Erweiterung der Sätze der linearen Elastizität auf Gradientenelastizität und deren Anwendung
Mechanische Eigenschaften von metallischen Werkstoffen und ihre mikrostrukturellen Ursachen
Zusammenfassung der Projektergebnisse
The project “Theorems of linear elasticity extended to gradient elasticity and their applications” dealt with the proof of fundamental theorems of coupled anisotropic strain gradient elasticity theory and the development of different models within it. Namely: conditions for positive definiteness of the potential energy density, and complementary strain energy density, have been derived and the uniqueness theorem, principles of minimum potential energy and minimum complementary energy, have been proven. To obtain inequalities for the positive definiteness including the coupling term and tensorial relations for the compliance tensors of fourth-, fifth- and sixth-rank, a diagonalization in terms of block matrices is given, such that the potential energy density is presented in an uncoupled quadratic form of a modified strain and the second gradient of displacement. Tensorial expressions for the conditions of positive definiteness of the stored energy density and for the compliance tensors are presented for arbitrary symmetry class of gradient materials. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem and the principles of minimum potential energy and minimum complementary energy have been proven for the case of coupled linear strain gradient elasticity for arbitrary symmetry class of gradient materials. To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of the displacement. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in a standard manner by considering the difference between the two solutions, similar to Kirchhoff’s proof in classical elasticity. The proofs of both principles of a minimum of potential and complementary energies are provided in the usual manner adopted in the classical theory of elasticity. Betti's reciprocal theorem for the case of coupled linear strain gradient elasticity has been proven using an equivalent transformation of the stain and strain gradient energy density. However, it has not been published because we have found, that these results have already been presented in the literature. Betti's reciprocal theorem is a key point for proof of the theorem of Clapeyron and Saint- Venant's principle which follow directly from this theorem. As an application of gradient theories, two models of coupled gradient thermo-hydrodynamics and heat balance and of coupled gradient thermo-elasticity and stationarity of thermal conductivity are developed jointly by Russian and German groups. As an application of strain gradient theory for the solution of boundary value problems a mixed variation formulation of the finite element method has been developed and applied to solve boundary value problems within the strain gradient elasticity theory. A set of modeling plane crack problems has been solved, namely, the tension of the plate with central, single-edge, and two (symmetrically placed) edge cracks. The first problem has been considered in the context of similar results available in the literature. It indicates good agreement with those obtained by FEM, by the boundary element method, and by an analytical/numerical technique based on hypersingular integral equations. To the best author’s knowledge, edge crack problems are considered within strain gradient elasticity theory for the first time. All three problems demonstrate specific qualitative features characterizing stain gradient solutions, increasing crack stiffness with length scale parameters, and cusp-like closure effect. Therefore, within the project, all the most important fundamental theorems of the coupled linear anisotropic strain gradient elasticity theory have been proven, different models within this theory have been developed and a set of modeling boundary value problems within the strain gradient elasticity theory has been solved.
Projektbezogene Publikationen (Auswahl)
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Positive definiteness in coupled strain gradient elasticity. Continuum Mechanics and Thermodynamics, 33(3), 713-725.
Nazarenko, Lidiia; Glüge, Rainer & Altenbach, Holm
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Effective properties of particulate nano-composites including Steigmann–Ogden model of material surface. Computational Mechanics, 68(3), 651-665.
Nazarenko, Lidiia; Stolarski, Henryk & Altenbach, Holm
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Generalized Brinkman-Type Fluid Model and Coupled Heat Conductivity Problem. Lobachevskii Journal of Mathematics, 42(8), 1786-1799.
Belov, P. A.; Altenbach, H.; Lurie, S. A.; Nazarenko, L. & Kriven, G. I.
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Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 101(9).
Nazarenko, Lidiia; Glüge, Rainer & Altenbach, Holm
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Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions. Continuum Mechanics and Thermodynamics, 34(1), 93-106.
Nazarenko, Lidiia; Glüge, Rainer & Altenbach, Holm
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Coupled problems of gradient thermoelasticity for periodic structures. Archive of Applied Mechanics, 93(1), 23-39.
Lurie, S.; Volkov-Bogorodskii, D.; Altenbach, H.; Belov, P. & Nazarenko, L.
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On variational principles in coupled strain-gradient elasticity, Mathematics and Mechanics of Solids 27(10) 2256–2274.
Lidiia Nazarenko, Rainer Glüge & Holm Altenbach
