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REFERENCES [1] Anlas G, Santare M.H., Lambros J., Numerical calculation of stress intensity factors in functionally graded materials, Int. J. Fract, 104 (2000) 131-143. [2] Rao B.N., Rahman S., Mesh-free analysis of cracks in isotropic functionally graded materials,Eng.Fract. Mech, 70(2003) 1-27. [3] Kim J.H., Paulino G.H., Mixed-mode fracture of orthotropic functionally graded materials using finite elements and the modified crack closure method,Eng.Fract. Mech, 69 (2002) 1557-1586. [4] Kim J.H., Paulino G.H., Mixed-mode J-integral formulation and

Abstract

The simplified approach to the modelling of low cycle fatigue (LCF) of functionally graded materials (FGM) based on the continuum mechanics is presented. The fatigue damage model takes into account the mechanical part of the load and a constant service temperature. The concept of FGM as a particle-reinforced metal-matrix composite with gradual change of the reinforcement fraction is used. The FGM is considered as a material consisting of homogeneous layers containing different volume fractions of the reinforcement. The variation of the reinforcement fraction changes the material properties for each layer. The different material properties are obtained according to modified rule of mixture. Since the fatigue damage of metal matrix composites is strongly influenced by the inelastic deformation of the metallic matrix, the constitutive equations of LCF damage model are taken into consideration. The combined isotropic/ kinematic hardening model with linear behaviour of isotropic and kinematic parts of hardening is adopted. The damage scalar parameter is associated with the plastic energy dissipation which is used to update the material properties. The fatigue damage model presented in this paper is applied to the fatigue damage analysis of the cooling channel of thruster used in space shuttles and rockets.

. and Gosz, M. (2002) On the computation of mixed-mode stress intensity factors in functionally graded materials. Int. J. Solids Structures, 39, 2557-2574. [6] Eischen, J.W. (1987) Fracture of nonhomogeneous materials. Int. J. Fracture 34, 3-22. [7] Erdogan, F. (1995) Fracture mechanics of functionally graded materials. Compos. Eng., 5, 753-770. [8] Eshelby, J.D; Read, W.T. and Shockley, W. (1953) Anisotropic elasticity with applications to dislocations. Acta Metallurgica, 1, 251-259. [9] Gao, H. and Chiu, C.H. (1992) Slightly curved or kinked cracks in anisotropic

References [1] Koizumi M. and Niino M. (1995): Overview of FGM research in Japan. – MRS Bulletin, vol.20, No.1, pp.19- 24. [2] Mortensen A. and Suresh S. (1995): Functionally graded materials and metal-ceramic composites. – Part I: Processing, International Materials Reviews, vol.40, No.6, pp.239-265. [3] Wang S.S. (1983): Fracture mechanics for delamination problems in composite materials. – Journal of Composite Materials, vol.17, No.3, pp.210-223. [4] Niino M., Hirai T. and Watanabe R. (1987): The functionally gradient materials. – Journal of the Japan

., & Zivelonghi, A. (2013). Recent progress in research on tungsten materials for nuclear fusion applications in Europe. J. Nucl. Mater. , 432 (1/3), 482–500. DOI: 10.1016/j.jnucmat.2013.03.062. 3. Missiaen, J. M., Raharijaona, J. J., Antoni, A., Pascal, C., Richou, M., & Magaud, P. (2011). Design of a W/steel functionally graded material for plasma facing components of DEMO. J. Nucl. Mater. , 416 (3), 262–269. DOI: 10.1016/j.jnucmat.2011.05.054. 4. Weber, T., Stueber, M., Ulrich, S., Vaßen, R., Basuki, W. W., Lohmiller, J., Sittel, W., & Aktaa, J. (2013). Functionally

misfitting inclusions. Acta Metallurgica, Vol. 21, pp. 571-574. Sladek, V. - Sladek, J. - Zhang, Ch. (2008) Computation of stresses in non-homogeneous elastic solids by local integral equation method: a comparative study. Computational Mechanics, Vol. 41, pp. 827-845. Sladek, V. - Sladek, J. - Sator, L. (2013) Physical decomposition of thin plate bending problems and their solution by mesh-free methods. Engineering Analysis with Boundary Elements, Vol. 37, pp. 348-365. Suresh, S. - Mortensen A. (1998) Fundamentals of Functionally Graded Materials. Institute of Materials

Functionally Graded Materials: Processing and Thermomechanical Behaviour of Graded Metals and Metal-Ceramic Composites , Cambridge University Press, Cambridge. 9. Wajand J.A., Wajand J.T. (2005), Middle- and high-speed combustion engines , WNT, Warszawa, (in Polish). 10. Życzkowski M. (1988), Strength of structural elements, PWN Warszawa (in Polish). 11. Material data Library: http://www.matweb.com

specific properties, which vary according to a known function. The new material was named as Functionally Graded Material (FGM), and as the name indicates, this material is generally associated with composite particles where the volume fraction of the particles varies in one or more directions [ 1 ]. The invention of the new material (FGM) opened up new avenues by increasing the performance of industrial machines due to its intrinsic qualities such as lightness (combined with high strength characteristics) and good resistance to corrosion. The industrial applications of

References [1] Birman, V., Byrd, L.W. Modeling and analysis of functionally graded materials and structures. Applied mechanics reviews (2007) 60: 195-216. [2] Ying, J., Lu, C.F., Chen, W.Q. Two -dimensional elasticity solutions for functionally graded beams resting on elastic foundation. Composite Structures (2008) 84: 209-219. [3] Benatta, M.A, Mechab, I., Tounsi, A., Adda Bedia, E.A. Static analysis of functionally graded short beams including warping and shear deformation effects. Computational Materials Science (2008)44: 765-773. [4] Kadoli, R., Akhtar, K

. 11, 385–414. 39. Yang F., Chong A.C.M., Lam D.C.C., Tong P., (2002), Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures , Vol. 39(10), 2731–2743. 40. Zhang K, Ge M.-H., Zhao C., Deng Z-C., Lu, X-J., (2019), Free vibration of nonlocal Timoshenko beams made of functionally graded materials by Symplectic method, Composites Part B: Engineering , 156, 174–184.