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Bouncing, rolling and sticking
- f stiff and soft bodies
Bouncing, rolling and sticking of stiff and soft bodies Amabile - - PowerPoint PPT Presentation
Bouncing, rolling and sticking of stiff and soft bodies Amabile - - PowerPoint PPT Presentation
Bouncing, rolling and sticking of stiff and soft bodies Amabile Tatone Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno Universit` a dellAquila - Italy INTAS Project: Some Nonclassical Problems For Thin Structures ,
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Micro switch
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repulsive force repulsive + adhesive forces
d
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Rigid body
A motion of the body B is described at each time t by a placement defined on the paragon shape D : p : D × I → E characterized by the following representation: p(x, t) = po(t) + R(t)(x − xo) where R(t) : V → V is a rotation in the translation space of E. Test velocity fields: w(x) = wo + WR(t)(x − xo), with sym W = 0
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Affine body
A motion of the body B is described at each time t by a placement defined on the paragon shape D : p : D × I → E characterized by the following representation: p(x, t) = po(t) + F(t)(x − xo) where F(t) : V → V is linear and such that det F(t) > 0 Test velocity fields: w(x) = wo + GF(t)(x − xo)
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Rigid body
Balance principle:
- D
b · w dV +
- ∂D
q · w dA = 0 ∀w, ∀t Equations of motion: −m ¨ po(t) − m g + f (t) = 0 skw
- − ¨
R(t) J R(t)T + M(t) R(t)T = 0
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Affine body
Balance principle: −S · GF vol D +
- D
b · w dV +
- ∂D
q · w dA = 0 ∀w, ∀t Equations of motion: −m ¨ po(t) − m g + f (t) = 0 −¨ F(t) J F(t)T +
- M(t) − S(t) vol D
- F(t)T = 0
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Mass and Euler tensor: m :=
- D
ρ dV ; J :=
- D
ρ(x − xo) ⊗ (x − xo) dV Total force and moment tensor: f (t) :=
- ∂D
q(x, t) dA; M(t) :=
- ∂D
(x − xo) ⊗ q(x, t)dA
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Piola stress and material properties
Frame indifference: S · WF = 0 ∀W | sym W = 0 ⇒ skw SF T = 0 Mooney-Rivlin strain energy (incompressible material): ϕ(F) := c10(ı1(C) − 3) + c01(ı2(C) − 3) Stress response (energetic + reactive + dissipative): SF T = S(F)F T − π I + µ ˙ FF −1
- S(F) · ˙
F = dϕ(F) dt ⇒
- S(F) F T = 2(c10F F T − c01F −TF −1)
Dissipation principle: S · ˙ F − dϕ(F)/dt ≥ 0 ⇒ µ ≥ 0
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Contact force constitutive laws
Repulsive force: qr(x, t) = αr d(x, t)−νr n Damping force: qd(x, t) = −βd d(x, t)−νd (n ⊗ n) ˙ p(x, t) Frictional force: qf (x, t) = −βf d(x, t)−νf (I − n ⊗ n) ˙ p(x, t) Adhesive force: qa(x, t) = −βa (d(x, t)−νaa − d(x, t)−νar ) n
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Contact force constitutive laws
The contact forces are surface forces per unit deformed area: q(x, t) =
- j
qj(x, t) k(x, t) Area change factor: k(x, t) := F(t)−Tn∂D(x) det(F(t)) n∂D(x) outward unit normal vector
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n
- d(x, t)
d(x, t) := (p(x, t) − o) · n
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n
- d0
d(x, t) := (p(x, t) − o) · n
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Contact force constitutive laws
Repulsive force: qr(x, t) = αr d(x, t)−νr n Damping force: qd(x, t) = −βd d(x, t)−νd (n ⊗ n) ˙ p(x, t) Frictional force: qf (x, t) = −βf d(x, t)−νf (I − n ⊗ n) ˙ p(x, t) Adhesive force: qa(x, t) = −βa (d(x, t)−νaa − d(x, t)−νar ) n
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repulsive force repulsive + adhesive forces repulsive + adhesive forces
d q νr = 8, νaa = 3, νar = 6
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Numerical simulations
L R
dL dR θ t t
001 011 002 012
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Numerical simulations
rocking on a sloping plane
021 022 023
bouncing and rolling
031 032 033 034 035
elastic bouncing and oscillations
041 112 200 214 215 216 217 318 319
adhesion and detachment
501 502 503 505
spinning top
3D-101 3D-111 3D-102 3D-112
dice throwing
3D-201 3D-211 3D-202 3D-212
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The end
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Appendix
Cauchy stress
T = SF T 1 det F
Pressure π
It is the reactive part of T. In an incompressible solid/fluid the velocity fields are said to be isochoric. The trace of the velocity gradient turns out to be zero. A reactive stress, whose power is zero for any isochoric velocity field, has to be a spherical tensor −π I: π I · G = π tr G = 0
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Appendix
Mooney-Rivlin
It is a hyperelastic material model used for rubber-like materials as well as for biological tissues. The principal invariants of C := FF T are defined as ı1(C) := F · F, ı2(C) := F ⋆ · F ⋆ where F ⋆ := F −T det F is the cofactor of F.
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References
◮ Nicola Pugno, Towards a Spiderman suit: large invisible cables and
self-cleaning releasable superadhesive materials, J. Phys.: Condens. Matter, 19, 2007.
◮ Alessandro Granaldi, Paolo Decuzzi, The dynamic response of
resistive microswitches: switching time and bouncing, J.
- Micromech. Microeng., 16, 2006.
◮ Jiunn-Jong Wu, Adhesive contact between a nano-scale rigid sphere
and an elastic half-space, J. Phys. D: Appl. Phys., 39, 2006
◮ Z. J. Guo, N. E. McGruer, G. G. Adams, Modeling, simulation and
measurement of the dynamic performance of an ohmic contact, electrostatically actuated RF MEMS switch, J. Micromech. Microeng., 17, 2007
◮ Makoto Ashino, Alexander Schwarz, Hendrik H¨
- lscher, Udo D.
Schwarz, and Roland Wiesendanger, Interpretation of the atomic scale contrast obtained on graphite and single-walled carbon nanotubes in the dynamic mode of atomic force microscopy, Nanotechnology, 16, 2005
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