Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials II: Advanced Anisotropy

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• 作者：Kostas P. Soldatos
• 关键词：Anisotropy ; Fibre ; reinforced materials ; Fibre bending stiffness ; Monoclinic materials ; Non ; elliptic equations ; Non ; symmetric stress ; Polar linear elasticity ; Orthotropic materials ; Second ; gradient linear elasticity ; Two families of fibres ; Weak discontinuity surfaces ; 74A10 ; 74A35 ; 74A40 ; 74A60 ; 74B99 ; 74E10
• 刊名：Journal of Elasticity
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：118
• 期：2
• 页码：223-242
• 全文大小：622 KB
• 参考文献：1. Soldatos, K.P.: Foundation of polar linear elasticity for fibre-reinforced materials. J. Elast. 114, 155-78 (2014) CrossRef
  2. Soldatos, K.P.: On loss of ellipticity in second-gradient hyper-elasticity of fibre-reinforced materials. Int. J. Non-Linear Mech. 47, 117-27 (2012) CrossRef
  3. Spencer, A.J.M., Soldatos, K.P.: Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int. J. Non-Linear Mech. 42, 355-68 (2007) CrossRef
  4. Ting, T.C.T.: Anisotropic Elasticity. Oxford University Press, New York (1996)
  5. Jones, R.M.: Mechanics of Composite Materials. Taylor & Francis, Washington (1998)
  6. Steward, I.W.: The Static and Dynamic Continuum Theory of Liquid Crystals. Taylor & Francis, London (2004)
  7. Zheng, Q.S.: Theory of representations for tensor functions: A unified invariant approach to constitutive equations. Appl. Mech. Rev. 47, 545-87 (1994) CrossRef
  8. Spencer, A.J.M.: Deformations of Fibre-Reinforced Materials. Clarendon Press, Oxford (1972)
  9. Soldatos, K.P.: Second-gradient plane deformations of ideal fibre-reinforced materials: implications of hyper-elasticity theory. J. Eng. Math. 68, 99-27 (2010) CrossRef
  10. Soldatos, K.P.: Towards a new generation of 2D mathematical models in the mechanics of thin-walled fibre-reinforced structural components. Int. J. Eng. Sci. 47, 1346-356 (2009) CrossRef
  11. Dagher, M.A., Soldatos, K.P.: On small azimuthal shear deformation of fibre-reinforced cylindrical tubes. J. Mech. Mater. Struct. 6, 141-68 (2011) CrossRef
  12. Farhat, A.F., Soldatos, K.P.: Cylindrical bending and vibration of polar material laminates. Mech. Adv. Mat. Struct. (2015, to appear). doi:10.1080/15376494.2013.864438
  13. Spencer, A.J.M.: Constitutive theory for strongly anisotropic solids. In: Spencer, A.J.M. (ed.) Continuum Theory of the Mechanics of Fibre-Reinforced Materials. CISM Courses and Lectures, vol.?282, pp.?1-2. Springer, Wien-New York (1984)
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-2681

文摘

This paper presents a complete formulation of the linear theory of elasticity for polar fibre reinforced materials with two families of embedded fibres by building upon previous theoretical background available for transversely isotropic fibre-reinforced materials (Soldatos in J. Elast. 114:155-78, 2014). Polar material behaviour stems from the postulate that fibres of either family are not perfectly flexible. During deformation of the fibrous composite, they instead act as slender Euler-Bernoulli beams and, therefore, their individual deformation generates couple-stress and non-symmetric stress. For simplicity, the outlined formulation is based on the plausible consideration that the fibre bending mode is the most dominant and, therefore, the most significant one to be accounted for when compared with its fibre stretching and fibre twist counterparts. The simplification achieved with this consideration becomes evident in the particular case of transversely isotropic materials, when comparisons are made against the relevant full developed theory (Soldatos in J. Elast. 114:155-78, 2014). Association of this simplifying consideration with the presence of two families of fibres resistant in bending furnishes the present model with the ability to embrace groups of advanced material anisotropy, such as the case of locally monoclinic materials which represent the most advanced group. The particular cases of local and plane orthotropy, where the two families of fibres are perpendicular to each other, are also deduced and considered in detail. Because of their importance in modelling thin- and/or moderately thick-walled structural components and laminates, particular attention is also given to the so-called cases of general and special orthotropy, where the fibres of both families are straight. The obtained governing equations are non-elliptic and this feature of the model is connected with the manner that solutions involving second-gradient weak discontinuity surfaces within the material may complement corresponding solutions associated with continuous displacements having their derivatives of all orders also continuous. The manner that second-gradient weak discontinuity surfaces are sought and found is finally demonstrated in detail for the case of specially orthotropic materials.