The theory of long-term damageability of a homogeneous material is generalized to the case of an orthotropic fibrous composite material with a stochastic structure. Equations of mechanics of microinhomogeneous media of this structure form the base of the theory. The process of damage of components of a composite is modeled by the formation of stochastically located micropores. The criterion of fracture of a unit microvolume is characterized by its long-term strength determined by the dependence of the time of brittle fracture on the degree of closeness of the equivalent stress to its limit value, which characterizes the short-term strength on the basis of the Huber–von Mises criterion accepted as an arbitrary function of coordinates. Efficient deformation properties and the stress-strain state of an orthotropic fibrous composite with microdamages in components are determined on the base of stochastic equations of elasticity of orthotropic media. For given macrostresses and macrostrains and an arbitrary moment of time, balance equations of damage (porosity) of components are formulated. On the basis of the iteration method, we construct algorithms for calculating dependences of microdamage of components of an orthotropic fibrous material on time and dependences of macrostresses or macrostrains on time and obtain the corresponding curves for the case of a bounded function of the long-term microstrength, which is approximated by an exponential law.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. M. Kachanov, Fundamentals of Fracture Mechanics [in Russian], Nauka, Moscow (1974).
L. V. Nazarenko, “Influence of microfractures on the deformation properties of anisotropic materials,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 63–67 (1999).
L. V. Nazarenko, “Deformation properties of granular–fibrous composites in the case of microdamageability of the matrix,” Prykl. Probl. Mekh. Mat., Issue 6, 146–153 (2008).
V. P. Tamuzh and V. S. Kuksenko, Micromechanics of Fracture of Polymeric Materials [in Russian], Zinatne, Riga (1978).
L. P. Khoroshun, B. P. Maslov, E. N. Shikula, and L. V. Nazarenko, Statistical Mechanics and Efficient Properties of Materials [in Russian], Naukova Dumka, Kiev (1993) (A. N. Guz’ (editor), Mechanics of Composites, Vol. 3).
F. Desrumaux, F. Meraghni, and L. Benzeggagh, “Generalised Mori–Tanaka scheme to model anisotropic damage using numerical Eshelby tensor,” J. Compos. Mater., 35, No. 7, 603–623 (2001).
L. P. Khoroshun, “Methods of the theory of random functions in determining the macroscopic properties of microheterogeneous media,” Prikl. Mekh., 14, No. 2, 3–17 (1978); English translation: Int. Appl. Mech., 14, No.2, 113–124 (1978).
L. P. Khoroshun, “Principles of the micromechanics of material damage. 1. Short-term damage,” Prikl. Mekh., 34, No. 10, 120–127 (1998); English translation: Int. Appl. Mech., 34, No. 10, 1035–1041 (1998).
L. P. Khoroshun, “Principles of the micromechanics of material damage. 2. Long-term damage,” Prikl. Mekh., 43, No. 2, 108–121 (2007); English translation: Int. Appl. Mech., 43, No. 2, 217–227 (2007).
L. P. Khoroshun and L. V. Nazarenko, “Deformation and microdamaging of a discrete-fibrous composites with transversally isotropic components,” Prikl. Mekh., 39, No. 6, 82–90 (2003); English translation: Int. Appl. Mech., 39, No. 6, 696–703 (2003).
L. P. Khoroshun and E. N. Shikula, ”Short-term microdamageability of laminated materials under thermal actions,” Prikl. Mekh., 38, No. 4, 60–68 (2008); English translation: Int. Appl. Mech., 38, No. 4, 432–439 (2008).
L. P. Khoroshun and E. N. Shikula, “Short-term microdamageability of fibrous materials with transversely isotropic fibers under thermal actions,” Prikl. Mekh., 38, No. 6, 74–83 (2002); English translation: Int. Appl. Mech., 38, No. 6, 701–709 (2002).
L. V. Nazarenko, “Deformation of composites with arbitrarily oriented orthotropic fibers under matrix microdamages,” Mat. Met. Fiz.-Mekh. Polya, 51, No. 4, 181–192 (2008).
L. V. Nazarenko, “Deformation of orthotropic composites with unidirectional ellipsoidal inclusions under matrix microdamages,” Mat. Met. Fiz.-Mekh. Polya, 51, No. 1, 121–130 (2008); English translation: J. Math. Sci., 160, No. 4, 429–442 (2009).
L. V. Nazarenko, “Elastic properties of materials with ellipsoidal pores,” Prikl. Mekh., 32, No. 1, 53–60 (1996); English translation: Int. Appl. Mech., 32, No. 1, 46–52 (1996).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 52, No. 1, pp. 122–131, January–March, 2009.
Rights and permissions
About this article
Cite this article
Nazarenko, L.V. Damageability of a material reinforced with unidirectional orthotropic fibers for an exponential function of long-term microstrength. J Math Sci 168, 653–664 (2010). https://doi.org/10.1007/s10958-010-0016-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-0016-9