In this work residual stresses in a stress lattice are studied. The residual stresses are both measured and simulated. The stress lattice is casted of low alloyed grey cast iron. In fact, nine similar lattices are casted and measured. The geometry of the lattice consists of three sections in parallel. The diameter of the two outer sections are thinner than the section in the middle. When the stress lattice cools down, this difference in geometry yields that the outer sections start to solidify and contract before the section in the middle. Finally, an equilibrium state, with tensile stresses in the middle and compressive stresses in the outer sections, is reached. The thermo-mechanical simulation of the experiments is performed by using Abaqus. The thermo-mechanical solidification is assumed to be uncoupled. First a thermal analysis, where the lattice is cooled down to room temperature, is performed. Latent heat is included in the analysis by letting the fraction of solid be a linear function of the temperature in the mushy zone. After the thermal analysis a quasi-static mechanical analysis is performed where the temperature history is considered to be the external force. A rate independent J2-plasticity model with isotropic hardening is considered, where the material data depend on the temperature. Tensile tests are performed at room temperature, 200°C, 400°C, 600°C and 800°C in order to evaluate the Young´s modulus, the yield strength and the hardening accurate. In addition, the thermal expansion coefficient is evaluated for temperatures between room temperature and 1000°C. The state of residual stresses is measured by cutting the mid section or the outer section. The corresponding elastic spring-back reveals the state of residual stresses. The measured stresses are compared to the numerical simulations. The simulations show good agreement with the results from the experiments.
In this work a hybrid method of a genetic algorithm and sequential linear programming is suggested to obtain a D-optimal design of experiments. Regular as well as non-regular design spaces are considered. A D-optimal design of experiments maximizes the determinant of the information matrix, which appears in the normal equation. It is known that D-optimal design of experiments sometimes include duplicate design points. This is, of course, not preferable since duplicates do not add any new information to the response surface approximation and the computational effort is therefore wasted. In this work a Bayesian modification, where higher order terms are added to the response surface approximation, is used in case of duplicates in the design of experiments. In such manner, the draw-back with duplicates might be eliminated. The D-optimal problem, which is obtained by using the Bayesian modification, is then solved by a hybrid method. A hybrid method of a genetic algorithm that generates a starting point for sequential linear programming is developed. The genetic algorithm performs genetic operators such as cross-over and mutation on a binary version of the design of experiments, while the real valued version is used to evaluate the fitness. Next, by taking the gradient of the objective, a LP-problem is formulated which is solved by an interior point method that is available in Matlab. This is repeated in a sequence until convergence is reached. The hybrid method is tested for four numerical examples. Results from the numerical examples show a very robust convergence to a global optimum. Furthermore, the results show that the problem with duplicates is eliminated by using the Bayesian modification.
In this work the robustness of residual stresses in finite element simulations with respect to deviations in mechanical parameters in castings is evaluated. Young's modulus, the thermal expansion coefficient and the hardening are the studied parameters. A 2D finite element model of a stress lattice is used. The robustness is evaluated by comparing purely finite element based Monte Carlo simulations and Monte Carlo simulations based on linear and quadratic response surfaces. Young's modulus, the thermal expansion coefficient and the hardening are assumed to be normal distributed with a standard deviation that is 10% of their nominal value at different temperatures. In this work an improved process window is also suggested to show the robustness graphically. By using this window it is concluded that least robustness is obtained for high hardening values in combination to deviations in Young's modulus and the thermal expansion coefficient. It is also concluded that quadratic response surface based Monte Carlo simulations substitute finite element based Monte Carlo simulations satisfactory. Furthermore, the standard deviation of the responses are evaluated analytically by using the Gauss formula, and are compared to results from Monte Carlo simulations. The analytical solutions are accurate as long as the Gauss formula is not utilized close to a stationary point.
In the past stamping dies have in principle been designed by rules of thumb and intuition. As the sheet metals in the vehicle industry have got increased mechanical properties in recent years the demands on the stamping dies have increased. For instance increase in stiffness is desirable in order to better control spring-back. The most simple way to satisfy this new demand would be to make the stamping dies even more heavy in order to be able to handle the new sheet metals. Since there are restrictions of the weight of the stamping dies in the stamping machines and since the overhead cranes usually have reached the limit of what they can handle, this is not a desirable solution. Another approach, in order to increase the stiffness without increasing the weight is to use topology optimization. Recently in a master thesis at Volvo Car Corporation a conceptual design of a stamping die has been done by topology optimization. In that work no consideration is taken to the fact that the stamping die is casted. Casting implies that residual stresses possibly are produced during the solidification and cooling process. The residual stresses might affect the fatigue life and the risk of failure of the stamping die.
In this work the residual stress state after casting is analyzed for the original stamping die as well as the optimized stamping die from the master thesis discussed above. The analyses are performed using an uncoupled approach, where one thermal analysis is followed by a quasi-static elasto-plastic analysis. The thermal analysis simulates the solidification and cooling during the casting process, while the quasi-static elasto-plastic analysis uses the temperature history, obtained from the thermal analysis, in order to build up residual stresses. The thermal analysis includes the release of latent heat. Furthermore, the material properties included in the heat equation (density, conductivity, specific heat) are given as temperature dependent properties for the mould as well as the casting. In the quasi-static elasto-plastic analysis the plasticity is described by the von Mises yield surface in combination with isotropic hardening and the mechanical properties (thermal expansion coefficient, Young's modulus, yield stress, hardening parameter, Poisson's ratio) are given as temperature dependent properties. The simulations show high levels of residual stresses.
The present theoretical note shows how a naturalobjective function in stiffness optimization, including bothprescribed forces and non-zero prescribed displacements,is the equilibrium potential energy. It also shows how theresulting problem has a saddle point character that may beutilized when calculating sensitivities.
Stiffness topology optimization is usually based on a state problem of linear elasticity, and there seems to be little discussion on what is the limit for such a small rotation-displacement assumption. We show that even for gross rotations that are in all practical aspects small (<3 deg), topology optimization based on a large deformation theory might generate different design concepts compared to what is obtained when small displacement linear elasticity is used. Furthermore, in large rotations, the choice of stiffness objective (potential energy or compliance), can be crucial for the optimal design concept. The paper considers topology optimization of hyperelastic bodies subjected simultaneously to external forces and prescribed non-zero displacements. In that respect it generalizes a recent contribution of ours to large deformations, but we note that the objectives of potential energy and compliance are no longer equivalent in the non-linear case. We use seven different hyperelastic strain energy functions and find that the numerical performance of the Kirchhoff–St.Venant model is in general significantly worse than the performance of the other six models, which are all modifications of this classical law that are equivalent in the limit of infinitesimal strains, but do not contain the well-known collapse in compression. Numerical results are presented for two different problem settings.
In this paper an efficient approach to simulate thermal stresses due to frictional heating of disc brakes is presented. Inthe approach thermal and stress analysis are performed sequentially. The frictional heat analysis is based on the Eulerianmethod, which requires significantly low computational time as compared to the Lagrangian approach. Completethree-dimensional geometries of a disc and a pad are considered for the numerical simulations. The contact forcesare computed at each time step taking the thermal deformations of the disc into account. The nodal temperaturehistory is recorded at each time step and is used in sequentially coupled stress analysis, where a temperature dependentelasto-plastic material model is used to compute the stresses in a disc brake. The results show that during hard braking,high compressive stresses are generated on the disc surface in circumferential direction which cause plastic yielding. Butwhen the disc cools down, the compressive stresses transform to tensile stresses. Such thermoplastic stress history maycause cracks on disc surface after a few braking cycles. These results are in agreement with experimental observationsavailable in the literature.
In this paper, an efficient sequential approach for simulating thermal stresses in brake discs for repeated braking is presented. First, a frictional heat analysis is performed by using an Eulerian formulation of the disc. Then, by using the temperature history from the first step of the sequence, a plasticity analysis with temperature dependent material data is performed in order to determine the corresponding thermal stresses. Three-dimensional geometries of a disc and a pad to a heavy truck are considered in the numerical simulations. The contact forces are computed at each time step taking the thermal deformations of the disc and pad into account. In such manner, the frictional heat power distribution will also be updated in each time step, which in turn will influence the development of heat bands. The plasticity model is taken to be the von Mises yield criterion with linear kinematic hardening, where both the hardening and the yield limit are temperature dependent. The results show that during hard braking, high compressive stresses are generated on the disc surface in the circumferential direction which cause yielding. But when the disc cools down, these compressive stresses transform to tensile residual stresses. For repeated hard braking when this kind of stress history is repeated, we also show that stress cycles with high amplitudes are developed which might generate low cycle fatigue cracks after a few braking cycles.
In this paper frictional heating of a disc brake is simulated while taking wear into account. By performing thermomechanical finite element analysis, it is studied how the wear history will influence the development of hot bands. The frictional heat analysis is based on an Eulerian formulation of the disc, which requires significantly lower computational time as compared to a standard Lagrangian approach. A real disc-pad system to a heavy truck is considered, where complete three-dimensional geometries of the ventilated disc and pad are used in the simulations. A sequential approach is adopted, where the contact forces are computed at each time step taking the wear and thermal deformations of the mating parts into account. After each brake cycle, the wear profile of the pad is updated and used in subsequent analysis. The results show that when wear is considered, different distributions of the temperature on disc are obtained for each new brake cycle. After a few braking cycles two hot bands appear on the disc surface instead of only one. These results are in agreement with experimental observations.
Thermal stresses as a result from frictional heating must be considered when designing disc brakes, clutches or other rotating machine components with sliding contact conditions. The rotational symmetry of the disc in these kind of applications makes it possible to model these systems using an Eulerian approach instead of a Lagrangian framework. In this paper such an approach is developed and implemented. The disc is formulated in an Eulerian frame where the convective terms are defined by the angular velocity. By utilizing the Eulerian framework, a node-to-node formulation of the contact interface is obtained, producing most accurate frictional heat power solutions. The energy balance of the interface is postulated by introducing an interfacial temperature. Both frictional power and contact conductances are included in this energy balance. The contact problem is solved by a non-smooth Newton method. By adopting the augmented Lagrangian approach, this is done by rewriting Signorini’s contact conditions to an equivalent semi-smooth equation. The heat transfer in the disc is discretized by a Petrov–Galerkin approach, i.e. the numerical difficulties due to the non-symmetric convective matrix appearing in a pure Galerkin discretization is treated by following the streamline-upwind approach. In such manner a stabilization is obtained by adding artificial conduction along the streamlines. For each time step the thermo-elastic contact problem is first solved for the temperature field from the previous time step. Then, the heat transfer problem is solved for the corresponding frictional power. In such manner a temperature history is obtained sequentially via the trapezoidal rule. In particular the parameter is set such that both the Crank–Nicolson and the Galerkin methods are utilized. The method seems very promising. This is demonstrated by solving a two-dimensional benchmark as well as a real disc brake system in three dimensions.
In this paper an implicit method for frictional contact, impact and rolling is suggested. A nonclassical formulation of a two-dimensional hyperelastic body unilaterally constrained to rigid supports is proposed by following the ideas of Moreau and Jean. A total Lagrangian formulation of the system is given. The elastic properties are defined by coupling the second Piola–Kirchhoff stress to the Green–Lagrange strain via the Kirchhoff–St. Venant law. The equation of motion is written in the spirit of Moreau by using the mean value impulses introduced by Jean. The mean value impulses appear explicitly in the equation of motion. In such manner the treatment of nonconstant kinematic transformation matrices becomes straightforward. The rigid supports are described by smooth functions. By utilizing these functions and the mean value impulses, new contact/impact laws of Signorini and Coulomb type are formulated. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. Three two-dimensional examples are solved by the method: a contact problem, an impact problem and a rolling contact problem.
A general sequential linear programming (SLP) approach for reliability based design optimization (RBDO) with non-Gaussian random variables is presented. The RBDO problems are formulated by using optimal regression models (ORM) as surrogate models and S-optimal design of experiments (DoE). The S-optimal DoE is obtained by maximizing the average mean of the distances between the nearest neighbors. Finite element simulations are performed for the S-optimal DoE and corresponding ORM are obtained by a genetic algorithm. In such manner not only optimal regression coefficients are generated but also optimal rational base functions. The RBDO problems are solved by introducing intermediate variables defined by the iso-probabilistic transformation at the most probable point. By using these variables in the Taylor expansions, a corresponding deterministic linear programming problem is derived, which is corrected by applying second order reliability methods (SORM) as well as Monte Carlo simulations. For low target values on the reliability crude Monte Carlo simulations are used, but for high targets a Latin hypercube sampling (LHS) approach is utilized. The implementation of the suggested sampling- and SORM-based SLP approach is efficient and robust. This is demonstrated by presenting trade-off curves between the objective function, constraints, variables and the target of reliability.