SHAKEDOWN ANALYSIS BY BEM

European Congress on Computational Methods in Applied Sciences and Engineering

ECCOMAS 2000

Barcelona, 11-14 September 2000

”ECCOMAS

SHAKEDOWN ANALYSIS BY XXX

X. Xxxxxxx, X. Xxxxxxx and X. Xxxxxxxxxxxx

Dipartimento di Ingegneria Strutturale e Geotecnica Universit`a degli Studi di Palermo

xxxxx xxxxx Xxxxxxx, 00000 Xxxxxxx, Xxxxx e-mail: xxxxxxx@xxxx.xxxxx.xxxxx.xx

Key words: BEM, SGBEM, Plasticity, Shakedown

Contract grant sponsor: Ministro dell’Universit`a e della Ricerca Scientifica e Tecnologica, Italy

Abstract. In the ambit of the symmetric Galerkin boundary element formulation the statical shakedown load multiplier and the limit analysis are reformulated making use of macrozone modelling. The subdivision of the domain into macroelements makes it pos- sible to deal with piecewise homogeneous materials of the body. For each macroelement a discretization of the boundary and a subdivision of the domain into portions called cells are performed in order to introduce the unknowns (i.e. traction and displacement dis- continuities) on the boundary and material plastic laws appropriately interpolated. The weighed regularity imposed between adjacent macroelements produces algebraic operators which are symmetric and sign-definite, thus preserving the meaningful properties of the continuum. The load multiplier computed by this strategy is only an approximation to the actual value due to the modelling of the boundary unknown functions and to the yield conditions of the material. The plastic collapse load multiplier is dealt with as a special case of shakedown in which a proportional load is applied in the body.

1 INTRODUCTION

The symmetric Galerkin boundary element method (SGBEM) has had a dramatic increase in attention by a lot of researches and has been applied to a great number of classes of engineering practice problems. A synthesis of the variety of engineering problems dealt with, as linear elastostatic, elastoplasticity, fracture, dynamics, FEM–

BEM coupling, has been recently provided by Xxxxxx et al8.

This method shows some remarkable advantages in comparison with the collocation approach, but the competitiveness of the method must also be tested, and mainly, in comparison with the FEM. It is well known that the FEM is very suitable for problems with nonlinearities and inhomogeneities, whereas the collocation BEM is more appro- priate for problems with unbounded domains or in the presence of stress concentration.

The overcoming of the difficulties in computing some coefficients of the solving system2−6 and the possibility of treating subdomains having different elastic constants in a single mathematical model permits an incisive use of the SGBEM in dealing with

structural systems, also in the presence of nonlinearities and inhomogeneities (piecewise homogeneity).

The latter idea of treating different subdomains in the SGBEM arises in a paper by Xxxxxxx et al5 who, in the ambit of FE–symmetric BE coupling, have proposed the

employment of a macroelement (ME) handled with BEM entirely surrounded by FEs and have associated an appropriate stiffness matrix with it.

Subsequently, the present authors11,12 proposed subdividing the continuum into MEs, each of which is embedded in an infinite domain with appropriate physical character-

istics. Through a variable condensation process an equation is written for every ME linking the weighed forces and the weighed displacements on the interface elements between MEs to some quantities characterizing the interface boundary forces, to the interface nodal displacements and to the external actions. Using the latter relationships for each ME, a mixed–value analysis method is derived. A similar approach has been

employed by Xxxxxx et al.14 for the analysis and homogeneization of periodic composites

through a subdivision of the continuum into two subdomains with different materials whose interface boundary can be a locus of a possible decohesion.

In order to find an elastic response by MEs, in the ambit of the SGBEM, the same authors13 have introduced a stiffness matrix and a flexibility matrix for MEs, thus deriving the displacement and traction analysis methods, respectively.

The SGBEM can be employed in a direct analysis1,3, for example in the shakedown

analysis, in order to obtain the shakedown load multiplier by using the statical or kine- matical approach recast in the language of the BEM. For a homogeneous body the established formulation rests on a boundary discretized by boundary elements and on a domain subdivided by cell elements (CEs). The mixed boundary variables are displace- ments and tractions whereas the domain variables are the plastic strains associated with the cells.

For a piecewise homogeneous body using the ME approaches permits to propose the

computation of the shakedown load multiplier by means of two different strategies:

a) The body domain is discretized by MEs, each of which takes on the role of a cell element; the discretization can be made by building a parent element with 4 or 8 nodes in a similar way as for the FEs.

b) In the body a sparse discretization by MEs is performed and a subsequent subdivision into CEs is made in every ME.

In both cases the use of the mixed–value analysis will have as variables some param- eters characterizing the interface boundary forces, the interface node displacements and also the plastic strains associated with each ME, assumed as CE (see case a) or with each CE inside the ME in the sparse discretization performed (see case b).

This paper is devoted to shakedown and limit analysis via SGBEM. To this purpose the crucial feature consists in the fact that self stress and compatible permanent strain fields must be assessed by means of suitable boundary integral equations. The self stresses are obtained as the elastic response of the body to imposed permanent strains and the compatible strains are those giving rise to identically vanishing elastic stresses in the body.

In the hypothesis of dealing with a multidomain approach by SGBEM the main aim of the paper is that to find the basic equation of every ME when in them a subdivision by CEs has been made. In these equations the unknowns will be the quantities connected to one’s boundary and the plastic strains in every CE.

Such a goal permits to provide an effective procedure for shakedown and limit analysis in a way quite similar as it is made by FEM. The statical approach to shakedown analysis has been dealt with only, whereas the limit analysis is viewed as a special case for a proportionally varying load.

Notation.

∫

The matrix notation is used, with matrices and vectors denoted by bold-face letters.

α

The symbols ∫ A and Ω A denote the integrals of A over the boundary Γ , (α = 0, 1, 2),

α

and over the domain Ω, respectively, and these symbols can be applied twice to indicate double integrations.

2 AN ELASTIC PERFECTLY PLASTIC MODELLING PROBLEM

Let us consider (Fig.1) an in–plane elastic perfectly plastic solid body Bˇ occupying

the (open, simply connected) domain Ω of boundary Γˇ = Γˇ1 ∪ Γˇ2, restrained on Γˇ1

against rigid motions and loaded by imposed displacements u¯1 on Γˇ1, forces ¯f2 on Γˇ2,

body forces b¯ and thermal–like strains ϑ¯ in Ω. The material has elastic plastic behavior with relevant plastic law of an associative type.

The external actions vary in time t slowly enough so that inertial and viscous forces are negligible. They constitute an n–parameter load family, with Q = Q(t) being the vector collecting the independent load parameters. In general the real load history Q = Q(t) is unknown but the loading domain Π can be assumed to have a convex hyperpolyhedral shape, characterized by ν load points Qi(i = 1, ··· , ν), vertices of

the hyperpolyhedron characterizing the so–called basic loads. Any load path inside Π represents a potential active load history.

In order to obtain the actual response of the body subjected to a specified load

history, we suppose the body is embedded in the infinite domain Ω∞ having the same

physical properties as Bˇ. The body boundary may be thought of as a boundary Γˇ of

domain Ω with normal n exiting to Ω or as a boundary Γˇ+ of domain Ω∞ \ Ω with

normal n+ entering Ω. In the presence of an unlimited domain the boundary quantities take on the role of

– layered forces f1 on Γˇ1, ¯f2 on Γˇ2;

− −

– layered relative displacements v¯1 = u¯1 on Γˇ1, v2 = u2 on Γˇ2.

Moreover, because of plastic behavior of the material in Ω, a permanent strain field

sp must be introduced.

f

Γ2

Γ1

Ω

ϑ

Γ2

f

b

Γ2 Γ1

u

u

Fig.1 - Plane solid body subjected to external actions

∞

In Ω the field equations are represented by the well–known Somigliana Identities (SI) shown in Appendix I, here rewritten in compact form as follows

u = u[f1, v2, sp]+ uˆ[v¯1, ¯f2, b¯, e¯ ]

t = t[f1, v2, sp]+ ˆt[v¯1, ¯f2, b¯, e¯ ]

ơ = ơ[f1, v2, sp]+ ơˆ[v¯1, ¯f2, b¯, e¯ ]

(1a − c)

In them the permanent strain vector sp takes on a similar meaning to the thermal–like strain vector e¯ .

The set of equations governing the elastic–plastic evolutive problem at time t is obtained by setting

u1 = u¯1 on

t2 = ¯f2 on

Γˇ1

Γˇ2

(2a − c)

sp =

∫ t ˙p

s

0

dt in Ω

where s˙p vector denotes the plastic strain rate field.

Substituting the SI into the boundary conditions (2a,b) gives the first kind integral equation system of Fredholm, i.e.

2

u1[f1, v2, sp]+ uˆ1c[v¯1, ¯f2, b¯, e¯ ]+ 1 u¯1 = u¯1 on

2

t2[f1, v2, sp]+ ˇt2c[v¯1, ¯f2, b¯, e¯ ]+ 1¯f2 = ¯f2 on to which the relevant plastic flow laws must be added

Γˇ1 Γˇ2

(3a, b)

≤

s˙p = λ˙ ∂φ; φ(ơ) 0;

∂ơ

with the stresses given by

λ˙ ≥ 0;

λ˙ φ(ơ)= 0 in Ω (4a − d)

ơ = ơ[f1, v2, sp]+ ơˆ[v¯1, ¯f2, b¯, e¯ ] in Ω (5) In them the following expressions

∫ ∫ ∫

u1(x)= Guu f1 + Guf (−u2)+ Guσ sp

∫ 1 ∫ 2 ∫ Ω

t2(x)= Gfuf1 + Gff (−u2)+ Gfσ sp

(6a − c)

∫ 1 ∫2 ∫Ω

ơ(x)=

1 Gσu f1 +

2 Gσf (−u2)+

Gσσ sp

Ω

represent the elastic response to the unknowns, whereas the expressions

1

uc (x)=

1

∫

Guf (−u¯1)+ 2

Guu ¯f2 +

∫

Guu b¯ +

Ω

∫

Guσ e¯

Ω

∫

tc (x)=

Gff (−u¯1)+

∫

Gfu ¯f2 +

∫

Gfu b¯ +

Gfσ e¯

(6d − f )

1

2

2

Ω

Ω

∫ ∫ ∫ ∫

ơˆ(x)=

1 Gσf (−u¯1)+

Gσu ¯f2 +

2

Gσu b¯ +

Ω

Gσσ e¯

Ω

represent the elastic response to the known external actions. The apices c in the vectors

uˆc and ˆtc introduced in eq.(3a,b) show the presence of integrals computed as Cauchy

1 2

1 ¯ 1 ¯

principal values (CPV) to which the free terms 2 u1(x) and 2 f2(x) must be added.

The previous two boundary conditions (2a,b) or also (3a,b) may be replaced by

equivalent conditions but written on the boundary Γˇ+, considering that the domain

Γ

Ω∞ \ Ω must be unstrained and unstressed. As a consequence

or in extensive form

u+ = 0 on ˇ+

1

2

Γ

1

2

t+ = 0 on ˇ+

(7a, b)

u+ ≡ u1[f1, v2, sp]+ uˆc [v¯1, ¯f2, b¯, e¯ ] − 1 u¯1 = 0 on

Γˇ+

1 1 2

1

(8a, b)

2

2

2

2

t+ ≡ t2[f1, v2, sp]+ ˆtc [v¯1, ¯f2, b¯, e¯ ] − 1¯f2 = 0 on

Γˇ+

the latter also obtainable directly using eqs.(3a,b).

3 A DISCRETE APPROACH

∪

Let us discretize the body boundary into boundary elements and model the unknowns (forces on Γˇ1, displacement discontinuities on Γˇ2) on the sides of the boundary by means of appropriate shape functions. Let be B the body and Γ = Γ1 Γ2 its polygonal boundary.

One sets

f1 = ψf F1 on Γ1, u2 = ψu U2 on Γ2 (9a, b)

−

where the reaction vector F1 is defined on the sides of the boundary Γ1, whereas the displacement vector U2 = V2 is relative to the nodes of the boundary Γ2.

In order to model2 the material plastic behavior within Ω, let us subdivide Ω into

n portions called cell–elements (CEs) within which the material plastic laws are appro- priately interpolated. Let us assume for each k-th CE

s˙kp = Mk p˙ k, λ˙ k = lk Λ˙ k, lk ≥ 0, Λ˙ k ≥ 0 in Ωk, (k = 1,..., n)

−

(10a d) where Ωk indicates the subdomain occupied by the typical CE, Mk is a suitable matrix

of shape functions, lk is a suitable weighting function, whereas p˙ k and Λ˙ k are unknown

vector and scalar parameter describing the plastic deformation rate of the k-th CE.

The relevant plastic flow laws (4a-d) are rewritten for each CE

x

x ∂ơ

s˙ p = λ˙

∂φk , φ

(ơ) ≤ 0,

λ˙ ≥ 0,

λ˙ φ

k

(ơ)= 0 in Ω

, (k = 1,..., n)

x

x

x

x

can be substituted by k-th CE global yielding laws

(11a − d)

p˙ k

= Λ˙

∂Φk , Φ

k ∂Pk k

(Pk

) ≤ 0,

Λ˙ k

≥ 0,

Λ˙ k

Φk(Pk

)= 0 with (k = 1,..., n)

(12a − d)

M

The latter equations refer to laws written for a convex yield function Φk and for a generalized internal force vector at plastic nodes, so defined

∫

Ω

Pk =

k

T ơ, Φk =

∫

Ω

lk φ(MkPk) (13a, b)

x

x

A piecewise linearized form of Φk can be obtained by a piecewise linear approximation

of Φk. Introducing the vectors P, p˙ , Λ˙ and Φ which collect quantities pertaining to all

the CEs, eqs.(12a-d) can be restored in a more compact form as

∂P

p˙ = Λ˙ ∂Φ ; Φ(P) ≤ 0;

Λ˙ ≥ 0;

Λ˙ Φ(P)= 0 (14a − d)

where the positions are assumed

∫

P = MT ơ,

Ω

s˙p = M p˙ ,

λ˙ = l Λ˙

(15a − c)

with MT a block row matrix.

When we introduce the modellings (9a,b) and (10a) into eqs.(8a,b) the following mixed–value equation system is obtained

∫ ∫ ∫

+

ˆc 1 ¯ +

u1 ≡ 1 Guu Ψf · F1 + 2 Guf Ψu · (−U2)+

Ω Guσ M · p + u1 − 2 u1 = 0 on Γ1

∫ ∫ ∫

+

ˆc 1¯ +

t2 ≡

1 Gfu Ψf · F1 +

2 Gff Ψu · (−U2)+

Ω Gfσ M · p + t2 − 2 f2 = 0 on Γ2

(16a, b)

with the stresses given by

∫ ∫ ∫

ơ = 1 Gσu Ψf · F1 +

2 Gσf Ψu · (−U2)+

Ω Gσσ M · p + ơˆ

in Ω (16c)

The latter expressions rewritten in compact form become

u+ ≡ u1[F1, V2, p]+ uˆc − 1 u¯1 = 0 on Γ+

1 1 2

1

(17a, b)

t+ ≡ t2[F1, V2, p]+ ˆtc − 1¯f2 = 0 on Γ+

2

and also

2 2 2

ơ = ơ[F1, V2, p]+ ơˆ

in Ω (17c)

Let us perform the weighing of the response on the discretized boundaries Γ+

and

2

Γ+, by using the shape functions ψf

1

f

1

2

∫

and ψu

1

respectively, i.e.

2

u

2

∫

1

W+ ≡

ψT u+ = 0, R+ ≡

ψT t+ = 0 (18a, b)

then the classic symmetric and defined solving system is obtained, that is

Buu F1 + Buf (−U2)+ Buσ p + Lu = 0 Bfu F1 + Bff (−U2)+ Bfσ p + Lf = 0

Moreover, the vector P (eq.15a) of the internal forces of the CEs becomes

(19a, b)

P = Bσu F1 + Bσf (−U2)+ Bσσ p + Lσ (20)

In the latter the following positions are valid

∫ ∫

Buu = ψT

∫ ∫

Guu ψf, Buf = ψT Guf ψu = BT ,

1

∫ f ∫1

1 f 2∫ ∫ fu

Buσ = ψT

Guσ M = BT , Bfσ = ψT

Gfσ M = BT ,

∫1 f ∫Ω

σu ∫ ∫ 2 u Ω

σf

(21 )

2 u

Bff = ψT Gff ψu,

2

Bσσ = Ω M Ω Gσσ M,

a − i

∫ T c 1

∫ T c 1¯ ∫ T

Lu =

1 ψf uˆ1 − 2 u¯1

, Lf =

2 ψu ˆt2 − 2 f2 , Lσ = Ω M ơˆ

The solving system becomes in compact form as follows

0 = BX + Bσ p + L

σ

P = BT X + Bσσ p + Lσ

(22a, b)

to which the relevant plastic flow laws (14a-d) of all the CEs must be appended. In eqs.(22a,b) we have set

F1

Buu B

Buσ

Lu

X =

−U2

, B =

uf

Bfu Bff

, Bσ =

Bfσ

, L = Lf

, (22c − f )

4 BASIC EQUATIONS OF THE MACROELEMENT IN A MULTIDO- MAIN APPROACH

We want to obtain the elastoplastic response by subdividing the body into two

∪ ∪

macroelements ΩA, ΩB, with Ω = ΩA ΩB Γˇ0 where interface between the two macroelements A and B.

For the bodies A and B a new notation is necessary:

Γˇ0 is the common regular

Γ

Γ

1

1

ˇA and ˇB represent the

constrained boundaries of A and B, just as ΓˇA and ΓˇB correspond to the free boundaries,

2 2

whereas ΓˇA and ΓˇB represent the common interface thought of as belonging to ΩA and

0 0

ΩB , respectively.

In order to study the elastoplastic problem, we should separately consider each macroelement11,12,14 by employing the formulation of the SGBEM. For each macroele- ment we suppose that the two subdomains ΩA and ΩB (Fig.2) are separately embedded

in the infinite domains ΩA and ΩB , having the same elastic properties as the subdo-

mains ΩA e ΩB, respectiv∞xxx. For s∞implicity the apices A, B are transitory suppressed.

f A

Ω∞

v A

Γ2 A

Γ1 A

ϑ

A

bA

Γ0 A

ΩA Γ2 A

Ω∞

Γ2 B

Γ B

ϑ B

0

Γ B

b

B

2

f B

ΩB Γ1 B

vB

Fig.2 - Embedding of the two macroelements of the solid body in the infinite domain

The object of this section is to obtain the basic equations for each macroelement connecting the boundary and field quantities, known and unknown, i.e. f1, v¯1 = −u¯1

on Γˇ1, ¯f2, v2 = −u2 on Γˇ2 and b¯, e¯ , p in Ω, defined in the macroelement, to the new

interface boundary Γˇ0 variables, i.e. the layered forces f0 = t0 and the layered dis-

−

placement discontinuities v0 = u0. In particular, the vector p collects the permanent strains of the CEs inside the macroelement.

\∞

On the basis of previous considerations, the domain Ω Ω of each macroelement must be unstrained and unstressed, thus the following conditions can be written

u+ = 0 on Γˇ1 , t+ = 0 on Γˇ2 , u+ = 0 and t+ = 0 on Γˇ0 (23a − d)

1 2 0 0

or also

1

1

2

u+ ≡ u1[f1, v2, t0, v0, sp]+ uˆc [v¯1, ¯f2, b¯, e¯ ] − 1 u¯1 = 0 on

Γˇ1

t+ ≡ t2[f1, v2, t0, v0, sp]+ ˆtc [v¯1, ¯f2, b¯, e¯ ] − 1¯f2 = 0 on

Γˇ2

2 2 2

(24a − d)

0

0

2

u+ ≡ uc [f1, v2, t0, v0, sp] − 1 u0 + uˆ0[v¯1, ¯f2, b¯, e¯ ]= 0 on

Γˇ0

0

0

2

t+ ≡ tc [f1, v2, t0, v0, sp] − 1 t0 + ˆt0[v¯1, ¯f2, b¯, e¯ ]= 0 on

Γˇ0

The boundary discretization is made by using the same shape functions (9a,b) for the boundary variables on Γ1 and Γ2 of the macroelement and by introducing new shape functions in order to model both the forces and the displacements of the interface boundary Γ0

t0 = ψf 0 T0 , u0 = ψu0 U0 (25a, b)

Employing the modellings (9), (10a) and (25) permits us to write the following relations for each macroelement

∫

1

≡

u+(x)

1

Guu ψf F1 +

∫ ∫

2 Guf ψu (−U2)+ 0

Guu ψf 0 T0 +

∫

0 Guf ψu0 (−U0)

∫

ˆc ¯

¯ ¯ ¯ 1 ¯ +

+ Ω Guσ Mp + u1[v1, f2, b, e] − 2 u1 = 0 on Γ1

∫

2

≡

t+(x)

1

Gfu ψf F1 +

∫ ∫

2 Gff ψu (−U2)+ 0

Gfu ψf 0 T0 +

∫

0 Gff ψu0 (−U0)

∫

+ ˆc ¯

¯ ¯ ¯ 1¯ +

Ω Gfσ Mp + t2[v1, f2, b, e] − 2 f2 = 0 on Γ2

∫

0

≡

u+(x)

1

∫ ∫

∫

Guu ψf F1 + 2 Guf ψu (−U2)+ 0 Guu ψf 0 T0 + 0 Guf ψu0 (−U0)

+ 1 ˆc ¯

¯ ¯ ¯ +

2 ψu0 (−U0)+ Ω Guσ Mp + u0[v1, f2, b, e]= 0 on Γ0

∫

0

≡

t+(x)

1

Gfu ψf F1 +

∫

2 Gff ψu (−U2)+ 0

Gfu ψf 0 T0 +

∫

0 Gff ψu0 (−U0)

1 ∫ ˆc ¯

¯ ¯ ¯ +

− 2 ψfu0 T0 +

Ω Gfσ Mp + t0[v1, f2, b, e]= 0 on Γ0

(26a − d)

where Ω, Γ1, Γ2 are the domain, the originally constrained and free boundaries of the macroelement, respectively. The stress ơ is defined through the boundary and field actions as follows

∫

ơ(x)= 1 Gσu ψf F1 +

∫

2 Gσf ψu (−U2)+

∫

0 Gσu ψf 0 T0 +

∫

Ω Gσσ Mp

(26e)

+ ơˆc[v¯1, ¯f2, b¯, e¯ ] in Ω

The same conditions may be written in a compact form as follows

1

1

2

1

u+ ≡ u1[F1, V2, T0, V0, p]+ uˆc [v¯1, ¯f2, b¯, e¯ ] − 1 u¯1 = 0 on Γ+

t+ ≡ t2[F1, V2, T0, V0, p]+ ˆtc [v¯1, ¯f2, b¯, e¯ ] − 1¯f2 = 0 on Γ+

2 2 2

2

0

(27a − d)

0

0

2

u

u+ ≡ uc [F1, V2, T0, V0, p]+ 1 ψ

0 (−U0)+ uˆ0[v¯1, ¯f2, b¯, e¯ ]= 0 on Γ+

t+ ≡ tc [F1, V2, T0, V0, p] − 1 ψ

0 T0 + ˆt0[v¯1, ¯f2, b¯, e¯ ]= 0 on Γ0+

0 0 2 f

and also

ơ = ơ[F1, V2, T0, V0, p]+ ơˆ[v¯1, ¯f2, b¯, e¯ ] in Ω (27e)

Here we note that in the equations the apices c in the vectors uc and tc , as well as in

uˆc

0 0

and ˆtc

1 2, show the presence of integrals computed as CPV, to which the free terms

1 ψu0 (−U0) and − 1 ψf 0 (T0) must be added.

2 2

The above equations are the basic expressions of the generic macroelement and may

be employed in structural analysis in order to obtain several approaches13. In each approach the variables are defined on the common interface of the macroelements and so we can have a mixed-value method with the unknowns U0 and T0, a displacement

method with the unknowns U0 only and a traction method with the unknowns T0 only. Let us apply the mixed-value approach as proposed11,12 by the present authors. In

order to obtain a relation connecting the unknown quantities to the loads, a weighing process of the boundary equations (27a-d) must be introduced using the above shape functions; each quantity must be weighed through the shape functions modelling the dual unknowns in an energetic meaning defined on the same boundary. So equation (27e), which defines the stress in Ω, must also be weighed using the shape function M modelling the permanent strains in the CEs. Global conditions of each macroelement can be obtained:

- In eqs.(27a,b) the shape functions ψf , ψu are employed, giving

1

∫ ∫

+ T T c

W1 ≡ 1 ψf u1[F1, V2, T0, V0, p]+ 1 ψf uˆ1 − 2 u¯1 = 0

(28 )

∫ ∫

+ T T

c 1¯

a, b

R2 ≡ 2 ψu t2[F1, V2, T0, V0, p]+ 2 ψu

ˆt2 − 2 f2 = 0

where W+ and R+ define the weighed quantities of the displacements on Γ+ and of

1 + 2 1

the forces on Γ2 , respectively.

- In order to utilize eqs.(27c,d), it is necessary to compute u0 and t0 on Γ0 instead of

u+ and t+ on Γ+, i.e.

0 0 0

0 2 u

u0 = uc [F1, V2, T0, V0, p] − 1 ψ

0 (−U0)+ uˆ0[v¯1, ¯f2, b¯, e¯ ] on Γ0

(29a, b)

t0 = tc [F1, V2, T0, V0, p]+ 1 ψf 0 T0 + ˆt0[v¯1, ¯f2, b¯, e¯ ] on Γ0

0 2

On Γ0 we operate the weighing process using the shape functions ψf 0, ψu0 respec- tively, thus obtaining the following relations

∫

= T c [

] 1 ∫ T

∫

( )+ T ˆ

W0 0 ψf 0 u0 F1, V2, T0, V0, p − 2

0 ψf 0 ψu0

−U0

0 ψf 0 u0

(30 )

∫ T c 1 ∫ T ∫ T ˆ

a, b

R0 =

0 ψu0 t0[F1, V2, T0, V0, p]+ 2

0 ψf 0 ψu0T0 +

0 ψu0 t0

where W0 and R0 define the weighed quantities of the displacements and of the forces on the interface Γ0 of the contiguous subdomains.

- In eqs.(27e) the shape function M is employed, giving

∫ ∫

P = MT ơ[F1, V2, T0, V0, p]+

Ω Ω

MT ơˆ

(31)

as the weighed stress of the CEs, also called internal force at the plastic nodes of the CEs.

The global conditions between the weighed quantities W0, R0 and P, the unknowns U0 and T0, the unknown permanent strains p and the external actions, all quantities defined in each macroelement, may be written summarizing the eqs.(28), (30) and (31) in the following way:

and

0 = BX + B0 X0 + Bσ p + L

0

Z0 = BT X + (B00 + C00) X0 + B0σ p + L0

(32a, b)

σ

P = BT X + Bσ0 X0 + Bσσ p + Lσ (32c)

to which we append the relevant plastic flow laws (14a-d) of all the CEs, included in the macroelement considered.

The following positions have been assumed:

X0 =

T0

, B0 =

Buu B

0

uf0

, B00 =

Bu u B

0

0

u0f0 ,

−U0

Bfu0 Bff0

Bf0u0 Bf0f0

C00 =

0 Cu0f0

Cf0u0 0

, B0σ =

Bu0σ Bf0σ

, L0 =

Lu0 Lf0

(33a − f )

The same positions introduced in eqs.(22d,e) and (21f) are valid for the matrices B, Bσ, Bσσ, whereas we must also define the following positions:

∫ ∫

1

0

Buu0 = ψT

Guu ψf

, Buf

∫ ∫

= ψT

Guf ψu

Bfu

f

∫

= ψT

0

∫

Gfu ψf

0

, Bff

∫1 f ∫0

2

= ψT

0

Gff ψu

2

0

∫

Bu0u0 = 0

u 0∫

ψ

T

f0 0

0

Guu ψf0

0

∫

, Bf0f0 = 0

u 0∫

ψ

T

u0 0

0

Gff ψu0

∫ T ∫ T 1 ∫ T

T (34a − n)

∫

∫

Bu0f0 =

∫

ψf0

0

∫

0 Guf ψu0 = Bf0u0 , Cu0f0 = − 2

0 ψf0 ψu0 = −Cf0u0

Bu σ = ψT

Guσ M, Bf σ = ψT

Gfσ M,

ψ

f0

0

0 ∫ 0

f0 Ω

0 ∫ 0

u0 Ω

Lu0 = 0

T uˆ0, Lf =

ψT ˆt0

u0

0

Let us consider eqs.(32). The matrix B is symmetric and definite, and therefore using eqs.(32a) let us express X as a function of X0, p and of the load vector. One obtains

X = −B−1 (B0 X0 + Bσ p + L) (35)

Substituting the latter equation into eqs.(32b,c) we can write

Z0 = (D00 + C00) X0 + D0σ , p + Lˆ0

P = Dσ0 X0 + Dσσ p + Lˆσ

(36a, b)

which together with the plastic laws (14), here rewritten

∂P

p˙ = Λ˙ ∂Φ ; Φ(P) ≤ 0;

Λ˙ ≥ 0;

Λ˙ Φ(P)= 0 (36c − f )

represent the basic equations of the macroelement with elastic-perfectly plastic material behavior, written in discrete form.

In eqs.(36a,b) the following positions are valid

D00 = B00 − (B0)T (B)−1 B0 , D0σ = B0σ − (B0)T (B)−1 Bσ = (Dσ0)T ,

Dσσ = Bσσ − (Bσ)T (B)−1 Bσ ,

Lˆ0 = L0 − (B0)T (B)−1 L , Lˆσ = Lσ − (Bσ)T (B)−1 L

The matrices

D00

and

Dσσ

are symmetric, whereas

C00

is hemisymmetric.

(37a − e)

5 MIXED-VALUE ELASTIC-PLASTIC ANALYSIS

−

For every macroelement of the discretized model eq.(36a) expresses at time t the weighed boundary quantities as a function of the boundary unknowns T0 and ( U0), of the permanent strains p and of the external actions.

For the body in fig.1 subdivided into two macroelements ΩA and ΩB as in fig.2, the

following relations are valid

ZA = DA

+ CA XA + DA pA + LˆA

0 00

00 0 0σ 0

(38a, b)

ZB = DB + CB XB + DB pB + LˆB

0 00 00 0 0σ 0

00

= C

00

It is worth noting that CA B because the interface boundary is the same (Γ0 ≡

ΓA = ΓB) and because we have employed the same modelling for the boundary variables

0 0

f0 and u0 on ΓA and ΓB .

0 0

If we introduce the partition of the matrices in the latter equations , they can be extended in the following way

WA = DA

TA + DA + C

UA + DA

pA + LˆA

0 00,11 0

00,12

00,12 − 0

0σ,1

0,1

(39a, b)

RA = A + C

TA + A

UA + DA

pA + LˆA

0 D00,21

00,21 0

D00,22 − 0

0σ,2

0,2

WB = DB

TB + DB + C

UB + DB

pB + LˆB

0 00,11 0

00,12

00,12 − 0

0σ,1

0,1

(40a, b)

RB = B + C

TB + B

UB + DB

pB + LˆB

0 D00,21

00,21 0

D00,22 − 0

0σ,2

0,2

The actual elastioplastic solution of the assembled structure as a response to the same permanent strains and to the external actions is obtained by imposing the global compatibility and equilibrium transmission conditions between the adjacent elements, i.e.

WA = WB

0 0 (41a, b)

0

0

RA = −RB

Moreover, the condensation of the variables is obtained by imposing

T0 = TA = −TB

0 0 (42a, b)

U0 = UA = UB

0 0

The solving equation system reads

K0 X0 + K0σ p + Lˆt = 0 (43)

The internal forces (36b) of the CEs, written for both of the macroelements of fig.2,

become

PA = DA TA + DA

−U + DA pA + LˆA

σ0,1 0

σ0,2

0 σσ σ

A

B

0

σ0,2

0

σσ

σ

(44a, b)

σ0,1

PB = DB

TB + DB

−U + DB

pB + LˆB

If we consider the previous positions (42), the latter expressions take on the following compact form

P = Kσ0 X0 + Kσσ p + Lˆσ (45)

D

D

D

The following positions are valid

D

K0 =

A 00,11 A 00,21

B 00,11 B

+ D

− D



00,21

A 00,12 A

D

D

00,22

B 00,12 B

− D

+ D

00,22

, K0σ =

A 0σ,1 A 0σ,2

B

−D

= K

,

0σ,1 T

D

B σ0

0σ,2

DA 0

 LˆA

LˆB

Kσσ =

B , Lˆt = 

Lˆ

σσ,

0,1 −

0,1 

The matrix

0 Dσσ

A

0,2

B

0,2

(46a − d)

+ ˆL

K0 is symmetric and definite. We note the absence of the submatrices

C00,12 and C00,21 which have cancelled each other out.

Eqs.(43), (45) and (36) represent the elastoplastic solving system. Thus if we know at time t the permanent strain vector p = [ pAT pBT ]T and the external loads, the

0

mixed-value vector X0 = [ TT (−U0)T ]T of the discretized interface variables rep-

resents the actual response of the structure. The vector P = [ PAT PBT ]T is the consistent internal force of the CEs in the two macroelements.

Now, let us consider two cases:

1) The elastoplastic response for each i-th basic load can be obtained setting p = 0 in

i

eqs.(43) and (45). Thus we can compute the internal force vector PE of the CEs in

i

0

each macroelement:

PE = Lˆσi − Kσ0 K−1 Lˆti (47)

2) When there is no load, the internal force vector PS can be regarded as the elastic response of the body to a strain vector pS . Thus it can be expressed through eqs.(43) and (45) after setting null the external actions:

K0 XS + K0σ pS = 0

which gives

0

0

PS = Kσ0 XS + Kσσ pS

(48a, b)

PS = K pS (49)

σ0

with K = Kσσ − KT K—0 1 K0σ being the structure stiffness matrix of the macroele-

ments subdivided into CEs.

6 THE SHAKEDOWN LIMIT LOAD

≤

The problem consists in computing the maximum load multiplier s, say ssh, in such a way that shakedown occurs for a load domain sΠ when s ssh. Here Π is a refer- ence load domain. The computing of ssh is obtained on the basis of the classic static shakedown theorem.

If we utilize the discrete approach to the shakedown problem, the load multiplier ssh

is expressed as

ssh = xxx s s.t. the conditions

S S S

s, P , p , X0

E S

(50a − d)

Φ s Pi + P ≤ 0 (i = 1,..., ν)

or more simply as

K0 XS + K0σ pS = 0 PS = Kσ0 XS + Kσσ pS

0

0

ssh = xxx s s.t. the conditions

S S

s, P , p

(51a − c)

i

Φ s PE + PS ≤ 0 (i = 1,..., ν)

PS = K pS

In the light of the discretization operated, it can be stated that ssh, provided by the above problem (50) or (51), is the shakedown load multiplier of the discretized model

obtained using the boundary macroelement approach, but whether ssh is an upper or lower bound to the real shakedown load multiplier cannot be assessed in general. In this respect, it has been remarked1,2 that on one hand eqs.(50b,c) imply the existence of stresses ơS related to PS , but on the other hand eq.(50a) does not guarantee that the

yield conditions are everywhere satisfied. Thus, the yield conditions required by Xxxxx’s theorem are not fully satisfied and ssh can only be expected to be an approximation to the real shakedown load multiplier.

The shakedown formulation presented is formally equivalent to the analogous one of the shakedown problem within FE context; it turns out to be a problem of mathematical programming and in particular of linear programming whenever the plastic yield surface collected in the vector Φ are piecewise linear.

The plastic collapse limit analysis problem can be approximated by using the previous discretization provided that ν = 1 in the shakedown formulation. So we obtain

sl = xxx s s.t. the conditions

S S

s, P , p

(52a − c)

Φ s PE + PS ≤ 0

PS = K pS

CONCLUSIONS

The discrete version of the shakedown load multiplier problem is addressed using the statical approach of the classical shakedown theory, but treated in the language of the SGBEM.

A first subdivision into MEs is performed in order to deal with bodies made of by piecewise homogeneous materials. For each ME an appropriate boundary discretization by BEs with the introduction of the boundary unknowns and a field discretization by CEs with the consequent approximation of the material plastic yield laws appropriately interpolated are performed. The constitutive model of the CEs is enforced in a weak sense and is expressed in terms of generalized quantities.

An important aspect of the SGBEM analysis using the macroelement strategy con- cerns the nonlinear process when it regards a limited zone at the body; in this case a condensation of variables reduces the unknowns regarding the ME, a locus of nonlinear phenomena, i.e. the boundary quantities and the generalized plastic strains of the CEs, giving a very simple analysis problem.

REFERENCES

[1] X. Xxxxx and X. Xxxxxxxxxx, On shakedown analysis by boundary elements, C.A. Massonet anniversary Volume Verba volant, scripta manent, Liege, 265-277 (1984).

[2] X. Xxxxxxxxxx, An energy approach to the boundary element method. Part II: Elasto- plastic solids, Comp. Meth. Appl. Mech. Engng., 69, 263-276 (1988).

[3] X. Xxxxxxx, Shakedown and limit analysis by the boundary integral equation method,

Xxx. X. Xxxx., A/ Solids, 5, 685-699 (1992).

[4] X. Xxxxxx, How to deal with hypersingular integrals in the symmetric BEM, Comm. Num. Meth. Engng., 9, 219-232 (1993).

[5] X. Xxxxxxx, X. Xxxxxxx, X. Xxxxxx, Symmetric coupling of finite elements and boundary elements. In: Xxxx, X.X., Xxxxx, G., Xxxxxx, N., Xxxxxx, X.X. (eds.): Advances in Boundary Element Techniques, 407-427, New York, Springer-Verlag, (1993) .

[6] X. Xxxxxx, Regularized direct and indirect symmetric variational BIE formulations for threedimensional elasticity, Engng. An. with Boundary Elements, 15, 93-102 (1995).

[7] X. Xxxxxx, X. Xxxxxx, Symmetric BE method in two–dimensional elasticity: evalua- tion of double integrals for curved elements, Comp. Mech., 19, 58-68 (1996).

[8] X. Xxxxxx, X. Xxxxx, X. Xxxxxxxxxx, Symmetric Galerkin boundary element method,

Appl. Mech. Rev., 51, 669-704 (1998).

[9] X.X. Xxxx, Evaluation of singular and hypersingular Galerkin integrals: direct limits and symbolic computation. In: Xxxxxx X. and Xxxxxx, X. (eds.): Singular Integrals in Boundary Element Method, 33-84, Southampton: Computation Mechanics Pub- blications, (1998) .

[10] X. Xxxxxxx, X. Xxxxxx Xxxxxxx, X. Xxxxxxx, Mathematical aspects and applications of the symmetric Galerkin Boundary Element Method. In: Xxxxxxxx, X., Xxxxx, X., Xxxxxxx, E.N. (eds.) - Computational Mechanics: New Trends and Applications, IV WCCM Symposium, Buenos Aires, (1998) .

[11] X. Xxxxxxx, X. Xxxxxxx,X. Xxxxxxxxxxxx, Impiego delle sottostrutture nella formu- lazione simmetrica del BEM, in CD-ROM XIV Congresso AIMETA, Facolt di Ingeg- neria, Como, Italy, (1999).

[12] X. Xxxxxxx, X. Xxxxxxx, Macroelements in the mixed boundary value problems, sub- mitted to Comp. Mech., (1999).

[13] X. Xxxxxxx, X. Xxxxxxx,X. Xxxxxxxxxxxx, Displacement and traction formulations in the symmetric BEM, submitted to Comp. Meth. Appl. Mech., (1999).

[14] X. Xxxxxx, X. Xxxxx, X. Xxxxx, Symmetric boundary element method for the analy- sis and homogenization of periodic composites, Preliminary Report n.988/MI/A6-1, Politecnico di Milano, Dip. Ing. Stru., (1999).

APPENDIX.

Somigliana Identities and fundamental solutions.

∞

In the infinite domain Ω with assigned phisical characteristics, the well-known Somigliana Identities (SI) represent the response (displacement, stress, traction acting on the surface element Γ having a slope defined by a unit normal n) in x due to external actions distributed on the boundary and in the same domain. One has

u(x)= u[f , b, v, e]=

∫

1 Guu f +

∫

2 Guf v +

∫

Ω Guu b +

∫

Ω Guσ e

t(x)= t[f , b, v, e]=

∫

1

Gfu f +

∫

∫

2

Gff v +

∫

∫

Ω

Gfu b +

∫

∫

Ω

Gfσ e

∫

(A1a − c)

ơ(x)= ơ[f , b, v, e]= These actions are

1 Gσu f +

2 Gσf v +

Ω Gσu b +

Ω Gσσ e

- the force layer f = f (xj) distributed along Γ1, making up the traction jump (f =

1

—

t+ + t) between the surface Γ1 and Γ+;

- the displacement discontinuity layer v = v(xj) distributed along Γ2, making up the

2

displacement jump (v = u+ — u) between the surface Γ2 and Γ+ with an external

unit vector nj normal to the surface Γ2;

- the body forces b = b(xj) and the thermal-like strains e = e(xj) in Ω.

Among the effects computed in x, the traction t(x) is defined on a surface which slope is characterized by an external unit vector n.

∈ ∞

The fundamental solutions Ghk(x, xj), present in the Somigliana identities, describe effects evaluated in x Ω (i.e. displacement for h = u, traction for h = f , stress for

∈ ∞

h = σ), caused by a unit singularity applied at xj Ω (i.e. a unit concentrated force for k = u, a unit concentrate relative displacement for k = f , a unit concentrated strain

for k = σ). As a result of the reciprocity theorems (Xxxxxxx, Colonnetti, Volterra), the symmetries hold:

kh

Ghk(x, xj)= GT (xj, x) (h, k = u, f, σ). (A2)

For bidimensional problems these matrices show singularities of different order when

x → xj, namely 0 (log r) for Guu, 0 (1 ) for Guf and Guσ, 0 ( 1 ) for Gff and Gσσ. All

the matrices G

r G . r2

hk can be expressed in terms of uu