Quasi-static evolution of delaminated structures: analysis of stability and bifurcation

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Quasi-static evolution of delaminated structures:

analysis of stability and bifurcation

Rachel-Marie Pradeilles-Duval

To cite this version:

Rachel-Marie Pradeilles-Duval. Quasi-static evolution of delaminated structures: analysis of stabil- ity and bifurcation. International Journal of Solids and Structures, Elsevier, 2004, 41, pp.103-130.

�10.1016/j.ijsolstr.2003.07.006�. �hal-00111398�

Quasi-static evolution of delaminated structures: analysis of stability and bifurcation

Rachel-Marie Pradeilles Duval

Laboratoire de Mécanique des Solides, Département de Mécanique, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Within the framework of dissipative systems with time-independent behavior, the study of the evolution of dela- minated structures modeled by frames of plates is considered via a global energetic analysis. Assuming the current equilibrium state is known, the governing rate problem for the instantaneous delamination is formulated as either a system of local equations or as a global variational inequality.

This global formulation enables to study stability and non-bifurcation of the evolution of a delaminated structure under quasi-static loading, corresponding to the statement of existence and uniqueness criteria for the rate solution.

Two analytical applications to simple structures are presented.

Keywords:Delamination; Plates; Kirchhoff–Love theory; Stability; Bifurcation; Propagation

1. Introduction

The study of fracture in composite structures has been widely investigated. The main issues are the nucleation of the damage and its evolution (Moon et al., 2002; La Saponara et al., 2002; Bruno and Greco, 2001; Zou et al., 2002). In the latter category, numerous papers deal with the modeling of delaminated structures where various viewpoints are considered. The delamination can be represented as an imperfect interface whose total damage represents the delamination (Perret et al., 1998; Borg et al., 2002; Greco et al., 2002; Qiu et al., 2001). This assumption is especially efficient to define the start of fracture. Another line of investigation considers the delamination as the propagation of a crack parallel to a plate or a shell which corresponds to the delaminated structure (Storakers and Anderson, 1988; Larson, 1991; Cochelin and Potier-Ferry, 1991; Pradeilles-Duval, 2001; Ousset, 1999). Analytical or numerical examples of propagation of delamination in such composite structures use various propagation criteria mainly based on energy criteria. Extended Griffith formulations, sometimes with decomposition into fracture modes, have been used as well (Hutchinson et al., 2000; Nilsson et al., 2001).

E-mailaddress:rachel@lms.polytechnique.fr(R.-M.PradeillesDuval).

This paper is devoted to the evolution of delaminated structures represented by a frame of plates. The goal is the introduction of a global formulation for the rate problem, when the propagation is governed by a Griffith criterion. This new formulation enables to analyze the existence and the uniqueness of the solution of this problem. In other words, the present global formulation allows the study of the stabilityand the bifurcation in the propagation of the delamination in the modeled delaminated plates, which, to our knowledge, has not been reported in the literature. The approach followed here is an extension of previous work on irreversible phase transformation in three-dimensional bodies (Pradeilles-Duval and Stolz, 1995).

In Section 2, given the kinematic theory chosen in each plate, the kinematic junctions between the plates along the delamination front is defined. Using a classical thermodynamic framework for time-independent behavior of structures (Nguyen, 1993), we derive the equations for the equilibrium problem without propagation (i.e. the delamination front given). Then the value of the total dissipation when the front propagates is obtained. This gives rise to the energy release rate associated to the propagation of the delamination front.

In Section 3, in order to define the quasi-static evolution of the system, one has to set the rate boundary value problem. Having chosen an extended GriffithÕs law as criterion of delamination and a normality rule as evolution law, the solution, in term of velocity and propagation of the front, is to be governed by either a system of local equations or a global formulation in the form of a variational in- equality, based on a potential on the rate mechanical quantities. For a given delamination propagation, the rate boundary value problem corresponds to a non-classical elastic boundary value problem with internal pre-stresses.

In Section 4, it is shown that the evolution of the system (i.e. propagation of front), assuming a classical criterion for the evolution of the front, is governed by a variational inequality. This formulation gives some conclusions on stability and bifurcation of the current state, without the need to actually determine the new geometry obtained by propagation.

Extensions of the above formulation given for Kirchhoff–Love plates are obtained for plates usually attributed to von Karman (i.e. when large transverse displacements and large deformations are considered) in Section 5.

Finally, analytical applications of these formulations to simple structures are proposed in Section 6.

A numerical procedure based on finite element method is outlined in Appendix A.

2. Problem settings

2.1. Geometry

Using classical modeling of delaminated structures (Cochelin, 1988; Larson, 1991; Storakers, 1991), the system, denoted byXt, is considered as three assembled plates, one of which represents the undamaged part (X⁰_t) and the two other plates correspond to the plates above (X¹_t) and below (X²_t) the delamination (see Fig.

1). Consequently,X_t¼X⁰_t [X¹_t [X²_t. The front of delamination along which the three plates are linked, is denoted byC_t. The external boundary ofX_tisoX.

The subscript t emphasizes the fact that the geometry is time-dependent during the evolution of the structure, the sub-domainsX⁰_t,X¹_t andX²_t as well as the delamination frontCtare expected to change due to the propagation of the delamination. On the other hand, the external boundaryoXis assumed to be un- changed.

The plates are parallel to the Cartesian orthonormal frame ðe₁;e₂Þ. Letting e₃¼e₁^e₂ denote the transverse direction, normal to the initial (undeformed) configuration of the plates, the following notation are introduced:

• hⁱ, the vertical position of the plateicompared with the mean plane of the undamaged plate (X⁰_t),

• n, the unit normal vector to oX, external toXt andt¼e₃^nthe tangent vector tooX,

• mands, the unit normal and tangent vectors toCt, withm^s¼e₃andmexternal to the undamaged part X⁰_t.

In addition, a summation convention is used, with a summation range of either 0–2 (Latin superscripts) or 1–2 (Greek subscripts). Latin superscripts refer to the plateX⁰_t, X¹_t, X²_t, supporting the corresponding quantity.

X²

a¼1

x_af_a¼x_af_a¼x₁f₁þx₂f₂ and X²

i¼0

fⁱgⁱ¼fⁱgⁱ¼f⁰g⁰þf¹g¹þf²g²:

Moreover, the plates X⁰_t, X¹_t and X²_t are parallel but not in the same plane. In the following, they are considered through their projection on the mean plane of X⁰_t. So, a point is defined by its horizontal positionðx1;x₂Þand by the plane it bellows which indicates its vertical position (x3¼hⁱ). One should note that, as shown in Fig. 1, a positionðx1;x₂Þcorresponds to two points, one inX¹_t and another inX²_t.

Consequently,Ct corresponds to the curve in the mean plane ofX⁰_t.

In the following,rf defines the gradient off, in the planeðe₁;e₂Þ. Two different kinds of jumps acrossCt

are defined:sft¼f⁰f¹f²andsftⁱ¼f⁰fⁱ. The gradient of quantityf in the directionbis denoted by f;b¼ rf e_b. If s denotes the curvilinear coordinate along C_t, associated to s, then, for instance, f;m¼ rf mandf_;s¼ rf s.

2.2. Kinematic description

The motion in the plateiis classically defined by

• the in-plane displacement,uⁱðx1;x₂Þ ¼uⁱ_aðx1;x₂Þe_a,

• the transverse displacement of the mean plane,wⁱðx1;x₂Þe₃,

• the rotation of a small segment initially normal to the plate, in other words, the flexural rotation, hⁱðx1;x₂Þ ¼hⁱ_aðx1;x₂Þe_a.

So, the displacementnⁱof the particle whose initial coordinates areðx1;x₂;x₃Þis:

nⁱðx1;x₂;x₃Þ ¼uⁱðx1;x₂Þ þwⁱðx1;x₂Þe₃x₃hⁱðx1;x₂Þ:

Fig. 1. Delamination modeled by plates frame: (a) top view, (b) transverse view.

The global strain tensor associated to this motion within the framework of small perturbations is given by:

eⁱðnⁱÞ ¼eⁱðuⁱÞ þ1

2½e₃cⁱðwⁱ;hⁱÞ þcⁱðwⁱ;hⁱÞ e₃ x₃jⁱðhⁱÞ;

with

eðuÞ ¼1

2ðruþ^TruÞ; jðhÞ ¼1

2ðrhþ^TrhÞ; cðw;hÞ ¼ rwh: ð1Þ e,jandcare respectively the plane strain of the mean plane, the gradient of the rotation of the normal and the distortion of the plate.

Along the delamination front, the motions of the three plates are linked. One can choose among several sets of conditions to formulate this link and this choice can strongly influence the predicted behavior of the whole structure (Anquez et al., 1990; Pradeilles-Duval, 1992; Roudolff and Ousset, 2002). In this paper, the continuity of plane displacement, of transverse displacement and of rotation alongCtare enforced through the following conditions:

sutⁱhⁱh⁰¼u⁰hⁱh⁰uⁱ¼0 swtⁱ¼w⁰wⁱ¼0 shtⁱ¼h⁰hⁱ¼0

9=

; 8i2 f1;2g: ð2Þ

These conditions imply the following relations concerning the tangential derivative of the displacement:

srutⁱ shⁱrh⁰ s¼0 srwtⁱ s¼0 srhtⁱ s¼0

9=

; 8i2 f1;2g:

2.3. Constitutive relations

Each plate is assumed to be elastic and the elastic energy is denoted byW, which is a function ofeðuÞ, jðhÞ,cðw;hÞ.

So, in the following, the generalized stresses are the in-plane stress tensor,N, the bending stress tensor, M, and the shear force,T. They are obtained in each plate through the constitutive relations:

N¼oW

oe ; M ¼oW

oj; T ¼oW

oc : ð3Þ

2.4. Equilibrium

The system of plates is loaded by prescribed displacements and forces. Classically, the latter are rep- resented by vector densities over the external boundary, assuming that no surface force is applied. Then, if the motion of the structure is defined by the generalized velocitiesðu;w;hÞ, the virtual power of external forces is reduced to:

Peðu;w;hÞ ¼ Z

oX

½F ðuþwe₃Þ þC hds:

In the previous equation,F denotes the external distributed force andCthe external distributed moment on oX.

Similarly, according to the previous definition of constitutive relations, the virtual power of internal forces is:

Piðu;w;hÞ ¼ Z

Xt

½N:eðuÞ þM:jðhÞ þT cðw;hÞdx:

By virtue of the virtual power principle, a state of equilibrium is governed by Piðu;w;hÞ þ Peðu;w;hÞ ¼0, for all fields ðu;w;hÞ which satisfy the continuity relations (2). This leads to the classical local equilibrium equations:

divN¼0 divMþT ¼0 divT ¼0

9=

; inXⁱ_t; withi2 f0;1;2g; ð4Þ

and the relations between generalized stresses and distributed forces on the boundaryoXare N n¼F_be_b; T n¼F₃; M n¼C:

Moreover, the continuity relations (2) alongCt induce static junctions:

sNt m¼0; sTt m¼0; sMt mþhⁱNⁱ m¼0:

2.5. Equations for Kirchhoff–Love plates

The main results of this article are established for Kirchhoff–Love plates (Timoshenko and Woinwsky- Krieger, 1959). Their extension to von Karman plates (Timoshenko and Woinwsky-Krieger, 1959, Chapter 13; Love, 1944) is then addressed in Section 5.

In the framework of Kirchhoff–Love plates, one has rw¼h, which corresponds to cðw;hÞ ¼0 and j¼ rrw. The equilibrium equations (4) reduce to:

divN¼0 div divM¼0

inXⁱ_t; withi2 f0;1;2g: ð5Þ

Thanks to relations betweenwandh alongoXandC_t, the boundary conditions onoXbecome:

N n¼Fbe_b; divM n o

osðn M tÞ ¼F₃oCs

os ; n M n¼C_n

ð6Þ

and the continuity conditions onCt become:

sNt m¼0;

sdivMt mþ o

osm fsMt

þhⁱNⁱg s

¼0;

m sMt mþhⁱm Nⁱ m¼0:

ð7Þ

Define the following notations on any line inXt(mis a normal vector to this line):

• MMeⁱ¼Mⁱ mhⁱNⁱ m, the moment on the line,

• Qⁱ¼Tⁱ m_os^oðMMeⁱ tÞ, the transverse force on the line.

Then, the boundary conditions onoXand the continuity relations onC_t are rewritten as:

• onoX,

N n¼Fbe_b; Q¼F3oCs

os ; MMe n¼Cn; ð8Þ

• on Ct,

sNt m¼0; sQt¼0; sMMet m¼0: ð9Þ

2.6. Prescribed loading and displacement

The virtual work associated to given external forces (resp. prescribed displacements) is denoted by kPextðu;wÞ(resp.kPkinðN;MÞ) and is written as

Pextðu;wÞ ¼X³

k¼1

Z

oX_Fk

F_k^d½e_k ðuþwe₃Þdsþ Z

oX_C

C^dðn rwÞds ð10Þ

and

PkinðN;MÞ ¼X²

b¼1

Z

oXub

u^d_b½e_b ðN nÞds Z

oXw

divM n

þo

os ðn M tÞ

w^ddsþ Z

oX_h

ðn MMeÞh^dds;

ð11Þ whereoX_f denotes the part of the total external boundaryoXwhere the quantityf is prescribed.

In the previous formula, k represents the loading parameter (i.e. the prescribed data on the boundary at time t are kðtÞF_k^dðx1;x2Þ on oXF_k, kðtÞC^dðx1;x2Þ on oXC, kðtÞu^d_bðx1;x2Þ on oXu_b, kðtÞw^dðx1;x2Þ on oXw,kðtÞh^dðx1;x2Þon oXh). The evolution of the structure is studied when this parameter increases.

Prescribed loading and generalized displacements are given so that the problem is ‘‘well-posed’’:

• oX¼oXu_b[oXF_bandoXu_b\oXF_b ¼ ; 8b2 f1;2g(complementarity of the prescribed in-plane displace- ment and the associated in-plane force),

• oX¼oXw[oXF3 and oXw\oXF3 ¼ ; (complementarity of the transverse displacement and the shear force),

• oX¼oXh[oXC andoXh\oXC ¼ ;(complementarity of the flexural rotation and the normal bending moment).

2.7. Equilibrium equations

The total potential energy of the system is E^potðu;w;Ct;kÞ ¼

Z

Xt

Wðu;wÞdxkPextðu;wÞ:

Using standard Lagrangian method, with the kinematic constraints defined by (2), the following system governs equilibrium for a given frontC_t:

• boundary conditions with prescribed displacement:

8b2 f1;2g; u_b¼ku^d_b on oXub; w¼kw^d on oX_w;

rw n¼kh^d_n on oXhn;

ð12Þ

• compatibility relations (1) on the plates,

• compatibility relations (2) along the delamination frontCt through kinematic junction,

• constitutive law (3) on each plate,

• classical local equilibrium equations (5) on generalized stresses,

• equilibrium equations (7) or (9) due to the junction alongCt,

• boundary conditions with given external loading:

8b2 f1;2g; e_b N n¼kF_b^d on oXFb; Q¼kF₃^d onoXF3;

e M

M n¼kC^d on oXC:

ð13Þ

The displacement, solution of this boundary value problem,ðuðk;C_tÞ;wðk;C_tÞÞis also characterized as the kinematically admissible minimizer of the potential energy. The equilibrium value of E^pot will play an important role in the analysis to follow and will be denoted byP, i.e.Pðk;C_tÞ ¼E^potðuðk;C_tÞ;wðk;C_tÞ;kÞ.

2.8. Propagation of the delamination: associated derivative

When the loading parameter increases, the delamination is expected to propagate. Let /m denote the normal velocity of the frontCtin the mean plane of plateX⁰_t, with/ðsÞP0,8s2Ct. So that, ifxonCt, thenx/mdt is onCtþdt.

For any quantityf defined alongCt, the time derivativeD_/ðfÞassociated to the propagation is defined as:

D_/ðfÞðxÞ ¼ lim

dt!0

fðx/mdt;tþdtÞ fðx;tÞ

dt ; ð14Þ

wherexis a point onCt. In particular, one finds that:

D_/m¼o/

oss¼/_;ss and D_/s¼ o/

osm¼ /_;sm:

For a functionf defined overX_t, one gets for the value off onC_t, D_/ðfÞ ¼ff_/rf m;

whereff_ is the partial time derivative off, i.e.^of_ot. Besides, the time derivative of a generic integral overXt

is given by:

d dt

Z

Xt

fdx¼ Z

Xt

ff_dx Z

Ct

sft/ds:

During any quasi-static evolution ofCt with velocity /, the relations (2) must be hold at any time. So, Hadamard jumprelations on displacement are verified:

D/ðsutⁱhⁱrw⁰Þ ¼s_uutⁱhⁱrww_⁰/srutⁱhⁱrrw⁰ m¼0 D_/ðswtⁱÞ ¼swwt_ ⁱ¼0 D/ðsrwtⁱÞ ¼srwwt_ ⁱ/srrwtⁱ m¼0

9>

>>

>=

>>

>>

;

8i2 f1;2g: ð15Þ

Likewise, enforcement of the static continuity relations (7) at any time implies:

sNNt_ m¼ o

osð/sNt sÞ; sdivMMt_ mþ o

osnm ðsMMt_

þhⁱNN_ⁱÞ so

¼ o os s o

os/ðsMt

þhⁱNⁱÞ s

;

m ðsMMt_ þhⁱNN_ⁱÞ m¼/½m frðsMtþhⁱNⁱÞ mg m ðsMtþhⁱNⁱÞ:ðsmþmsÞo/

os;

ð16Þ

or

sNNt_ m¼ o

osð/sNt sÞ;

sQQt_ ¼ o os s o

osf/ðsMt

þhⁱNⁱÞg s

; sMMMMee_t m¼/½m frðsMtþhⁱNⁱÞ mg m o/

osðsMMet sÞ:

2.9. Total dissipation

When the loading parameterkincreases or the delamination frontCtpropagates, the evolution along the equilibrium path of the potential energy at equilibriumP is given by:

dP dt ¼ d

dt½E^potðuðCt;kÞ;wðCt;kÞ;Ct;kÞ ¼ d dt

Z

X_t

WðuðCt;kÞ;wðCt;kÞÞdx

kPextðuðCt;kÞ;wðCt;kÞÞ

¼E^pot_;u d

dtuþE_;w^potd

dtwþE^pot_;Ct d

dtCtþE_;k^potkk:_

Thanks to the behavior relation (3), to the continuity equation onC_t (7) and to virtual power principle, Z

X_t

N:rduþM:rrdwda

¼ Z

oX

du ðN nÞ

divM n

þ o

osðn M tÞ

dwþ ðn M nÞðrdw nÞ

ds

Z

Ct

fm Nⁱ ðsdutⁱhⁱdrw⁰Þ þm Mⁱ srdwtⁱ ðdivMⁱ mÞsdwtⁱgds ð17Þ thus, using the Hadamard jumprelations (15), ones obtains:

E_;u^potd

dtuðCt;kÞ þE^pot_;w d

dtwðCt;kÞ ¼ Z

Xt

N :ruu_þM:rrww_dakPextðuu;_ wwÞ_

¼kkP_ kinðN;MÞ þ

Z

Ct

fm Nⁱ ðsrutⁱhⁱrrw⁰Þ þm Mⁱ srrwtⁱg m/ds: ð18Þ The total time derivative of the potential energy at equilibrium is thus found to be given by:

dP

dt ¼kk_PkinðN;MÞ kk_Pextðu;wÞ Z

Ct

sWt/dsþ Z

Ct

fm Nⁱ ðsrutⁱhⁱrrw⁰Þ þm Mⁱ srrwtⁱg m/ds:

ð19Þ

The total dissipated power of the systemDis given by the difference between the power of external forces and the rate of total free energy of the system, i.e. by:

D¼ Z

oX

½F ðuu_þwwe_ ₃Þ þCrww_ ndsd dt

Z

X_t

Wðu;wÞdx¼ dP

dt kkP_ extðu;wÞ þkkP_ kinðN;MÞ

¼ P;C_t

dCt

dt : ð20Þ

The total dissipated power is thus found, by virtue of (19), to be of the form:

D¼ Z

C_t

GðsÞ/ðsÞds; ð21Þ

where the energy release rateGðsÞis given by:

G¼sWtm Ni ðsrutⁱh_irrw₀Þ mm Mi srrwtⁱ m: ð22Þ

3. Quasi-static evolution

In this section, the quasi-static evolution is presented. It includes the definition of the criterion which governs the propagation of the delamination, then the local equations as well as the global formulation of the rate boundary value problem are presented.

3.1. Propagation criterion

Taking into account the form of the dissipation, the governing law for the evolution ofCtis introduced according to the framework of the generalized standard materials (Nguyen, 1987, 1993).

Accordingly, the existence of a convex threshold functiongðGÞis postulated, so that the propagation of the delamination front is subject to the following law:

• ifgðGðsÞÞ<0, then/ðsÞ ¼0, i.e. no propagation at points,

• ifgðGðsÞÞ ¼0, then/ðsÞP0, i.e. the propagation ofCtat point sis possible, which is equivalent to

/ðsÞ ¼cðsÞog

oG withcðsÞgðGðsÞÞ ¼0ðconsistency relationÞandcðsÞP0: ð23Þ A generalized Griffith criterion can be considered by introducing a threshold energy to be reached for propagation of the delamination front to occur. In this case, one would havegðGÞ ¼GG_C.

In the following,C^rupt_t is the subset of the front Ct where the criterion is reached and the propagation is possible.

C^rupt_t ¼ fs2Ct such thatgðGðsÞÞ ¼0g:

3.2. Consistency condition and evolution law

When the front is moving, Eq. (23) implies that, during the actual propagation ofC^rupt_t ,gðGðsÞÞ ¼0 must be maintained, leading to the consistency equation:

if c>0 onC^rupt_t ; D_/ðgðGÞÞ ¼ og

oGD_/ðGÞ ¼0:

In the present formulation,

D/ðGÞ ¼GGðe NN;_ MMÞ þ_ GGð_b uu;wwÞ þ_ /G; ð24Þ where

e

GGðNN;_ MMÞ ¼ ðm_ NN_iÞ ðsrutⁱh_ij0Þ m ðm MM_iÞ ðsjtⁱ mÞ;

b

GGð_uu;wwÞ ¼ sNt_ ruu_₀þ ðsMtþh_iNiÞ ðjj_0Þ;

G¼ sM rruþM rrrwt mþ ðm rNi mÞ ðsrutⁱh_ij0Þ m

þ ðm rMi mÞ sjtⁱ mþs Ni ½ðsrrutⁱh_irj0Þ m sþs Mi ðsrjtⁱ mÞ s:

Finally, the evolution law of the delamination front is:

c2V=8c⁰2V; D/ðGÞðc⁰cÞog

oG60 ð25Þ

with

V ¼ fcðsÞwith cP0 onC^rupt_t andc¼0 onCtnC^rupt_t g: ð26Þ

3.3. Local equations of the rate problem

The evolution law forCt being chosen, the instantaneous evolution of the global system induced by a loading evolution with velocity given bykk_is now investigated. This evolution turns out to be governed by the following rate problem for the rates of fields variablesðuu;_ ww_Þ:

• boundary conditions with prescribed displacements:

8b2 f1;2g; uu__b¼kku_ ^d_b on oX_ub; _

w

w¼kkw_ ^d on oXw; rww_ n¼kkh_ ^d_n on oXhn;

ð27Þ

• compatibility:

in each plate Xⁱ_t: ee_ⁱ¼1

2ðruu_ⁱþ^Truu_ⁱÞ; jj_ⁱ¼ rrww_ⁱ; ð28Þ

on Ct verification of the Hadamard relations (15),

• constitutive relations:

N_ Nⁱ M_ Mⁱ

" #

¼

o²Wⁱ oe²

o²Wⁱ oeoj o²Wⁱ ojoe

o²Wⁱ oj²

0

@

1 A ee_ⁱ

j_ jⁱ

" #

¼Wⁱ⁰⁰ ee_ⁱ j_ jⁱ

" #

in Xⁱ_t; withi2 f0;1;2g; ð29Þ

• local equilibrium equations for the generalized stresses:

divNN_ ¼0 div divMM_ ¼0

inXⁱ_t; withi2 f0;1;2g; ð30Þ

• boundary conditions with prescribed forces:

8b2 f1;2g; e_b NN_ n¼kkF_ _b^d on oXFb; Q_

Q¼ divMM_ n o

osðn MM_ tÞ ¼kkF_ ₃^d on oXF3; C_

C n¼n MM_ n¼kkC_ ^d onoXC;

ð31Þ

• equilibrium conditions in rate form (16) along the moving delamination frontC_t,

• evolution law ofCt given by (25).

The rate boundary value problem (15), (16) and (27)–(31) onuu_ andww_ is similar to the original equilibrium system (1)–(3), (5), (7), (12) and (13), but, here the right-hand sides of equations depend on a prescribed evolution of the delamination front. The rate boundary value problem with a prescribed evolution ofCt(i.e.

/orcgiven) is similar to a problem of elasticity with non-classical conditions (15) and (16) localized onCt. 3.4. Global formulation of the rate value boundary problem

In order to have a formulation better suited to the study of stability and bifurcation, a global functional is defined:

Fð_uu;ww;_ c;kkÞ ¼_ X²

i¼0

Z

Xⁱ_t

1

2½_ee;jjW_ ⁱ⁰⁰ ee_ j_

j dxkkP_ extð_uu;wwÞ _ Z

Ct

b

GGð_uu;wwÞc_ og oGds1

2 Z

Ct

Gc² og oG

2

ds;

ð32Þ whereee_andjj_ are defined by (28).

Theorem 1.ð_uu;ww;_ cÞis the solution of the system of local equations (15), (16), (25), (27), (30), (31)with the compatibility relations(28)and the constitutive equations(29)if and only if

ð_uu;ww;_ cÞ 2K oF ouu_ð~uu

uuÞ þ_ oF

oww_ ð~wwwwÞ þ_ oF

ocð~cccÞP0; 8ð~uu;ww;~ ~ccÞ 2K; ð33Þ whereK is the convex set of admissible fields with given boundary conditions and evolution criterion on del- amination front and is defined by:

K¼ ð_uu;ww;_ cÞj _ u

u;ww;_ rwwC_ ⁰ðXⁱ_tÞ;C¹_mðXⁱ_tÞ i2 f1;2g _

u

u_b¼kku_ ^d_b onoXub; b2 f1;2g _

w

w¼kkw_ ^d onoXw

rww_ n¼kkh_ ^d_n onoXhn

suut_ ⁱhⁱrww_⁰¼c_oG^ogðsrutⁱhⁱrrw⁰Þ m i2 f1;2g

swwt_ ⁱ¼0 on Ct

srrwt_ ⁱ¼c_oG^ogsjtⁱ m i2 f1;2g

cP0 on C^rupt_t

c¼0 onCtnC^rupt_t

0 BB BB BB BB BB BB BB BB BB

@ 8>

>>

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9>

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; :

Proof.In the following,^ee¼eð^uuÞ ¼eð~uuuuÞ,_ jj^¼ rr^ww¼ rrðww~wwÞ,_ //^¼^cc_oG^og¼ ð~cccÞ_oG^og. Becauseð_uu;ww;_ cÞ 2K, one gets relations (15) and (27), if/¼c_oG^og.

Consider the inequality oF

ouu_^uuþoF oww_ ww^þoF

oc^ccP0;

withðuu;_ ww;_ cÞ 2K and for anyð^uu;ww;^ ^ccÞ, admissible with vanishing displacement on the external boundary (where displacement are prescribed) then, by virtue of the notations due to the compatibility relations (28) and the constitutive equations (29),

06X²

i¼0

Z

Xⁱ_t

½NN_ : ^eeþMM_ : ^jjdakkP_ extð^uu;wwÞ ^ Z

C_t

fr^uu₀:sNtþ^jj0:ðsMtþhiNiÞg/ds

Z

Ct

b

GGð_uu;wwÞ_ //^ds Z

Ct

G///^ds:

Using the divergence theorem and the properties of admissible fieldsð^uu;wwÞ^ on the external boundary, 06X²

i¼0

Z

Xⁱ_t

½divNN_ þdiv divMMe_ ₃ ð^uuþwwe^ ₃Þdaþ Z

oXFb

½e_b NN_ nkkF_ _b^d^vv_bds

þ Z

oXF3

"

divMM_ non MM_ t os kkF_ ₃^d

#

^ w wdsþ

Z

oX_C

½n MM_ nkkC_ ^dðn r^wwÞds

Z

C

fm sNN_ ^uutþm sMM_ r^wwtm sdivMMt^_ wwþ r^uu₀:sNt/þ^jj0:ðsMtþh_iNiÞ/gds:

At this step, this inequality must be satisfied for any admissible fieldð^uu;ww;^ ^ccÞ. Consequently, the first four terms, in the previous inequality imply that the rate solution must satisfy Eqs. (30) and (31). The inequality is thus reduced to:

06 Z

Ct

m sNNt_ ^uu₀

n þm sMMt_ r^ww0m sdivMMt^_ ww0

o ds

Z

Ct

s r^uu₀ sNt s/

n þjj^0:ðsMtþh_iNiÞ/o ds

Z

Ct

m NN_i s^uutⁱ

n þm MM_i srwwt^ ⁱþGGðb uu;_ wwÞ_ //^þG///^o ds:

Finally, using the compatibility of the velocities ð^uu;wwÞ^ with the front velocity //^¼^cc_oG^og on Ct (because ðuu;_ ww;_ cÞandð~uu;ww;~ ~ccÞare elements ofK), one obtains

06 Z

C

m sNNt_

þ o

osð/sNt sÞ

^ u u₀ds þ

Z

C

m sMMt_ m

n þh_im NN_i m/m ðsrMtþh_irNiÞ mo

½r^ww₀ mds þ

Z

C

o/

os½ðsMt

þh_iNiÞ:ðsmþmsÞ

½rww^₀ mds

Z

C

sdivMMt_ m

þ o os

nm ðsMM_ thim NN_iÞ so þo

os s o

os½/ðsMt

þhiNiÞ s

w^ w0ds

Z

C

D_/ðGÞ//ds:^

This inequality must hold for any admissibleð^uu;ww;^ ^ccÞ, which implies that local equilibrium equations in rate form onCt (16) are fulfilled as well the evolution law (25).

To summarize, the variational inequality (33) has been shown to imply the rate boundary problem ((15), (16), (25), (27), (30) and (31)) assuming the compatibility relations (28) and the constitutive equations (29).

The converse implication is straightforward and Theorem 1 follows. h

Rewriting of the rate problem: The setKof admissible fields with the kinematic boundary conditions, the Hadamard jumprelations (15) and the evolution law ofCtis convex.

In the following paragraphs, for givenkk,_ /(i.e. for givenc), ifðuu;_ ww_Þis solution to Eqs. (15), (16), (27), (30), (31), with (28) and (29) then, let

Hðc;kkÞ ¼_ Fðuuðc;_ kkÞ;_ wwðc;_ kkÞ;_ c;kkÞ_

denote the value ofF for the solution of the rate problem, when the front velocity and the loading evolution are prescribed viacandkk. The evolution of the delamination front is governed by_

oH

oc½c;kkð~_ cccÞP0; 8~ccadmissible with evolution lawð23Þand criterion ð25Þ:

In the sequel, we denote by HHe the second-order derivative of functionalH with respect toc. Because the functionalH is a quadratic function of ðc;kkÞ,_

oH

oc½c;kk ¼_ o²H

oc²cþo²H ocokk_

k_ k

and the evolution inequality can be written as:

findc2V admissible with (23) and (25) such as ð~cccÞ HH ce

þ o²H ocokk_

k_ k

P0; 8~ccadmissible withð23Þandð25Þ: ð34Þ

The functional HHe is defined on the current configuration. It is a function of the displacement and of the position of delamination front at timet. So, the instantaneous evolution of the structure from its configu- ration at timetdepends solely on the current geometry and the state of equilibrium at time t.

4. Characterization of the current state

In this section, we investigate whether there exists one or more admissible solutions to the quasi-static instantaneous evolution, when the loading parameter increases. To achieve this goal, general results from Duvaut and Lions (1976) and the same framework as in Nguyen (1993) are used.

In what follows, ‘‘stability’’ refers to the existence of a solution to the rate problem and ‘‘non-bifur- cation’’ to the uniqueness of the solution of the same problem. More precisely, ‘‘non-bifurcation’’ means that the solution to the system of local equations (15), (16) and (25)–(31) is unique. This definition of non- bifurcation thus does not consider bifurcations of higher order in time.

4.1. Stability of the evolution

Theorem 2.The evolution of the delaminated structure from a state, characterized by a displacementðu;wÞand a delamination frontCt, is stableðhas at least one solutionÞif and only if

cHH ce >0; 8c2V f0g;

whereV is the set of admissible fields with evolution laws, defined by(26).

Proof.The evolution of the system is associated to the solution c2V of Eq. (34). Taking into account different admissible fieldsc⁰, it leads to:

c2V and HH ce þ o²H ocokk_

k_

k¼0: ð35Þ

Then, to solve (35) for a prescribed load increasekk, one has to invert_ HHe. IfHHe is positive definite onV, then (35) can be solved for any givenkk. Obviously, if one is able to find_ cassociated to anykk_by Eq. (35), then the functionc!HH ce is surjective on V.

Moreover, let the dead load be defined as the first value of the loading parameter that enables delam- ination propagation without a loading parameter evolution i.e. such that there existsc2V f0g such as

e H

H c¼0. Consequently, the dead load indicates the possibility of an unstable evolution where a delamin- ation propagation may occur without a load increase. h

4.2. Non-bifurcation of evolution

Theorem 3.There is no bifurcationðat most one solutionÞin the evolution of the delaminated structure if and only if

cHH ce >0 8c2spanðVÞ f0g;

where spanðVÞis the vector space generated byV.

Proof.Consider two different solutionsc₁andc₂ to (34). Then, we get:

ðc₁c₂Þo²H ocokk_

k_

kPðc₂c₁ÞHH ce ₂ and ðc₂c₁Þo²H ocokk_

k_

kPðc₁c₂ÞHH ce ₁: ð36Þ Consequently, it gives:ðc1c₂ÞHHeðc1c₂Þ ¼0, whereðc1c₂Þbelongs to spanðVÞ. IfHHe is definite positive on spanðVÞ, then the previous equation impliesc₁¼c₂, which contradicts the hypothesis of two distinct solutions.

Otherwise,ðc1c2Þbelongs to KerðHHeÞ, subset of spanðVÞwhereHHe is not definite. h

5. Application to von Karman plate theory

Application to von Karman plates (with large transverse displacement and large deformation) (Love, 1944; Timoshenko and Woinwsky-Krieger, 1959) is important because it allows coupling between del- amination and buckling. The approach used in Sections 3 and 4 for Kirchhoff–Love plates can be extended to von Karman plates in a straightforward way. The difference between the Kirchhoff–Love and von Karman theories is mainly due to the non-linearity of the plane strain with respect to the transverse dis- placement in von Karman plates:

eðu;wÞ ¼1

2ðruþ^Truþ rw rwÞ; jðhÞ ¼ rrw: ð37Þ

This enables to take buckling into account in von Karman theory.

Upon following the same steps as in Sections 3 and 4, one obtains the corresponding evolution for- mulation for von Karman plates, whose most important aspects are summarized in this section.

5.1. Study of the equilibrium state 5.1.1. Equilibrium equations

Stationarity of the Lagrangian with kinematic constraints described by (2) onCtgives, in addition to the kinematic relations onoX(12), the compatibility relations inXt (37) and the constitutive relation (3), the following equilibrium equations:

• inXⁱ_twithi2 f0;1;2g, divN¼0;

div divMdivðN rwÞ ¼0; ð38Þ

• alongCt, sNt m¼0;

sdivMt mþo

osm sMt

þhⁱNⁱ s

¼0;

m ðsMtþhⁱNⁱÞ m¼0;

ð39Þ

• on the external boundary,

e_b N n¼kF_b^d onoXF_b; 8b2 f1;2g;

divM no

osðn M tÞ ¼kF₃^d on oXF₃; n M n¼kC^d on oX_C:

ð40Þ

5.1.2. Energy release rate

Analyzing the total dissipation of the system, we get:

G¼sWtm ðNⁱ srutⁱþMⁱ srrwtⁱhⁱNⁱ rrw0Þ m: ð41Þ

5.2. Evolution of delamination front 5.2.1. Rate boundary value problem

The evolution problem for the instantaneous delamination induced by a load increment kk_ is defined again by (15), (16), (23), (25), (27), (28), (29), (30), (31), except for the following differences:

• in platei, (28) is changed in ee_ⁱ¼1

2ðruu_ⁱþ^Truu_ⁱþ rwⁱ rww_ⁱþ rww_ⁱ rwⁱÞ; jj_ⁱ¼ rrww_ⁱ; ð42Þ

• in platei, (30) is replaced by divNN_ ¼0;

div divMM_ divðNN_ rwþN rwwÞ ¼_ 0; ð43Þ

• on Ct, the equilibrium conditions in rate form (16) along the moving delamination front Ct

become

sNNt_ m¼ o

osf/sNt sg; sdivMMt_ mþ o

osfm ðsMMt_ þhⁱNN_ⁱÞ sg ¼ o os s o

osð/½sMt

þh_iNiÞ s

þ/sN:rrwt;

m ðsMMt_ þhⁱNN_ⁱÞ m¼/½m rðsMtþhⁱNⁱÞ m fsMtþhⁱNⁱg:ðsmþmsÞo/

os:

ð44Þ

During the propagation of the delamination front, the criterion gðGÞ ¼0 has to be maintained. So,

og

oGD_/ðGÞ ¼0 where

D_/ðGÞ ¼GGðe NN;_ MMÞ þ_ GGð_b uu;wwÞ þ_ /G with now

e

GGðNN;_ MM_ Þ ¼ m NN_ ðsrutⁱhⁱrrw0Þ mm MM_ srrwtⁱ m;

b

GGðuu;_ wwÞ ¼_ sNt:ðruu_₀þ rw0 rww_₀Þ þ ðsMtþhⁱNⁱÞ:rrww_₀;

G¼ sN:ðrruþ rw rrwÞ þM:rrrwt mþm rNⁱ m ðsutⁱhⁱrrw0Þ m þm rMⁱ m srrwtⁱ mþNⁱ:ðsrrutⁱhⁱrrrw0Þ þMⁱ:srrrwtⁱ:

5.2.2. Global formulation

As in the Kirchhoff–Love case, the local equations of the rate problem are equivalent to a variational inequality of the form (33), with the functionalF now given by:

Fðuu;_ ww;_ c;kkÞ ¼_ X²

i¼0

Z

Xⁱ_t

1

2½_ee;jjW_ ⁱ⁰⁰ ee_ j_

j dxkkP_ extðuu;_ wwÞ _ Z

Ct

b

GGðuu;_ wwÞc_ og oGdsþ1

2 Z

Ct

Gc² og oG

2

ds;

whereee_ andjj_ are given by (42).

6. Applications

Two analytical applications are presented in this section.

The first one (Section 6.1) corresponds to the bending of a delaminated plate. This structure is an ex- tension of the double cantilever beam considered in Roudolff and Ousset (2002), La Saponara et al. (2002), Greco et al. (2002). Here, analytical results concerning the energy release rate are given as well as some conclusions on the stability of the evolution.

The second example (Section 6.2) is a delaminated structure under compression. It uses the non-linear von Karman plate theory and we study the propagation of delamination on a frame of plates after buckling of the plates below and above the fracture. This phenomenon induces coupling between delamination and buckling as shown in Moon et al. (2002), Nilsson et al. (2001) and Storakers and Nilsson (1993). In what follows, we first obtain an analytical form of the energy release rate with or without buckling. Then, due to the global functional introduced to this end in the previous section, the linear propagation of the delam- ination front is discussed in terms of stability and bifurcation.

6.1. Propagation of delamination in a double cantilever beam 6.1.1. Definition of the structure

The structure considered is a rectangular plate (thickness 2e) with a fracture in its median plane. The constitutive material is assumed to be homogeneous, isotropic and elastic with characteristics denoted byE, the Young modulus,m, the Poisson ratio.

The rectangular plateXt is defined by:

Xt¼ ðx1;x₂Þjx12 ½0;L; x₂2

b 2;b

2

:

It is delaminated throughout the whole axise₂. The delamination front is located on the line defined by x₁¼a, x₂2 ½^b₂;^b₂ ða<LÞ.a_ini denotes the initial location of the delamination front.

LetX¹_t (respectivelyX²_tÞdenote the plate above (resp. below) the delamination and X⁰_t the undamaged plate.

The loading of the structure is the following (see Fig. 2):

• the edgex₁¼L is clamped (displacements prescribed to zero),

• onx₂¼ ^b₂, the normal and shear stresses vanish and the rotation is given equal to 0, i.e.

N e₂¼0; divM e₂þ ðe₁ M e₂Þ_;1¼0; w_;2¼0;

• on the remaining boundary (x1¼0), if 2wd denotes the crack opening displacement for the delaminated structure,

F₁^d¼F₂^d¼0; C^d¼0; w²¼ wd and w¹¼w_d:

6.1.2. Equilibrium

The displacement vector and the stress tensor are easily found to be independent on thex₂ coordinate.

The configuration is symmetrical with respect to thex₁axis. So, we get for the solution:

Fig. 2. Double cantilever beam and loading: (a) initial and undeformed configuration, (b) deformation of the structure under loading with propagation of the delamination front.

u¹ðx1;x2Þ ¼u²ðx1;x2Þ ¼0;

w¹ðx1;x2Þ ¼w²ðx1;x2Þ ¼wd 1

3x1

2a þ x³₁ 2a³

; u⁰ðx1;x₂Þ ¼0;

w⁰ðx1;x2Þ ¼0 and

N⁰¼N¹¼N²¼0;

M⁰¼0;

M¹¼ M²¼3Dwdx₁

a³ ðe₁e₁þme₂e₂Þ withD¼ Ee³ 12ð1m²Þ: The potential energy at a state of equilibrium is Pðwd;aÞ ¼^3bDw_a3 ^2d. 6.1.3. Energy release rate and linear propagation

The energy release rate associated to a delamination whose current length is denoted bya is:

G¼9Dw²_d a⁴ :

If a Griffith-like criterion is adopted, such as /ðGGCÞ ¼0 with /P0 and G6GC on Ct, the critical loading is reached forw_d ¼w_CðaÞ ¼

ffiffiffiffiffiffiffiffi

GCa⁴ 9D

q .

Considering only uniform propagations of delamination, which means that the normal velocity of propagation is constant along C_tð/ðsÞ ¼/ðx2Þ ¼aaÞ, the analysis of evolution consists in studying_ Gas a function ofa. Then, the normal velocity of delamination front is given byD/ðGÞ ¼0, which is equivalent to

oG

ow_dww_dþ^oG_oa/¼0. This leads to^aa_a^_¼¹₂^ww_w^_d

d.

Consequently, the evolution with uniform propagation is always stable, when the loading is driven by displacement, as shown on Figs. 3, 4 and 5.

0 a_ini/L 1

0 1

a/L wd /wmax

evolution of structure G=G_C

without propagation with linear propagation

Fig. 3. Evolution of the delamination length with loading in DCB experiment (ainidenotes initial length of the delamination).

This result could have been established from the global functionalH which here takes the following form:

Hðww_d;aaÞ ¼_ 3bDðww__dÞ²

a³ þ18bDðw_dÞ² a³

_ a a a

"

ww__d wd

#aa_ a:

00

w_d

F

without propagation

with linear propagation F_max

wC(a

ini) w

max

Fig. 4. Load/deflection curve with linearly propagating delamination (DCB experiment).

0 a_ini/L 1

0

a/L F

without propagation with linear propagation F_max

Fig. 5. Resistance curves: transverse force versus delamination length (case of linear propagation) (DCB experiment).

6.1.4. Analysis of evolution and general propagation

Let now consider general evolutions of the delamination front when the propagation criterion is reached.

Any normal velocity is then sought in the form:

/ðsÞ ¼/ðx2Þ ¼a0þX¹

n¼1

a_ncos 2npx2

b

" # þX¹

n¼1

b_nsin 2npx2

b

" #

and the uniqueness of theðan;b_nÞcoefficients is investigated.

Analysis of the rate boundary value problem ((15), (16), (27)–(31)) implies that the propagation velocity necessarily takes the form /ðsÞ ¼/ðx2Þ ¼P1

n¼0a_ncosð2np^x_b²Þ.

Then, we get the following displacement rate:

_ u

u¹¼uu_²¼0 and uu_⁰¼0;

_ w w⁰¼0;

_ w

w¹¼ ww_²¼ww_d 13x1

2a

þ x³₁ 2a³

þ3a0wd

2a x1

a

x²₁ a²

þ3wd

a² X¹

n¼1

a_n

asinh%^2npx_b¹&

cosh%^2npa_b &

x₁cosh%^2npx_b¹&

sinh%^2npa_b &

h i

cosh%^2npa_b &

sinh%^2npa_b &

2np^a_b cos 2npx2

b

:

As introduced in Section 4, stability and non-bifurcation are characterized by positive definiteness of the functional H over the setV of admissible delamination front normal velocities or over the vector space spanðVÞ, respectively.

Here,H corresponds to:

Hð/;ww_dÞ ¼Hða0;a1;. . .;a_n;. . .;ww_dÞ

¼3bDww_²_d

a³ 18bDw_dww__d

a⁴ þ18bDw²_da²₀

a⁵ þ9bDw²_d a⁵

X¹

n¼1

a²_n 1

"

þ2

2npa

b sinh²%^2npa_b &

sinh%^2npa_b &

cosh%^2npa_b &

^2npa_b

# :

Consequently, the second derivative ofH with respect to/is positive definite for all aspect ratiosa=band there is no bifurcation in the evolution of the delamination front.

6.1.5. Discussion

The foregoing analysis of the Double Cantilever Beam gives results similar to experimental observations in La Saponara et al. (2002) and Roudolff and Ousset (2002). One should note that this modeling cannot deal with either the start of delamination (aL) or the caseaL, i.e. when the delamination front reaches the clamped edge, because edge effects are not taken into account.

6.2. Delaminated plate under compression 6.2.1. Definition of the structure

A rectangular plate

Xt¼ ðx1;x2Þ withx12 ½

L;L; x22

b 2;b

2

;

assumed to be homogeneous, isotropic and elasticðE;mÞ, is considered in this example.

It is delaminated throughout the whole axise₂. The delamination is in the middle of the plate. Its front is initially located on the lines defined by

x₁¼a₁; x₂2

b 2;b

2

and

x₁¼a₂; x₂2

b 2;b

2

ðjaij<Land a₁>a₂Þ:

Let 2ldenote to the length of the delaminated part (i.e.a₁a₂¼2l) (see Fig. 6).

The loading of the structure is as follows (see Fig. 6):

• on the transverse boundary (i.e..x_2:¼ ^b₂),N e₂¼0, divM e₂þ ðe₁ Me₂Þ_;1¼0,w_;2¼0,

• along the edgesx₁¼ L the boundary conditions are:

u e₁¼ ud; F₂^d ¼F₃^d¼0; C^d¼0 on x₁¼L;

u e₁¼u_d; F₂^d¼F₃^d¼0; C^d ¼0 on x₁¼ L forx22

b 2;b

2

; ð45Þ

• there is no surface force.

In order to take buckling into account, the von Karman plate theory is used and the evolution problem is therefore formulated based on the results of Section 5.

Let Cand D denote the classical ratios _ð1m^Ee2Þand _12ð1m^Ee32Þ, wheree is the thickness of the plates above and below the delamination.

6.2.2. Equilibrium

Using the local equations given in paragraph 5.1.1, two possibilities arise:

• Ifu_d6u_B with u_B¼_l2Cð1m^Lp2D2Þ¼_12l^Le2ð1m^2p22Þ, then the displacement and the stresses are:

uðx1;x₂Þ ¼u_d

Lðx1e₁mx2e₂Þ; w¼0;

N¼Eeu_d

L e₁e₁; M¼0:

The potential energy at equilibrium is in this case independent ofl:

Pðud;lÞ ¼2CLbð1m²Þ u_d L

" #2

:

Fig. 6. Delaminated plate under axial compression: (a) initial and undeformed configuration, (b) post-buckling of the loaded structure with propagation of the delamination front.