Stabilization of quasistatic evolution of elastoplastic systems - PowerPoint PPT Presentation

Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading Oleg Makarenkov Department of Mathematical Sciences University of Texas at Dallas in cooperation with Ivan Gudoshnikov

A parallel network of elastoplastic springs − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 r 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ spring stress Elastoplastic spring: + c 1 − c + a 1 [ c , ] ξ 1 1 1 1 2 − c 1 elastic component e 1 plastic component p 1 (relaxed length) spring length

A parallel network of elastoplastic springs − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 r 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ spring stress >  = p 0 Elastic deformatio n : s Ae + c 1 ∈  = = Plastic deformatio n : p N ( s )   p 0 p 0 C ∞ = +  [ 0 , ), if s c , <  p 0 = − + × 1  − C [ c , c ] ... c = ∈ − + 1 1  N ( s ) { 0 }, if s ( c , c ), 1 − + × − + 1 1 [ c , c ] ... [ c , c ]  1 1 −∞ = − m m  ( , 0 ], if s c . spring length 1

Initial system of variational inequalities − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 l 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ = Elastic deformatio n : s Ae , ∈  Plastic deformatio n : p N ( s ), C + ∈ ℜ n Geometric constraint : e p D , + = T Enforced constraint : R ( e p ) l ( t ), + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 .

Tension/compression law − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 l 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ = Elastic deformatio n : s Ae , ∈  Plastic deformatio n : p N ( s ), C + ∈ ℜ n Geometric constraint : e p D , + = T Enforced constraint : R ( e p ) l ( t ), + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 .

Tension/compression law − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 l 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ = Elastic deformatio n : s Ae , ∈  Plastic deformatio n : p N ( s ), C + ∈ ℜ n Geometric constraint : e p D , + = T Enforced constraint : R ( e p ) l ( t ), + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 .

Tension/compression law l 1 ( t ) − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) e 7 + p 7 ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ = Elastic deformatio n : s Ae , For enforced constraint 1 : ∈  Plastic deformatio n : p N ( s ), ( e 4 + p 4 )+( e 7 + p 7 )+( e 5 + p 5 )-( e 1 + p 1 )= l 1 ( t ) C + ∈ ℜ n Geometric constraint : e p D , + = T Enforced constraint : R ( e p ) l ( t ), + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 .

Static balance law − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 l 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ = Elastic deformatio n : s Ae , ∈  Plastic deformatio n : p N ( s ), C + ∈ ℜ n Geometric constraint : e p D , + = T Enforced constraint : R ( e p ) l ( t ), + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 .

Static balance law − c + a 1 [ c , ] ξ 1 ξ 2 ξ 3 l 1 ( t ) 1 1 1 2 3 ( i 1 , j 1 ) = (1,2) f 2 ( t ) f 3 ( t ) ( i 2 , j 2 ) = (4,5) − c − c − c − c + + + + a 2 a 3 a 4 a 5 ξ 4 ξ 5 ξ 6 ξ 7 [ c , ] [ c , ] [ c , ] [ , ] f 6 ( t ) c 5 6 7 4 4 4 ( i 3 , j 3 ) = (5,1) 2 2 3 3 5 5 ( i 4 , j 4 ) = (1,6) − c − c + + a 7 a 6 ξ 8 [ c , ] [ c , ] f 8 ( t ) f 4 ( t ) f 5 ( t ) 8 7 7 ( i 5 , j 5 ) = (7,3) 6 6 f 7 ( t ) ( i 6 , j 6 ) = (5,8) − c + a 8 [ c , ] l 2 ( t ) 8 8 ( i 7 , j 7 ) = (6,7) ( i 8 , j 8 ) = (8,6) − c + a 9 [ c , ] 9 9 ( i 9 , j 9 ) = (4,7) ξ = Elastic deformatio n : s Ae , For node 2 : ∈  Plastic deformatio n : p N ( s ), - s 1 + r 1 + f 2 ( t )=0 C + ∈ ℜ n Geometric constraint : e p D , + = T Enforced constraint : R ( e p ) l ( t ), + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 .

Moreau sweeping process = Elastic deformatio n : s Ae , ∈ + ∈ +  Plastic deformatio n : p N ( s ), e p U g ( t ) C { } + ∈ ℜ = ∈ ℜ = n n T Geometric constraint : e p D ( ), U x D ( ) : R x 0 ( ) + = = ξ T Enforced constraint : R ( e p ) l ( t ), g ( t ) D l ( t ) V − ⊥ = 1 V A U + ... + = − + ... + = − 1 q T 1 m T + ∈ Graph theory: r r D Rr s s D s e h ( t ) V = − T f ( t ) D h ( t ) + + + + + + = 1 m 1 q Static balance : s ... s r ... r f ( t ) 0 ( ) − = 1 ⊥ ⊂ ( ) h ( t ) A h ( t ) = ⊥ T n U Algebra : Ker D D R U − ∈  A ∈ = + − y N ( ) ( y )  c p N ( Ae ) y e h ( t ) g ( t ) − + − 1  ( A C h ( t ) g ( t ) V ) C + − + ∈ ∈ +  A  e p g ( t ) h ( t ) U  z N ( ) ( y ) y U − 1 + − A C h ( t ) g ( t ) + − ∈ = + + − e h ( t ) g ( t ) V z e p h ( t ) g ( t ) ∈ z ( 0 ) U