modeling of winding aiming at improving knowledge of residual stress evolution in steel strips and therefore their flatness during the coiling process. An exact analytical solution of an elasto-plastic strip with isotropic hardening at finite strains under an imposed transformation of curvature is developed. Issues related to flow rules for non-differentiable yield functions (Tresca) have been broached and a unique solution is obtained. The equivalence for this transformation, between Von Mises and Tresca yield functions is demonstrated. This solution contributes to an efficient model by terms of computation times that aims at simulating coiling by taking into account inelastic deformations and enabling parametric studies in order to improve the process

Energetic approach coupled with analytic solution for the evaluation of residual stress

This paper is part of a more general mixed analytical/numerical strategy aiming at computing residual stresses of metallic strips after coiling process. Multiphase transitions and transformation induced plasticity occur during coil cooling. Thus, each layer of coil is subjected to an overall eigenstrain that can be sufficient to generate macroscopic plastic deformations. For each layer, a solution of the problem of an elastic-plastic hollow cylinder undergoing an arbitrary eigenstrain is derived. Mathematical developments relies on the linear inhomogeneous Navier equation by dealing with plasticity through the introduction of a deviatoric unknown plastic strain. An analytical solution is obtained in the form of series expansion, for any trial plastic strain. Then, an energetic principle enables to determine the plastic strain chosen as a solution of the problem. Practically, a numerical optimization procedure is performed directly on coefficients of the plastic strain series expansion

Evaluation of temperature field and heat flux by inverse analysis during steel strip rolling

International audienceKnowledge of the temperature field in the roll is a critical factor of modern, high-speed rolling mills. In this paper, an inverse analytical method is developed to determine the temperature field and especially the temperature (and heat flux) at the surface of the roll by measuring the temperature with a thermocouple (fully embedded) at only one point inside the roll. Iterative methods are not studied because short computation times are desired. Some assumptions are done to resolve analytically the unsteady heat equation, taking into account the restrictions of the measurement system (e.g., measurement according to successive times). The solution is validated by comparing the outputs of the method and prescribed analytical temperature fields. Good agreement is obtained. Noise sensitivity is estimated by adding artificial random numbers to the inputs. Good accuracy is observed. A 10 \% error of the temperature sensor depth is also considered and does not compromise the method. On the other hand, the computation time (around 0.05 second by cycle) is studied to rapidly optimise the industrial parameters during the rolling process

Analytical inverse solution for coupled thermoelastic problem for the evaluation of contact stress during steel strip rolling

International audienceKnowledge of the contact stress between roll and strip is a critical factor in modern, high-speed rolling mills. Previously two inverse analytical methods have been developed to determine the elastic contact stress on the one hand and the heat flux or the temperature in the whole roll (and especially at the surface) on the other hand, by measuring the stress tensor inside the roll body with fibre optics and by measuring the temperature with a thermocouple fully embedded at only one point inside the roll. However measurements done by fibre optics take into account the elastic stress and the thermal stress. However the contact stress was determined under isothermal assumption, which is strongly incorrect for hot rolling conditions. In this paper, the coupled thermoelastic problem is solved analytically using the theorem of superposition and the expression of the temperature field exhibited previously. A significant improvement of the accuracy of the inverse method for reconstructing the contact stress is observed by taking into account thermal stress. Hot rolling simulation is given to demonstrate this result. The computation time is studied to rapidly optimise the industrial parameters during the rolling process, and considering that both inverse methods have been run, the computation of thermal stress does not cost significant additional CPU times

Temperature and heat flux fast estimation during rolling process

International audienceMonitoring and controlling flatness during the rolling process becomes critical for ensuring the product quality. Flatness defects are due to highly three-dimensional phenomena. Indeed, strips with different widths are rolled during the same campaign and cooling systems are heterogeneous along the axial direction to modify the thermal expansion of the roll. Therefore this paper presents a fully three-dimensional inverse analytical method to determine the temperature field and heat fluxes (especially at the surface of the roll) by interpreting measurements of temperature done with several thermocouples fully embedded in the roll body and aligned along the axial direction. Since the method is dedicated to on-line interpretation and designed as a tool for adapting the rolling parameters during the rolling process, iterative methods are not studied to avoid long computation times, which justifies the development of an analytical solution of the problem. The computation time displayed by Scilab 5.3 with a quadcore 2.8 GHz is around 0.5 second by cycle for accurate computation and 0.07 second by cycle for rough computation. This paper improves a previous work (2D and relying on four assumptions designed for the prediction of wear). In the present contribution the 3D unsteady heat equation of the rotating roll is solved analytically with only one assumption in order to deal with the restriction of the measurement system (i.e., measurement according to successive times). Therefore not only radial and tangential heat fluxes are taken into account but also axial heat flux. The solution is validated by comparing the outputs of the method and some prescribed analytical temperature fields. Good agreement is obtained. Noise sensitivity is estimated by adding artificial random numbers to the inputs, and good accuracy is observed. Moreover sensitivity to sensor depth is estimated and demonstrated to be not compromising