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Optimizing the composition profile of a functionally graded interlayer using a direct transcription method

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Abstract

We consider a three-layer cylinder consisting of a functionally graded interlayer sandwiched between a metallic layer and a ceramic layer, and address the problem of finding the interlayer composition profile minimizing the stresses resulting from material property mismatch and induced in the cylinder by temperature and pressure loading. First, we present various ways of formulating the continuous minimization problem as a thermoelastic optimal control problem, where the control is the volume fraction of either of the constituents of the interlayer, the metal or the ceramic. A direct transcription method is then used to discretize the continuous problem into a finite dimensional nonlinear optimization problem. Lastly, we present a numerical solution to the discretized problem for a specific case. Interestingly, this solution is such that the overall material distribution contains a discontinuity in volume fraction, and therefore in material properties, at the interface between the metal and the graded interlayer

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Abbreviations

Latin letters:

 

G:

Shear modulus

K:

Bulk modulus

p:

Pressure

Q:

Heat rate (per unit length and angle)

r:

Radial coordinate

T:

Temperature

T ref :

Reference temperature for zero thermal stress

u:

Radial displacement

Greek letters:

 

α:

Coefficient of thermal expansion

σ r :

Radial stress

σθ :

Hoop stress

σ z :

Axial stress

ρ i :

Inner radius of layer i+1, for i=0,1,2, and outer radius of layer i, for i=1,2,3

λ:

Coefficient of thermal conductivity

ξ:

Volume fraction of ceramic

μ:

Set of material properties ={G,K,α,λ}

Ψ:

Vector function representing the micromechanical model

Superscript:

 

′:

Differentiation with respect to γ

Subscripts:

 

+:

Right limit

−:

Left limit

c:

Ceramic

m:

Metal

g:

Graded

ext:

External

int:

Internal

References

  1. Suresh S, Mortensen A (1998) Fundamentals of functionally graded materials, processing and thermomechanical behaviour of graded metals and metal–ceramic composites. In: Institute of materials (London England), no 698. IOM Communications Ltd, London

    Google Scholar 

  2. Torquato S (2002) Random heterogenous materials: microstructure and macrostructure properties. In: Interdisciplinary applied mathematics, vol 16. Springer, Berlin Heidelberg New York

    Google Scholar 

  3. Markworth AJ, Ramesh KS, Parks WP (1995) Modelling studies applied to functionally graded materials. J Mater Sci 30(9):2183–2193

    Article  Google Scholar 

  4. Ravichandran KS (1995) Thermal residual stresses in a functionally graded material system. Mater Sci Eng A 201(1–2):269–276

    Article  Google Scholar 

  5. Lee WL, Stinton DP Berndt CC, Erdogan F, Lee Y-D, Mutasim Z (1996) Concept of functionally graded materials for advanced thermal barrier coating applications. J Am Ceramics Soc 79(12):3003–3012

    Article  Google Scholar 

  6. Tanaka K, Tanaka Y, Enomoto K, Poterasu VF, Sugano Y (1993) Design of thermoelastic materials using direct sensitivity and optimization methods. Reduction of thermal stresses in functionally gradient materials. Comput Meth Appl Mech Eng 106:271–284

    Article  Google Scholar 

  7. Tanaka K, Watanabe H, Sugano Y, Poterasu VF (1996) A multicriterial material tailoring of a hollow cylinder in functionally gradient materials: scheme to global reduction of thermoelastic stresses. Comput Meth Appl Mech Eng 135:369–380

    Article  MATH  Google Scholar 

  8. Cho JR, Ha DY (2002) Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM. Comput Meth Appl Mech Eng 191:3195–3211

    Article  MATH  Google Scholar 

  9. Turteltaub S (2002) Optimal control and optimization of functionally graded materials for thermomechanical processes. Int J Solids Struct 39(12):3175–3197

    Article  MATH  Google Scholar 

  10. Ootao Y, Tanigawa Y, Nakamura T (1999) Optimization of material composition of FGM hollow circular cylinder under thermal loading: a neural network approach. Compos Part B 30:415–422

    Article  Google Scholar 

  11. Ootao Y, Kawamura R, Tanigawa Y, Imamura R (1999) Optimization of material composition of nonhomogeneous hollow sphere for thermal stress relaxation making use of neural network. Comput Meth Appl Mech Eng 180:185–201

    Article  MATH  Google Scholar 

  12. Ootao Y, Tanigawa Y, Ishimaru O (2000) Optimization of material composition of functionally graded plate for thermal stress relaxation using a genetic algorithm. J Therm Stresses 23(3):257–271

    Article  Google Scholar 

  13. Boussaa D. (2000) Optimizing a compositionally graded interlayer to reduce thermal stresses in a coated tube. Comptes rendus de l’Académie des sciences, Série IIb 238:209–215

    MATH  Google Scholar 

  14. Bamberger Y (1997) Mécanique de l’ingénieur III, solides déformables. Hermann Paris, pp 68–71 (in French)

  15. Zuiker J, Dvorak G (1994) The effective properties of functionally graded composites – I. Extension of the Mori–Tanaka method to linearly varying field. Compos Eng 4(1):19–35

    Google Scholar 

  16. Aboudi J, Pindera M-J, Arnold SM (1994) Elastic response of metal matrix composites with tailored microstructures to thermal gradients. Int J Solids Struct 31(10):1393–1428

    Article  MATH  Google Scholar 

  17. Zuiker JR (1995) Functionally graded materials: choice of micromechanics model and limitations in property variation. Compos Eng 5(7):807–819

    Article  Google Scholar 

  18. Grujicic M, Zhang Y (1998) Determination of effective elastic properties of functionally graded materials using Voronoi cell finite element method. Mater Sci Eng A 251(1–2):64–76

    Article  Google Scholar 

  19. Reiter T, Dvorak GJ, Tvergaard V (1997) Micromechanical models for graded composite materials. J Mech Phys Solids 45(8):1281–1302

    Article  Google Scholar 

  20. Reiter T, Dvorak GJ (1998) Micromechanical models for graded composite materials: II. Thermomechanical loading. J Mech Phys Solids 46(9):1655–1673

    Article  MATH  Google Scholar 

  21. Bendsøe MP, Sigmund O (2003) Topology optimization: theory methods and applications, 2nd edn. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  22. Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390

    Article  Google Scholar 

  23. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech (Ingenieur Archiv) 69(9–10):635–654

    Article  MATH  Google Scholar 

  24. Allaire G, Kohn RV (1993) Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur J Mech A/Solids 12(6):839–878

    MathSciNet  MATH  Google Scholar 

  25. Allaire G, Bonnetier E, Francfort G, Jouve F (1997) Shape optimization by the homogenization method. Numer Math 76(1):27–68

    Article  MathSciNet  MATH  Google Scholar 

  26. Williamson RL, Robin BH, Drake JT (1994) Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part II. Interface optimization for residual stress reduction. J Appl Phys 74(2):1321–1326

    Google Scholar 

  27. Shaw LL (1998) Thermal residual stresses in plates and coatings composed of multi-layered and functionally graded materials. Compos Part B Eng 29(3):199–210

    Article  Google Scholar 

  28. Ootao Y, Tanigawa Y (1999) Three-dimensional transient thermal stresses of functionally graded rectangular plate due to partial heating. J Ther Stresses 22(1):35–55

    Article  Google Scholar 

  29. Praveen GN, Chin CD, Reddy JN (1999) Thermoelastic Analysis of Functionally graded Ceramic-Metal Cylinder. J Eng Mech 125(11):1259–1267

    Article  Google Scholar 

  30. Cheng Z-Q, Batra RC (2000) Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Compos Part B Eng 31(2):97–106

    Article  Google Scholar 

  31. Elperin T, Rudin G (2000) Thermal reliability testing of functionally gradient materials using thermal shock method. Heat Mass Transf 36(3):231–236

    Article  Google Scholar 

  32. Cho JR, Oden JT (2000) Functionally graded material: a parametric study on thermal-stress characteristics using the Crank–Nicolson–Galerkin scheme. Comput Meth Appl Mech Eng 188:17–38

    Article  MATH  Google Scholar 

  33. Berryman JG, Berge PA (1996) Critique of two explicit schemes for estimating elastic properties of multiphase composites. Mech Mater 22:149–164

    Article  Google Scholar 

  34. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic, London, New York

    MATH  Google Scholar 

  35. Betts JT (2001) Practical methods for optimal control using nonlinear programmig. In: Advances in design and control. SIAM, Philadelphia

    Google Scholar 

  36. Betts JT, Erb SO (2003) Optimal low thrust trajectories to the moon. SIAM J Appl Dyn Sys 2(2):144–170

    Article  MathSciNet  MATH  Google Scholar 

  37. Maurer H, Mittelmann HD (2000) Optimization techniques for solving elliptic control problems with control and state constraints: part 1. Boundary control. Comput Optim Appl 16:29–55

    Article  MathSciNet  MATH  Google Scholar 

  38. Maurer H, Mittelmann HD (2001) Optimization techniques for solving elliptic control problems with control and state constraints: part 2. Distributed control. Comput Optim Appl 18:141–160

    Article  MathSciNet  MATH  Google Scholar 

  39. Betts JT (1998) Survey of numerical methods for trajectory optimization. J Guid Control Dyn 21(2):193–207

    Article  Google Scholar 

  40. Biegler LT, Cervantes AM, Wächter A (2002) Advances in simultaneous strategies for dynamic process optimization. Chem Eng Sci 57:575–593

    Article  Google Scholar 

  41. Conn AR, Gould NIM, Toint PL (1992) LANCELOT A Fortran Package for Large-Scale Nonlinear Optimization (Release A). Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  42. Bui HD (1994) Inverse problems in the mechanics of materials : an introduction. CRC, Boca Raton, FL

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mécanique et d’Acoustique, CNRS, 31, chemin Joseph Aiguier, F-13402, Marseille cedex 20, France

    Djaffar Boussaa

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  1. Djaffar Boussaa
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Boussaa, D. Optimizing the composition profile of a functionally graded interlayer using a direct transcription method. Comput Mech 39, 59–71 (2006). https://doi.org/10.1007/s00466-005-0008-7

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  • DOI: https://doi.org/10.1007/s00466-005-0008-7

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