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Introduction

v. X =(t , R ). x =(t, r ). Material manifold. y. Z. x.

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Introduction

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  1. v X=(t,R) x=(t,r) Material manifold y Z x Objectivity and continuum mechanicsCs. Asszonyi1, T. Fülöp1 and P. Ván1,21MONTAVID Thermodynamic Research Group, Budapest, Hungary; 2Dept. 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. Problems: too restrictive, paradoxes, compatibility with kinetic theory. Suggestion: observer independent treatment (4malism). velocity deformation gradient Derivative of a vector: Problem with the formulation of Noll Rigid rotating frames: →inertial forces brought in Noll (1958) upper convected Four-transformations, four-Jacobian: Tensorial property – form of the derivative. Constitutive theory – nonequilibrium thermo [3] where  four-velocity is an objective vector. force flux objective time derivative Simple shear with a single internal variable: Non-relativistic spacetime [1] spacetime ≠space + time viscosity and viscometric functions Absolute time. Conclusions Space-time M: four dimensional affine space (over the vector spaceM), Time I: is a one-dimensional affine space, Time evaluation : MI: is an affine surjection. Distance: Euclidean structure on E=Ker() Objectivity is best formulated objectively, without introducing reference frames. Four-quantities and a non-relativistic 4malism is elaborated. Kinematical consequence(Fülöp in GS-CM10): no reference configuration for fluidsdistinguished deformation and strain for solids Constitutive theory:frame independent Liu procedure (spacetime-nonlocal)[4] rheological models (preliminary) Open questions: Relativistic correspondence Mass-momentum or energy-momentum?  TIME CANNOT BE NEGLECTED! Consequences: 4-quantities: Restrictions: vectors and covectors Four manifolds - objective derivatives [2] References [1] T. Matolcsi. Spacetime Without Reference Frames. Akadémiai Kiadó(Publishing House of theHungarian 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. [4] P. Ván. Internal energy in relativistic dissipative fluids, Journal of Mechanics and Materials and Structures, 2008, 3/6, 1161-1169. Kinematics: deformation gradient – existence gradient Id. 804

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