POLITECNICO DI TORINO Department of Mechanical Engineering

1. POLITECNICO DI TORINO Department of Mechanical Engineering Thermo-mechanical problems solved with Cell Method Carlo Rosso Department of Mechanical Engineering Tel. +39 011 090 6917, e-mail: carlo.rosso@polito.it Short course on Cell Method, Turin, 2-3-4, 16-17-18 october 2007

2. Cell Method in TM analysis: overview • What Thermo-Mechanical Analysis is and what it provides • Application of CM to the Thermo-Mechanical Analysis • Bidimensional case: plate • Results • Conclusion

3. What TM Analysis is and what it provides • Two different loads on mechanical structures: Generalized forces: force, pressure, moment Temperature They cause structure deformation It causes structure deformation Strain and stress field Strain field Strains and stresses are related and they cause the structure failure

4. What TM Analysis is and what it provides The relationship between strain and stress is a feature of the material and it depends on: Temperature Time Strain rate Kelvin – Voigt model hardening relaxation creep

5. What TM Analysis is and what it provides In addition, strain is provided by stress (previous section) and by temperature. Due to the stress Due to the temperature The evaluation of strains and stresses is quite difficult when the mechanical and the thermal contributions vary in the time and in magnitude. That evaluation is useful for assessing the life duration of a component.

6. What TM Analysis is and what it provides If temperature is constant and lower that 1/3 of the melting temperature of the material, the main problems are given by stress. Normally, components under those conditions are designed for High Cycle Fatigue (more than 104 cycles). When temperature rises up and it varies with respect to time, the main problems are provided by strain. Components under high temperatures present low life duration, so it is analysed with Low Cycle Fatigue approach (lower that 104 cycles). For both cases the quantities an engineer must know are stresses and strains!

7. What TM Analysis is and what it provides At the present, commercial FE softwares do not investigate time evolution of stress and strain They solve the thermal problem (given a heat flux they compute a temperature distribution on component) then by using the output of that analysis they solve the mechanical problem. What the drawback? It is impossible to reproduce a thermo-mechanical dynamics of the system. A one shot solution is required!! ERGO USE CELL METHOD!

8. CM applied to Thermo-mechanical problems As first step, the thermal problem must be solved TONTI’s Diagram for thermal conduction

9. CM applied to Thermo-mechanical problems Boundary conditions are temperatures (on primal nodes) or heat source (on dual face) Topological equation for temperature Balance heat source equation Constitutive equation Known quantities Unknown quantities

10. F CM applied to Thermo-mechanical problems Second step, applied a mechanical load to the structure Elasticity theory has been shown in the previous lectures, but how the forces can be taken into account? Force is a source variable, so it is linked to the dual face, but equilibrium is needed: F that loads the dual face of the dual cell can be applied on the primal node that is included in the loaded dual cell.

11. Bidimensional case: plate Plate behaviour is analysed, no flexural action are considered • L is the length of the plate • h is the height of the plate • u is the horizontal displacement of the plate • v is the vertical displacement of the plate

12. Bidimensional case: plate Hooke equation is valid for linear behaviour of material and it relates stress and mechanical strain Strains are provided by temperature and mechanical loads

13. Bidimensional case: plate Stress is related to force by means of: So the external force can be written as: Unknown quantities Known quantities

14. Bidimensional case: plate For tria cell,

15. Bidimensional case: plate For tria cell, So thermal additional forces are:

16. Bidimensional case: plate Summarising the process: 1) Create the model, set the thermal and mechanical boundary conditions 2) Obtained the primal temperatures, compute the thermal force vector 3) Solve the thermo-mechanical problem: 4) Compute thermal and mechanical strains and hence mechanical stresses

17. Where we are going? If the temperature and force changes with respect to time are known: It is possible to evaluate step by step the displacements and hence the strains Second phase: integrate mechanical property variation with respect to strain, strain rate, temperature and time In such a way it is possible to simulate the behaviour of a component under actual condition and have an accurate estimation of its life.

18. Conclusions Cell Method allows: • Easy approach to multiphysics problem • Development non linear approach to thermo-mechanical problems • Easy implementation of time evolution Results achieved • Instrument to analyse in one step thermal diffusion and mechanical problem • Opportunity of computing separately the Thermal and the Mechanical strains

19. Any Questions? Thank you for your kind attention