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

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

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 Tatone Soft and rigid impact

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 Tatone Soft and rigid impact

A toy model for contact simulations Contact between a body and a rigid flat support ◮ rigid body ◮ affine body (homogeneous deformations) ◮ contractile affine body Tatone Soft and rigid impact

Contact force constitutive laws Repulsive force: q r ( x , t ) = α r d ( x , t ) − ν r n Damping force: q d ( x , t ) = − β d d ( x , t ) − ν d ( n ⊗ n ) ˙ p ( x , t ) Frictional force: q f ( x , t ) = − β f d ( x , t ) − ν f ( I − n ⊗ n ) ˙ p ( x , t ) Adhesive force: q a ( x , t ) = − β a ( d ( x , t ) − ν aa − d ( x , t ) − ν ar ) n Tatone Soft and rigid impact

Contact force constitutive laws q repulsive force repulsive + adhesive forces repulsive + adhesive forces d ν r = 8 , ν aa = 3 , ν ar = 6 Tatone Soft and rigid impact

Contact force constitutive laws n o d ( x , t ) d ( x , t ) := ( p ( x , t ) − o ) · n Tatone Soft and rigid impact

Contact force constitutive laws n o d 0 d ( x , t ) := ( p ( x , t ) − o ) · n Tatone Soft and rigid impact

Rigid block Tatone Soft and rigid impact

Rigid disk Tatone Soft and rigid impact

Numerical simulations (rigid body) 001 011 R 002 012 L d L d R θ t t Tatone Soft and rigid impact

Numerical simulations (rigid body) rocking on a sloping plane spinning top 3D-101 3D-111 021 022 023 3D-102 3D-112 bouncing dice throwing 031 3D-201 3D-211 rolling 3D-202 3D-212 032 033 034 035 adhesion and detachment 501 502 503 505 Tatone Soft and rigid impact

Affine body F = ∇ p Tatone Soft and rigid impact

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 ) = p 0 ( t ) + ∇ p ( t )( x − x 0 ) 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 ) = v 0 + ∇ v ( x − x 0 ) Along a motion at time t v 0 = ˙ p 0 ( t ) , ∇ v = ∇ ˙ p ( t ) Tatone Soft and rigid impact

Affine body Balance principle: � � b ( x , t ) · v dV + q ( x , t ) · v dA − S ( t ) · ∇ v vol( D ) = 0 , ∀ v D ∂ D Balance equations: − m ¨ p 0 ( t ) − m g + f ( t ) = 0 −∇ ¨ p ( t ) J + M ( t ) − S ( t ) vol( D ) = 0 Tatone Soft and rigid impact

Mass and Euler tensor: � m := ρ dV D � J := ρ ( x − x 0 ) ⊗ ( x − x 0 ) dV D Total force and moment tensor: � f ( t ) := q ( x , t ) dA ∂ D � M ( t ) := ( x − x 0 ) ⊗ q ( x , t ) dA ∂ D Tatone Soft and rigid impact

Material constitutive characterization Frame indifference: skw SF T = 0 S · W F = 0 ∀ W | sym W = 0 ⇒ Dissipation inequality: F − d S · ˙ dt ϕ ( F ) ≥ 0 Reduced dissipation inequality: S + F T · ˙ FF − 1 ≥ 0 S + := S − � S ( F ) Tatone Soft and rigid impact

Material constitutive characterization Hyperelastic stress: F = d ϕ ( F ) S ( F ) · ˙ � dt Mooney-Rivlin strain energy (incompressible material): ϕ ( F ) := c 1 ( ı 1 ( C ) − 3) + c 2 ( ı 2 ( C ) − 3) . � � ı 2 ( C ) := 1 tr( C ) 2 − tr( C 2 ) ı 1 ( C ) := tr ( C ) , . 2 Tatone Soft and rigid impact

Material constitutive characterization Reduced dissipation inequality: S + F T · ˙ FF − 1 ≥ 0 S + := S − � S ( F ) The simplest way to satisfy a-priori the dissipation inequality: S + F T = µ sym ( ˙ FF − 1 ) , µ ≥ 0 Stress response (dissipative + energetic + reactive): FF − 1 )( F T ) − 1 + � S = µ sym ( ˙ S 0 ( F ) − π ( F T ) − 1 Tatone Soft and rigid impact

Contact forces Surface forces per unit deformed area: � q ( x , t ) = q j ( x , t ) k ( x , t ) j Area change factor: k ( x , t ) := �∇ p ( t ) − T n ∂ D ( x ) � det ∇ p ( t ) n ∂ D ( x ) outward unit normal vector Tatone Soft and rigid impact

Numerical simulations (elastic body) elastic bouncing, rolling and oscillations 041 112 200 214 215 216 217 318 319 Tatone Soft and rigid impact

Affine contractile body ∇ p G F Tatone Soft and rigid impact

Affine contractile body ∇ p G F Kr¨ oner-Lee decomposition: F ( t ) := ∇ p ( t ) G ( t ) − 1 Contraction velocity: V = ˙ GG − 1 Tatone Soft and rigid impact

Affine contractile body Balance principle: � � b ( x , t ) · v dV + q ( x , t ) · v dA − S ( t ) · ∇ v vol( D ) D ∂ D � � + Q ( t ) · V − A ( t ) · V vol( D ) = 0 , ∀ ( v , V ) Balance equations: − m ¨ p 0 ( t ) − m g + f ( t ) = 0 −∇ ¨ p ( t ) J + M ( t ) − S ( t ) vol( D ) = 0 Q ( t ) − A ( t ) = 0 Tatone Soft and rigid impact

Material constitutive characterization Frame indifference: skw S ∇ p T = 0 S · W ∇ p = 0 ∀ W | sym W = 0 ⇒ Dissipation inequality: � � p − d GG − 1 + S · ∇ ˙ A · ˙ ϕ ( F ) det G ≥ 0 dt Tatone Soft and rigid impact

Material constitutive characterization Reduced dissipation inequality: S + ∇ p T · ˙ FF − 1 + A + · ˙ GG − 1 ≥ 0 S + := S − � A + := A + F T SG T − (det G ) ϕ ( F ) I S ( F ) , Hyperelastic stress: F = d ϕ ( F ) S ( F ) G T · ˙ � dt Tatone Soft and rigid impact

Material constitutive characterization The simplest way to satisfy a-priori the dissipation inequality: S + ∇ p T = µ sym ( ˙ FF − 1 ) , µ ≥ 0 A + = µ γ ˙ GG − 1 , µ γ ≥ 0 Stress characterization: FF − 1 )( ∇ p T ) − 1 + � S = µ sym ( ˙ S 0 ( F ) − π ( ∇ p T ) − 1 � � GG − 1 − F T SG T − (det G ) ϕ ( F ) I A = µ γ ˙ Tatone Soft and rigid impact

Material constitutive characterization Equations of motion: − m ¨ p 0 − m g + f = 0 −∇ ¨ p J + M − S vol( D ) = 0 GG − 1 = F T SG T − (det G ) ϕ ( F ) I + Q µ γ ˙ Tatone Soft and rigid impact

Numerical simulations (contractile body) oscillating driving Q 12g1 12g2 12g3 Tatone Soft and rigid impact

Numerical simulations (contractile body) oscillating driving G Tatone Soft and rigid impact

Numerical simulations (contractile body) oscillating driving G Tatone Soft and rigid impact

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). Tatone Soft and rigid impact

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). Tatone Soft and rigid impact

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 Tatone Soft and rigid impact