Stabilization of quasistatic evolution of elastoplastic systems - - PowerPoint PPT Presentation

Stabilization of quasistatic evolution

• f 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

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) r1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

1 2 3 4 5 6 7 8

Elastoplastic spring:

a1

] , [

1 1 + − c

c

ξ1

1 2

elastic component e1 plastic component p1 (relaxed length)

− 1

c

+ 1

c

spring stress spring length

A parallel network of elastoplastic springs

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) r1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

1 2 3 4 5 6 7 8

− 1

c

+ 1

c

spring stress spring length

) ( : n deformatio Plastic : n deformatio Elastic s N p Ae s

C

∈ =  = p  = p  > p  < p  . if ), , ( if , if ], , ( }, { ), , [ ) ( ] , [ ... ... ] , [

1 1 1 1 ] , [ 1 1

1 1

− + − + + − + −

= ∈ =      −∞ ∞ = × × =

+ −

c s c c s c s s N c c c c C

c c m m

Initial system of variational inequalities

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) l1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C

 1 2 3 4 5 6 7 8

Tension/compression law

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) l1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C

 1 2 3 4 5 6 7 8

Tension/compression law

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) l1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C

 1 2 3 4 5 6 7 8

Tension/compression law

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) l1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C

 1 2 3 4 5 6 7 8 For enforced constraint 1: (e4+p4)+(e7+p7)+(e5+p5)-(e1+p1)=l1(t) e7 + p7

Static balance law

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) l1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C

 1 2 3 4 5 6 7 8

Static balance law

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) l1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C

 1 2 3 4 5 6 7 8 For node 2:

• s1+r1+f2(t)=0

) ( ... ... : balance Static ), ( ) ( : constraint Enforced ), ( : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ } ( )

⊥ −

= = = ℜ ∈ = + ∈ + U A V t l D t g x R D x U t g U p e

V T n 1

) ( ) ( : ) ( ) ( ξ

( )

U T

t h A t h t h D t f V t h e ) ( ) ( ) ( ) ( ) (

1 −

= − = ∈ +

Moreau sweeping process

V t g t h e U t h t g p e Ae N p

C

∈ − + ∈ + − + ∈ ) ( ) ( ) ( ) ( ) (  ) ( ) ( ) ( ) ( t g t h p e z t g t h e y − + + = − + =

( ) ( )

( )

U z U y y N z y N y

A t g t h C A A V t g t h C A

∈ + ∈ ∈ −

− + − +

− −

) ( ) ( ) (

) ( ) ( ) ( ) (

1 1

   



c s D s s

T m

− = + +...

1

Rr D r r

T q

− = + +...

1

Graph theory:

( )

⊥ ⊥ ⊂

= U D D

n T

R Ker : Algebra

Moreau sweeping process

ξ6 ξ3 ξ2 ξ8 ξ5 a4 a2 a3 a7 a5 a6 l2(t) a9

] , [

4 4 + − c

c ] , [

7 7 + − c

c ] , [

5 5 + − c

c

(i1, j1) = (1,2) (i2, j2) = (4,5) (i3, j3) = (5,1) (i4, j4) = (1,6) (i5, j5) = (7,3) (i6, j6) = (5,8) (i7, j7) = (6,7) (i8, j8) = (8,6) (i9, j9) = (4,7) r1(t)

] , [

6 6 + − c

c ] , [

3 3 + − c

c ] , [

2 2 + − c

c ] , [

9 9 + − c

c

f4(t) ξ a1

] , [

1 1 + − c

c

ξ1 ξ4 ξ7 a8

] , [

8 8 + − c

c

f8(t) f7(t) f6(t) f5(t) f2(t) f3(t)

1 2 3 4 5 6 7 8

( )

1 dim ), (

) ( ) (

1

+ + − = ∈ −

− +

−

q n m V y N y

A V t g t h C A 



{ }

⊥ −

= = ℜ ∈ = U A V x R D x U

T n 1

: ) (

( ) ( )

U V

t h A t h t l D t g ) ( ) ( ) ( ) (

1 −

= = ξ ) ( ) ( ) ( ) ( t g t h e y t h D t f

T

− + = − =

Geometry of the moving constraint

s constraint

• mpression

tension/c

• f

number springs

• f

number nodes

• f

number 1 dim 1 dim ) ( ) ( , ) , ( R ) ( ) ( ) ( ) ( ) ( ), (

1 ) (

= = = + + − = − − = ∈ ∈ = = ⊗ Π = − + = Π ∈ −

−

q m n q n m V q n U V t g U t h Av u v u V U V t t C t g t h C A t y N y

A m A t C

 

A criterion for the safe load condition to hold

s constraint

• mpression

tension/c

• f

number springs

• f

number nodes

• f

number 1 dim 1 dim ) ( ) ( , ) , ( R ) ( ) ( ) ( ) ( ) ( ), (

1 ) (

= = = + + − = − − = ∈ ∈ = = ⊗ Π = − + = Π ∈ −

−

q m n q n m V q n U V t g U t h Av u v u V U V t t C t g t h C A t y N y

A m A t C

 

Proposition 1 (safe load):

–Ah(t)∈C ⇒ C(t) ≠ ∅ C(t) = ∅ ⇒ plastic collapse

A criterion for plastic shakedown to occur

Proposition 2 (plastic shakedown): Assume that the safe load condition holds. If then the sweeping process doesn’t have any solutions that are constant

• n the interval [t1, t2].

Dynamics under T-periodic loading

Existence of a periodic attractor

Theorem 1 (existence of periodic attractor, Krejci): If C(t) is T-periodic, then the set of all T-periodic solutions is a global attractor. For each fixed t∈[0,t], the active set J(t,x(t)) is the same for all x∈ri(X).

) (

) (

y N y

A t C

∈ − 

x(0) y(0) x(t0) J(t0,x(t0)) = {5}, J(t0,y(t0)) = ∅ x(t) is T-periodic ⇒ ⇒ y(t) is not T-periodic and z(t) is not T-periodic if x(0) is an initial condition of a T- periodic solution x(t) then the set of initial conditions of all other T- periodic solutions is a straight line. z(0) x(0) 1 2 3 4 5 C(t)

Uniqueness of non-constant T-periodic solutions

Theorem 2 (uniqueness of T-periodic solutions): Let C(t) ⊂ Rm be T-periodic. Assume that any m vectors out the collection {ni} are linearly independent and the number of adjoin facets doesn’t exceed m. Then the sweeping process has at most one non-constant T-periodic solution.

) (

) (

y N y

A t C

∈ − 

n1 n2 n3 n4 n5 no no yes

Uniqueness of non-constant T-periodic solutions

Theorem 3 (uniqueness of non-constant T-periodic solutions): Let h(t) and g(t) be T-periodic. Assume that dimV = m-1. If V is either between the blue vertex and the blue triangle or between the green vertex and the green triangle, then C(t) is a simplex and the sweeping process has at most one non-constant T-periodic solution.

( )

V t g t h C A t C y N y

A t C

  ) ( ) ( ) ( ), (

1 ) (

− + = ∈ −

−

V

C(t)

Structurally stable family of periodic solutions

dimU = 3 dimV = 2

Thank you for your attention !!!

References: [1] S. Adly, M. Ait Mansour, L. Scrimali, Sensitivity analysis of solutions to a class of quasi-variational inequalities. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (2005), no. 3, 767-771. [2] I. Gudoshnikov, O. Makarenkov, Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading, submitted, https://arxiv.org/abs/1708.03084 [3] P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gattotoscho, 1996. [4] J. J. Moreau, On unilateral constraints, friction and plasticity. New variational techniques in mathematical physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973), pp. 171–322. Edizioni Cremonese, Rome, 1974.

Static balance law

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



Geometric constraint and enforced constraint combined

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ }

l x R D x U U p e

T n l t l

= ℜ ∈ = ∈ + : ,

) (

Geometric constraint and enforced constraint combined

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced , : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ }

l x R D x U U p e

T n l t l

= ℜ ∈ = ∈ + : ,

) ( V U q q T

x x x I D R + = =

×

ξ

{ } ( )

⊥ −

= = = ℜ ∈ = + ∈ + U A V t l D t g x R D x U t g U p e

V T n 1

, ) ( ) ( , : ), ( ξ

Static balance

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced ), ( : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ }

l x R D x U U p e

T n l t l

= ℜ ∈ = ∈ + : ) ( ,

) ( V U q q T

x x x I D R + = =

×

ξ

{ } ( )

⊥ −

= = = ℜ ∈ = + ∈ + U A V t l D t g x R D x U t g U p e

V T n 1

, ) ( ) ( , : ) ( ), ( ξ

j-th spring

                =

T

D

i-th node

+1

1 j-th spring

                =

T

D

i-th node

• 11

aj ξi aj ξi

T ij

D

T ij

D

s D s s

T m

− = + +...

1

Static balance

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced ), ( : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ }

l x R D x U U p e

T n l t l

= ℜ ∈ = ∈ + : ) ( ,

) ( V U q q T

x x x I D R + = =

×

ξ

{ } ( )

⊥ −

= = = ℜ ∈ = + ∈ + U A V t l D t g x R D x U t g U p e

V T n 1

, ) ( ) ( , : ) ( ), ( ξ

j-th spring

                =

T

D

i-th node

+1

1 aj ξi

T ij

D

s D s s

T m

− = + +...

1

Rr D r r

T q

− = + +...

1

Graph theory: – DTRk=(0,…,0,1,0,…,0,-1,0,…,0)T Ik-th component Jk-th component

Static balance

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced ), ( : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ }

l x R D x U U p e

T n l t l

= ℜ ∈ = ∈ + : ) ( ,

) ( V U q q T

x x x I D R + = =

×

ξ

{ } ( )

⊥ −

= = = ℜ ∈ = + ∈ + U A V t l D t g x R D x U t g U p e

V T n 1

, ) ( ) ( , : ) ( ), ( ξ

j-th spring

                =

T

D

i-th node

+1

1 aj ξi

T ij

D

s D s s

T m

− = + +...

1

) ( = + − − t f Rr D s D

T T

Rr D r r

T q

− = + +...

1

Graph theory: – DTRk=(0,…,0,1,0,…,0,-1,0,…,0)T Ik-th component Jk-th component

Static balance

. ) ( ... ... : balance Static ), ( ) ( : constraint Enforced ), ( : constraint Geometric ), ( : n deformatio Plastic , : n deformatio Elastic

1 1

= + + + + + + = + ℜ ∈ + ∈ = t f r r s s t l p e R D p e s N p Ae s

q m T n C



{ }

l x R D x U U p e

T n l t l

= ℜ ∈ = ∈ + : ) ( ,

) ( V U q q T

x x x I D R + = =

×

ξ

{ } ( )

⊥ −

= = = ℜ ∈ = + ∈ + U A V t l D t g x R D x U t g U p e

V T n 1

, ) ( ) ( , : ) ( ), ( ξ s D s s

T m

− = + +...

1

Rr D r r

T q

− = + +... : ry Graph theo

1

) ( = + − − t f Rr D s D

T T

) ( ) ( Ker ) ( t h D t f D t h Rr s

T T

− = ∈ + +

⊥

∈ + U t h s ) (

( )

U

t h A A t h U t h s ) ( ) ( ) (

1 − ⊥

= ∈ +

1

applying By

−

A V t h A e ∈ +

−

) (

1