A symmetric boundary integral formulation for ... - Semantic Scholar

IABEM 2002, International Association for Boundary Element Methods, UT Austin, TX, USA, May 28-30, 2002

A symmetric boundary integral formulation for cohesive interface problems. A. Salvadori1 Dept. of Civil Engng., University of Brescia, via Branze 38, 25123 Brescia, ITALY

Abstract An incremental symmetric boundary integral formulation for the problem of many domains connected by non-linear cohesive interfaces is here presented. The problem of domains with traction-free cracks and/or rigid connections are particular instances of the proposed cohesive formulation. The numerical approximation of the considered problem is achieved by the symmetric Galerkin boundary element method.

1

Introduction

The present work deals with isotropic linear elastic bodies linked to each other by interfaces. In particular, M non linear cohesive interfaces are considered to connect N domains made of different materials. Under the assumption of small displacements and strains, the response of such a system to quasi-static external actions is studied. The problem of N domains with M tractionfree or pressurized cracks and/or rigid connections are particular instances of the proposed cohesive formulation. The subject of the present work is significant to predict the mechanical behavior and to assess the safety factor of structures. The interface between different materials is indeed one of the most important regions governing the strength and stability of structures (Chandra Kishen, 1996) and plays a major role in fracturing of quasi brittle materials (Salvadori, 1999), polymer (Lauke and Schueller, 2001), ceramics and composites (Smith and Teng, 2001), bioengineering materials, biological solids and tissues (Middleton et al., 1996). Investigations over interface constitutive laws have been undergoing a great development in the last years. Starting from pioneering works, cohesive interfaces are often modelled by a (holonomic) nonlinear elastic relation between cohesive tractions p and opening displacement w (Hillerborg et al., 1976). This approach is meaningful only when local unloading can be reasonably assumed as negligible. During the last decade, various authors proposed non-associated elastoplastic cohesive models to describe the interface behavior under combined normal and shear stresses in the presence of local unloading. A literature review on the subject can be found in (Salvadori, 1999). For the problem of N domains connected by M cohesive non linear interfaces, a boundary integral incremental formulation is given (section 2), in terms of the displacement fields u, v, the traction field t and the displacements discontinuity field w along the interfaces. The integral operator that governs the problem, in presence of a holonomic interface law, is proved to be linear 1

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1


with respect to the rate unknown fields and admits a variational formulation, its solution being a critical point of a (quadratic) functional. Existence and uniqueness of the solution basically depend on the adopted cohesive interface law. Interface constitutive equations typically present a softening branch and this feature can cause bifurcations in the sense of multiplicity of solutions to the rate problem. This important issue has been analyzed by various authors (among others: (Maier and Frangi, 1998), (Carini and Salvadori, 2001)). In the context of the present work, uniqueness and stability issues are marginally dealt with and reference is made to interface models that are stable in the second-order work sense. The numerical approximation of the considered problem can be achieved by different techniques. Despite the finite element method is the most widely used, boundary element methods (Bonnet et al., 1998) are very attractive for this class of problems because all non linearities are localized on the boundary of linear elastic domains. The Galerkin approximation scheme, applied to the symmetric integral formulation, ensures uniqueness, stability and convergence of the numerical solution in suitable functional spaces (McLean, 2000).

2

Single-zone formulation

Consider a homogeneous isotropic solid in a Cartesian reference system, with domain Ω ⊂ R d , d = 2, 3 and with boundary Γ = Γu ∪ Γp . Assuming small strains and displacements, consider ¯ (x) on Γu and its response to quasi-static external actions: tractions ¯t(x) on Γp , displacements u domain forces ¯f (x) in Ω. The symmetric Galerkin boundary integral formulation of the single-zone linear elastic 2 problem rests on Green’s functions with a weakly singular (Guu ), strongly singular (Gup and Gpu ) and hypersingular (Gpp ) behaviour. If the field point x is moved to the boundary in a limit process, “integrals” involving strongly singular kernels may be understood in their distributional nature of the Cauchy Principal Value (CPV). Similarly, the integral involving the hyper singular kernel G pp can be understood in its distributional nature of Hadamard’s finite part. The boundary integral formulation of the problem formulated above reads as follows (Hong and Chen, 1988) on smooth boundaries: Z

Z

(1)

C(x)u(x) + − Gup (r; l(y))u(y) dy + − Gup (r; l(y))¯ u(y) dy = Z

Γu

Γp

Guu (r)t(y) dy + Z

Z

Γp

Γu

Guu (r)¯t(y) dy +

Z

Z

Ω

Guu (r)¯f (y) dy ,

x∈Γ

D(x)t(x)+ = Gpp (r; n(x); l(y))u(y) dy+ = Gpp (r; n(x); l(y))¯ u(y) dy = Z

Γp

Z

Γu

Z

− Gpu (r; n(x))t(y) dy + − Gpu (r; n(x))¯t(y) dy + − Gpu (r; n(x))¯f (y) dy Γu

Γp

Ω

(2) x∈Γ

where r = x − y. Problem (1)-(2) admits a variational formulation, i.e. it can be obtained from the stationarity of a given functional Ψ(u, t); the solution of the problem is shown to be a saddle-point (Polizzotto, 1988) for Ψ. Many papers have been devoted to the numerical approximation of equations (1)-(2): for a review on all related issues see (Bonnet et al., 1998), (McLean, 2000). 2

Equations (1)-(2) can be easily arranged to describe a larger class of engineering problems.

2


3

Interfaces def

Consider N domains Ωn , n ∈ IN = {1, .., N } connected to each other by M interfaces (figure 1). Let Γm,n ⊂ ∂Ωn indicate the generic boundary pertainw def ing to the interface m ∈ IM = {1, 2, ..., M }. If a domain Ωn has nothing to do with the m-th interface, Γm,n = ∅. Denoting with Ωn1 and Ωn2 w (n1 , n2 ∈ IN ) the two domains connected by the m1 and th interface, the two smooth boundaries Γm,n w m,n2 Γw define the lips of the interface m (see figure 2 1 outward normals, say nn1 , nn2 , , Γm,n 2). On Γm,n w w are defined as usual. The hypothesis of small displacements and strains implies: Figure 1: Domains connected by cohesive interfaces

def

def

nn1 = n(xn1 ) = −n(xn2 ) = −nn2

(3)

and the equilibrium conditions read: def

def

tn1 = σ (xn1 ) nn1 = −σ (xn2 ) nn2 = −tn2

(4)

It is our goal to define a reference surface, say Γm w , that will be identified with the interface between domains Ωn1 and Ωn2 . To this aim, adopting a non linear continuum mechanics terminology, fixed t in a time interval T , two one-to-one applications u n1 , un2 between the reference surface Γm w 1 2 and the boundaries Γm,n , Γm,n are set such that, w w n1 n1 1 ∃! x ∈ Γm ∀xn1 (t) ∈ Γm,n w w : u (t) = x (t) − x n2 n2 2 ∀xn2 (t) ∈ Γm,n ∃! x ∈ Γm w w : u (t) = x (t) − x n1 n2 m with Γm w defined by the property u (0) = u (0) = 0. In a less sophisticated way, Γw = m m m,n2 m,n1 from the geometrical point of view. A normal n (x) along Γw still must be = Γw Γw defined, consistently with the definition of the relative opening displacement w m : def

wm (x) = un1 − un2 The power due to cohesive tractions, ˙ = tn1 · u˙ n1 + tn2 · u˙ n2 = tn1 · (u˙ n1 − u˙ n2 ) = tn1 · w ˙m δW naturally lead us to define: def

pm (x) = tn1

def

nm (x) = nn1

x ∈ Γm w

(5)

As a further assumption in the problem formulation, cohesive tractions p m and relative opening displacements wm are related by a (nonlinear) cohesive law pm (wm (x)), ∀x ∈ Γm w . Cracks are 3


a) Geometry

b) Static

Figure 2: Description of the interface here formulated as interfaces with vanishing tangential stiffness, i.e. p = 0 ∀w. On the contrary, rigid connections are here formulated as interfaces with rigid constitutive law, i.e. w = 0 ∀p. By definition (5), pm is the traction on a domain due to the interface. Considering for instance a linear elastic spring-type interface, the constitutive law reads: (6)

pm (wm (x)) = −Km wm

with Km positive definite. The stored energy in the spring amounts to 12 wm · Km wm and equals R ˙ dt. If the interface merely causes a dissipation the energy “lost” by the elastic domains − δ W def in the system, the instantaneous mechanical dissipation (Simo and Hughes, 1998) is D mech = ˙ m ≥ 0. −pm · w

4

Incremental multi-zone formulation def

Consider N domains Ωn , n ∈ IN = {1, 2, ..., N } connected to each other by M interfaces (figure 1). Quasi-static external tractions ¯tn are imposed on the Neumann boundary Γnp of each domain ¯ n are imposed on the Dirichlet boundary Γnu of each domain Ωn . By Ωn , whereas displacements u m,n the definition of Γw , the incremental form of equations (1)-(2) on the Dirichlet and Neumann boundaries of the n-th domain read as follows: f˙ u,n =

Z

Z

Γn u

Z

Guu t˙ dy − − Gup u˙ dy + Z

Γn p

f˙ p,n = = Gpp u˙ dy − − Gpu t˙ dy + Γn p

Γn u

m=1 M Z X

Z

Γn p

Z

Z

G t˙ n dy − −m,nGup u˙ n dy m,n uu

Γw

Z

Γw

=m,nGpp u˙ dy − −m,nGpu t˙ n dy

m=1

where: def ¯˙ − f˙ u,n = C(x)u

M Z X

Z

n

Γw

¯˙ dy − Guu¯t˙ dy + − Gup u Γn u

Z

Z

Ωn

Z

Γw

Guu¯f˙ dy

def ¯˙ dy + − Gpu¯f˙ dy f˙ p,n = −D(x)¯t˙ + − Gpu¯t˙ dy− = Gpp u

Γn p

Γn u

Ωn

x ∈ Γnu x ∈ Γnp

(7) (8)

x ∈ Γnu x ∈ Γnp

i Equations (1)-(2) on the two faces Γl,n i = 1, 2 of the l-th interface produce the four vector w equations:

g˙ u,ni =

−δ2i I + (−1)i C(x) u˙ ni +

Z

Z

Guu t˙ dy − −ni Gup u˙ dy ni

Γu

4

Γp

(9)


+

M Z X

m=1

Z

Guu t˙ ni dy − −m,ni Gup u˙ ni dy m,ni

Γw

Z

Γw

i x ∈ Γl,n w

i = 1, 2

g˙ p,ni = (D(x) − δ2i I) t˙ ni − −ni Gpu (x − y; (−1)i+1 nni (x); l(y)) t˙ dy Z

(10)

Γu

+ =ni Gpp (x − y; (−1)i+1 nni (x); l(y)) u˙ dy − +

Γp

M Z X

−m,ni Gpu (x − y; (−1)i+1 nni (x)) t˙ ni dy

m=1 Γw M Z X

m=1

=m,ni Gpp (x − y; (−1)i+1 nni (x); l(y)) u˙ ni dy

l,ni x ∈ Γw

Γw

i = 1, 2

where δij is the usual Kronecker symbol and def

g˙ u,ni = − Z

Z

Z

¯˙ dy − Guu¯t˙ dy + −ni Gup u ni

Γp

Z

Γu

Z

Z

Ω ni

Guu¯f˙ dy

i = 1, 2

¯˙ dy + −n Gpu¯f˙ dy g˙ p,ni = −ni Gpu¯t˙ dy− =ni Gpp u def

Γp

Γu

Ω

i = 1, 2

i

When not explicitly indicated, normals are taken as outward with respect to the surface they are referred to. In equations (10) the normal at the field point x has been chosen as inward when i = 2. Define vectors v˙ m and z˙ m on the m-th interface as follows: def

v˙ m =

1 n1 (u˙ + u˙ n2 ) 2

def

z˙ m =

,

1 m 1 n1 ˙ (u˙ − u˙ n2 ) = w 2 2

(11)

It is worthy to stress that once a domain n is selected whose boundary forms a lip of the interface l, say the lip n1 , within the interface m 6= l the domain can form no lips, the lip n 1 or the lip n2 as well. From equation (11) one has therefore: u˙ n (y) = v˙ m (y) ± z˙ m (y) u˙ ni (y) = v˙ l (y) − (−1)i z˙ l (y)

l t˙ n (y) = ±p˙ m (y) y ∈ Γm w 6= Γw y ∈ Γlw t˙ ni (y) = −(−1)i p˙ l (y)

; ;

i = 1, 2

Subtracting equations (91 ) and (92 ) one obtains: −

2 X

i

(−1) g˙

u,ni

l

l

= (I − 2C(x))v˙ − z˙ −

2 X

(−1)

i=1

+

(−1)

Z

i

i=1

i=1

−

2 X

2 Z X

l i=1 Γw

i

M X

±

m=1;m6=l

Gnuui p˙ l dy −

Z

Γm w

Gnuui p˙ m

2 Z X

Z

dy − −

Γm w

− Gnupi v˙ l dy +

i=1

Γlw

n

Γu i

Gnuui t˙

Gnupi v˙ m

2 X i=1

Z

dy − −ni Z

dy ∓ − Z

Γm w

Γp

Gnupi u˙

Gnupi z˙ m

(−1)i − Gnupi z˙ l dy Γlw

!

dy +

(12)

!

dy + x ∈ Γlw

The apex ni on Gnrsi indicates that the elastic properties of kernels refer to the domain Ω ni whose boundary forms a lip of the interface l. It is implicitly assumed in equation (12) that when the boundary of the domain Ωni does not form a lip of the interface m 6= l, integrals of the form R Gnrsi (·)m dy vanish. Γm w 5


Subtracting eq. (101 ) from eq. (102 ) and adding eq. (101 ) from eq. (102 ) the following two vector equations come out: −

2 X

(−1)i g˙ p,ni = −(I − 2D(x))p˙ l −

Z

2 X

+

(−1)

M X

i

i=1

Z

±−

m=1;m6=l

2 Z X

− Gnpui p˙ l dy +

−

i=1

g˙

p,ni

Γlw

= p˙ +

2 X

Z

M X

Z

±−

i=1 m=1;m6=l

2 X

+

i

Z

(−1) −

i=1

Γlw

Γm w

Gnpui

Z

dy− =

Γm w

= Gnppi v˙ l dy −

− −ni

i=1

i=1

−

Z 2 X i=1

l

2 X

Γm w

Gnpui p˙ m

Γu

Γlw

l

p˙ dy −

Γu

Z

Gnppi v˙ m 2 X

dy∓ =

Z

dy− = 2 X

Gnppi u˙

dy+ =ni Z

Γm w i

Γp

Z

(−1) =

i=1

Γlw

Gnppi z˙ m

(13)

!

dy + x ∈ Γlw

Γlw

(14)

dy + Gnppi z˙ m

dy∓ =

Gnppi

Γm w

Γp

!

Z

Gnppi v˙ m Z

!

(−1)i = Gnppi z˙ l dy

i=1

Gnpui t˙

Gnpui p˙ m

Z

− −ni Gnpui t˙ dy+ =ni Gnppi u˙ dy +

(−1)i

i=1

i=1

2 X

2 X

Γm w

Z 2 X

l

!

dy +

= Gnppi z˙ l dy

v˙ dy +

i=1

Γlw

x ∈ Γlw

˙ = DT z˙ and Denote with DT the tangent stiffness matrix of the cohesive law, i.e. p˙ = DT w consider smooth boundaries, whence (Salvadori, 2002b): I − 2C(x) = I − 2D(x) = 0. Equations (7), (8), (13) take the final form: f˙ u,n =

Z

+

Γn u

Z

Guu t˙ dy − − Gup u˙ dy

M X

Z

Z

± −m,n (Gnuu DT − Gnup ) z˙ m dy − −m,n Gnup v˙ m dy

m=1

Z

(15)

Γn p

Γw

Z

Γw

x ∈ Γnu

f˙ p,n = = Gpp u˙ dy − − Gpu t˙ dy Γn p

−

M X

±

m=1

−

2 X

i

(−1) g˙

Z

p,ni

=m,n (Gnpu DT Γw =−

2 X

−

(−1)

Gnpp ) z˙ m Z

i

− −ni

i=1

i=1

+

2 X i=1

−

M X

(−1)i

Z

Γu

Z

dy− =m,n

Z

Gnpui t˙

dy+ =ni Γp

v˙

m

dy

Gnppi u˙ Z

Γm w

= (Gnpui DT + (−1)i Gnppi ) z˙ l dy +

i=1

Γw

Gnpp

x ∈ Γnp

!

(17)

dy + !

± = (Gnpui DT − Gnppi ) z˙ m dy− = Gnppi v˙ m dy +

m=1;m6=l

Z 2 X

(16)

Γn u

Γlw

Z 2 X

Γm w

= Gnppi v˙ l dy

i=1

x ∈ Γlw

Γlw

Finally consider the following linear combinations of equations (12) and (14): −

2 X

(−1)i DTT g˙ u,ni − g˙ p,ni

i=1

= DT −

DTT

l

z˙ −

2 X i=1

(−1)

(18) i

Z

n

Γu i

DTT

Gnuui t˙ 6

Z

dy − −ni Γp

DTT

Gnupi u˙

!

dy +


−

2 X

(−1)

i=1

+ + −

M X

i

±−

m=1;m6=l

2 Z X

Γm w

DTT

(Gnuui DT

−

Gnupi ) z˙ m

− DTT (Gnuui DT + (−1)i Gnupi ) z˙ l dy −

l i=1 Γw 2 X

i=1 2 X

Z

Z

!

Z

dy − −

Γm w

DTT

Gnupi v˙ m

!

dy +

− DTT Gnupi v˙ l dy Γlw

− −ni Gnpui t˙ dy+ =ni Gnppi u˙ dy +

Z 2 X

M X

Γu

±

= ((−1)

i=1

2 Z X i=1

i=1 m=1;m6=l

+

Z

Γlw

i

Γp

Z

= (Gnpui DT Γm w

Gnpui DT

+

Gnppi ) z˙ m

−

Gnppi ) z˙ l

dy −

Z

dy− = 2 X

Γm w

Z

Gnppi v˙ m

!

dy +

(−1) = Gnppi v˙ l dy

i=1

i

Γlw

x ∈ Γlw

The incremental problem of domains connected by cohesive interfaces stems from equations (15)(18): it may be rewritten in the operatorial form M† u˙ = p˙ †

(19)

The unknown vector u˙ is made of tractions t˙ on the Dirichlet boundaries Γnu , displacements u˙ on the Neumann boundaries Γnp , mean displacements v˙ and relative half opening displacements z˙ on † the cohesive interfaces Γm w , having set n ∈ IN nd m ∈ IM . Let us indicate the domain of M as the space U and let the space P denote the range of the operator M† . Let the classical bilinear form A(u0 , p0 ) put into duality the two spaces U and P : 0

0

def

A(u , p ) =

Z

Γ

u0 · p0 dx

(20)

The following variational statement holds. Proposition - If DT = DTT the integral operator M† is symmetric with respect to the bilinear form (20) A(M† u, ˙ v) ˙ = A(u, ˙ M† v) ˙ (21) Proof: Making use (Sirtori et al., 1992) of the reciprocity properties of kernels G rs and of the symmetry of DT , the thesis follows immediately.

Corollary - The solution of (19) is a critical point of the functional 1 Ψ† [u] ˙ = A(u, ˙ M† u) ˙ − A(u, ˙ p˙ † ) 2

(22)

Remarks 1. The formulation (19) is a sound extension of the elasticity BIE (1)-(2). In fact, in case of vanishing Γm w , the linear elastic integral equations of n unconnected domains is recovered.

7


˙ = z˙ = 0 2. The incremental problem of N domains connected by M rigid connections (i.e. w imposed a priori) stems from equations (7), (8) (12) and (13). The problem (19) becomes: M‡ u˙ = p˙ ‡

(23)

The operator M‡ is symmetric with respect to the bilinear form (20) and the solution is the critical point of the functional 1 ˙ M‡ u) ˙ − A(u, ˙ p˙ ‡ ) Ψ‡ [u] ˙ = A(u, 2

(24)

Functional (24) has a sign definiteness property. The solution of problem (23) is the solution of the min-max problem: ˙ (25) min max Ψ‡ [u] ˙ p˙ t,

˙ v˙ u,

3. The incremental problem of a single domain with M cracks stems from equations (15), (16) and (18) by setting N = 1, DT = 0. It may be rewritten as: M∗ u˙ = p˙ ∗

(26)

The formulation is symmetric and the solution is the critical point of the functional 1 Ψ∗ [u] ˙ = A(u, ˙ M∗ u) ˙ − A(u, ˙ p˙ ∗ ) 2

(27)

Functional (27) has a sign definiteness property. The solution of problem (26) is the solution of the min-max problem: min max Ψ∗ [u] ˙ (28) t˙

˙ z˙ u,

4. The incremental problem of N domains connected by M R rigid connections and M C cohesive interfaces is a combination of problems (19) and (23), that will be denoted as: Mu˙ = p˙

(29)

The problem admits of a variational formulation and the solution is the critical point of the functional 1 ˙ Mu) ˙ − A(u, ˙ p) ˙ (30) Ψ[u] ˙ = A(u, 2 that, generally speaking, does not possess a sign definiteness property. If the interface law is stable in a second order sense (DT negative definite), the solution of problem (29) is the solution of the min-max problem: min max Ψ[u] ˙ ˙ p˙ t,

˙ v, ˙ z˙ u,

(see also (Carini and Salvadori, 2001)).

8

(31)


5. Many holonomic cohesive models are characterized by the symmetry of their operator D T (see e.g. (Xu and Needleman, 1994), (Camacho and Ortiz, 1996)). Such models, often used when local unloading are negligible, stem from the assumption of existence of a free energy density φ per unit undeformed area. Denoting with q a suitable collection of internal variables (to describe the inelastic processes attendant to decohesion), the first and second laws of thermodynamics imply the general form of the cohesive law: (32)

p = gradw [φ]

Equation (32) has a symmetry implication. In fact, when its rate form is written, one obtains: ˙ ˙ p˙ = grad w [φ] = DT w with the matrix DT =

∂2φ ∂wi ∂wj

(33)

(34)

symmetric by construction. 6. Elastic plastic based models (e.g. (Lotfi and Shing, 1994), (Carol et al., 1997), (Jefferson, 1998), (Salvadori et al., 2002)) make use of non associated potentials to model dilatancy. Non associativity implies unsymmetry for the tangent operator D T . To give a variational formulation for such models, one can apply a suitable symmetrization strategy (Carini and Salvadori, 2001). Structure of operator M - It is easy to see the symmetry of operator M in view of the symmetry property of the involved kernels (Sirtori et al., 1992). Operator M is made of four blocks: def

M=

"

MD MID

MDI MI

#

(35)

Block MD contains the integrals pertaining to the boundaries Γnu and Γnp of each domain Ωn . Most of this block is made of null terms: the higher the number of domains, the larger the sparsity of the block. In case of n unconnected domains, the block M D coincides with the whole operator M. Blocks MDI and MID describe the interactions between the domains and the interfaces. A 0 entry in the n, m position of block MDI implies that the boundary of domain n has empty intersection with the support of interface m. Block MI is due to the interaction between the interfaces. A 0 entry in the m1 , m2 position of block MI signifies that domains n1 and n2 that form the lips of interface m1 have an empty intersection with domains n3 , n4 that form the lips of interface m2 . Symmetric Galerkin approximation - Let h > 0 be a parameter and let U h denote a family of finite dimensional subspaces of U such that ∀u˙ ∈ U , inf ||u˙ − u˙ h || → 0 u˙ h ∈Uh

9

as h → 0

(36)


The symmetric Galerkin approximation of (29) consists in finding u˙ h ∈ Uh critical point of the functional 1 ˙ (37) Ψ[u˙ h ] = A(u˙ h , Mu˙ h ) − A(u˙ h , p) 2 From the algebraic point of view, let ψ u (y), ψ t (y), ψ p (y), ψ v (y), ψ z (y) be matrices of shape ˆ , t˙ h (y) = ψ t (y) ˆt, p˙ h (y) = ψ p (y) p ˆ , v˙ h (y) = ψ v (y) v ˆ , z˙ h (y) = functions and u˙ h (y) = ψ u (y) u ˙ ˙ ˙ ˙ ˙ ψ z (y) zˆ be discrete approximations of u(y), t(y), p(y), v(y), z(y), respectively. Let the vector u˙ ˆ , ˆt, p ˆ, v ˆ , zˆ. Let the matrix ψ h collect the shape functions matrices ψ u (y), collect the unknowns u ψ t (y), ψ p (y), ψ v (y), ψ z (y). From the stationarity of (37), the linear system Mu˙ = p˙

(38)

comes out, where M = A(ψ h , Mψ h )

p˙ = A(ψ h , p) ˙

The (symmetric) Galerkin approximation scheme ensures stability and convergence of the numerical solution in suitable functional spaces (for further details see (McLean, 2000)). Literature review - To the best of my knowledge, the first non symmetric multi-domain formulation was devised by (Blandford et al., 1981) as a method for dealing with two co-planar crack surfaces. The multi-region method introduced artificial boundaries into the homogeneous body, connecting the cracks to the boundary. Real multi-bodies problems with interfaces have been analyzed in (Lee and Choi, 1988), (Tan and Gao, 1990) via the multi-region method. The use of a special fundamental solution to avoid modelling the interface of two different materials was introduced in (Yuuki and Cho, 1988). Different strategies pertaining to the collocation method have been implemented in (He, 1994), (Sladek and Sladek, 1995). A symmetric formulation for the problem of rigidly connected domains in dynamics was given by (Maier et al., 1991). Linear (spring-type) cohesive interfaces (6) were also considered in homogeneous materials. In the formulation, free-terms were not taken into account. The structure of operators (35) arose also in the elasto-dynamic context. A more recent work (Gray and Paulino, 1997) a symmetric formulation for the problem of rigidly connected domains was proposed. Free-terms were not taken into account again, “considering the boundary integral equations in a limit to the boundary sense”. The structural property (35) was clearly stated for rigid connections with reference to a Laplace equation. The there proposed method was actually general and the extension to elastostatic should coincide with operator (35) for rigid connections. In the same year another work (Layton et al., 1997) considered a symmetric Galerkin BEM formulation for the problem of rigidly connected domains. The final matrix results only partially symmetric. For a criticism on such a work see (Gray and Paulino, 1997).

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Concluding remarks

In the present work a symmetric boundary integral formulation of multi-domain problems with non linear cohesive interfaces has been proposed, which is suitable for any engineering application with different materials and interfaces between them. An object oriented implementation of the 10


present formulation, defining suitable topological structures (“stars” and “berths” of interfaces) to enforce the compatibility and the local equilibrium conditions at any corner or edge, will be subject of a further publication. Many complementary issues have not been discussed in the present work. Among them: The singularity of the integral operator M , that requires the evaluation of double “integrals” containing singular and hypersingular kernel functions. Three main techniques (regularization methods e.g. (Krishnasamy et al., 1991), numerical approximations e.g. (Aimi et al., 1999) and analytical integrations e.g. (Salvadori, 2002a)) have been proposed for the evaluation of singularities. Fast integral operators techniques. From the computational point of view, a major drawback of discrete integral operator M rests in its non-localness, that lead to a linear system with a symmetric but full matrix. Many different techniques have been recently proposed to reduce the computational cost of the matrix evaluation and of the system solution for potential problems (panel clustering (Hackbush and Nowak, 1989), wavelets (Dorobantu, 1984), fast multipole method (Yoshida et al., 2001), H-matrices (Hackbusch, 1999)). Only a few works appeared on the panel clustering in 3D elasticity, see e.g. (Hayami and S.A., 1997) and none of them exploits the sparsity of the discrete integral operator M. Another open issue, strictly related to fast integral operators technique, is the analysis and implementation of an efficient iterative solver, that fully exploits the symmetry, sign definiteness and sparsity of the discrete integral operator M. Existence and uniqueness of the solution of equation (29) basically depend on the adopted cohesive interface laws. Non linear interface constitutive equations typically present a softening branch and this peculiarity can cause multiplicity of solutions to the rate problem (29). This important issue has been deeply analyzed by various authors (Carini and Salvadori, 2001), (Maier and Frangi, 1998), in different contexts adopting different methodologies. Non-associated elastoplastic cohesive models , whose basic property is the capability of reproducing the path-dependence of cohesive tractions versus opening displacements, have been proposed during the last decade by various authors. The elastoplastic problem can be formulated as a linear complementarity problem (LCP) in rates, following (Maier, 1969). In the presence of non-associated flow rules, the operator that governs the LCP is not symmetric (DT 6= DTT ) and results (21)-(22) do not hold. Suitable symmetrization strategies have been recently proposed (Salvadori, 1999), (Carini and Salvadori, 2001) (but not implemented, yet), to give the problem a variational formulation. The time integration procedure, for instance, can be performed by classical finite difference schemes. However, it will be of great interest to provide a variational formulation for the whole timespace problem. Some works in this context have been put forward in (Brun et al., 2001) and are very promising especially for softening interface laws that can present a snap-back behavior, for which methods for tracing structural responses beyond critical points must be considered (in this frame, the Arc-Length method seems to be the preferable candidate).

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