Soft and rigid impact Amabile Tatone Dipartimento di Ingegneria - - PowerPoint PPT Presentation

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Soft and rigid impact Amabile Tatone Dipartimento di Ingegneria - - PowerPoint PPT Presentation

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Soft and rigid impact Amabile Tatone Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno Universit` a dellAquila - Italy 1st International Conference on Computational Contact Mechanics Lecce, Italy, Sept. 16-18, 2009

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Soft and rigid impact

Amabile Tatone

Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno Universit` a dell’Aquila - Italy

1st International Conference on Computational Contact Mechanics Lecce, Italy, Sept. 16-18, 2009

Tatone Soft and rigid impact

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Based on a joint work with:

Alessandro Contento and Angelo Di Egidio

Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno Universit` a dell’Aquila - Italy

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References

◮ Alessandro Granaldi, Paolo Decuzzi, The dynamic response of

resistive microswitches: switching time and bouncing, J.

  • Micromech. Microeng., 16, 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

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A toy model for contact simulations

Contact between a body and a rigid flat support

◮ rigid body ◮ affine body (homogeneous deformations) ◮ contractile affine body

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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

repulsive force repulsive + adhesive forces repulsive + adhesive forces

d q νr = 8, νaa = 3, νar = 6

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Contact force constitutive laws

n

  • d(x, t)

d(x, t) := (p(x, t) − o) · n

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Contact force constitutive laws

n

  • d0

d(x, t) := (p(x, t) − o) · n

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Rigid block

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Rigid disk

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Numerical simulations (rigid body)

L R

dL dR θ t t

001 011 002 012

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Numerical simulations (rigid body)

rocking on a sloping plane

021 022 023

bouncing

031

rolling

032 033 034 035

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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Affine body

F = ∇p

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Affine body

The motion of a body B is described at each time t by a transplacement p(·, t) defined on the reference shape D : p : D × I → E characterized by the following representation: p(x, t) = p0(t) + ∇p(t)(x − x0) where ∇p(t) : V → V is a tensor such that det ∇p(t) > 0. An affine velocity field v at time t has the representation: v(x) = v0 + ∇v(x − x0) Along a motion at time t v0 = ˙ p0(t), ∇v = ∇˙ p(t)

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Affine body

Balance principle:

  • D

b(x, t) · v dV +

  • ∂D

q(x, t) · v dA − S(t) · ∇v vol(D) = 0 , ∀v

Balance equations:

−m ¨ p0(t) − m g + f(t) = 0 −∇¨ p(t) J + M(t) − S(t) vol(D) = 0

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Mass and Euler tensor:

m :=

  • D

ρ dV J :=

  • D

ρ(x − x0) ⊗ (x − x0) dV

Total force and moment tensor:

f(t) :=

  • ∂D

q(x, t) dA M(t) :=

  • ∂D

(x − x0) ⊗ q(x, t)dA

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Material constitutive characterization

Frame indifference:

S · W F = 0 ∀W | sym W = 0 ⇒ skw SFT = 0

Dissipation inequality:

S · ˙ F − d dt ϕ(F) ≥ 0

Reduced dissipation inequality:

S+FT · ˙ FF−1 ≥ 0 S+ := S − S(F)

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Material constitutive characterization

Hyperelastic stress:

  • S(F) · ˙

F = dϕ(F) dt

Mooney-Rivlin strain energy (incompressible material):

ϕ(F) := c1(ı1(C) − 3) + c2(ı2(C) − 3) . ı1(C) := tr (C), ı2(C) := 1 2

  • tr(C)2 − tr(C2)
  • .

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Material constitutive characterization

Reduced dissipation inequality:

S+FT · ˙ FF−1 ≥ 0 S+ := S − S(F)

The simplest way to satisfy a-priori the dissipation inequality:

S+FT = µ sym ( ˙ FF−1) , µ ≥ 0

Stress response (dissipative + energetic + reactive):

S = µ sym ( ˙ FF−1)(FT)−1 + S0(F) − π (FT)−1

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Contact forces

Surface forces per unit deformed area:

q(x, t) =

  • j

qj(x, t) k(x, t)

Area change factor:

k(x, t) := ∇p(t)−Tn∂D(x) det ∇p(t) n∂D(x) outward unit normal vector

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Numerical simulations (elastic body)

elastic bouncing, rolling and oscillations

041 112 200 214 215 216 217 318 319

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Affine contractile body

∇p G F

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Affine contractile body

∇p G F

Kr¨

  • ner-Lee decomposition:

F(t) := ∇p(t) G(t)−1

Contraction velocity:

V = ˙ GG−1

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Affine contractile body

Balance principle:

  • D

b(x, t) · v dV +

  • ∂D

q(x, t) · v dA − S(t) · ∇v vol(D) +

  • Q(t) · V − A(t) · V
  • vol(D) = 0 ,

∀(v, V)

Balance equations:

−m ¨ p0(t) − m g + f(t) = 0 −∇¨ p(t) J + M(t) − S(t) vol(D) = 0 Q(t) − A(t) = 0

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Material constitutive characterization

Frame indifference:

S · W ∇p = 0 ∀W | sym W = 0 ⇒ skw S∇pT = 0

Dissipation inequality:

A · ˙ GG−1 + S · ∇˙ p − d dt

  • ϕ(F) det G
  • ≥ 0

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Material constitutive characterization

Reduced dissipation inequality:

S+∇pT · ˙ FF−1 + A+ · ˙ GG−1 ≥ 0 S+ := S − S(F) , A+ := A + FTSGT − (det G)ϕ(F)I

Hyperelastic stress:

  • S(F)GT · ˙

F = dϕ(F) dt

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Material constitutive characterization

The simplest way to satisfy a-priori the dissipation inequality:

S+∇pT = µ sym ( ˙ FF−1) , µ ≥ 0 A+ = µγ ˙ GG−1 , µγ ≥ 0

Stress characterization:

S = µ sym ( ˙ FF−1)(∇pT)−1 + S0(F) − π (∇pT)−1 A = µγ ˙ GG−1 −

  • FTSGT − (det G)ϕ(F)I
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Soft and rigid impact

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Material constitutive characterization

Equations of motion:

−m ¨ p0 − m g + f = 0 −∇¨ p J + M − S vol(D) = 0 µγ ˙ GG−1 = FTSGT − (det G) ϕ(F)I + Q

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Numerical simulations (contractile body)

  • scillating driving Q

12g1 12g2 12g3

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Numerical simulations (contractile body)

  • scillating driving G

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Numerical simulations (contractile body)

  • scillating driving G

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References

◮ Simo J.C., Wriggers P. and Taylor R.L., “A perturbed Lagrangian

formulation for the finite element solution of contact problems,”

  • Comp. Methods Appl. Mech. Engrg., 51, 163–180 (1985).

◮ Wriggers-Za Wriggers P. and Zavarise G., “Chapter 6,

Computational Contact Mechanics,” in Encyclopedia of Computational Mechanics, Stein E., de Borst R., Hughes T.J.R., editors, John Wiley & Sons (2004).

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References

◮ Di Carlo A. and Quiligotti S., “Growth and Balance,” Mech. Res.

Comm., 29, 449–456 (2002).

◮ Di Carlo A., “Surface and bulk growth unified,” in Mechanics of

Material Forces, Steinmann P. and Maugin G. A., editors, Springer, New York, 53–64 (2005).

◮ Nardinocchi P. and Teresi L., “On the active response of soft living

tissues,” J. Elasticity, 88, 27–39 (2007).

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Supplementary references

◮ Gianfranco Capriz, Paolo Podio-Guidugli, Whence the boundary

conditions in modern continuum physics?, Atti Convegni Lincei n. 210, 2004

◮ Antonio Di Carlo, Actual surfaces versus virtual cuts, Atti Convegni

Lincei n. 210, 2004

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