ON DEVELOPING OF A THERMOELASTIC CONTINUUM DAMAGE MODEL FOR DIELECTRIC COMPOSITE MATERIALS

Öz

This paper deals with developing a continuum damage mechanics model belonging to constitutive equations which represent linear electro-thermo-elastic behavior of a composite material, where the material was reinforced with arbitrarily distributed single fiber family and which have micro-cracks. The composite medium is assumed to be dielectric, incompressible, homogeneous, and dependent on temperature gradient. The matrix material made of elastic material involving an artificial anisotropy because of fibers reinforcing by arbitrary distributions and the existence of micro-cracks, has been assumed as an isotropic medium. It is accepted that the fiber family is inextensible. Using the basic laws, of continuum damage mechanics and continuum electrodynamics and the equations belonging to kinematic of fiber, the constitutive functionals have been obtained. It has been detected as a result of the thermodynamic constraints that stress potential function depends on two symmetric tensors and two vectors, and the heat flux vector function depends on two symmetric tensors and three vectors. To determine arguments of the constitutive functionals, findings relating to the theory of invariants have been used as a method because of that isotropy constraint is imposed on the matrix material. Finally, the constitutive equations of symmetric stress, polarization field, asymmetric stress, heat flux vector and strain-energy density release rate have been written in material coordinates. 

Anahtar Kelimeler

• Electro-thermoelastic behavior,Continuum damage mechanics,Constitutive equations,Composite materials,Invariants

DİELEKTRİK KOMPOZİT MALZEMELER İÇİN BİR TERMOELASTİK SÜREKLİ ORTAM HASAR MODELİNİN GELİŞTİRİLMESİ ÜZERİNE

Öz

Bu makale, keyfi dağılımlı tek fiber ailesi ile takviyeli ve mikro çatlaklara sahip bir kompozit malzemenin lineer elektro-termo elastik davranışını temsil eden kurucu denklemlere ait bir sürekli ortam hasar mekaniği modeli geliştirmeyi ele almaktadır. Kompozit ortamın dielektrik, sıkıştırılamaz, homojen olduğu ve sıcaklık gradyanına bağlı olduğu varsayılmaktadır. Keyfi dağılımlı fiber takviyesi ve mikro çatlakların varlığı nedeniyle yapay bir anizotropi içeren elastik malzemeden yapılmış matris malzemesi izotropik bir ortam olarak kabul edilmiştir. Fiber ailesinin uzatılmaz olduğu kabul edilmektedir. Sürekli ortam hasar mekaniğinin ve sürekli ortam elektrodinamiğinin temel kanunları ve süreklilik fiber kinematiğine ait denklemleri kullanılarak bünye fonksiyonelleri elde edilmiştir. Termodinamik kısıtlamaların sonucu olarak, gerilme potansiyeli fonksiyonunun iki simetrik tensör ve iki vektöre bağlı olduğu ve ısı akısı vektör fonksiyonunun ise iki simetrik tensör ve üç vektöre bağlı olduğu belirlenmiştir. Bünye fonksiyonellerinin argümanlarını belirlemek için, invaryantlar teorisine ilişkin bulgular, matris malzemesine uygulanan izotropi kısıtlaması nedeniyle bir yöntem olarak kullanılmıştır. Sonunda, simetrik gerilmenin, polarizasyon alanının, asimetrik gerilmenin, ısı akısı vektörünün ve gerinme-enerjisi yoğunluğunun değişim hızının bünye denklemleri maddesel koordinat sisteminde yazılmıştır. 

Anahtar Kelimeler

• : Elektro-termoelastik davranış,Sürekli Ortam Hasar Mekaniği,Bünye denklemleri,Kompozit malzemeler,İnvaryantlar

Kaynakça

1. 1. Iannucci, L., Ankersen, J., (2006). An energy based damage model for thin laminated composites. Comp. Sci. and Tech., 66, 934–951.
2. 2. Tay,T. E., Lıu, G., Yudhanto, A., Tan, V. B. C., (2008). A Micro–Macro Approach to Modeling Progressive Damage in Composite Structures. Int. J. Damage Mechanics, 17 (1), 5-28.
3. 3. Tay, T.E., (2003). Characterization and Analysis of Delamination Fracture in Composites – An Overview of Developments from 1990 to 2001. Appl. Mechs. Revs., 56(1), 1-32.
4. 4. Zenkour, A. M., Kafr El-Sheikh, (2004). Thermal effects on the bending response of fiber-reinforced viscoelastic composite plates using a sinusoidal shear deformation theory. Acta Mechanica 171, 171–187.
5. 5. Clayton, J. D., (2009). A non-linear model for elastic dielectric crystals with mobile vacancies. International journal of Non-Linear Mechanics, 44 (6), 675-688.
6. 6. Nowacki, W., (1962). Thermoelasticity. Pergamon Press, New York.
7. 7. Nowinski, J.L., (1978). Theory of Thermoelasticity with Applications. Sijthoff & Noordhoff International Publishers, The Netherlands.
8. 8. Sih, G.C., (1962). On the singular character of thermal stress near a crack tip. Journal of Applied Mechanics, 29, 587–589.

9. 9. Wilson, W.K., Yu, I.W., (1979). The use of J-integral in thermal stress crack problems. International Journal of Fracture, 15, 377–387.
10. 10. Prasad, N.N.V., Alibadi, M.H., Rooke, D.P., (1994). The dual boundary element method for thermoelastic crack problems. International Journal of Fracture, 66, 255–272.
11. 11. Georgiadis, H.G., Brock, L.M., Rigatos, A.P., (1998). Transient concentrated thermal/mechanical loading of the faces of a crack in a coupled-thermoelastic solid. International Journal of Solids and Structures 35, 1075–1097.
12. 12. Kotousov, A.G., (2002). On a thermo-mechanical effect and criterion of crack propagation. International Journal of Fracture 114, 349–358.
13. 13. Chen, W.Q., Ding, H.J., Ling, D.S., (2004). Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution. International Journal of Solids and Structures, 41, 69–83.
14. 14. Usal, M., Usal, M.R. and Esendemir, Ü., (2008). A Continuum Formulation for FiberReinforced Viscoelastic Composite Materials With Microstructure Part - II: Isotropic Matrix Material. Sci. Eng. Composite Mater. 15(3), 235-247.
15. 15. Usal, M.R., Usal, M. and Esendemir, Ü., (2006). A Mathematical Model for Thermomechanical Behavior of Arbitrary Fiber Reinforced Viscoelastic Composites – II. Sci. Eng. Composite Mater. 13(4), 301-311.
16. 16. Erdem, A.Ü., Usal, M. and Usal, M.R., (2005). A Mathematical Model For The Electro Thermomechanical Behavior Of Fiber-Reinforced Dielectric Viscoelastic Composites With Isotropic Matrix Material. J. Fac. Eng. Arch. Gazi Univ. 20(3), 321-334.
17. 17. Usal, M., Usal, M.R. and Erdem, A.Ü., (2009). On Magneto-Viscoelastic Behavior of Fiber-Reinforced Composite Materials Part - II: Isotropic Matrix Material. Sci. Eng. Composite Mater. 16(1), 57-71.
18. 18. Usal, M.R., (2007). A Constitutive Formulation of Arbitrary Fiber-Reinforced Viscoelastic Piezoelectric Composite Materials – II. Inter. J. Non-Lin. Sci. Numer. Simul. 8(2), 275-293.
19. 19. Usal, M.R., (2011). On Constitutive Equations for Thermoelastic Dielectric Continuum in Terms of Invariants. Int. J. Eng. Sci. 49(7), 625-634.
20. 20. Usal, M., (2010). A Constitutive Formulation for the Linear Thermoelastic Behavior of Arbitrary Fiber-Reinforced Composites. Mathem. Problems Eng. (2010), Article ID 404398, 19 pages.
21. 21. Usal, M., Usal, M. R., Esendemir, Ü., (2013). On constitutive equations for thermoelastic analysis of fiber-reinforced composites with isotropic matrix material. Continu. Mech. Thermdyn., 25(1), 77-88.
22. 22. Usal M.R., Korkmaz E., Usal M., (2006). Constitutive Equations for an Elastic Media with Micro-voids. J. App. Sci., 6(4), 843-853.
23. 23. Usal, M., Usal, M.R., Erçelik, A.H., (2014). A Constitutive Model for Arbitrary FiberReinforced Composite Materials Having Micro-cracks Based on Continuum Damage Mechanics. ScientificDigest: JOAE (Journal of Applied Engineering). 2(4), paper ID: JOAE04201424-19.
24. 24. Usal, M., (, 2015). On continuum damage modeling of fiber Reinforced viscoelastic composites with micro-cracks in terms of invariants, Math. Prob. in Engin., vol. 2015, Article ID 624750, 15 pages.
25. 25. Şuhubi, E. S., (1994). Continuum Mechanics- Introduction. I.T.U., Fac. of Arts and Sci. Publication, Istanbul, Turkey.
26. 26. Weitsman, Y., (1988). A continuum damage model for viscoelastic materials. J. App. Mech., 55, 773–780.
27. 27. Weitsman, Y., (1988). Damage Coupled With Heat Conduction in Uniaxially Reinforced Composites. J. App. Mech., 55, 641-647.
28. 28. Spencer, A.J.M., (1972). Deformations of Fibre-Reinforced Materials. Clarendon press, Oxford.
29. 29. Spencer, A.J.M., (1984). Continuum Theory of the Mechanics of Fibre Reinforced Composites. Int. Cent. for Mech. Sci., Course and Lect., nr.282, ed. by Spencer A.J.M., Springer Verlag, Wien-New York.
30. 30. Maugin, G. A., (1988). Continuum Mechanics of Electromagnetic Solids, Elsevier Science, North-Holland.
31. 31. Eringen, A. C. and Maugin, G.A., (1990). Electrodynamics of Continua, Vol.I, Fondations and Solid Media, North-Holland.
32. 32. Simo, J.C., and Ju, J.W., (1987). Strain and Stress Based Continuum Damage ModelFormulation, International Journal of Solid and Structures, 23 (7), 821-840.
33. 33. Spencer, A.J.M., (1971). Theory of invariants. In. Eringen, A. C. (Ed.), Continuum Physics, vol. 1. Academic Press, New York, 239–353.
34. 34. Eringen, A. C., (1980). Mechanics of Continua. Robert E. Krieger Publishing, Hungtington, NY, USA.
35. 35. Eringen, A.C., (1971). Continuum Physics – Mathematics. vol. 1, Acad. Press, New York.
36. 36. Zheng, Q.S. and Spencer, A.J.M., (1993). Tensors Which Characterize Anisotropies. Int. J. Eng. Sci. 31, 679 – 693.
37. 37. Zheng, Q.S., (1993). On Transversely Isotropic, Orthoropic and Relative Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. Part V: The Irreducibility of the Representations for Three Dimensional Orthoropic Functions and the Summary. Int. J. Eng. Sci. 31, 1445-1453.