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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

Mn×n

sym = Mn×n D

⊕ RI A = AD + 1

n(tr A)I

with tr AD = 0 reference configuration: Ω ⊂ Rn bounded domain with ∂Ω ∈ C2 displacement: u : Ω → Rn linearized strain: Eu = 1

2(∇u + ∇uT )

(mechanically questionable) additive decomposition: Eu = e + p (mechanically questionable) elastic strain: e: Ω → Mn×n

sym

plastic strain: p: Ω → Mn×n

D

stress: σ = 2 µ eD + κ(tr e)I (µ, κ > 0) stress constraint: σD(x) ∈ K for a.e. x ∈ Ω , where K ⊂ Mn×n

D

is closed, convex, and B(0, r) ⊂ K ⊂ B(0, R) yield surface: ∂K

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 1/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Classical Formulation

Assume ∂Ω = Γ0 ∪ Γ1, with Γ0, Γ1 relatively open, Γ0 ∩ Γ1 = Ø, and ∂Γ0 = ∂Γ1 ∈ C2 . Given f(t, x) body force on Ω, g(t, x) surface force on Γ1, and w(t, x) boundary displacement on Γ0,

Ω

f(t) g(t)

Γ Γ

1

g(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, June 27 - July 2, 2005, Cetraro (Cosenza) Page 2/43

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 eD(t, x) + κ(tr e(t, x))I

• equilibrium conditions

−div

xσ(t, x) = f(t, x) on Ω,

σ(t, x) ν(x) = g(t, x) on Γ1

• boundary conditions

u(t, x) = w(t, x) on Γ0

• stress constraint

σD(t, x) ∈ K

• Prandtl-Reuss flow rule

˙ p(t, x) ∈ NK(σD(t, x)) , NK(ξD) = normal cone to K at ξD

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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.

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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 Γ0. Let Q(A) := µ|AD|2 + κ

2 (tr A)2

for every A ∈ Mn×n

sym .

Elastic energy: Q(e(t)) :=

• Ω

Q(e(t, x)) dx. Stress: σ(t, x) = ∇Q(e(t, x)), i.e., σ(t) = ∇Q(e(t)). Deviatoric part: σD(t, x) = ∇Q(eD(t, x)), i.e., σD(t) = ∇Q(eD(t)). Let us fix p(t) and consider the minimum problem min

Eu=e+p(t) on Ω u=w(t) on Γ0

Q(e) .

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Let us fix p(t) and consider the minimum problem min

Eu=e+p(t) on Ω u=w(t) on Γ0

Q(e) . Suppose that (u(t), e(t)) is a minimizer. Considering the variation (u(t) + εϕ, e(t) + εEϕ), with ϕ smooth and vanishing on Γ0, 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 Γ0. As σ(t) = ∇Q(e(t)), we conclude that σ(t), Eϕ = 0 for any such ϕ, hence div

xσ(t) = 0 on Ω ,

σ(t)ν = 0 on Γ1 .

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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 f ∗(x∗) := sup

x∈X

{x∗, x − f(x)} for every 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 χC is defined by χC(x) = 0 for x ∈ C , and χC(x) = +∞ for x / ∈ C . We have ∂χC(x) = NC(x), the normal cone to C at x, i.e., x∗ ∈ ∂χC(x) ⇐ ⇒ x∗, y − x ≤ 0 for every y ∈ C .

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Incremental Problems: the Flow Rule

The flow rule ˙ p(t, x) ∈ NK(σD(t, x)) = ∂χK(σD(t, x)) can be written as σD(t, x) ∈ ∂H( ˙ p(t, x)) , where H(ξ) := χ∗

K(ξ) = sup η∈K

ξ : η is the support function of K .

• H is convex and positively homogeneous of degree 1;
• B(0, r) ⊂ K ⊂ B(0, R)

⇐ ⇒ r|ξ| ≤ H(ξ) ≤ R|ξ|. As σD(t, x) = ∇Q(eD(t, x)), the flow rule can be rewritten as 0 ∈ −σD(t, x) + ∂H( ˙ p(t, x)) 0 ∈ −∇Q(eD(t, x)) + ∂H( ˙ p(t, x)) 0 ∈ ∂pQ(EDu(t, x) − p)|p=p(t,x) + ∂H( ˙ p(t, x)) 0 ∈ ∂pQ(Eu(t, x) − p)|p=p(t,x) + ∂H( ˙ p(t, x))

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Discretize time: 0 = t0

k < t1 k < · · · < tk k = T with max 1≤i≤k(ti k − ti−1 k

) → 0. Discretizing 0 ∈ ∂pQ(Eu(t, x) − p)|p=p(t,x) + ∂H( ˙ p(t, x)) we get 0 ∈ ∂pQ(Eu(ti

k, x) − p)|p=p(ti

k,x) + ∂H(p(ti

k, x) − p(ti−1 k

, x)) 0 ∈ ∂pQ(Eu(ti

k, x) − p)|p=p(ti

k,x) + ∂pH(p − p(ti−1

k

, x))|p=p(ti

k,x)

so p(ti

k, x) minimizes Q(Eu(ti k, x) − p) + H(p − p(ti−1 k

, x)) . Let H(p) :=

• Ω

H(p(x)) dx. Then for a given u(ti

k)

p(ti

k) minimizes Q(Eu(ti k) − p) + H(p − p(ti−1 k

)) , while for a given p(ti

k)

(u(ti

k), e(ti k)) minimizes Q(e) under the condition

Eu = e + p(ti

k) on Ω , u = w(ti k) on Γ0. Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 9/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Incremental Problems

In other words for a given u(ti

k)

p(ti

k) minimizes Q(e) + H(p − p(ti−1 k

)) under the condition Eu(ti

k) = e + p on Ω,

while for a given p(ti

k)

(u(ti

k), e(ti k)) minimizes Q(e) + H(p(ti k) − p(ti−1 k

)) under the condition Eu = e + p(ti

k) on Ω , u = w(ti k) on Γ0.

This leads to define by induction (ui

k, ei k, pi k) as a solution to

min

• Q(e) + H(p − pi−1

k

) : (u, e, p) ∈ A(w(ti

k))

• ,

where A(w) := {(u, e, p) : Eu = e + p, u = w on Γ0}. We set (u0

k, e0 k, p0 k) equal to the initial condition (u0, e0, p0). Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 10/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

The Space BD(Ω)

Since H(p) :=

• Ω

H(p(x)) dx and r|ξ| ≤ H(ξ) ≤ R|ξ|, in general the minimum problem min

• Q(e) + H(p − pi−1

k

) : (u, e, p) ∈ A(w(ti

k))

• has no solution, if we interpret

A(w) := {(u, e, p) ∈ W 1,1(Ω; Rn)×L2(Ω; Mn×n

sym)×L1(Ω; Mn×n D

) : Eu = e + p, u = w on Γ0} . Use the space BD(Ω), defined as {u ∈ L1(Ω; Rn) : Eu ∈ Mb(Ω; Mn×n

sym)}

(Temam R.: Mathematical problems in plasticity, 1985).

• Weak∗ compactness.
• Trace on ∂Ω.
• If u0 = u on Ω and u0 = 0 on Ωc , then Eu0 = −u ⊙ ν Hn−1 on Γ0.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Relaxed Incremental Problems

For p ∈ Mb(Ω ∪ Γ0; Mn×n

D

) we define H(p) :=

• Ω∪Γ0

H dp d|p|

• d|p|.

It is possible to prove (see Goffman-Serrin 1964 or Temam 1985) that H is weakly∗ lower semicontinuous on Mb(Ω ∪ Γ0; Mn×n

D

). For w ∈ H1(Ω; Rn) we define A(w):= {(u, e, p) ∈ BD(Ω)×L2(Ω; Mn×n

sym)×Mb(Ω ∪ Γ0; Mn×n D

) : Eu = e + p on Ω, p = (w − u) ⊙ ν Hn−1 on Γ0}. Define (ui

k, ei k, pi k) by induction as a solution to

min

• Q(e) + H(p − pi−1

k

) : (u, e, p) ∈ A(w(ti

k))

• .

Existence of solutions follows now from the direct methods.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Euler Conditions

Since (ui

k, ei k, pi k) is a solution to

min

• Q(e) + H(p − pi−1

k

) : (u, e, p) ∈ A(w(ti

k))

• ,

by the triangle inequality for H the triple (ui

k, ei k, pi k) is a solution to

min

• Q(e) + H(p − pi

k) : (u, e, p) ∈ A(w(ti k))

• .

We assume that this is true also for the initial condition (u0, e0, p0). Considering the variation (ui

k + εϕ, ei k + εEϕ, pi k) ∈ A(w(ti k)), with ϕ

smooth and vanishing on Γ0, we get Q(ei

k + εEϕ) ≥ Q(ei k) for every ε.

Then σi

k, Eϕ = ∇Q(ei k), Eϕ = 0 for every such function. This

implies div σi

k = 0 on Ω ,

σi

kν = 0 on Γ1 . Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 13/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Since (ui

k, ei k, pi k) is a solution to

min

• Q(e) + H(p − pi

k) : (u, e, p) ∈ A(w(ti k))

• ,

considering the variation (ui

k, ei k − εq, pi k + εq) ∈ A(w(ti k)), with

q ∈ L2(Ω; Mn×n

D

), we get Q(ei

k − εq) + H(εq) ≥ Q(ei k) for every ε > 0.

Taking the right derivative with respect to ε at ε = 0, we get −∇Q(ei

k), q + H(q) ≥ 0 ,

which is equivalent to (σi

k)D, q ≤ H(q)

for every q ∈ L2(Ω; Mn×n

D

). By localizing we obtain for a.e. x ∈ Ω (σi

k)D(x) : ξ ≤ H(ξ)

for every ξ ∈ Mn×n

D

. This implies (σi

k)D(x) ∈ ∂H(0) = K for a.e. x ∈ Ω (use H = χ∗ K ).

We have already obtained div σi

k = 0 on Ω, and σi kν = 0 on Γ1.

By convexity the conditions in red are equivalent to minimality.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Discrete Energy Estimate

Piecewise constant interpolation: uk(t) := ui

k, ek(t) := ei k, pk(t) := pi k, σk(t) := σi k for ti k ≤ t < ti+1 k

. We assume that t → w(t) is absolutely continuous from [0, T] into H1(Ω; Rn). Let us define wi

k := w(ti k).

We now prove that there exists a sequence δk → 0+ such that Q(ei

k) + i

• j=1

H(pj

k − pj−1 k

) ≤ Q(e0) + ti

k

σk(t), E ˙ w(t) dt + δk for every k and every i = 1, . . . , k.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Proof of the Discrete Energy Estimate

Given j with 1 ≤ j ≤ i, let us compare the minimizer (uj

k, ej k, pj k) with

(uj−1

k

− wj−1

k

+ wj

k, ej−1 k

− Ewj−1

k

+ Ewj

k, pj−1 k

) ∈ A(wj

k). By minimality

Q(ej

k) + H(pj k − pj−1 k

) ≤ Q(ej−1

k

+ Ewj

k − Ewj−1 k

) . The right-hand side can be developed as Q(ej−1

k

+ Ewj

k − Ewj−1 k

) = = Q(ej−1

k

) + σj−1

k

, Ewj

k − Ewj−1 k

+ Q(Ewj

k − Ewj−1 k

) . From the absolute continuity of w with respect to t we have Ewj

k − Ewj−1 k

= tj

k

tj−1

k

E ˙ w(t) dt , hence σj−1

k

, Ewj

k − Ewj−1 k

= tj

k

tj−1

k

σk(t), E ˙ w(t) dt .

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Moreover Q(Ewj

k − Ewj−1 k

) ≤ C tj

k

tj−1

k

E ˙ w(t)2 dt 2 . Therefore Q(ej

k) + H(pj k − pj−1 k

) ≤ ≤ Q(ej−1

k

) + σj−1

k

, Ewj

k − Ewj−1 k

+ Q(Ewj

k − Ewj−1 k

) ≤ ≤ Q(ej−1

k

) + tj

k

tj−1

k

σk(t), E ˙ w(t) dt + C tj

k

tj−1

k

E ˙ w(t)2 dt 2 ≤ ≤ Q(ej−1

k

) + tj

k

tj−1

k

σk(t), E ˙ w(t) dt + ρk tj

k

tj−1

k

E ˙ w(t)2 dt , where ρk := C max

1≤j≤k

tj

k

tj−1

k

E ˙ w(t)2 dt → 0 by the absolute continuity of the integral.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

For every 1 ≤ j ≤ i we have obtained Q(ej

k) + H(pj k − pj−1 k

) ≤ ≤ Q(ej−1

k

) + tj

k

tj−1

k

σk(t), E ˙ w(t) dt + ρk tj

k

tj−1

k

E ˙ w(t)2 dt . Summing with respect to j we get Q(ei

k) + i

• j=1

H(pj

k − pj−1 k

) ≤ Q(e0) + ti

k

σk(t), E ˙ w(t) dt + δk , where δk := ρk T

0 E ˙

w(t)2 dt → 0. From this we obtain sup

0≤t≤T

ek(t)2

2 ≤ Q(e0) + sup 0≤t≤T

ek(t)2 T E ˙ w(t)2 dt + δk , hence ek(t) is bounded in L2(Ω, Mn×n

sym) uniformly in k and t.

In particular the right-hand side of the energy inequality is bounded.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Dissipation and Helly Theorem

For every t → p(t) from [0, T] into Mb(Ω ∪ Γ0; Mn×n

D

) we define the dissipation of p in the interval [s, t] ⊂ [0, T] by D(p; s, t) := sup

s=s0<s1<···<sm=t m

• i=1

H(p(si) − p(si−1)) . For the piecewise constant interpolations pk(t) of pi

k we have

D(pk; 0, ti

k) = i

• j=1

H(pj

k − pj−1 k

) , so that Q(ek(t)) + D(pk; 0, t) ≤ Q(e0) + t σk(s), E ˙ w(s) ds + δk . Discrete Energy Estimate = ⇒ D(pk; 0, T) is bounded uniformly in k. Helly Theorem = ⇒ there exists a subsequence, independent of t, such that pk(t) ⇀ p(t) weakly∗ in Mb(Ω ∪ Γ0; Mn×n

D

) for every t ∈ [0, T].

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Minimality of the Limit Functions

Extracting a further subsequence (depending on t), ukj(t) ⇀ u(t) weakly∗ in BD(Ω) and ekj(t) ⇀ e(t) weakly in L2(Ω; Mn×n

sym).

It is easy to prove that (u(t), e(t), p(t)) ∈ A(w(t)), i.e., Eu(t) = e(t) + p(t) on Ω, p(t) = (w(t) − u(t)) ⊙ ν Hn−1 on Γ0 . By the Euler conditions div

xσkj(t) = 0 on Ω ,

σkj(t)ν = 0 on Γ1 , (σkj)D(t) ∈ K a.e. on Ω . As σkj(t) ⇀ σ(t) := ∇Q(e(t)) weakly, we pass to the limit and get div

xσ(t) = 0 on Ω ,

σ(t)ν = 0 on Γ1 , σD(t) ∈ K a.e. on Ω . Since Euler conditions imply minimality, we obtain that (u(t), e(t), p(t)) solves min

• Q(e) + H(p − p(t)) : (u, e, p) ∈ A(w(t))
• .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 20/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Convergence of the Solutions of the Incremental Problems

Using the strict convexity of Q, it is easy to prove that for any given p(t) ∈ Mb(Ω ∪ Γ0; Mn×n

D

) there exists at most one pair (u(t), e(t)) such that (u(t), e(t), p(t)) is a solution to the minimum problem min

• Q(e) + H(p − p(t)) : (u, e, p) ∈ A(w(t))
• .

Therefore, the limit functions u(t) and e(t) do not depend on the

• subsequence. This implies that the whole sequence converges, i.e.,

uk(t)⇀u(t) weakly∗ in BD(Ω) and ek(t)⇀e(t) weakly in L2(Ω; Mn×n

sym).

A similar argument shows that t → e(t) is weakly continuous from [0, T] into L2(Ω; Mn×n

sym) at those points t where t → p(t) is continuous

from [0, T] into Mb(Ω ∪ Γ0; Mn×n

D

). As t → p(t) has bounded variation, this implies that the set of discontinuity points of t → e(t) is at most countable.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Energy Inequality

Since pk(t) ⇀ p(t) weakly∗ in Mb(Ω ∪ Γ0; Mn×n

D

) for every t ∈ [0, T], and H is lower semicontinuous with respect to weak∗ convergence, we get D(p; 0, t) ≤ lim inf

k→∞ D(pk; 0, t) .

As σk(t) := ∇Q(ek(t)) ⇀ σ(t) := ∇Q(e(t)) weakly in L2(Ω; Mn×n

sym) for

every t ∈ [0, T], we can pass to the limit in the discrete energy estimate and we obtain Q(e(t)) + D(p; 0, t) ≤ Q(e(0)) + t σ(s), E ˙ w(s) ds .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 22/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

The Opposite Energy Inequality

We now prove that the fact that (u(t), e(t), p(t)) solves min

• Q(e) + H(p − p(t)) : (u, e, p) ∈ A(w(t))
• implies the opposite energy inequality

Q(e(t)) + D(p; 0, t) ≥ Q(e(0)) + t σ(s), E ˙ w(s) ds . Let us fix t ∈ (0, T] and let si

k = i kt.

Let us compare the minimizer (u(si−1

k

), e(si−1

k

), p(si−1

k

)) with (u(si

k)−w(si k)+w(si−1 k

), e(si

k)−Ew(si k)+Ew(si−1 k

), p(si

k)) ∈ A(w(si−1 k

)). By minimality Q(e(si−1

k

)) ≤ Q(e(si

k) − (Ew(si k) − Ew(si−1 k

))) + H(p(si

k) − p(si−1 k

)) .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 23/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

We have obtained Q(e(si−1

k

)) ≤ Q(e(si

k) − (Ew(si k) − Ew(si−1 k

))) + H(p(si

k) − p(si−1 k

)) . The first term in the right-hand side can be written as Q(e(si

k) − (Ew(si k) − Ew(si−1 k

))) = = Q(e(si

k)) − σ(si k), Ew(si k) − Ew(si−1 k

) + Q(Ew(si

k) − Ew(si−1 k

)) . Now, arguing as in the proof of the discrete energy estimate we obtain that there exists a sequence ρk → 0+ such that Q(e(si−1

k

)) ≤ Q(e(si

k)) + H(p(si k) − p(si−1 k

)) − − si

k

si−1

k

σ(si

k), E ˙

w(s) ds + ρk si

k

si−1

k

E ˙ w(s)2 ds . We define the piecewise constant function σk : [0, t] → L2(Ω; Mn×n

sym) by

setting σk(s) := σ(si

k) for si−1 k

< s ≤ si

k . Therefore Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 24/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Q(e(si−1

k

)) ≤ Q(e(si

k)) + H(p(si k) − p(si−1 k

)) − − si

k

si−1

k

σk(s), E ˙ w(s) ds + ρk si

k

si−1

k

E ˙ w(s)2 ds . Since

k

• i=1

H(p(si

k) − p(si−1 k

)) ≤ DH(p; 0, t), summing in i we obtain Q(e(0)) ≤ Q(e(t)) + DH(p; 0, t) − t σk(s), E ˙ w(s) ds + δk , where δk := ρk T E ˙ w(s)2 ds → 0. As the set of discontinuity points

• f the function s → σ(s) is at most countable, σk(s) ⇀ σ(s) weakly in

L2(Ω; Mn×n

sym) for a.e. s ∈ [0, t]. Therefore we can pass to the limit and

we obtain Q(e(t)) + D(p; 0, t) ≥ Q(e(0)) + t σ(s), E ˙ w(s) ds .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 25/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Quasistatic Evolution

The results obtained so far lead to the following definition. A quasistatic evolution is a map t → (u(t), e(t), p(t)) from [0, T] into BD(Ω)×L2(Ω; Mn×n

sym)×Mb(Ω ∪ Γ0; Mn×n D

) such that for every t ∈ [0, T] we have:

• global stability: (u(t), e(t), p(t)) ∈ A(w(t)) and

Q(e(t)) ≤ Q(e) + H(p − p(t)) for every (u, e, p) ∈ A(w(t));

• energy balance: s → p(s) has bounded variation from [0, T] into

Mb(Ω ∪ Γ0; Mn×n

D

) and Q(e(t)) + D(p; 0, t) = Q(e(0)) + t σ(s), E ˙ w(s) ds , where σ(t) := ∇Q(e(t)).

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 26/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Existence Theorem

Let us summarize the result obtained so far.

• Theorem. Let t → w(t) be an absolutely continuous function from

[0, T] into H1(Ω; Rn) and let (u0, e0, p0) be globally stable at time t = 0, i.e., (u0, e0, p0) ∈ A(w(0)) such that Q(e0) ≤ Q(e) + H(p − p0) for every (u, e, p) ∈ A(w(0)). Then there exists a quasistatic evolution t → (u(t), e(t), p(t)) such that (u(0), e(0), p(0)) = (u0, e0, p0). A similar result can be obtained with external body and surface forces f(t) and g(t), provided a suitable safe load condition is satisfied.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Notation

We recall that the elastic energy was defined by Q(e(t)) :=

• Ω

Q(e(t, x)) dx , where Q(A) := µ|AD|2 + κ 2 (tr A)2 for every A ∈ Mn×n

sym .

For w ∈ H1(Ω; Rn) we define A(w):= {(u, e, p) ∈ BD(Ω)×L2(Ω; Mn×n

sym)×Mb(Ω ∪ Γ0; Mn×n D

) : Eu = e + p on Ω, p = (w − u) ⊙ ν Hn−1 on Γ0}.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Quasistatic Evolution

Given an absolutely continuous function t → w(t) from [0, T] into H1(Ω; Rn), a quasistatic evolution is a map t → (u(t), e(t), p(t)) from [0, T] into BD(Ω)×L2(Ω; Mn×n

sym)×Mb(Ω ∪ Γ0; Mn×n D

) such that for every t ∈ [0, T] we have:

• global stability: (u(t), e(t), p(t)) ∈ A(w(t)) and

Q(e(t)) ≤ Q(e) + H(p − p(t)) for every (u, e, p) ∈ A(w(t));

• energy balance: s → p(s) has bounded variation from [0, T] into

Mb(Ω ∪ Γ0; Mn×n

D

) and Q(e(t)) + D(p; 0, t) = Q(e(0)) + t σ(s), E ˙ w(s) ds , where σ(t) := ∇Q(e(t)).

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 29/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Existence Theorem

Let us summarize the result obtained so far.

• Theorem. Let t → w(t) be an absolutely continuous function from

[0, T] into H1(Ω; Rn) and let (u0, e0, p0) be globally stable at time t = 0, i.e., (u0, e0, p0) ∈ A(w(0)) such that Q(e0) ≤ Q(e) + H(p − p0) for every (u, e, p) ∈ A(w(0)). Then there exists a quasistatic evolution t → (u(t), e(t), p(t)) such that (u(0), e(0), p(0)) = (u0, e0, p0).

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 30/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Properties of Quasistatic Evolutions

Let t→(u(t), e(t), p(t)) be a quasistatic evolution and let σ(t):=∇Q(e(t)). For t ∈ [0, T] the global stability is equivalent to the Euler conditions div

xσ(t) = 0 on Ω ,

σ(t)ν = 0 on Γ1 , σD(t) ∈ K a.e. on Ω (same argument used for the incremental problems). In particular σ(t), e ≥ −H(q) for every (v, e, q) ∈ A(0).

• Theorem. The functions t → e(t) and t → σ(t) are absolutely

continuous from t ∈ [0, T] into L2(Ω; Mn×n

sym) and

˙ e(t)2 ≤ C E ˙ w(t)2 , ˙ σ(t)2 ≤ C E ˙ w(t)2 for a.e. t ∈ [0, T].

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 31/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Proof of the Absolute Continuity of t → e(t)

As Q(e(t)) = 1

2σ(t), e(t) and H(p(t2) − p(t1)) ≤ D(p; t1, t2), by the

energy equality

1 2σ(t2), e(t2) − 1 2σ(t1), e(t1) + H(p(t2) − p(t1)) ≤

t2

t1

σ(s), E ˙ w(s) ds for every t1 , t2 ∈ [0, T] with t1 < t2 . By the Euler condition we have −σ(t1), e ≤ H(q) for every (v, e, q) ∈ A(0). Taking (v, e, q) equal to (u(t2)−u(t1)−(w(t2)−w(t1)), e(t2)−e(t1)−(Ew(t2)−Ew(t1)), p(t2)−p(t1)) we obtain −σ(t1), e(t2)+σ(t1), e(t1)+σ(t1), Ew(t2)−Ew(t1) ≤ H(p(t2)−p(t1)) , which can be written as −σ(t1), e(t2) + σ(t1), e(t1) + t2

t1

σ(t1), E ˙ w(s) ds ≤ H(p(t2) − p(t1)) .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 32/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Adding the inequalities

1 2σ(t2),e(t2) − 1 2σ(t1),e(t1) + H(p(t2)−p(t1)) ≤

t2

t1

σ(s),E ˙ w(s)ds −σ(t1),e(t2) + σ(t1),e(t1) + t2

t1

σ(t1),E ˙ w(s)ds ≤ H(p(t2)−p(t1)) , we obtain

1 2σ(t2), e(t2) − 1 2σ(t1), e(t1) − σ(t1), e(t2) + σ(t1), e(t1) ≤

≤ t2

t1

σ(s) − σ(t1)|E ˙ w(s) ds . Putting together the terms in red we get

1 2σ(t2) − σ(t1), e(t2) − e(t1) ≤

t2

t1

σ(s) − σ(t1), E ˙ w(s) ds .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 33/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

The inequality

1 2σ(t2) − σ(t1), e(t2) − e(t1) ≤

t2

t1

σ(s) − σ(t1), E ˙ w(s) ds implies that e(t2) − e(t1)2

2 ≤ C

t2

t1

e(s) − e(t1)2 E ˙ w(s)2 ds . By the Gronwall lemma we deduce e(t2) − e(t1)2 ≤ C t2

t1

E ˙ w(s)2 ds . This implies that the function t → e(t) is absolutely continuous from [0, T] into L2(Ω; Mn×n

sym) and that

˙ e(t)2 ≤ C E ˙ w(t)2 .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 34/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Absolute Continuity of t → p(t) and t → u(t)

• Theorem. Let t → (u(t), e(t), p(t)) be a quasistatic evolution. Then

the functions t → p(t) and t → u(t) are absolutely continuous on [0, T] with values in Mb(Ω ∪ Γ0; Mn×n

D

) and BD(Ω), respectively.

• Proof. As H(p(t2) − p(t1)) ≤ D(p; t1, t2), by the energy equality

Q(e(t2)) − Q(e(t1)) + H(p(t2) − p(t1)) ≤ t2

t1

σ(s), E ˙ w(s) ds . Therefore r p(t2) − p(t1)1 ≤ Q(e(t1)) − Q(e(t2)) + t2

t1

σ(s), E ˙ w(s) ds . Since t → Q(e(t)) is absolutely continuous on [0, T], the previous inequality yields the absolute continuity of t → p(t). As Eu(t) = e(t) + p(t) on Ω and p(t) = (w(t) − u(t)) ⊙ ν Hn−1 on Γ0, the absolute continuity of t → u(t) follows.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Mb(Ω ∪ Γ0; Mn×n

D

)-Valued AC Functions

Using standard methods as in (*) we can prove that, if t → p(t) is absolutely continuous from [0, T] with values in Mb(Ω ∪ Γ0; Mn×n

D

), then the weak∗ -limit ˙ p(t) := w∗- lim

s→t

p(s) − p(t) s − t exists for a.e. t ∈ [0, T], and H( ˙ p(t)) = lim

s→t H

p(s) − p(t) s − t

• for a.e. t ∈ [0, T]. Moreover, the function t → H( ˙

p(t)) is measurable and D(p; a, b) = b

a

H( ˙ p(t)) dt for every a, b ∈ [0, T] with a ≤ b.

(*) Appendix of Brezis: Op´ erateurs maximaux monotones . . . , 1973.

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 36/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Balance of Powers

Let t→(u(t), e(t), p(t)) be a quasistatic evolution and let σ(t):=∇Q(e(t)). Then the energy balance Q(e(t)) + D(p; 0, t) = Q(e(0)) + t σ(s), E ˙ w(s) ds ∀t ∈ [0, T] is equivalent to the balance of powers σ(t), ˙ e(t) + H( ˙ p(t)) = σ(t), E ˙ w(t) for a.e. t ∈ [0, T] . Indeed, since all functions involved are absolutely continuous, the second formula is obtained from the first formula by differentiation, using the equality d dtD(p; 0, t) = H( ˙ p(t)) for a.e. t ∈ [0, T] .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 37/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Stress-Strain Duality

Let Σ(Ω) be the space of admissible stresses, defined by Σ(Ω) := {σ ∈ L2(Ω; Mn×n

sym) : div σ ∈ Ln(Ω; Rn), σD ∈ L∞(Ω; Mn×n D

)} . Given σ ∈ Σ(Ω) and u ∈ BD(Ω) with div u ∈ L2(Ω) we define the distribution [σD : EDu] on Ω by [σD : EDu]|ϕ := −div σ|ϕ u − 1

ntr σ|ϕ div u − σ|u ⊙ ∇ϕ

for every ϕ ∈ C∞

c (Ω). It is proved in (*) that [σD : EDu] is a bounded

measure on Ω whose variation satisfies |[σD : EDu]| ≤ σD∞|EDu| in Ω . It is also proved in (*) that, if σ ∈ Σ(Ω), then [σν]tan ∈ L∞(∂Ω; Rn).

(*) Kohn-Temam: Dual spaces of stresses ... Appl. Math. Optim. 1983.

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Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Stress-Plastic Strain Duality

Let ΠΓ0(Ω) be the space of admissible plastic strains, defined as the set

• f all p ∈ Mb(Ω ∪ Γ0; Mn×n

D

) for which there exist u ∈ BD(Ω), w ∈ H1(Ω; Rn), and e ∈ L2(Ω; Mn×n

sym) satisfying

p = Eu − e = ED − eD on Ω , p = (w − u) ⊙ ν Hn−1 on Γ0 . Given σ ∈ Σ(Ω) and p ∈ ΠΓ0(Ω), we fix u ∈ BD(Ω), e ∈ L2(Ω; Mn×n

sym),

and w ∈ H1(Ω; Rn) satisfying the previous equality. Then we define a measure [σD : p] ∈ Mb(Ω ∪ Γ0) by setting [σD : p] := [σD : EDu] − σD : eD

• n Ω ,

[σD : p] := [σν]tan · (w − u)

• n Γ0 .

It is possible to prove that [σD : p] does not depend on u, w, and e. We define σD, p := [σD : p](Ω ∪ Γ0).

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 39/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Integration by Parts

Let σ ∈ Σ(Ω), let w ∈ H1(Ω; Rn), and let (u, e, p) ∈ A(w), so that Eu = e + p on Ω and p = (w − u) ⊙ ν on Γ0. Then σD, p + σ, e − Ew + div σ, u − w = [σν]tan, u − wΓ1 . Let us define K(Ω) := {σ ∈ L2(Ω; Mn×n

sym) : div σ ∈ Ln(Ω; Rn), σD(x) ∈ K a.e. x ∈ Ω} .

The definition of H H(ξ) := sup

η∈K

η : ξ leads to the equality H(p) := sup

σ∈K(Ω)

σD, p for every p ∈ ΠΓ0(Ω).

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 40/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Further Properties of Quasistatic Evolutions

Let t → (u(t), e(t), p(t)) be a quasistatic evolution with boundary condition t → w(t), and let σ(t) := ∇Q(e(t)). Since (u(t), e(t), p(t)) ∈ A(w(t)) for every t ∈ [0, T], it follows that ( ˙ u(t), ˙ e(t), ˙ p(t)) ∈ A( ˙ w(t)) for a.e. t ∈ [0, T]. Using the formula σD, p + σ, e − Ew + div σ, u − w = [σν]tan, u − wΓ1 , with σ = σ(t), (u, e, p) = ( ˙ u(t), ˙ e(t), ˙ p(t)), and w = ˙ w(t), we obtain σD(t), ˙ p(t) + σ(t), ˙ e(t) − E ˙ w(t) = 0 for a.e. t ∈ [0, T] . On the other hand the balance of powers gives σ(t), ˙ e(t) + H( ˙ p(t)) = σ(t), E ˙ w(t) for a.e. t ∈ [0, T] . Adding these equalities we get H( ˙ p(t)) = σD(t), ˙ p(t) for a.e. t ∈ [0, T] .

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 41/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Flow Rule: Weak Formulation

The equality just proven H( ˙ p(t)) = σD(t), ˙ p(t) for a.e. t ∈ [0, T] , together with the inequality H( ˙ p(t)) ≥ τD, ˙ p(t) for a.e. t ∈ [0, T] , valid for any τ ∈ K(Ω), implies that for a.e. t ∈ [0, T] τD − σD(t), ˙ p(t) ≤ 0 for every τ ∈ K(Ω) . This inequality says that ˙ p(t) belongs to the normal cone to the convex set K(Ω) in the sense of the duality product between stresses and plastic

• strains. It can be considered as a weak formulation of the flow rule

˙ p(t, x) ∈ NK(σD(t, x)).

Calculus of Variations and Nonlinear Partial Differential Equations, June 27 - July 2, 2005, Cetraro (Cosenza) Page 42/43

Gianni Dal Maso, SISSA, Trieste Rate-Independent Evolution Problems in Elasto-Plasticity

Flow Rule: Strong Formulation

If ˙ p(t) ∈ L2(Ω; Mn×n

D

), the classical formulation of the flow rule is ˙ p(t, x) | ˙ p(t, x)| ∈ NK(σD(t, x)) for Ln-a.e. x ∈ Ω. If ˙ p(t) ∈ Mb(Ω ∪ Γ0; Mn×n

D

), we expect ˙ p(t) | ˙ p(t)|(x) ∈ NK(ˆ σD(t, x)) for | ˙ p(t)|-a.e. x ∈ Ω, (*) where ˙ p(t) / | ˙ p(t)| is the Radon-Nikodym derivative of ˙ p(t) w.r.t. | ˙ p(t)|. We need a precise representative ˆ σD(t, x) of σD(t, x) defined a.e. w.r.t. Ln + | ˙ p(t)|. Theorem. If K is strictly convex, then (*) is satisfied with ˆ σD(t, x) defined as suitable limit of the averages of σD(t, y) on balls B(x, ρ).

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