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Robustness Analysis of Residual Stresses in Castings
Linköping University, Department of Management and Engineering. Linköping University, The Institute of Technology.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is about robustness analysis of residual stresses in castings. This topic includes the analysis of residual stresses in castings and the robustness analysis itself, both covered in the thesis.

Residual stresses are important when designing casted components. For instance, the residual stress state after casting might affect the fatigue life, facilitate crack propagation and cause spring-back related problems when a casted component is machined or used. Examples of components where such problems are recognized are stamping dies and brake discs, both considered in the thesis. Residual stresses in castings are simulated by finite element analysis in this thesis. A sequential un-coupled approach is used where a thermal analysis of the solidification and cooling generates a temperature history. Then a quasi-static structural analysis is performed, driven by the temperature history. During the structural analysis residual stresses are developed due to different cooling rates in combination with plasticity. For comparison, measurements of residual stresses in castings have also been performed. The agreement between analyses and measurements is satisfactory.

In a residual stress analysis there are several random variables such as process, geometrical and material parameters. Usually those random variables are assumed to be deterministic and their nominal values are used. It can be beneficial to include the variation of the random variables in analysis of residual stresses. For that purpose robustness analysis of the residual stresses are performed in this thesis. In some of the appended papers the robustness is evaluated with respect to variation in e.g. Young’s modulus, yield strength and hardening, thermal expansion coefficient, geometric dimensions and time in mould of the casting. The robustness analyses are performed by using metamodels as surrogates to the finite element model, due to the computational expensiveness of the residual stress analyses. Conventional regression models, Kriging approximations and an optimal polynomial regression model, proposed in one of the appended papers, are metamodels used in the thesis. When a metamodel is established the choice of the design of experiments can be crucial. The generation of the design of experiments is also investigated in the thesis. For instance, a hybrid method constituted by a genetic algorithm and sequential linear programming is proposed for the generation of optimal design of experiments. A-, D-, I- and S-optimal design of experiments are generated by the developed  hybrid method. Those design of experiments as well as Latin  Hypercube sampled design of experiments are used throughout the thesis. Since residual stress analysis, robustness analysis and metamodeling are considered in the thesis, more or less all parts required to perform robustness analysis of residual stresses in castings are covered.

Results in the thesis show that the level of residual stresses in castings can be high due to the casting process. Thus, crack development and spring-back related problems might be influenced by those stresses. Results also show that the level of residual stresses can be very dependent on the variation in certain random variables such as the thickness of the casting, hardening and Young’s modulus. Therefore, it can be of importance to include the variations of the random variables in order to accurately predict the residual stresses when designing castings.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press , 2012. , 44 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1415
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-72354ISBN: 978-91-7393-002-4 (print)OAI: oai:DiVA.org:liu-72354DiVA: diva2:459301
Public defence
2012-01-20, E1405, Tekniska högskolan, Jönköping, 10:00 (Swedish)
Opponent
Supervisors
Available from: 2011-11-25 Created: 2011-11-25 Last updated: 2012-04-02Bibliographically approved
List of papers
1. Residual stresses in a stress lattice: experiments and finite element simulations
Open this publication in new window or tab >>Residual stresses in a stress lattice: experiments and finite element simulations
2009 (English)In: Journal of Materials Processing Technology, ISSN 0924-0136, E-ISSN 1873-4774, Vol. 209, no 9, 4320-4328 p.Article in journal (Refereed) Published
Abstract [en]

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 midsection 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.

Place, publisher, year, edition, pages
Elsevier, 2009
Keyword
Residual stresses; Stress lattice; Grey iron; Finite elements; Thermomechanics; Casting
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-72345 (URN)10.1016/j.jmatprotec.2008.11.025 (DOI)
Available from: 2011-11-25 Created: 2011-11-25 Last updated: 2017-12-08Bibliographically approved
2. Simulation of residual stresses in stamping dies
Open this publication in new window or tab >>Simulation of residual stresses in stamping dies
2008 (English)In: Proceedings of the IDDRG 2008 Conference : Best in class stamping, 16-18 June 2008, Olofström, Sweden / [ed] Nader Asnafi, Olofström: Industriellt utvecklingscentrum i Olofström AB , 2008, 765-776 p.Conference paper, Published paper (Refereed)
Abstract [en]

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.

 

Place, publisher, year, edition, pages
Olofström: Industriellt utvecklingscentrum i Olofström AB, 2008
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-72348 (URN)978-91-633-2948-7 (ISBN)
Conference
IDDRG 2008, Best in class stamping, 16-18 June 2008, Olofström, Sweden
Available from: 2009-06-15 Created: 2011-11-25 Last updated: 2011-11-25Bibliographically approved
3. D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method
Open this publication in new window or tab >>D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method
2010 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 42, no 1, 73-88 p.Article in journal (Refereed) Published
Abstract [en]

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.

Keyword
D-optimality, Design of experiments (DoE), Sequential linear programming (SLP), Genetic algorithms (GA), Response surface methodology (RSM), Bayesian modification (BM)
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-72347 (URN)10.1007/s00158-009-0464-3 (DOI)
Projects
MERA
Available from: 2009-06-15 Created: 2011-11-25 Last updated: 2017-12-08Bibliographically approved
4. Design of Experiments - A- D- I- S-optimality
Open this publication in new window or tab >>Design of Experiments - A- D- I- S-optimality
2010 (English)In: Proceedings of the 2nd International Conference on Engineering Optimization, 2010Conference paper, Published paper (Refereed)
Abstract [en]

A metamodel approximates an original model with a model that is more efficient and yields information about the response. Response surfaces and Kriging approximations are such metamodels. A metamodel is based on evaluations of the original function at some design points, where the choice of design points is crucial. The design points constitute the design of experiments (DoE). There are many methodologies of how to chose the DoE. In this work A-, D-, I- and S-optimal DoEs are generated and evaluated. The optimal DoEs are obtained by solving the following mathematical optimization problems:

  • A-otimality. Minimize the average variance of the model coefficient estimates.
  • D-otimality. Minimize the generalized variance of the model coefficient estimates.
  • I-otimality. Minimize the average of the expected variance (taken as an integral)over the region of prediction.
  • S-otimality. Maximize the geometric mean of the distances between nearest neighborsof the design points.

The optimization problems are solved by a hybrid method which consists of a genetic algorithm and sequential linear programming. The different optimality criteria are evaluated for a number of test cases in order to show the characteristics of each criteria. Regular as well as non-regular design spaces are considered. Furthermore, Kriging approximations of the well known Rosenbrock’s banana function are generated to evaluate the accuracy of a resulting metamodel based on the different DoEs. Results from the test cases show that D-optimal DoEs tend to place more design points close to the boundary of the design space compared to A- and I-optimality. It is also shown that A- D- and I-optimal DoEs often include duplicate design points which is not beneficial for a deterministic response, but might be beneficial for non-deterministic responses. Concerning S-optimal DoEs the design points are evenly distributed over the entire design space and no duplicates occur. Furthermore, the S-optimal DoE generates the best fitted Kriging approximation of the Rosenbrock’s banana function.

Keyword
A- D- I- S-optimality, Design of experiments (DoE), Response surface, Kriging, Genetic algorithm, Sequential linear programming
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-72346 (URN)
Conference
2nd International Conference on Engineering Optimization. September 6 - 9, Lisbon, Portugal
Available from: 2011-01-17 Created: 2011-11-25 Last updated: 2011-11-25Bibliographically approved
5. Optimal Polynomial Regression Models by using a Genetic Algorithm
Open this publication in new window or tab >>Optimal Polynomial Regression Models by using a Genetic Algorithm
2011 (English)In: Proceedings of the Second International Conference on Soft ComputingTechnology in Civil, Structural and Environmental Engineering Conference, (Crete,Greece), 2011009, 2011Conference paper, Published paper (Other academic)
Abstract [en]

Different regression models are commonly used to approximate the behavior of an unknown response in a given design domain. The regression models are usually obtained from a design of experiments, the corresponding responses and the constitution of the regression model. In this work a new approach is proposed, where the constituents of a polynomial regression model are of arbitrary order. A genetic algorithm is used to find the optimal terms to be included in the so-called optimal polynomial regression model. The objective for the genetic algorithm is to minimize the sum of squared errors of the predicted responses. In practice the genetic algorithm generates an optimal set of exponents of the design variables for the specified number of terms in the regression model, where each term is a product of a regression coefficient and the design variables. Several example problems are presented to show the performance and accuracy of the optimal polynomial regression model. Results show an improved performance for optimal polynomial regression models compared to traditional regression models.

Keyword
Polynomial regression model, Metamodeling, Design of experiments (DoE)
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-72350 (URN)
Conference
The Second International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering Conference, 6-9 September, Chania, Crete, Greece
Projects
MERA
Available from: 2011-11-25 Created: 2011-11-25 Last updated: 2011-11-25Bibliographically approved
6. Robustness of residual stresses in castings and an improved process window
Open this publication in new window or tab >>Robustness of residual stresses in castings and an improved process window
2009 (English)In: Proceedings of the 35th Design Automation Conference, August 30-September 2, San Diego, USA 2009, 2009Conference paper, Published paper (Refereed)
Abstract [en]

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.

Keyword
Robustness, Finite element method, Monte Carlo, Response surface, Casting, Process window
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-72349 (URN)
Conference
The 35th Design Automation Conference, August 30-September 2, San Diego, USA
Projects
MERA
Available from: 2009-06-15 Created: 2011-11-25 Last updated: 2011-11-25Bibliographically approved
7. Robustness of residual stresses in brake discs by metamodeling
Open this publication in new window or tab >>Robustness of residual stresses in brake discs by metamodeling
2011 (English)In: proceedings of the ASME IDETC/CIE Conference, (Washington D.C., USA),2011, 2011Conference paper, Published paper (Other academic)
Abstract [en]

During casting residual stresses are developed due to the solidification and cooling. In this work the robustness of residual stresses in casted brake discs with respect to variations in four parameters is evaluated. The parameters are Young’s modulus, yield strength and hardening, time of breaking the mould and the thickness of the brake disc. The robustness analysis is performed by Monte Carlo simulation of metamodels which are surrogates to a finite element model. Quadratic response surfaces and Kriging approximations are considered. Those are based on finite element analyses defined by a Latin hypercube sampled design of experiments. In the finite element analyses an uncoupled approach is utilized where a thermal analysis generates a temperature history of the solidification and cooling. Then follows a structural analysis which is driven by the temperature history. After casting the machining of the brake disc is analyzed by gradually removing elements in the finite element model. The results show that the variation in the studied parameters yield large variation in residual stresses. The thickness of the brake disc is the parameter that has largest influence to the variation in residual stresses. Furthermore, the level of the residual stresses are in general high and might influence the fatigue life of the brake disc.

Keyword
Robustness analysis, Metamodeling, Design of experiments (DoE), Residual stresses
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-72351 (URN)
Conference
The ASME 2011 International Design Engineering Technical Conferences (IDETC) and Computers and Information in Engineering Conference (CIE), August 28-31, Washington, DC, USA
Available from: 2011-11-25 Created: 2011-11-25 Last updated: 2011-11-25Bibliographically approved

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