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Continuum mechanics0. Introduction and motivation
Ales Janka
office Math [email protected]
http://perso.unifr.ch/ales.janka/mechanics
September 22, 2010, Universite de Fribourg
Continuum mechanics 0. Introduction
What is Continuum mechanics?
Continuum mechanics = domain of physics and engineering describing:elasticity and plasticity of solids anddynamics of fluids (liquids or gases).
Continuum mechanics 0. Introduction
What is a continuum?
A continuum = a physical object with mass which can be mappedonto points of a subdomain Ω ⊂ IR3. Mass distribution in Ω issupposed to be continuous:
for any subdomain ω ⊂ Ω, the mass mω of ω is calculated by
mω =
∫ω
dm
and small changes in the size of ω produce small changes in mω.
Continuum mechanics 0. Introduction
What is a continuum?
Modeling objects as continua neglects atomic, molecular andcrystal structure of mass.
The continuum approach is nevertheless a good approximation onlength 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 toderive differential equations describing their behavior.Information about the particular material of the continua is addedthrough an empiric constitutive relation / law.
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in engineering: crash-tests
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in engineering: crash-tests
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in engineering: crash-tests
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in engineering: aeronautics
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in engineering: aeronautics
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. weather forecasts
Meteosuisse forecast for Sep 22, 2010, temperatures
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in natural sciences: formation of galaxies (fluid dynamics)
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in natural sciences: formation of galaxies (fluid dynamics)
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in medicine (bone and tissue mechanics, blood flow)
from Arbenz, van Lenthe, Mennel, Muller 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. 927–947
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in medicine (bone and tissue mechanics, blood flow)
from Arbenz, van Lenthe, Mennel, Muller 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. 927–947
Continuum mechanics 0. Introduction
Continuum mechanics in practice.. in biology (tissue mechanics and growth)
from Schmundt et al. 2006
Continuum mechanics 0. Introduction
Programme
Kinematic description of a continuum: deformation andmotion of Ω.
Mechanical equilibria and conservation laws.
Constitutive laws of materials: elastic and visco-elasticmaterials, Newtonian fluids.
Typical problems of continuum mechanics: analytical andnumerical solution of elasto-statics/dynamics, compressibleand incompressible elasticity, Newtonian fluids.
Continuum mechanics 0. Introduction
Necessary mathematical techniques
Mechanical state and properties of a continuum areindependent of the choice of a coordinate system.
We will introduce and use tensor calculus: covariant andcontravariant tensors and basic tensor operations, tensor fields ineuclidean space, derivatives of tensors.
Solutions of differential equations will be calculated analytically(on simple problems) or numerically: we will (re)-introduce thebasics of a finite element method.
Continuum mechanics 0. Introduction
The beauty of simple analytical formulas: rubber baloonGreat 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?
Continuum mechanics 0. Introduction
The beauty of simple analytical formulas: rubber baloonGreat deal of understanding through a simple toy model
σ
r
r
p
0rt
Force equilibrium on the cut:
|Fσ| = |Fp|2πrtσ = πr2p
elastic stress: σ = Eε =2πr − 2πr0
2πr0E
rubber incompressibility: 4πr2t = 4πr20 t0
p(r) = 2Et0 r2
0
r3
r − r0
r0
Continuum mechanics 0. Introduction
The beauty of simple analytical formulas: rubber baloonGreat deal of understanding through a simple toy model
Force equilibrium on the cut:
|Fσ| = |Fp|2πrtσ = πr2p
elastic stress: σ = Eε =2πr − 2πr0
2πr0E
rubber incompressibility: 4πr2t = 4πr20 t0
p(r) = 2Et0 r2
0
r3
r − r0
r0
Continuum mechanics 0. Introduction
The beauty of mathematical analysis: singularitiesWhy it breaks always at a kink?
von Mieses stress: indicator of plastic deformation and rupture
Mathematical analysis of the solution of elasticity equationspredicts, that rupture occurs in re-entrant corners (kinks)!
Continuum mechanics 0. Introduction
The beauty of mathematical analysis: (in)stabilityBuckling phenomenon
The nightmare of civil engineers and architects
Continuum mechanics 0. Introduction
The beauty of mathematical analysis: (in)stabilityBuckling phenomenon
The nightmare of civil engineers and architects
Continuum mechanics 0. Introduction
The beauty of mathematical analysis: (in)stabilityBuckling phenomenon
The nightmare of civil engineers and architects
Continuum mechanics 0. Introduction
The beauty of mathematical analysis: (in)stabilityBuckling phenomenon
But it can also be exploited to our advantage!shock absorber for road safety
Continuum mechanics 0. Introduction
