Rate-independent damage in thermo-viscoelastic materials with inertia

Rate-independent damagein thermo-viscoelastic materials with inertia

Giuliano Lazzaroni

SISSA, Trieste, Italy

STAMM 2014 at Poitiers

Joint work with R. Rossi (Brescia), M. Thomas (WIAS Berlin), and R. Toader (Udine)

Giuliano Lazzaroni (SISSA) Rate-independent damage in thermo-viscoelastic materials 1 / 14

Damage in thermo-visco-elastodynamics

Aim: Existence result for evolutionary systems including • partial damageAim: Existence result for evolutionary systems including • viscosity & inertiaAim: Existence result for evolutionary systems including • thermal effects

8

>

>

<

>

>

:

for u : damped equation of elastodynamics

for z : rate-independent, unidirectional flow rule

for ✓ : heat equation coupled in a consistent way

u = displacementz = damage (internal variable), z 2 [0, 1] ✓= temperature

⇢

z= 1 : sound materialz= 0 : most damaged state

Modelling: Frémond’s approach, Generalised Standard Materials

Difficulty: Interplay of rate-independent and rate-dependent phenomena (Roubícek)


The model: 1. Thermo-viscoelasticity⌦⇢R3 bounded, Lipschitz, u(t, x)2R3, e(u)=ru+ruT

2 , z(t, x)2[0, 1], ✓(t, x)2R+

thermo-visco

elasticity: ⇢ u � div �(u) = F(t)

where �(u) =C e(u)

+ D e(u)� ✓B

(z)(z, ✓)

✓ � div�

K(z, ✓)r✓�

= |z|+⇥

D(z, ✓)e(u)�✓B⇤

: e(u) + H(t)

• Kelvin-Voigt rheology & inertia• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation

Assumptions: C 2 C0,1(R;R3⇥3⇥3⇥3), D 2 C0(R⇥ R;R3⇥3⇥3⇥3)

Assumptions: C, D are bounded, symmetric, positive definite, uniformly in (z, ✓)Assumptions: C is nondecreasing: 8 z1 z2 C(z1)A : A C(z2)A : A



2 , z(t, x)2[0, 1], ✓(t, x)2R+

thermo-

viscoelasticity: ⇢ u � div �(u) = F(t)

where �(u) =C e(u) + D e(u)

� ✓B

(z)(z, ✓)

✓ � div�

K(z, ✓)r✓�

= |z|+⇥

D(z, ✓)e(u)�✓B⇤

: e(u) + H(t)

• Kelvin-Voigt rheology & inertia

• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation





2 , z(t, x)2[0, 1], ✓(t, x)2R+

thermo-

viscoelasticity: ⇢ u � div �(u) = F(t)

where �(u) =C(z)e(u) + D(z, ✓)e(u)

� ✓B✓ � div

�

K(z, ✓)r✓�

= |z|+⇥

D(z, ✓)e(u)�✓B⇤

: e(u) + H(t)

• Kelvin-Voigt rheology & inertia

• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation



• Uniform ellipticity of C: Damage is partial• Uniform ellipticity of D: Dynamics is damped• Monotonicity of C: Increase of damage reduces the stored elastic energy



2 , z(t, x)2[0, 1], ✓(t, x)2R+

thermo-viscoelasticity: ⇢ u � div �(u) = F(t)

where �(u) =C(z)e(u) + D(z, ✓)e(u)� ✓B✓ � div

�

K(z, ✓)r✓�

= |z|+⇥

D(z, ✓)e(u)�✓B⇤

: e(u) + H(t)

• Kelvin-Voigt rheology & inertia• Temperature changes produce additional stresses Thermal expansion term, coupling terms in the heat equation


Assumptions: C, D are bounded, symmetric, positive definite, uniformly in (z, ✓)Assumptions: C is nondecreasing: 8 z1 z2 C(z1)A : A C(z2)A : AAssumptions: B 2 R3⇥3 constant, symmetricAssumptions: K 2 C0(R⇥ R;R3⇥3), symmetric, and 9 2 (1, 5

3 ) such thatAssumptions: 8 z, ✓, ⇠ : c1(|✓|+1) |⇠|2 K(z, ✓)⇠ · ⇠ c2(|✓|+1) |⇠|2


Some remarks on the heat equation

cv(✓)| {z }

⌘1

✓ � div�

K(z, ✓)r✓�

= |z|+⇥

D(z, ✓)e(u)�✓B⇤

: e(u) + H(t)

• K = heat conductivity with subquadratic growth condition:

9 2 (1, 53 ) 8 z, ✓, ⇠ : c1(|✓|+1) |⇠|2 K(z, ✓)⇠ · ⇠ c2(|✓|+1) |⇠|2

? Borrowed from Rocca-Rossi, in the spirit of Feireisl-Petzeltová-Rocca? Needed in the proof of a-priori estimates? Applies to polymers such as PMMA

• cv(✓) = heat capacity, constant because of constitutive assumptions

? Allows us to avoid a so-called enthalpy transformation? Compatible with regimes where ✓ � ✓D (Debye temperature)? Indeed we show that temperature is > of a (tunable) threshold for suitable data


The model: 2. Damage flow rule

@R1(z) + @zG(z,rz)� div�

D⇠G(z,rz)�

+ 12C

0(z)e(u) : e(u) 3 0

Assumptions: R1 is a 1-homogeneous dissipation potential of the form

R1(v) :=

(

|v| if v 0+1 otherwise

• Rate-independent: energy necessary to damage is independent of velocity• Enforces monotonicity (z 0): unidirectional damage, no healing

Assumptions: G 2 C0(R⇥R3;R[{1}), G(0, 0) = 0Assumptions: G(z, ⇠)<1 ) z2 [0, 1]Assumptions: G(z, ·) is convex and has q-growth from above and below, q > 1




D⇠G(z,rz)�

+ 12C

0(z)e(u) : e(u) 3 0


R1(v) :=

(






D⇠G(z,rz)�

+ 12C

0(z)e(u) : e(u) 3 0


R1(v) :=

(



Semistability

z(t) 2 argmin⇣

n

E(t, u(t), ⇣) +Z

⌦R1(⇣�z(t)) dx

o

where E(t, u, ⇣) :=Z

⌦

h

12C(⇣)e(u) : e(u) + G(⇣,r⇣)� F(t) u

i

dx


Example: Ambrosio-Tortorelli modelPhase-field approximation

Regularisation of a sharp crack:The internal variable interpolates continuouslybetween sound and fractured material

Figure: "Phase field order parameter" by Cenna - Own work. Licensed under Public domain viaWikimedia Commons http://commons.wikimedia.org/wiki/File:Phase_field_order_parameter.jpg

Prototype of the time-discrete problem: given u, z0

, min0⇣z0

⇢

Z

⌦

12C(⇣)e(u) : e(u) dx +

Z

⌦G(⇣,r⇣) dx +

Z

⌦R1(⇣�z0) dx

�

Setting C(⇣) := (⇣2+�) I with � > 0, and G(⇣,r⇣) := |r⇣|2 + 12 (1+⇣2) + I[0,1](⇣)

, min0⇣z0

⇢

Z

⌦( 1

2 (⇣2+�) |e(u)|2 dx +

Z

⌦

12 (1�⇣)2 dx +

Z

⌦|r⇣|2 dx

�

Ambrosio-Tortorelli (without passage to brittle limit)


The PDE system

Problem: Given F 2 H1(0, T; L2(⌦;R3)), H 2 L2(0, T; L2(⌦)), prove existence for8

>

>

>

<

>

>

>

:

⇢ u � div⇣

D(z, ✓)e(u) + C(z)e(u)� ✓B⌘

= F(t)


D⇠G(z,rz)�

+ 12C

0(z)e(u) : e(u) 3 0

✓ � div�

K(z, ✓)r✓�

= R1(z) +⇥

D(z, ✓)e(u)�✓B⇤

: e(u) + H(t)

+ initial Cauchy conditons on u(0), u(0), z(0), ✓(0)

+ natural boundary conditions: Dirichlet/Neumann on u and Neumann on ✓

+ (all boundary conditions will be homogeneous in this talk)

Challenges: • Interplay of rate-independent| {z }

damage

and of rate-dependent| {z }

viscosity, heat

processes

Challenges: • Highly nonlinear coupling of the systemChallenges: • Dissipation rates as heat sources, L1 right-hand side in heat equation


Literature

(Incomplete) List of related works:

• Roubícek: Coupling of rate-independent processes with viscosity, inertia, heat.We extend his ansatz to a unidirectional process, with D(z, ✓) nonconstant

• Feireisl-Petzeltová-Rocca and Rocca-Rossi: Treatment of the heat equation.They consider a rate-dependent setting for phase transitions and damage

• Larsen-Ortner-Süli: Dynamic evolution for unidirectional damage, isothermal.Ambrosio-Tortorelli setting with C(z) = D(z) = (z2 + �) eC(x)

• Vast literature on (fully) rate-independent processes, Mielke’s abstract theory.For damage: Mielke-Roubícek, Thomas-Mielke, Fiaschi-Knees-Stefanelli, . . .

• About rate-dependent damage (without inertia, with or without heat):Frémond, Bonetti, Bonfanti, Heinemann, Kraus, Nedjar, Schimperna, Segatti. . .

• Modelling and numerics, e.g. Miehe-Welschinger-Hofacker (inertia + heat)


Existence result for the energetic formulationTheorem (L.-Rossi-Thomas-Toader):Given loading F , heat source H � 0 , and initial data u0 , u0 , z0 2 [0, 1] , ✓0 � ✓⇤ > 0 ,there exists an energetic solution (u, z, ✓) such that the initial conditions hold and

• unidirectionality and semistability: z(·, x) nonincreasing and for all t

8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z

⌦(z(t)�⇣) dx

where ⇣ 2 W1,q and z2 L1(W1,q)\ L1 \BV(L1)

• momentum equation in weak form for all t

• mechanical energy balance for all t

• heat equation in weak form for all t




8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z

⌦(z(t)�⇣) dx


� ⇢

Z t

0

Z

⌦u · v +

Z t

0

Z

⌦

⇣

D(z, ✓)e(u) + C(z)e(u)� ✓B⌘

: e(v)

= ⇢

Z

⌦u0 · v(0)� ⇢

Z

⌦u(t) · v(t) +

Z t

0

Z

⌦F v

8 v2 L2(H1)\W1,1(L2), where u2H1(H1D)\W1,1(L2)






8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z

⌦(z(t)�⇣) dx



⇢2

Z

⌦|u(t)|2 +

Z

⌦(z0�z(t)) + E(t, u(t), z(t)) +

Z t

0

Z

⌦

⇥

D(z, ✓)e(u)�✓B⇤

: e(u)

= ⇢2

Z

⌦|u0|2 + E(0, u0, z(0))�

Z t

0

Z

⌦F u





8 ⇣ z(t) : E(t, u(t), z(t)) E(t, u(t), ⇣) +Z

⌦(z(t)�⇣) dx




h✓(t), ⌘(t)i �Z

⌦✓0 ⌘(0)�

Z t

0

Z

⌦✓ ⌘ +

Z t

0

Z

⌦K(✓, z)r✓·r⌘

=

Z t

0

Z

⌦

⇥

D(z, ✓)e(u)�✓B⇤

: e(u) ⌘ +

Z t

0

Z

⌦⌘ |z|+

Z t

0

Z

⌦H ⌘

8 ⌘ 2H1(L2)\C0(W2,3+�), where ✓2 L2(H1)\ L1(L1)\BV((W2,3+�)⇤)


Positivity of the temperature

• Under the previous assumptions (H � 0 , ✓0 � ✓⇤ > 0), there exists e✓ > 0 with

✓(t, x) � e✓ > 0 for a.a. (t, x)

• Furthermore, if in addition

9H⇤ > 0 : H(t, x) � H⇤ for a.a. (t, x) and ✓0(x) �p

H⇤/c for a.a. x

where c is a constant depending on B and D, then

✓(t, x) � maxn

e✓,p

H⇤/co

for a.a. (t, x)

We can tune the constant in such a way that ✓ � ✓D (Debye model)


Strategy: 1. Time discretisation0 = t0

n < · · · < tnn = T with tk

n � tk�1n = T

n =: ⌧n

1. zkn 2 argmin

⇢

E(tkn, uk�1

n , z) +Z

⌦(zk�1

n �z) dx : z zk�1n

�

2. Solve the system in the unknowns (ukn, ✓

kn) :

⇢

Z

⌦

ukn�2uk�1

n +uk�2n

⌧ 2n

· v +Z

⌦⌧n|e(uk

n)|��2e(ukn) : e(v)

+

Z

⌦

⇣

D(zk�1n , ✓k�1

n )e⇣

ukn�uk�1

n⌧n

⌘

+ C(zkn)e(u

kn)� ✓k

n B⌘

: e(v) =Z

⌦Fk

n v

Z

⌦

✓kn�✓k�1

n⌧n

⌘ +

Z

⌦K(zk

n, ✓kn)r✓k

n ·r⌘ �Z

⌦

zk�1n �zk

n⌧n

⌘

=

Z

⌦

h

D(zk�1n , ✓k�1

n )e⇣

ukn�uk�1

n⌧n

⌘

� ✓kn B

i

: e⇣

ukn�uk�1

n⌧n

⌘

⌘ +

Z

⌦Hk

n ⌘

• Minimisation decoupled from the system of momentum & heat equation• Regularisation by �-Laplacian (� > 4), disappearing as n ! 1• Existence by direct method and theory of pseudomonotone operators


Strategy: 2. Time-discrete to continuous

• Discrete energy inequality A priori estimates

• Enhanced estimates for ✓ by positivity (cf. Feireisl-Petzeltová-Rocca)

• Convergence by compactness and Helly’s selection principle

• Semistability in the limit via “mutual recovery sequence” (cf. Thomas-Mielke)

• Passage to the limit in the discrete energetic formulation

(cf. Mielke’s scheme for rate-independent processes(cf. and Roubícek’s one for coupling with rate-dependent effects)


Vanishing viscosity & inertiaVanishing viscosity & inertia (cf. Roubícek, Dal Maso-Scala, isothermal)

In the limit we seek a rate-independent & temperature-independent model

Assumptions: Homogeneous Dirichlet problem, q-growth of G with q>3 , H" ' " ,

Assumptions: and K"(z, ✓) = 1"2 K(z, ✓) Heat conducted with infinite speed

8

>

>

>

<

>

>

>

:

"2⇢u" � div⇣

"D(z", ✓")e(u") + C(z")e(u")� ✓" B⌘

= F"

@R1(z") + @zG(z",rz")� div(D⇠G(z",rz")) + 12C

0(z")e(u") : e(u") 3 0

"✓" �

1"2

div(K"(z", ✓")r✓") = "R1(z") +⇥

"2D(z", ✓")e(u")� "✓" B⇤

: e(u") + H"

In the limit " ! 0 : Unidirectionality, semistability, � div�

C(z)e(u)�

= F(t) , and

E(t, u(t), z(t)) +Z

⌦(z(s)�z(t)) dx E(s, u(s), z(s)) +

Z t

s@tE(r, u(r), z(r)) dr

for all t and a.a. s 2 (0, t). Notion of local solution (Mielke, Roubícek, Stefanelli)

Moreover, ✓" * ✓? depending on time, constant in space


Vanishing viscosity & inertiaVanishing viscosity & inertia (cf. Roubícek, Dal Maso-Scala, isothermal)

In the limit we seek a rate-independent & temperature-independent model

Assumptions: Homogeneous Dirichlet problem, q-growth of G with q>3 , H" ' " ,

Assumptions: and K"(z, ✓) = 1"2 K(z, ✓) Heat conducted with infinite speed

8

>

>

>

<

>

>

>

:

"2⇢u" � div⇣

"D(z", ✓")e(u") + C(z")e(u")� ✓" B⌘

= F"

@R1(z") + @zG(z",rz")� div(D⇠G(z",rz")) + 12C

0(z")e(u") : e(u") 3 0

"✓" � 1"2 div(K

"

(z", ✓")r✓") = "R1(z") +⇥

"2D(z", ✓")e(u")� "✓" B⇤

: e(u") + H"

In the limit " ! 0 : Unidirectionality, semistability, � div�

C(z)e(u)�

= F(t) , and

E(t, u(t), z(t)) +Z

⌦(z(s)�z(t)) dx E(s, u(s), z(s)) +

Z t

s@tE(r, u(r), z(r)) dr

for all t and a.a. s 2 (0, t). Notion of local solution (Mielke, Roubícek, Stefanelli)

Moreover, ✓" * ✓? depending on time, constant in space


Conclusion: Results and open problems

• Existence of evolutions for rate-independent, unidirectional, partial damagewith viscosity, inertia, and thermal effects basing on time-discretisation

• We can account for nonhomogeneous, time-dependent, Neumann conditions

• We cannot take nonhomogeneous, time-dependent, Dirichlet conditions for u(There are problems if D is not constant)

• We consider the limit for vanishing viscosity & inertia if K" ' "�2,obtaining a temperature-independent model

• Open problem: Vanishing viscosity & inertia without assuming K" ' "�2

• Open problem: Passage to the limit from Ambrosio-Tortorelli to a sharp crack


Rate-independent damage in thermo-viscoelastic materials with inertia

Documents

Transcript of Rate-independent damage in thermo-viscoelastic materials with inertia