Objectivity and continuum mechanics Cs. Asszonyi 1, T. Fülöp 1 and P. Ván 1, 2 1 MONTAVID Thermodynamic Research Group, Budapest, Hungary; 2 Dept. of Theoretical Physics, KFKI, RMKI, Budapest, Hungary, asszonyi@gmail. com, tamas. fulop@gmail. com, vpet@rmki. kfki. hu Introduction Principle: Material is independent of the observer (reference frame) → material frame indifference, → objective time derivatives. Consequences: deformation concept, material manifold concept, restricted constitutive relations, rheology. velocity deformation gradient Derivative of a vector: Problems: too restrictive, paradoxes, compatibility with kinetic theory. Suggestion: observer independent treatment (4 malism). Problem with the formulation of Noll Rigid rotating frames: → inertial Noll (1958) forces brought in Four-transformations, four-Jacobian: upper convected Tensorial property – form of the derivative. Constitutive theory – nonequilibrium thermo [3] where four-velocity is an objective vector. Non-relativistic spacetime [1] force flux objective time derivative Simple shear with a single internal variable: v y Z spacetime ≠ space + time x viscosity and viscometric functions Absolute time. four dimensional affine space (over the vector space M), Time I: is a one-dimensional affine space, Time evaluation : M I: is an affine surjection. Distance: Euclidean structure on E=Ker( ) Conclusions Space-time M: TIME CANNOT BE NEGLECTED! Consequences: Objectivity is best formulated objectively, without introducing reference frames. Four-quantities and a non-relativistic 4 malism is elaborated. Kinematical consequence (Fülöp in GS-CM 10): no reference configuration for fluids distinguished deformation and strain for solids Constitutive theory: frame independent Liu procedure (spacetime-nonlocal) [4] rheological models (preliminary) 4 -quantities: Restrictions: vectors and covectors Open questions: Four manifolds - objective derivatives [2] Relativistic correspondence Mass-momentum or energy-momentum? References X=(t, R) Material manifold x=(t, r) [1] T. Matolcsi. Spacetime Without Reference Frames. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1993. [2] T. Matolcsi and P. Ván. Absolute time derivatives. Journal of Mathematical Physics, 2007, 48: 053507– 19, (math-ph/0608065). [3] P. Ván. Objective time derivatives in non-equilibrium thermodynamics, Proceedings of the Estonian Academy of Sciences, 2008, 57/3, 127– 131. Kinematics: TEMPLATE DESIGN © 2009 www. Poster. Presentations. com deformation gradient – existence gradient [4] P. Ván. Internal energy in relativistic dissipative fluids, Journal of Mechanics and Materials and Structures, 2008, 3/6, 1161 -1169. Id. 804
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