Rate-Independent Evolution Problems in Elasto-Plasticity: a - PowerPoint PPT Presentation

Rate-Independent Evolution Problems in Elasto-Plasticity: a Variational Approach Main reference: G. Dal Maso, A. DeSimone, M.G. Mora: Quasistatic evolution problems for linearly elastic - perfectly plastic materials. Preprint SISSA, Trieste, 2004, available at http://www.sissa.it/fa/ http://www.math.unifi.it/ ~cime/ Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Mechanical Preliminaries sym = M n × n A = A D + 1 M n × n ⊕ R I n (tr A ) I with tr A D = 0 D reference configuration: Ω ⊂ R n bounded domain with ∂ Ω ∈ C 2 u : Ω → R n displacement: 2 ( ∇ u + ∇ u T ) Eu = 1 linearized strain: (mechanically questionable) additive decomposition: Eu = e + p (mechanically questionable) p : Ω → M n × n e : Ω → M n × n elastic strain: plastic strain: sym D stress: σ = 2 µ e D + κ (tr e ) I ( µ, κ > 0) σ D ( x ) ∈ K for a.e. x ∈ Ω , where K ⊂ M n × n stress constraint: is D closed, convex, and B (0 , r ) ⊂ K ⊂ B (0 , R ) yield surface: ∂K Calculus of Variations and Nonlinear Partial Differential Equations, Page 1/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Classical Formulation Assume ∂ Ω = Γ 0 ∪ Γ 1 , with Γ 0 , Γ 1 relatively open, Γ 0 ∩ Γ 1 = Ø, and ∂ Γ 0 = ∂ Γ 1 ∈ C 2 . Given f ( t, x ) body force on Ω, g ( t, x ) surface force on Γ 1 , and w ( t, x ) boundary displacement on Γ 0 , g(t) Γ 1 Ω Γ 0 g(t) f(t) find u ( t, x ), e ( t, x ), p ( t, x ), and σ ( t, x ) such that Calculus of Variations and Nonlinear Partial Differential Equations, Page 2/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity • additive decomposition Eu ( t, x ) = e ( t, x ) + p ( t, x ) • constitutive equation σ ( t, x ) = 2 µ e D ( t, x ) + κ (tr e ( t, x )) I • equilibrium conditions − div x σ ( t, x ) = f ( t, x ) on Ω, σ ( t, x ) ν ( x ) = g ( t, x ) on Γ 1 • boundary conditions u ( t, x ) = w ( t, x ) on Γ 0 • stress constraint σ D ( t, x ) ∈ K • Prandtl-Reuss flow rule p ( t, x ) ∈ N K ( σ D ( t, x )) , N K ( ξ D ) = normal cone to K at ξ D ˙ Calculus of Variations and Nonlinear Partial Differential Equations, Page 3/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Variational Method for Rate-Independent Processes • Discrete Time: incremental formulation based on energy minimization. • Continuous Time: pass to the limit and obtain ◦ global minimality at each time; ◦ energy balance in each time interval: increment of stored energy + dissipated energy = = work done by the external forces. Main reference: Mainik-Mielke ( Calc. Var. 2004). Applications: finite plasticity and plasticity with hardening (Mielke, Levitas, Mainik, Roub´ ıˇ cek, Theil), crack growth (Chambolle, DM, Francfort, Toader). In this course we apply this method to linearized perfect plasticity. Suquet ( J. M´ ecanique 1981): same problem with different formulation. Calculus of Variations and Nonlinear Partial Differential Equations, Page 4/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Incremental Problems: the Elastic Part To simplify the exposition, we assume f ( t, x ) = 0 and g ( t, x ) = 0. So there are no external forces. The body is driven only by the boundary displacement w ( t, x ) imposed on Γ 0 . Let Q ( A ) := µ | A D | 2 + κ 2 (tr A ) 2 for every A ∈ M n × n sym . � Elastic energy: Q ( e ( t )) := Q ( e ( t, x )) dx . Ω Stress: σ ( t, x ) = ∇ Q ( e ( t, x )), i.e., σ ( t ) = ∇Q ( e ( t )). Deviatoric part: σ D ( t, x ) = ∇ Q ( e D ( t, x )), i.e., σ D ( t ) = ∇Q ( e D ( t )). Let us fix p ( t ) and consider the minimum problem min Q ( e ) . Eu = e + p ( t ) on Ω u = w ( t ) on Γ 0 Calculus of Variations and Nonlinear Partial Differential Equations, Page 5/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Let us fix p ( t ) and consider the minimum problem min Q ( e ) . Eu = e + p ( t ) on Ω u = w ( t ) on Γ 0 Suppose that ( u ( t ) , e ( t )) is a minimizer. Considering the variation ( u ( t ) + εϕ, e ( t ) + εEϕ ), with ϕ smooth and vanishing on Γ 0 , we get Q ( e ( t ) + εEϕ ) ≥ Q ( e ( t )) for every ε . Taking the derivative with respect to ε at ε = 0, we get �∇Q ( e ( t )) , Eϕ � = 0 for every smooth function ϕ vanishing on Γ 0 . As σ ( t ) = ∇Q ( e ( t )), we conclude that � σ ( t ) , Eϕ � = 0 for any such ϕ , hence div x σ ( t ) = 0 on Ω , σ ( t ) ν = 0 on Γ 1 . Calculus of Variations and Nonlinear Partial Differential Equations, Page 6/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Basic Convex Analysis We recall that, if X is a Hilbert space and f : X → ] −∞ , + ∞ ] is a lower semicontinuous proper convex function, its subdifferential ∂f ( x ) at a point x ∈ X is defined as the set of all x ∗ in X such that f ( y ) ≥ f ( x ) + � x ∗ , y − x � for every y ∈ X . The conjugate f ∗ of f is defined by for every x ∗ ∈ X . f ∗ ( x ∗ ) := sup {� x ∗ , x � − f ( x ) } x ∈ X It is easy to prove that x ∗ ∈ ∂f ( x ) ⇐ ⇒ x ∈ ∂f ∗ ( x ∗ ). If C is a nonempty closed convex subset of X , its indicator function χ C is defined by χ C ( x ) = 0 for x ∈ C , and χ C ( x ) = + ∞ for x / ∈ C . We have ∂χ C ( x ) = N C ( x ), the normal cone to C at x , i.e., x ∗ ∈ ∂χ C ( x ) ⇐ ⇒ � x ∗ , y − x � ≤ 0 for every y ∈ C . Calculus of Variations and Nonlinear Partial Differential Equations, Page 7/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Incremental Problems: the Flow Rule The flow rule p ( t, x ) ∈ N K ( σ D ( t, x )) = ∂χ K ( σ D ( t, x )) can be written as ˙ σ D ( t, x ) ∈ ∂H ( ˙ p ( t, x )) , where H ( ξ ) := χ ∗ K ( ξ ) = sup ξ : η is the support function of K . η ∈ K • H is convex and positively homogeneous of degree 1; • B (0 , r ) ⊂ K ⊂ B (0 , R ) ⇐ ⇒ r | ξ | ≤ H ( ξ ) ≤ R | ξ | . As σ D ( t, x ) = ∇ Q ( e D ( t, x )), the flow rule can be rewritten as 0 ∈ − σ D ( t, x ) + ∂H ( ˙ p ( t, x )) 0 ∈ −∇ Q ( e D ( t, x )) + ∂H ( ˙ p ( t, x )) 0 ∈ ∂ p Q ( E D u ( t, x ) − p ) | p = p ( t,x ) + ∂H ( ˙ p ( t, x )) 0 ∈ ∂ p Q ( Eu ( t, x ) − p ) | p = p ( t,x ) + ∂H ( ˙ p ( t, x )) Calculus of Variations and Nonlinear Partial Differential Equations, Page 8/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity k − t i − 1 Discretize time: 0 = t 0 k < t 1 1 ≤ i ≤ k ( t i k < · · · < t k k = T with max ) → 0. k Discretizing 0 ∈ ∂ p Q ( Eu ( t, x ) − p ) | p = p ( t,x ) + ∂H ( ˙ p ( t, x )) we get k , x ) − p ( t i − 1 0 ∈ ∂ p Q ( Eu ( t i k ,x ) + ∂H ( p ( t i k , x ) − p ) | p = p ( t i , x )) k k ,x ) + ∂ p H ( p − p ( t i − 1 0 ∈ ∂ p Q ( Eu ( t i k , x ) − p ) | p = p ( t i , x )) | p = p ( t i k ,x ) k k , x ) − p ) + H ( p − p ( t i − 1 p ( t i k , x ) minimizes Q ( Eu ( t i so , x )) . k � H ( p ( x )) dx . Then for a given u ( t i Let H ( p ) := k ) Ω k ) − p ) + H ( p − p ( t i − 1 p ( t i k ) minimizes Q ( Eu ( t i )) , k while for a given p ( t i k ) ( u ( t i k ) , e ( t i k )) minimizes Q ( e ) under the condition Eu = e + p ( t i k ) on Ω , u = w ( t i k ) on Γ 0 . Calculus of Variations and Nonlinear Partial Differential Equations, Page 9/43 June 27 - July 2, 2005, Cetraro (Cosenza)

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity Incremental Problems In other words for a given u ( t i k ) k ) minimizes Q ( e ) + H ( p − p ( t i − 1 p ( t i )) under k the condition Eu ( t i k ) = e + p on Ω, while for a given p ( t i k ) k ) − p ( t i − 1 ( u ( t i k ) , e ( t i k )) minimizes Q ( e ) + H ( p ( t i )) under k the condition Eu = e + p ( t i k ) on Ω , u = w ( t i k ) on Γ 0 . This leads to define by induction ( u i k , e i k , p i k ) as a solution to � � Q ( e ) + H ( p − p i − 1 ) : ( u, e, p ) ∈ A ( w ( t i min k )) , k where A ( w ) := { ( u, e, p ) : Eu = e + p, u = w on Γ 0 } . We set ( u 0 k , e 0 k , p 0 k ) equal to the initial condition ( u 0 , e 0 , p 0 ). Calculus of Variations and Nonlinear Partial Differential Equations, Page 10/43 June 27 - July 2, 2005, Cetraro (Cosenza)