Continuum mechanics 0. Introduction and motivation
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1 and motivation Aleš Janka office Math September 22, 2010, Université de Fribourg What is? = domain of physics and engineering describing: elasticity and plasticity of solids and dynamics of fluids (liquids or gases).
2 What is a continuum? A continuum = a physical object with mass which can be mapped onto points of a subdomain Ω IR 3. Mass distribution in Ω is supposed to be continuous: for any subdomain ω Ω, the mass m ω of ω is calculated by m ω = dm and small changes in the size of ω produce small changes in m ω. ω What is a continuum? Modeling objects as continua neglects atomic, molecular and crystal structure of mass. The continuum approach is nevertheless a good approximation on length scales much greater than the molecular scale. What does continuum mechanics do? Applies fundamental physical laws (conservation of mass, momentum and energy, force equilibrium... ) to continua to derive differential equations describing their behavior. Information about the particular material of the continua is added through an empiric constitutive relation / law.
3 in practice.. in engineering: crash-tests in practice.. in engineering: crash-tests
4 in practice.. in engineering: crash-tests in practice.. in engineering: aeronautics
5 in practice.. in engineering: aeronautics in practice.. weather forecasts Meteosuisse forecast for Sep 22, 2010, temperatures
6 in practice.. in natural sciences: formation of galaxies (fluid dynamics) in practice.. in natural sciences: formation of galaxies (fluid dynamics)
7 in practice.. in medicine (bone and tissue mechanics, blood flow) from Arbenz, van Lenthe, Mennel, Müller and Sala: A scalable multi-level preconditioner for matrix-free µ-finite element analysis of human bone structures, Int. J. Numer. Meth. in Engineering 73 (2008), pp in practice.. in medicine (bone and tissue mechanics, blood flow) from Arbenz, van Lenthe, Mennel, Müller and Sala: A scalable multi-level preconditioner for matrix-free µ-finite element analysis of human bone structures, Int. J. Numer. Meth. in Engineering 73 (2008), pp
8 in practice.. in biology (tissue mechanics and growth) from Schmundt et al Programme Kinematic description of a continuum: deformation and motion of Ω. Mechanical equilibria and conservation laws. Constitutive laws of materials: elastic and visco-elastic materials, Newtonian fluids. Typical problems of continuum mechanics: analytical and numerical solution of elasto-statics/dynamics, compressible and incompressible elasticity, Newtonian fluids.
9 Necessary mathematical techniques Mechanical state and properties of a continuum are independent of the choice of a coordinate system. We will introduce and use tensor calculus: covariant and contravariant tensors and basic tensor operations, tensor fields in euclidean space, derivatives of tensors. Solutions of differential equations will be calculated analytically (on simple problems) or numerically: we will (re)-introduce the basics of a finite element method. The beauty of simple analytical formulas: rubber baloon Great deal of understanding through a simple toy model Inflate a rubber party-balloon with an internal gas pressure p Initially, the baloon stretches to two different diameters, why?
10 The beauty of simple analytical formulas: rubber baloon Great deal of understanding through a simple toy model Force equilibrium on the cut: σ t r r0 r p F σ = F p 2πrtσ = πr 2 p elastic stress: σ = Eε = 2πr 2πr 0 2πr 0 E rubber incompressibility: 4πr 2 t = 4πr 2 0 t 0 p(r) = 2E t 0 r 2 0 r 3 r r 0 r 0 The beauty of simple analytical formulas: rubber baloon Great deal of understanding through a simple toy model Force equilibrium on the cut: F σ = F p 2πrtσ = πr 2 p elastic stress: σ = Eε = 2πr 2πr 0 2πr 0 rubber incompressibility: 4πr 2 t = 4πr 2 0 t 0 p(r) = 2E t 0 r 2 0 r 3 r r 0 r 0 E
11 The beauty of mathematical analysis: singularities Why it breaks always at a kink? von Mieses stress: indicator of plastic deformation and rupture Mathematical analysis of the solution of elasticity equations predicts, that rupture occurs in re-entrant corners (kinks)! The beauty of mathematical analysis: (in)stability Buckling phenomenon The nightmare of civil engineers and architects
12 The beauty of mathematical analysis: (in)stability Buckling phenomenon The nightmare of civil engineers and architects The beauty of mathematical analysis: (in)stability Buckling phenomenon The nightmare of civil engineers and architects
13 The beauty of mathematical analysis: (in)stability Buckling phenomenon But it can also be exploited to our advantage! shock absorber for road safety
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