Size effects on quasistatic growth of fractures

0.46 MB·English

by  Alessandro Giacomini

SIZE EFFECTS ON QUASISTATIC GROWTH OF FRACTURES
ALESSANDROGIACOMINI
4
0
Abstract. We perform an analysis of the size effect for quasistatic growth of fractures
0
in linearly isotropic elastic bodies under antiplanar shear. In the framework of the
2
variational model proposed byG.A.Francfort and J.-J. Marigo in[14],weprovethatif
n
thesize ofthebodytendstoinfinity,andevenif thesurface energy isof cohesiveform,
a
under suitable boundary displacements the fracture propagates following the Griffith’s
J
functional.
7
2
Keywords: variational models, energy minimization, freediscontinuityproblems, crack

propagation, quasistatic evolution, brittle fracture.
]
P 2000 Mathematics Subject Classification: 35R35, 35J85, 35J25, 74R10.
A
.
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Contents
[
1. Introduction 1

1 2. Preliminaries 5
v 3. Discrete in time evolution of fractures in the cohesive case 7
4
4. The main results 11
8
5. Proof of Theorem 4.1 13
3
6. Proof of Theorem 4.2 23
1
7. Proof of Theorem 4.3 25
0
4 8. A relaxation result 27
0 9. Two auxiliary propositions 32
/ References 34
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m
:
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1. Introduction
i
X
A well known fact in fracture mechanics is that ductility is also influenced by the size of the
r
a structure, and in particular the structure tends to become brittle if its size increases (see for
example [8], and referencestherein). The aim of this paper is to capture this fact for the problem
of quasistatic growth of fractures in linearly elastic bodies in the framework of the variational
theory of crack propagation formulated by Francfort and Marigo in [14].
Themodelproposedin[14]isinspiredto classicalGriffith’scriterionanddeterminesthe evolu-
tion of the fracture through a competition between volume and surface energies. Let us illustrate
it and the variant we investigate in the case of generalized antiplanar shear.
Let Ω RN be open, bounded, and with Lipschitz boundary. A fracture Γ Ω is any
⊆ ⊆
rectifiable set, and a displacement u is any function defined almost everywhere in Ω whose set of
discontinuitiesS(u)iscontainedinΓ(wewillmakeprecisethefunctionalsettinglater). Thetotal
energy of the configuration (u,Γ) is given by
(1.1) u2dx+ N 1(Γ).
−
|∇ | H
ZΩ\Γ
The first term in (1.1) implies that we assume to apply linearized elasticity in the unbroken part
of Ω. The second term can be considered as the work done to create Γ.
1
2 A.GIACOMINI
As suggestedin [14], more generalfracture energies canbe consideredin (1.1), especially those
of Barenblatt’s type [5], and here we consider energies of the form
(1.2) ϕ([u](x))d N 1(x),
−
| | H
ZΓ
where [u](x) := u+(x) u (x) is the difference of the traces of u on both sides of Γ, and ϕ :
−
−
[0,+ [ [0,+ [ (which depends on the material) is such that ϕ(0) = 0. In order to get a
∞→ ∞
physical interpretation of (1.2), let us set σ :=ϕ: we interpret σ([u](x)) as density of force in x
′
| |
thatactbetweenthetwolipsofthecrackΓwhosedisplacementareu+(x)andu (x)respectively.
−
Typicallyσ isdecreasing,andσ(s)=0fors s¯: this meansthatthe interactionbetweenthe two
≥
lips of the fracture decreases as the opening increases, and disappear when the opening is greater
than a critical length s¯. As a consequence, ϕ is increasing and concave, and ϕ(s) is constant
for s s¯. We will then consider ϕ increasing, concave, with ϕ(0) = 0, c = ϕ(0) < + , and
′
≥ ∞
lim ϕ(s)=1. We can interpret
s +
→ ∞
ϕ([u](x))d N 1(x)
−
| | H
ZΓ
as the workmade to createΓ with anopening givenby [u]. Assuming linearizedelasticity to hold
in Ω Γ, we consider a total energy of the form
\
(1.3) u 2+ ϕ([u])d N 1,
−
k∇ k | | H
ZΓ
where denotesthe L2 norm. The problemofirreversiblequasistaticgrowthoffracturesinthe
k·k
cohesive case can be addressedthrough a time discretization process in analogy to what proposed
in [14] for the energy (1.1).
Letg(t)be a time dependent boundary displacementdefinedon∂ Ω ∂Ω with t [0,T]. Let
D
⊆ ∈
δ >0andletI := 0=tδ <tδ < <tδ =T beasubdivisionof[0,T]withmax(tδ tδ)<δ,
δ { 0 1 ··· Nδ } i+1− i
and let gδ := g(tδ). Let us consider a preexisting crack configuration (Γ¯,ψ¯), with ψ¯ a positive
i i
functiononΓ¯: thismeans that ϕ(ψ¯)d N 1 isthe workdone onthe fractureΓ¯ beforeweapply
Γ¯ H −
the boundary displacement g(t). At time t=0 we consider uδ as a minimum of
R 0
(1.4) u 2+ ϕ([u] ψ¯)d N 1.
−
k∇ k ZSg(0)(u)∪Γ¯ | |∨ H
Here Sg(0)(u) := S(u) x ∂ Ω : u(x) = g(0)(x) , and for all x ∂ Ω we consider [u](x) :=
D D
g(x) u˜(x), whereu˜ is∪th{e t∈raceofuon∂Ω6 . Moreov}erwe intendtha∈tψ¯=0outside Γ¯. We define
the f−racture Γδ at time t = 0 as Sg(0)(uδ) Γ¯. We also set ψδ := ψ¯ [uδ] on Γδ. The presence
0 0 ∪ 0 ∨| 0 | 0
of Sg(0)(u) in (1.4) indicates that the points at which the boundary displacement is not attained
are considered as a part of the fracture. Notice moreover that problem (1.4) takes into account
anirreversibility condition assumptioninthe growthofthe fracture. Indeed,while onSg(0)(u) Γ¯
the surface energy which comes in minimization of (1.4) is exactly as in (1.2), on Sg(0)(u) \Γ¯
the surface energy involved takes into account the previous work made on Γ¯. The surface ene∩rgy
is of the form of (1.2) only if [u] > ψ¯, that is only if the opening is increased. If [u] ψ¯ no
| | | | ≤
energyisgained,thatisdisplacementsofthisformalongthefractureareinasensesurfaceenergy
free. Notice finally that the irreversibility condition involves only the modulus of [u]: this is an
assumption which is reasonable since we are considering only antiplanar displacement. Clearly
more complex irreversibility conditions can be formulated, involving for example a partial release
of energy: the one we study is the first straightforward extension of the irreversibility condition
given in [14] for the energy (1.1).
SupposingtohaveconstructedΓδ andψδ attimetδ,weconsideraminimumuδ oftheproblem
i i i i+1
(1.5) u 2+ ϕ([u] ψ¯δ)d N 1
k∇ k ZSg(tδi+1)(u)∪Γδi | |∨ i H −
SIZE EFFECTS ON QUASISTATIC GROWTH OF FRACTURES 3
and define Γδi+1 := Γδi ∪Sg(tδi+1)(uδi+1) and ψiδ+1 := ψiδ ∨|[uδi+1]| on Γδi+1. As noted previously,
problem(1.5)involvesanirreversibilitycondition: the surfaceenergydensity onΓδ increasesonly
i
where [u] >ψδ.
| | i
The discrete in time evolution of the fracture relative to the boundary datum g(t), the subdi-
vision I , the initial crack configuration (Γ¯,ψ¯) is given by (uδ,Γδ,ψδ) : i=0,...,N .
δ { i i i δ}
The irreversible quasistatic evolution of fracture relative to the boundary datum g(t) and the
initial crack configuration (Γ¯,ψ¯) is obtained as a limit for δ 0 of (uδ(t),Γδ(t),ψδ(t)), where
→
uδ(t):=uδ, Γδ(t):=Γδ and ψδ(t):=ψδ for tδ t<tδ .
i i i i ≤ i+1
This program has been studied in detail in several papers in the case ϕ 1, that is for energy
≡
of the form (1.1). A first mathematical formulation has been given in [12], where the authors
considerthe case ofdimensionN =2and fractureswhichare compactand witha uniform bound
on the number of connected components. This analysis has been extended to the case of plane
elasticity in [9]. In [13] the authors consider the general dimension N, and remove the bound on
the number of the connected components of the fractures: the key point is to introduce a weak
formulation of the problem considering displacements in the space SBV (see Section 2). Finally
in[11]the authorstreatthecaseoffinite elasticitynotrestrictedtoantiplanarshear,withvolume
energy depending on the full gradientunder suitable growthcondition, and in presence of volume
forces and traction forces: the appropriate functional space for the displacements is now GSBV
(see for example [4] for a precise definition).
In all these papers ([12],[9],[13],[11]), the analysis of the limit reveals three basic properties
(irreversibility,minimality and nondissipativity,see Theorem2.2) which aretakenas definition of
irreversiblequasistatic growthof brittle fractures: the time discretizationprocedure is considered
as a privileged way to get an existence result.
In the case of energy (1.3), several difficulties arise in the analysis of the discrete in time
evolution, and in the analysis as δ 0. In Section 3, we prove that the functional space we need
→
forthestepbystepminimizationisthespaceoffunctionswithboundedvariationBV (seeSection
2): moreover we prove that a relaxed version of (1.3) has to be employed, namely
(1.6) f( u)dx+ ϕ([u] ψ)d N 1+aDcu(Ω),
−
∇ | |∨ H | |
ZΩ ZΓ
where a = ϕ(0), f is defined in (3.5), and Dcu indicates the Cantorian part of the derivative
′
of u. An existence result for discrete in time evolution in this context of BV space is given in
Proposition 3.1.
The analysis for δ 0 presents several difficulties, the main one being the stability of the
→
minimality property of the discrete in time evolutions. The main purpose of this paper is to
provethatthese difficulties disappearas the sizeofthe referenceconfigurationincreases,thank to
the fact that the body response tends to become more and more brittle in spite of the presence
of cohesive forces on the fractures given by the function ϕ. More precisely we consider a crack
configuration (Γ¯,ψ¯) in Ω with N 1(Γ¯)<+ , and prove this fact for the discrete evolutions in
−
Ω :=hΩ with preexisting cracHk configuration∞(Γ¯ ,ψ¯ ) of the form Γ¯ :=hΓ¯ and ψ¯ (x):=ψ¯(x),
h h h h h h
under suitable boundary displacements. The idea is to rescale displacements and fractures to the
fixed configuration Ω, and take advantage from the form of the problem in this new setting. The
boundary displacements on ∂ Ω :=h∂ Ω will be taken of the form
D h D
x
g (t,x):=hαg t, , g AC([0,T];H1(Ω)), g(t) C, t [0,T], x Ω ,
h h
h ∈ k k∞ ≤ ∈ ∈
(cid:16) (cid:17)
where α > 0 and C > 0. We indicate by (uδ,h(t),Γδ,h(t),ψδ,h(t)) the piecewise constant interpo-
lation of the discrete in time evolution of fracture in Ω relative to the boundary displacementg
h h
and the preexisting crack configuration (Γ¯ ,ψ¯ ). Let us moreover set for every t [0,T]
h h
∈
δ,h(t):= f( uδ,h(t))dx+ ϕ(ψδ,h(t))d N 1+aDcuδ,h (Ω ).
− h
E ∇ H | |
ZhΩ ZΓδ,h(t)
4 A.GIACOMINI
In the case α= 1, we make the following rescaling
2
1 1
vδ,h(t,x):= uδ,h(t,hx), Kδ,h(t):= Γδ,h(t), x Ω.
√h h ∈
The main result of the paper is the following (see Theorem 4.1 for a more precise statement).
Theorem 1.1. If δ 0 and h + , there exists a quasistatic evolution of brittle fractures
t (v(t),K(t)) in→Ω relative t→o the∞preexisting crack Γ¯ and boundary displacement g in the
{ → }
sense of [13] (see Theorem 2.2) such that for all t [0,T] we have
∈
(1.7) vδ,h(t)⇀ v(t) weakly in L1(Ω;RN),
∇ ∇
Moreover for all t [0,T] we have
∈
1
(1.8) δ,h(t) v(t) 2+ N−1(K(t));
hN 1E →k∇ k H
−
in particular h N+1 Dcuδ,h(t)(Ω ) 0,
− h
| | →
1
(1.9) f( uδ,h(t))dx v(t) 2,
hN 1 ∇ →k∇ k
− ZΩh
and
1
(1.10) ϕ(ψδ,h(t))d N 1 N 1(K(t)).
hN 1 H − →H −
− ZΓδ,h(t)
Theorem 1.1 proves that as the size of the reference configuration increases, the response of
the body in the problemofquasistatic growthoffracturestends to become brittle, so thatenergy
(1.1)canbeconsidered. Moreoverwehaveconvergenceresultsforthevolumeandsurfaceenergies
involved.
The particular value α = 1 comes out because a problem of quasistatic evolution has been
2
considered. In fact if we consider an infinite plane with a crack-segment of length l and subject
to a uniform stress σ at infinity, following Griffith’s theory the crack propagatesquasistatically if
σ = K√IπCl, where KIC is the criticalstress intensity factor. So if the crackhas length hl,the stress
rescale as 1 . This is precisely what we are prescribing in the case α= 1: in fact the stress that
√h 2
intuitively we prescribe at the boundary can be reconstructed from u and rescales precisely as
h
∇
1 .
√h
For the proof of Theorem 1.1, the first step is to recognize that (vδ,h(t),Kδ,h(t),ψδ,h(t)) is
a discrete in time evolution relative to the boundary displacement g and the preexisting crack
configuration (Γ¯,ψ¯) for a total energy of the form
f ( u)dx+ ϕ ([u] ψ)d N 1+a√hDcu(Ω),
h h −
∇ | |∨ H | |
ZΩ ZΓ
where ϕ (s) 1 for all s [0,+ [, and f (ξ) ξ 2 for all ξ RN. From the fact that ϕ 1
h h h
ր ∈ ∞ ր| | ∈ ր
we recognize that the structure tends to become brittle. Bound on total energy for the discrete
in time evolution is available, so that compactness in the space BV can be applied: it turns out
that the limits of the displacements are of class SBV with gradient in L2(Ω;RN). Limits for the
fractures are constructed through a Γ-convergence procedure (see Lemma 5.4). Now the main
point is to recover the minimality property (see point (c) of Theorem 2.2)
v(t) 2 v 2+ N−1(Sg(t)(v) K(t)), v SBV(Ω)
k∇ k ≤k∇ k H \ ∈
fromthe minimality propertyof(vδ,h(t),Kδ,h(t),ψδ,h(t)). This isdone inLemma 5.5by meansof
a refined versionof the Transfer of Jump Lemma of [13]: the main difference here is that we have
to deal with BV functions and we have to transfer the jump on the part of Kδ,h(t) where ψδ,h(t)
is greater than a given small constant.
We also consider the cases α ]0,1[ and α > 1. It turns out that in the case α ]0,1[, the
∈ 2 2 ∈ 2
body is not solicited enough to make the preexisting crackΓ¯ propagate,and Ω tends to behave
h h
SIZE EFFECTS ON QUASISTATIC GROWTH OF FRACTURES 5
elastically in the complement of Γ¯ : more precisely we prove that (Theorem 4.2) in the case
h
ψ¯>ε>0, setting
1
(1.11) vδ,h(t,x):= uδ,h(t,hx),
hα
for all t [0,T]we have that vδ,h(t) convergesto the displacement of the elastic problemin Ω Γ¯
∈ \
under boundary displacement given by g(t).
In the case α> 1 we have that the preexisting fracture Γ¯ tends to propagate brutally toward
2 h
rupture: in fact in Theorem 4.3 we prove that vδ,h(0) given by (1.11) converges to a piecewise
constant function v in Ω, so that Γ¯ Sg(0)(v) disconnects Ω. This phenomenon is a consequence
∪
of the variationalapproachbased on the searchfor globalminimizers: as the size of Ω increases,
h
fractures carry an energy of order hN 1, while non rigid displacements carryan energy of greater
−
order: in this way fracture is preferred to deformation.
The paper is organized as follows: in Section 2 we recall some basic definitions and introduce
the functional setting for the problem. In Section 3 we deal with the problem of discrete in time
evolutions for fractures in the cohesive case. The main theorems are listed in Section 4, and
Sections 5, 6 and 7 are devoted to their proofs. In Section 8 we prove a relaxationresult which is
used in the problem of discrete in time evolution of fractures, while in Section 9 we prove some
auxiliary results employed in the study of the asymptotic behavior of the evolutions.
2. Preliminaries
In this section we state the notation and prove some preliminary results employed in the rest
of the paper.
Basic notation. We will employ the following basic notation:
- Ω is an open and bounded subset of RN with Lipschitz boundary;
- ∂ Ω is a subset of ∂Ω open in the relative topology;
D
- N 1 is the (N 1)-dimensional Hausdorff measure;
−
H −
- we say that A ˜ B if A B up to a set of N 1-measure zero;
−
- Γ Ωisrectifi⊆ableifthe⊆reexistsasequenceHofC1 manifolds(Mi)i N suchthatΓ ˜ iMi;
- for⊆all A RN, A denotes the Lebesgue measure of A; ∈ ⊆∪
- for all A⊆RN, | |denotes the characteristic function of A;
A
- if µ is a m⊆easure1on RN and A is a Borel subset of RN, µ A denotes the restriction of µ
to A, i.e. (µ A)(B):=µ(B A) for all Borel sets B RN;
∩ ⊆
- u and u denote the sup-norm and the L2 norm of u respectively;
- ikf uk,∞g BVk(Ωk;Rm), Sg(u):=S(u) x ∂ Ω : u(x)=g(x) ;
D
- if a,b ∈R, a b:=max a,b and a∪{b:=∈min a,b . 6 }
∈ ∨ { } ∧ { }
Functions of bounded variation. For the generaltheory of functions of bounded variation,
we refer to [4]; here we recall some basic definitions and theorems we need in the sequel. We say
that u BV(A) if u L1(A), and its distributional derivative Du is a bounded vector-valued
∈ ∈
Radon measure on A. In this case it turns out that the set S(u) of points x A which are not
∈
Lebesgue points of u is rectifiable, that is there exists a sequence of C1 manifolds (Mi)i N such
that S(u) M up to a set of N 1-measure zero. As a consequence S(u) admits a∈normal
i i −
ν (x) at ⊆N∪1-a.e. x S(u). MoHreover for N 1 a.e. x S(u), there exist u+(x),u (x) R
u − − −
H ∈ H ∈ ∈
such that
1
lim u(y) u±(x) dy =0,
r→0|Br±(x)|ZBr±(x)| − |
where B (x):= y B (x) : (y x) ν (x)≷0 , andB (x) is the ball with center x and radius
r± { ∈ r − · u } r
r. It turns out that Du can be represented as
Du(A)= u(x)dx+ (u+(x) u (x))ν (x)d N 1(x)+Dcu(A),
− u −
∇ − H
ZA ZA∩S(u)
6 A.GIACOMINI
where u denotes the approximate gradient of u and Dcu is the Cantor part of Du. BV(A) is a
∇
Banach space with respect to the norm u := u + Du(A).
BV(A) L1(A)
k k k k | |
Wewilloftenusethefollowingresult: ifAisboundedandLipschitz,andif(uk)k N isabounded
∈
sequence in BV(A), then there exists a subsequence (ukh)h∈N and u∈BV(A) such that
(2.1) u u strongly in L1(A),
kh →
Dukh ⇀∗ Du weakly∗ in the sense of measures.
We say that uk ⇀∗ u weakly∗ in BV(A) if (2.1) holds.
We say that u SBV(A) if u BV(A) and Dcu= 0. The space SBV(A) is called the space
∈ ∈
of special functions of bounded variation. Note that if u SBV(A), then the singular part of Du
∈
is concentrated on S(u).
The space SBV is very useful when dealing with variational problems involving volume and
surface energies because of the following compactness and lower semicontinuity result due to
L.Ambrosio ([1], [3]).
Theorem 2.1. Let A be an open and bounded subset of RN, and let (uk)k N be a sequence in
∈
SBV(A). Assume that there exists q >1 and c [0;+ [ such that
∈ ∞
u qdx+ N 1(S(u ))+ u c
k − k k
ZA|∇ | H || ||∞ ≤
for every k ∈N. Then there exists a subsequence (ukh)h∈N and a function u∈SBV(A) such that
u u strongly inL1(A),
kh →
(2.2) u ⇀ u weakly inL1(A;RN),
∇ kh ∇
N 1(S(u)) liminf N 1(S(u )).
H − ≤ h H − kh
In the rest of the paper, we will say that u ⇀ u weakly in SBV(A) if u and u satisfy
k k
(2.2). It will also be useful the following fact which can be derived from Ambrosio’s Theorem:
if uk ⇀ u weakly in SBV(A) and if N−1 S(uk) ⇀∗ µ weakly∗ in the sense of measures, then
H
N 1 S(u) µ as measures.
−
H ≤
Finally in the contextof fractureproblems we will use the followingnotation: if A is Lipschitz,
and if ∂ A ∂A, then for all u,g BV(A) we set
D
⊆ ∈
(2.3) Sg(u):=S(u) x ∂ A : u(x)=g(x) ,
D
∪{ ∈ 6 }
where the inequality on ∂ A is intended in the sense of traces. Moreover,we set for all x S(u)
D
∈
[u](x):=u+(x) u−(x),
−
and for all x ∂ A we set [u](x):=u(x) g(x), where the traces of u and g on ∂A are used.
D
∈ −
Quasi-static evolution of brittle fractures. Let Ω be an open bounded subset of RN
with Lipschitz boundary, and let ∂ Ω be a subset of ∂Ω open in the relative topology. Let
D
g : [0,T] H1(Ω) be absolutely continuous (see [7] for a precise definition); we indicate the
→
gradientof g at time t by g(t), and the time derivative of g at time t by g˙(t). For u SBV(Ω),
let Sg(t)(u) be defined as i∇n (2.3), and for everyA,B RN, let A ˜ B mean A B up∈to a set of
⊆ ⊆ ⊆
N 1-measure zero. The main result of [13] is the following theorem.
−
H
Theorem 2.2. Let Γ¯ be a rectifiable set in Ω ∂ Ω such that N 1(Γ¯) < + . There exists
D −
∪ H ∞
(u(t),Γ(t)) : t [0,T] withΓ(t) ˜ Ω ∂ Ωrectifiableandu(t) SBV(Ω)withSg(t)(u(t)) ˜ Γ(t)
D
{ ∈ } ⊆ ∪ ∈ ⊆
such that:
(a) Γ¯ ˜ Γ(s) ˜Γ(t) for all 0 s t T;
⊆ ⊆ ≤ ≤ ≤
(b) u(0) minimizes
v 2+ N 1(Sg(0)(v) Γ¯)
−
k∇ k H \
among all v SBV(Ω);
∈
SIZE EFFECTS ON QUASISTATIC GROWTH OF FRACTURES 7
(c) for t ]0,T], u(t) minimizes
∈
v 2+ N−1 Sg(t)(v) Γ(t)
k∇ k H \
(cid:16) (cid:17)
among all v SBV(Ω).
∈
Furthermore, the total energy
(t):= u(t) 2+ N−1(Γ(t))
E k∇ k H
is absolutely continuous and satisfies
t
(2.4) (t)= (0)+2 u(τ) g˙(τ)dxdτ
E E ∇ ∇
Z0 ZΩ
for every t [0,T].
∈
Condition (a) stands for the irreversibility of the crack propagation, conditions (b) and (c) are
minimality conditions, while (2.4) stands for the nondissipativity of the process.
Γ-convergence. LetusrecallthedefinitionandsomebasicpropertiesofDeGiorgi’sΓ-convergence
inmetricspaces. Wereferthereaderto[10]foranexhaustivetreatmentofthissubject. Let(X,d)
beametricspace. WesaythatasequenceF :X [ ,+ ]Γ-convergestoF :X [ ,+ ]
h
→ −∞ ∞ → −∞ ∞
(as h + ) if for all u X we have
→ ∞ ∈
(i) (Γ-liminf inequality) for every sequence (uh)h N converging to u in X,
∈
liminfF (u ) F(u);
h h
h + ≥
→ ∞
(ii) (Γ-limsup inequality) there exists a sequence (uh)h N converging to u in X, such that
∈
limsupF (u ) F(u).
h h
≤
h +
→ ∞
ThefunctionF iscalledtheΓ-limitof(Fh)h N (withrespecttod),andwewriteF = Γ limhFh.
∈ −
Γ-convergence is a convergence of variational type as explained in the following proposition.
Proposition 2.3. Assume that the sequence (Fh)h N Γ-converges to F and that there exists a
compact set K X such that for all h N ∈
⊆ ∈
inf F (u)= inf F (u).
h h
u K u X
∈ ∈
ThenF admitsaminimumonX,infXFh minXF,andanylimitpointofanysequence(uh)h N
→ ∈
such that
lim F (u ) inf F (u) =0,
h h h
h + −u X
is a minimizer of F. → ∞(cid:16) ∈ (cid:17)
Moreoverthe following compactness result holds.
Proposition2.4. If(X,d)isseparable, and(Fh)h N isasequenceoffunctionals onX,thenthere
exists asubsequence(Fhk)k∈N anda function F : X∈ →[−∞;+∞]suchthat (Fhk)k∈N Γ-converges
to F.
3. Discrete in time evolution of fractures in the cohesive case
In this section we are interested in generalized antiplanar shear of an elastic body Ω in the
context of linearized elasticity and in presence of cohesive fractures.
The notion of discrete in time evolution for fractures relative to time dependent boundary
displacementg(t)andpreexistingcrackconfiguration(Γ¯,ψ¯)hasbeendescribedintheIntroduction.
It relies on the minimization of functionals of the form
(3.1) u 2+ ϕ([u] ψ)d N 1,
−
k∇ k | |∨ H
ZΓ∪Sg(t)(u)
8 A.GIACOMINI
with ψ positive function on Γ. We now define rigorously the functional space to which the dis-
placements belong,andthe properties ofΩ,Γ, ψ andg(t) inorderto proveanexistence resultfor
the discrete in time evolution of fractures.
Let Ω be an open bounded subset of RN with Lipschitz boundary. Let ∂ Ω ∂Ω be open
D
⊆
in the relative topology, and let ∂ Ω := ∂Ω ∂ Ω. Let ϕ : [0,+ [ [0,+ [ be increasing and
N D
\ ∞→ ∞
concave, ϕ(0)=0 and such that lim ϕ(s)=1. If a:=ϕ(0)<+ , we have
s + ′
→ ∞ ∞
(3.2) ϕ(s) as for all s [0,+ [.
≤ ∈ ∞
LetT >0,andletusconsideraboundarydisplacementg AC([0,T];H1(Ω))suchthat g(t)
∈ k k∞ ≤
C for all t [0,T]. We discretize g in the following way. Given δ > 0, let I be a subdivision of
δ
∈
[0,T] of the form 0=tδ <tδ < <tδ = T such that max (tδ tδ ) <δ. For 0 i N we
set gδ :=g(tδ). 0 1 ··· Nδ i i − i−1 ≤ ≤ δ
i i
Let Γ¯ ˜ Ω be rectifiable, and let ψ¯ be a positive function on Γ¯ such that
⊆
(3.3) ϕ(ψ¯)d N 1 <+ .
−
ZΓ¯ H ∞
Let us extend ψ¯ to Ω setting ψ¯=0 outside Γ¯. As for the space of the displacements, it would be
naturalfollowing[13]toconsideru SBV(Ω). Sincea=ϕ(0)<+ ,wehaveunfortunatelythat
′
∈ ∞
theminimizationof (3.1)isnotwellposedinSBV(Ω). Letusinfactconsider(un)n N minimizing
∈
sequencefor(3.1): itturnsoutthatwemayassume(un)n N boundedinBV(Ω). Asaconsequence
(un)n N admits a subsequence weakly∗ convergent in BV∈(Ω) to a function u BV(Ω). Then we
∈ ∈
have that minimizing sequences of (3.1) converge (up to a subsequence) to a minimizer of the
relaxationof (3.1) with respect to the weak∗ topology of BV(Ω). By Proposition8.1, the natural
domain of this relaxed functional is BV(Ω), and that its form is
(3.4) f( u)dx+ ϕ([u] ψ)d N 1+aDcu(Ω),
−
∇ | |∨ H | |
ZΩ ZΓ∪Sg(t)(u)
where
ξ 2 if ξ a
| | | |≤ 2
(3.5) f(ξ):=

a2 +a(ξ a) if ξ a.
4 | |− 2 | |≥ 2
In view of these remarks, we consider BV(Ω) as the space of displacements u of the body Ω,
and a total energy of the form (3.4). The volume part in the energy (3.4) can be interpreted as
the contribution of the elastic behavior of the body. The second term represents the work done
to create the fracture Γ Sg(t)(u) with opening given by [u] ψ. The new term aDcu can be
∪ | |∨ | |
interpretedasthecontributeofmicrocracksintheconfigurationwhichareconsideredasreversible.
Letusdefinethediscreteevolutionofthefractureinthisnewsetting. Fori=0,letuδ BV(Ω)
0 ∈
be a minimum of
(3.6) min f( u)dx+ ϕ([u] ψ¯)d N 1+aDcu .
−
u∈BV(Ω)(ZΩ ∇ ZSg0δ(u)∪Γ¯ | |∨ H | |)
We set Γδ0 :=Sg0δ(uδ0)∪Γ¯.
Supposing to have constructed uδ and Γδ for all j =0,...,i 1, let uδ be a minimum of
j j − i
(3.7) min f( u)dx+ ϕ([u] ψδ )d N 1+aDcu,
u∈BV(Ω)(ZΩ ∇ ZSgiδ(u)∪Γδi−1 | |∨ i−1 H − | | )
whTerheeψfioδ−ll1ow:=inψg¯∨pr|o[upδ0o]s|i∨tio·n··e∨st|a[ubδil−is1h]|.thWeeexsiestteΓnδice:=ofΓtδih−i1s∪diSscgriδe(tueδi)e.volution.
Proposition 3.1. Let I = 0 = tδ < < tδ = T be a subdivision of [0,T] such that
δ { 0 ··· Nδ }
max(tδ tδ ) < δ, let Γ¯ be a preexisting crack, and ψ¯ a positive function on Γ¯ satisfying (3.3)
i − i−1
SIZE EFFECTS ON QUASISTATIC GROWTH OF FRACTURES 9
and extended to zero outside Γ¯. Then for all i = 0,...,N there exists uδ BV(Ω) such that
setting Γδ :=Γ¯, ψδ :=ψ¯ and δ i ∈
1 1
− −
i
(3.8) Γδi :=Γ¯∪ Sgjδ(uδj), ψiδ(x):=ψ¯(x)∨|[uδ0]|(x)∨···∨|[uδi]|(x)
j=0
[
the following holds:
(a) uδ gδ C;
k ik∞ ≤k ik∞ ≤
(b) for all v BV(Ω) we have
∈
(3.9) f( uδ)dx+ ϕ(ψδ)d N 1+aDcuδ (Ω)
∇ i i H − | i|
ZΩ ZΓδi
≤ZΩf(∇v)dx+ZSgiδ(v)∪Γδi−1ϕ(|[v]|∨ψiδ−1)dHN−1+a|Dcv|(Ω),
where a=ϕ(0) and f is defined in (3.5);
′
(c) we have that
(3.10) f(∇uδi)dx+ ϕ(ψiδ)dHN−1+a|Dcuδi|(Ω)
ZΩ ZΓδi
=v∈SiBnVf(Ω)(k∇vk2+ZSgiδ(v)∪Γδi−1ϕ(|[v]|∨ψiδ−1)dHN−1).
Proof. Wehavetoprovethatproblems(3.6)and(3.7)admitsolutions. Letusconsiderforexample
problem (3.7), the other being similar. Let (un)n N be a minimizing sequence for problem (3.7).
By a truncation argumentwe may assume that u∈ gδ . Comparingu with gδ, we get for
k nk∞ ≤k ik n i
n large
(3.11) ZΩf(∇un)dx+ZSgiδ(un)∪Γδi−1ϕ(|[un]|∨ψiδ−1)dHN−1+a|Dcun|(Ω)
f( gδ)dx+ ϕ(ψδ )d N 1+1 C ,
≤ZΩ ∇ 0 ZΓδi 1 i−1 H − ≤ ′
−
with C independent of n. Since there exists d > 0 such that aξ d f(ξ) for all ξ RN, we
′
deducethat(∇un)n∈N isboundedinL1(Ω;RN). Moreoverifs¯is|s|u−chth≤atϕ(s¯)= 21 and∈a¯ issuch
that s a¯ϕ(s) for all s [0,s¯], we have
≤ ∈
(3.12) [u ] d N 1 = [u ] d N 1+ gδ N 1( [u ] >s¯ )
ZS(un)| n | H − Z{|[un]|≤s¯}| n | H − k ik∞H − {| n | }
a¯ ϕ([u ])d N 1+2 gδ ϕ([u ])d N 1
≤ Z|[un]|≤s¯ | n | H − k ik∞Z|[un]|>s¯ | n | H −
(a¯+2 gδ )C .
≤ k ik∞ ′
Finally for all n
C
Dcu ′.
n
| |≤ a
We conclude that (un)n N is bounded in BV(Ω). Then there exists u BV(Ω) such that up to a
∈ ∈
subsequence un ⇀∗ u weakly∗ in BV(Ω) and pointwise almost everywhere. Let us set uδi :=u. By
10 A.GIACOMINI
Lemma 8.3 we deduce that
(3.13) f( u)dx+ ϕ([u] ψδ )d N 1+aDcu(Ω)
ZΩ ∇ ZSgδ−i(u)∪Γδi−1 | |∨ i−1 H − | |
liminf f( u )dx+ ϕ([u ] ψδ )d N 1+aDcu (Ω).
≤ n ZΩ ∇ n ZSgiδ(un)∪Γδi−1 | n |∨ i−1 H − | n|
Setting ψδ :=ψδ [uδ], we have that point (b) holds. Moreover uδ gδ C, so that
point (a)iholds.i−F1in∨al|lyip|oint (c) is a consequence of Proposition 8.1k. ik∞ ≤k 0k∞ ≤ (cid:3)
Let us consider now the following piecewise constant interpolation in time:
(3.14) uδ(t):=uδ, Γδ(t):=Γδ, ψδ(t):=ψδ, gδ(t):=gδ tδ t<tδ
i i i i i ≤ i+1
with uδ(T):=uδ , Γδ(T):=Γδ , ψδ(T):=ψδ , and gδ(T):=g(T).
Nδ Nδ Nδ
For every v BV(Ω) and for every t [0,T] let us set
∈ ∈
(3.15) δ(t,v):= f( v)dx+ ϕ([v] ψδ(t))d N 1+aDcv (Ω).
−
E ZΩ ∇ ZSgδ(t)(v)∪Γδ(t) | |∨ H | |
Then the following estimate holds.
Lemma 3.2. There exists eδ 0 for δ 0 and a + such that for all t [0,T] we have
a → → → ∞ ∈
tδ
i
(3.16) δ(t,uδ(t)) δ(0,uδ(0))+ f ( uδ(τ)) g˙(τ)dxdτ +eδ,
E ≤E ′ ∇ ∇ a
Z0 ZΩ
where tδ is the step discretization point such that tδ t<tδ .
i i ≤ i+1
Proof. Comparing uδ with uδ +gδ gδ by means of (3.9) we obtain
i i−1 i − i−1
δ(tδ,uδ) f( uδ + gδ gδ )dx+ ϕ(ψδ )d N 1+aDcuδ (Ω).
E i i ≤ZΩ ∇ i−1 ∇ i −∇ i−1 ZΓδi 1 i−1 H − | i−1|
−
Notice that by the very definition of f the following hold:
1) if uδ + gδ gδ a and uδ a
|∇ i−1 ∇ i −∇ i−1|≥ 2 |∇ i−1|≥ 2
f ( uδ + gδ gδ )=f ( uδ );
′ ∇ i−1 ∇ i −∇ i−1 ′ ∇ i−1
2) if uδ + gδ gδ < a and uδ a
|∇ i−1 ∇ i −∇ i−1| 2 |∇ i−1|≥ 2
f( uδ + gδ gδ ) f( uδ );
∇ i−1 ∇ i −∇ i−1 ≤ ∇ i−1
3) if uδ + gδ gδ a and uδ < a
|∇ i−1 ∇ i −∇ i−1|≥ 2 |∇ i−1| 2
f( uδ + gδ gδ ) f( uδ )+2( uδ , gδ gδ )+ gδ gδ 2;
∇ i−1 ∇ i −∇ i−1 ≤ ∇ i−1 ∇ i−1 ∇ i −∇ i−1 |∇ i −∇ i−1|
4) if uδ + gδ gδ < a and uδ < a
|∇ i−1 ∇ i −∇ i−1| 2 |∇ i−1| 2
f( uδ + gδ gδ )=f( uδ )+2( uδ , gδ gδ )+ gδ gδ 2.
∇ i−1 ∇ i −∇ i−1 ∇ i−1 ∇ i−1 ∇ i −∇ i−1 |∇ i −∇ i−1|
Then by convexity of f we deduce
δ(tδ,uδ) δ(tδ ,uδ )+ f ( uδ )( gδ gδ )dx+Rδ,a ,
E i i ≤E i−1 i−1 ZΩ ′ ∇ i−1 ∇ i −∇ i−1 i−1
where
Rδ,a := gδ gδ 2dx+ f ( uδ ) gδ gδ dx.
i−1 ZΩ|∇ i −∇ i−1| Z{|∇uδi−1|≥a2}| ′ ∇ i−1 ||∇ i −∇ i−1|
Then summing up from tδ to tδ, and taking into account (3.14) we get
i 0
tδ tδ
i i
δ(t,uδ(t)) δ(0,uδ(0))+ f ( uδ(τ)) g˙(τ)dxdτ + Rδ,a(τ)dτ,
′
E ≤E ∇ ∇
Z0 ZΩ Z0