Bem Project Must Use

7/31/2019 Bem Project Must Use

1/22

A boundary element methodology for viscoelastic analysis:Part I with cells

A.D. Mesquita, H.B. Coda *

Department of Structural Engineering, Sao Carlos School of Engineering, Sao Paulo University,

Av. Trabalhador Sao Carlense 400, 13566-960 Sao Paulo, Sao Carlos, Brazil

Received 1 March 2003; received in revised form 1 December 2005; accepted 12 April 2006Available online 7 July 2006

Abstract

In this study Kelvin and Boltzmann viscoelastic models are implemented in a two-dimensional boundary element atmo-sphere. This general methodology is based on differential constitutive relations for viscoelasticity, avoiding the use of relax-ation functions. Part I describes a methodology using internal cells. This methodology makes it possible to consider viscousparameters, which are not proportional to elastic tensor. From the kinematical relation between material and strain velo-cities at the approximation level a simple time marching process is achieved. At the end of Part I, numerical examplesare provided to validate the methodology. The BEM viscoelastic formulation without using cells is carefully describedin Part II.

2006 Elsevier Inc. All rights reserved.

1. Introduction

One can find in the literature a great amount of computational techniques to solve various similar mechan-ical problems. It is not different when viscoelasticity is the subject of study. A brief description of some well-known procedures is given below.

The first classic procedure for solving viscoelastic problems is the use of mathematical transformations. Theviscoelastic equation can be transformed into pseudo-elastic one by means of LaplaceCarson transforms.After solving the transformed problem, a numerical inversion can be performed recovering the desired time

domain behaviour [14]. This procedure presents some difficulties when viscous parameters vary along time,or when complicated time dependent boundary conditions are imposed.

Incremental schemes are also available in the literature. This kind of procedure uses relaxation functions totransform the convolutional aspect of the viscous behaviour in discrete contributions added to the elasticresponse [59]. In some studies viscoelastic problems are solved as viscoplastic ones, i.e., introducing the

0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2006.04.006

* Corresponding author. Tel.: +55 16 33739482; fax: +55 16 2739482.E-mail addresses: [email protected] (A.D. Mesquita), [email protected] (H.B. Coda).

Applied Mathematical Modelling 31 (2007) 11491170

www.elsevier.com/locate/apm
mailto:[email protected]:[email protected]:[email protected]:[email protected]


2/22

viscous behaviour by means of residuals calculated at the effective stress level [1013]. The complete informa-tion about creep (and relaxation) approaches can be seen in Refs. [14,15].

Recently, the authors developed an alternative FEM methodology to analyse bodies following the Kelvinand Boltzmann models [16,18]. In this study this new methodology is extended to BEM formulation. Thedevelopment of such subject follows two different ways. The first, described in Part I, follows exactly the same

idea of the FEM approach (respecting the particularities of BEM), i.e., the kinematical relation between thematerial velocity and the strains velocity at the approximation level is used resulting in BEM cells. In Part II amore elaborated procedure is described avoiding the use of cells for both Kelvin and Boltzmann models. Asimple coupling between FEM and BEM is also described in Part II to make it possible to analyse reinforcedmedia, as for example tunnels and sandwich panels.

The division of the subject into two parts is necessary as the first shows the basis of the methodology andgives a background to the proposed strategy, while the second, avoiding the use of cells, makes it possible toanalyse infinite and half-space regions.

It is important to observe that both formulations are able to consider the material with time varying viscousproperties, the so-called ageing viscoelasticity. At the end of each part examples are provided in order to val-idate the technique and check its accuracy and stability.

2. Definition of the problem

Before describing the numerical procedure to analyse the mechanical behaviour of solids presenting visco-elastic characteristics it is necessary to define the related differential equilibrium equation and boundary con-ditions. A general solid X subjected to prescribed surface forces p and displacement restrictions u can be seenin Fig. 1.

From Fig. 1 one observes that the boundary of the solid is divided into two parts C1 and C2. Over C1 and C2one prescribes surface forces and displacement restrictions, respectively. This can be written in a more formalway defining the boundary conditions of the body as follows:

uix; t uix; t over C1 1

pix; t pix; t over C2 2

where x is a point over the boundary and t is time. The desired solutions of the problem are the displacementfield u(x, t) "x 2 X and the stress field r(x, t) "x 2 X.

The usual way to write the differential equation that governs the solid is the study of the equilibrium of aninfinitesimal part of it, see Fig. 2.

From Fig. 2, following Einstein notation, one writes:

rij;i bj quj c _uj 3

1

2

1

2

n

Fig. 1. Problem description.

1150 A.D. Mesquita, H.B. Coda / Applied Mathematical Modelling 31 (2007) 11491170


3/22

where _ui is the velocity, ui is the acceleration, c is the friction parameter (see Section 2), q is the mass densityand bi is the body force. One should use the Cauchy formulae to relate directly stress to surface force, as,

pi rijnj 4

where nj is the surface outward unit vector.To complete the description of the problem one applies the kinematical hypothesis that relates displacement

and velocity to strain and strain rate. After that one applies the constitutive equation that relates strain andstrain rate to stress, transforming Eq. (3) as a function of displacement and its derivatives only. These relationsare shown in the following sections, but the numerical strategy to solve the physical problem is based on theequilibrium equation written for stress, i.e., Eq. (3).

3. Difference among viscosity, friction and inertia

In order to develop a methodology to solve viscoelastic problems based on differential constitutive relationsfor viscoelasticity (carrying out the time integration after introducing the space approximations), one has tounderstand the equilibrium relations for an infinitesimal portion of the studied body, briefly introduced in theachievement of the differential equation in the previous section. In Fig. 3ac dashed arrows represent therequired forces to support the effects indicated by solid arrows. Therefore, taking into account Fig. 3a, onecan realise that the force qui dV is required to impose acceleration ui to an infinitesimal part of the bodydV with mass density q. If the internal friction c is assumed independent of elastic relations, it is necessaryto impose a force c _ui dV to support a movement with constant velocity _ui (see Fig. 3).

Assuming the viscous effect g also independent of the elastic relations, opposite forces equal to g_ei dA (sim-

plified notation) are necessary to support the relative velocity between neighbouring points, Fig. 3c. Inertialand friction forces are volume-type forces, while elastic and viscoelastic forces are stress-type ones.

dx2

1xd

12 +12

x1

dx1

11+ d

1x

11

1x

21+ d

2x

21

2x

22 d+ 2x

22

2

x

11

12

22

21

b2

b1

c 1

1

2

1c

u

u

u

u

Fig. 2. Equilibrium of an infinitesimal part of the solid.

uudv

vddv

uvducu

u+u

xdx

dAAd

du

xA du

xA

(a) Acceleration (b) Friction (c) Viscosity

Fig. 3. One-dimensional representation of a moving point.

A.D. Mesquita, H.B. Coda / Applied Mathematical Modelling 31 (2007) 11491170 1151


4/22

Considering continuity properties, the viscous stresses appearing along the right side of an infinitesimal ele-ment should be generated by the neighbouring infinitesimal element placed on its right side. Along the bound-ary this compatibility is naturally assured by boundary conditions, Eqs. (1) and (2) and Cauchy formulae (4).

At this point, viscous effects should be incorporated into the global equilibrium equation of the body takinginto account strain velocity. Moreover, the viscous characteristics of the body must satisfy boundary condi-

tions together with the elastic ones. In order to fulfill these requirements the following sections describe theintroduction of viscous stress into Eq. (3).

4. Basic relations for viscoelasticity

This section is divided into two main parts, one related to the Kelvin model and the other related to theBoltzmann standard relations.

4.1. Kelvin model

Using rheological models defined in the uniaxial space is the usual way adopted to describe the viscoelasticbehaviour of solids. A simple representation, usually adopted to describe this kind of behaviour is the Kelvin

Voigt viscoelastic model, Fig. 4.From this model the following relations are stated:

eij eeij e

vij 5

rij reij r

vij 6

where eij and rij are the strain and stress tensors. Superscripts v and e represent viscous and elastic parts,respectively.

The elastic stress can be written in terms of strain components, as follows:

reij Cijlmeelm Cijlmelm 7

Similarly, the following relation gives the viscous stress components, see Fig. 3c,

rvij gijlm _evlm gijlm _elm 8

In Eqs. (7) and (8), Cijlm and gijlm are the elastic tensor and the viscous constitutive parameters, respectively,defined as follows [17]:

Cijlm kdijdlm ldildjm dimdjl 9

gijlm hkkdijdlm hlldildjm dimdjl 10

where k and l are the Lames constants, given by:

k mE

1 m1 2m11

l GE

21 m 12

E

Fig. 4. KelvinVoigt viscoelastic model (uniaxial representation).



5/22

in which Eand m are the Young modulus and Poisson ratio, respectively, while hk and hl are the viscosity coef-ficients. It is worth noting that these coefficients can vary along time, as described in Appendix A, and that theformulation presented here is valid for any time variation. As hk is not necessarily equal to hl, the viscousparameter tensor is not proportional to the elastic compliance tensor.

Replacing Eqs. (7) and (8) into Eq. (6) gives

rij Cijlmelm gijlm_elm 13

This is the general Kelvin constitutive relation used to generate the BEM formulation proposed here.Substituting Eq. (6) into Eq. (3) and neglecting the dynamic terms, one has

reij;i rvij;i bj 0 14

or simply

rij;i bj 0 15

Note that Eq. (14) explicitly exhibits the viscous stress term which plays an important role in the studiedproblem.

4.2. Boltzmann model

Another representation employed to describe the mechanical behaviour of viscoelastic materials, stress/strain constitutive relation, is the so-called standard Boltzmann model. This model is more general thanthe previous one and can be described in a uniaxial representation as illustrated in Fig. 5.

This model is represented by a serial arrangement of a KelvinVoigt model and an elastic relation. It canreproduce both instantaneous and viscous behaviours of a specific material.

It is easy to observe (see Fig. 5) that the stress level for each part of the model, elastic and viscoelastic, is thesame:

rij reij r

veij 16

where rij; reij and rveij are, respectively, total, elastic and viscoelastic stress parts. The total strain can be

decomposed into its elastic and viscoelastic parts, i.e.:elm e

elm e

velm 17

From Fig. 5, one can observe that the viscoelastic stress is the summation of a viscous and an elastic part, asfollows:

rveij relij r

vij 18

where rvij is the viscous part and relij is the elastic part of the stress developed in the KelvinVoigt fragment of

the Boltzmann model.From the previous equations, one is able to define the differential viscoelastic constitutive relation used here

to build the desired boundary integral equations.

rqs EveEeEve Ee

~Cqsckeck EeEveEve Ee ~gqsck_eck EveEve Ee ~gqsij ~Dijck _rck 19

Ee

Eve

e ve

Fig. 5. Boltzmann viscoelastic model (uniaxial representation).

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


6/22

where ~Cijlm; ~gqsck and ~Dijck are dimensionless elastic tensor, dimensionless viscoelastic compliance tensor andthe inverse of the dimensionless elastic tensor, respectively (see Appendix B).

In order to transform the stress rate into surface force rate it is necessary to impose the simplificationhk = hl = c only on boundary transformations. Therefore, expression (19) turns into

rij

EveEe

Eve Ee ~

C

lm

ij elm

cEveEe

Eve Ee ~

C

lm

ij_

elm

cEve

Eve Ee _

rij 20

This is the desired rheological differential relation for the Boltzmann model. More complicated rheologicalmodels can be introduced following similar procedures, as the one employed in Appendix B, to achieve Eq.(19). In Part I of the paper, for hk5 hl, Eq. (19) is used changing only the last term to be equal to the lastterm of Eq. (20), see Example 2.

Eq. (18) can be introduced into Eq. (3) resulting in an equilibrium equation similar to Eq. (15).

5. Integral equations and the BEM

As it is well known, the boundary element method is based on boundary integral equations. In this sectionthe divergence theorem is applied once to generate the starting displacement integral equation for both finiteand boundary elements [18,16]. After that the divergence theorem is applied again in order to obtain theboundary integral equation considering the Kelvin and Boltzmann models, respectively.

The viscoelastic integral equation for boundary or internal points is obtained here using the weightingresidual technique in the differential equilibrium equation (15) written in the following form:

raij;j bi raij;j rij;j errori 21

The error present in Eq. (21), when an approximate raij;j solution is adopted, can be weighted by a properfunction. In this paper, the Kelvin fundamental solution for elastic infinite body is adopted. Eq. (21) isweighted over the analysed domain X, as follows:

ZX

ukirij;j bi dX 0 22

where uki is the Kelvin fundamental solution. It represents the effect of a unit concentrated load applied to apoint located inside an infinite domain. From Eq. (22) superscript a is omitted for simplicity. Applying thedivergence theorem to the first term of Eq. (22), one achieves:Z

C

ukirijnj dC

ZX

uki;jrij dX

ZX

ukibi dX 0 23

As previously mentioned, C is the boundary of the analysed body and nj is its outward normal vector. Know-ing that rijnj= pi, Eq. (4), and that u

ki;jrij e

kijrij, where e

kij is the strain fundamental term, Eq. (23) turns

into:

ZC u

kipi dC Z

X e

kijrij dX Z

X u

kibi dX 0 24

This equation is the starting point to obtain the viscoelastic integral representations for BEM and FEM con-sidering both Kelvin and Boltzmann viscoelastic relations. In this part of the development the differential vis-cous relations, Eqs. (13) and (20), are used in the second term of Eq. (24) in order to derive a general BEMformulation for viscoelastic analysis.

5.1. Displacement integral equation-Kelvin model

The boundary integral equation is developed replacing the total stress present in Eq. (24) by relation (6).

ZC

piuki dC Z

X

biuki dX Z

X

reijekij dX Z

X

rvijekij dX 25



7/22

Replacing reij and rvij, given by Eqs. (7) and (8), in the last two terms of Eq. (25), gives:Z

C

piuki dC

ZX

biuki dX

ZX

elmClmij e

kij dX

ZX

_elmglmij e

kij dX 26

At this point it is necessary to remember the kinematical relation for small strain, as follows:

eij 12

ui;j uj;i 27

This relation can be derived with respect to time, resulting in the kinematical relation for strain velocity,

_eij 1

2 _ui;j _uj;i 28

Taking into account the stress and strain tensors symmetry, the kinematical relations and the symmetry ofCijlm and gijlm, one writes

ekijCijlmelm rklmelm r

klmul;m r

kijui;j 29

ekijgijlm _elm ekijgijlm _elm e

kijgijlm _ul;m e

klmglmij _ui;j 30

By introducing these values into Eq. (26) and applying again the divergence theorem, the following integralequation is found [16,17].Z

C

piuki dC

ZX

biuki dX

ZC

pkiui dC

ZX

eklmglmij _ui;j dX

ZX

rkij;jui dX 31

As usual in BEM formulations, due to the fundamental solution properties, the last integral of Eq. (31)gives an independent term, leading to the following integral representation:

Ckiui

ZC

pkiui dC

ZX

eklmglmij _ui;j dX

ZC

piuki dC

ZX

biuki dX 32

where the free term Cki is equal to dki (Kronecker delta) when the unit load is applied at internal points anddepends upon the geometry of the body if the collocation is chosen along the boundary [17,19].

The boundary of the analysed body, C, is discretized using ne boundary elements, Ce. The domain isapproached by (nc) internal cells (Xc), as shown in Fig. 6.

The variables are approximated by the following expressions:

uin; t /anUai t 33

pin; t /anPai t 34

bin; g; t /bn; gbbi t 35

glmij _ui;jn; g; t glmij/b;jn; g _Ubi t 36

where / and / are shape functions adopted for boundary and domain values, respectively andUai ; P

ai ; b

bi and _U

bi are the nodal (b) displacement, surface force, body force and velocity values, respectively.

CellBoundary

Element

Fig. 6. Problem discretization.



8/22

Replacing these approximations into Eq. (32) results in

Ckiui

ZCe

pki/ean dCU

ai t

ZX

eklmgijlm/cb;jn; g dX _U

bi t

ZXc

eklmglmij/cb;jn; g dX _U

bi t

ZCe uki/

ean dCP

ai t ZXc u

ki/cbn dXb

bi t 37

where superscripts e and c represent boundary elements and internal cells, respectively.Performing all spatial integrals and adopting the number of collocation points equal to the number of

nodes, the following system of time differential equations is found:

HUt V _Ut GPt Bbt 38

where the new matrix Vgives the viscosity contribution to the global movement; the other matrices and nodalvalues are the same that appear in the standard static BEM approach [17,19].

The time differential equation (38) can be integrated in time following a simple linear time approximation.

_Us1 Us1 Us=Dt 39

where s + 1 represents the present instant and Dt is the adopted time step. From this approximation one writes

the time marching process:H Us1 GPs1 Fs 40

where

H H V

Dt

41

and

Fs Bbs1 V

DtUs 42

5.2. Stress integral equation Kelvin model

As previously mentioned, to completely solve the problem it is necessary to calculate the stress field for anypoint of the analysed body. One can write the strain field representation by replacing the displacement integralrepresentation, Eq. (32), into the kinematical relation (27), as follows:

ekes; t

ZC

ekiepi dC

ZC

^pkieui dC

ZX

ekiebi dX

ZX

ekijegijlm _elm dX ^gkijegijlm _elms; t 43

where the derivatives of the kinematical relation were performed with respect to the source point (s)co-ordinates.

The last two terms present in Eq. (43) come from the derivative of the last integral of the left side of Eq.(32). Following standard textbooks on BEM [17,19], this derivative results in an integral in the sense of

Cauchy and a free term ^gkije written together with the new fundamental values as follows:

^pkie 1

4p1 mr22m dkir;e deir;k 2dekr;i 8r;kr;ir;ef g

or

on 1 2m dkine dienk dkeni 2r;kr;eni

2mr;ir;enk r;ir;kne

!44

ekije 1

8p1 mGr22m dkir;jr;e dkjr;ir;e deir;jr;k dejr;ir;k

1 2m dkidje dkjdie

2 dker;ir;j dijr;er;k 4r;kr;ir;jr;e

dijdke

45

^gkije 1

81 tG3 4tdikdje

1

2

dkedij& ' 46



9/22

The mentioned singular term of Eq. (43) is ekijeglmij _elm. From the integral representation of strains, one can

easily derive the elastic stress integral equation by applying the elastic constitutive relation (7), as follows:

rekls

ZC

rkilpi dC

ZC

pkilui dC dX

ZX

ekijlgijlm _elm dX gkijlgijlm _elmp

ZX

rkilbi dX 47

where the new fundamental values are given by

pkil 2G

4p1 mr22or

on1 2m dklr;i m dkir;l dqir;q

4r;kr;ir;l

2m nkr;lr;i nlr;kr;i 1 2m nir;kr;l nldki nkdli 1 4m nidkl

!48

ekijl 1

4p1 mr21 2m dkidlj dkidlj dkldji 2dklr;jr;i

2m dkjr;lr;i dlir;kr;j dkir;lr;j dljr;kr;i

2djir;kr;l 8r;kr;ir;jr;l

49

gkijl 1

81 m

8 6mdkidlj 1 4mdkldij 50

Adding rv to both sides of Eq. (47) and taking into account Eqs. (8) and (6), the final expression of the totalstress is

rkls

ZC

rkilpi dC

ZC

pkilui dC dX

ZX

ekijlgijlm _elm dX gkijlgijlm _elmp

ZX

rkilbi dX 51

where the new free term is given by

gkijl 1

81 m2dkidlj 1 4m dkldij 52

As the total and elastic stresses are given by Eqs. (51) and (47), respectively, Eq. (6) must be adopted to

evaluate the viscous stress.The same approximations, already defined to approximate the displacement equation, are adopted to

approach strain and stress integral equations. Thus, for any internal collocation point, strain, elastic stressesand total stresses are expressed, respectively, by

ekls; t

ZCe

ekil/a dCePai t

ZCe

^pkil/a dCeUai t

ZXc

ekijlgijlm/;cbm dXc _U

bl t

Xnpc1

^gkije

npgijlm

/b;ms _Ubl

ZXe

ekie/b dXcbbi 53

rekls; t ZCe rkil/a dCeP

ai t ZCe p

kil/a dCeU

ai t ZXc e

kijlgijlm

/;bm dXc _Ubl t

Xnpc1

gkije

npgijlm

/b;ms _Ubl

ZXe

rkie/b dXcbbi 54

rkls; t

ZCe

rkil/a dCePai t

ZCe

pkil/a dCeUai t

ZXc

ekijlgijlm/;bm dXc _U

bl t

Xnpc1

gkije

npgijlm

/b;ms _Ubl

ZXe

rkie/b dXcbbi 55

where np is the number of cells connected to the collocation point s. It means that, for this case, the free term iscalculated here as the average of the terms calculated for each common cell. After performing the integralsover all boundary elements and internal cells and writing Eqs. (53)(55) for the same number of source points

as the number of discretization nodes [17,19], one has

http://-/?-http://-/?-


10/22

es1 G0Ps1 H

0Us1 B0bs1 V

0 _Us1 56

rs1 G0Ps1 H

0Us1 B0bs1 V

0 _Us1 57

res1 G0Ps1 H

0Us1 B0bs1 V

0 _Us1 58

where subscript s + 1 means that the analysed instant is ts+1 = (s + 1)Dt. For simplicity, the presented BEM

formulation was derived for 2D plane strain problems. In plane stress situations one has to replace the Poissonratio m by m/(1 + m).

5.3. Displacement integral equation Boltzmann model

Eq. (24) is also the starting point for Boltzmann BEM viscoelastic formulation. Introducing Eq. (19) or (20)into expression (24) one writes:Z

C

ukipi dC EeEve

Ee Eve

ZX

ekij~Cijlmelm dX

cEeEve

Ee Eve

ZX

ekij~Cijlm _elm dX

cEve

Ee Eve

ZX

ekij _rij dX

ZX

ukibi dX 0 59

Considering that ekijEeCijlmelm rklmelm r

klmul;m r

kijui;j and e

kij _rij u

ki;j _rij, Eq. (59) is rewritten asZ

C

ukipi dC Eve

Ee Eve

ZX

rkijui;j dX Ee

Ee Eve

ZX

ekijgijlm _elm dX cEve

Ee Eve

ZX

uki;j _rij dX

ZX

ukibi dX 0 60

Applying the divergence theorem to the second and fourth integrals of Eq. (60), results in

ZC ukipi dC

Eve

Ee Eve ZC rkijnjui dC ZX r

kij;jui dX !

Ee

Ee Eve ZX ekijgijlm _elm dX

cEve

Ee Eve

ZC

uki _rijnj dC

ZX

uki _rij;j dX

!

ZX

ukibi dX 0 61

The third and the sixth integrals can be simplified by considering the differential equilibrium equations ofthe fundamental and real problems, i.e.:

rkij;j dp;sdki 62

_rij;j _bi 63

where _bi is the body force derived with respect to time and d(p, s) is the Diracs delta distribution. In Eq. (62) sand p are, respectively, source and field points.

Replacing Eqs. (62) and (63) into expression (61) and taking into account the Cauchy formula in its differ-ential form _rijnj _pj results,

Ckiuis Ee Eve

Eve

ZC

ukipi dC

ZC

pkiui dC Ee

Eve

ZX

ekijgijlm _elm dX c

ZC

uki _pi dC

ZX

uki_bi dX

!

Ee Eve

Eve

ZX

ukibi dX 64

The free term Cki is the same that appears in the Kelvin viscoelastic model formulation. Eq. (64) is the inte-gral representation for displacements based on Boltzmanns standard viscoelastic model. The first, second andsixth integrals are the same ones present in elastostatic problems [17,19].

In general the body force does not vary in time and is usually present before the analysis, therefore one may

consider_bi 0, simplifying Eq. (64). The fourth integral presents no difficulties, as the kernel singularity is the



11/22

same as the first one. Finally, the third integral is similar to the residual stress one present in non-linear anal-ysis and presents only a weak singularity. The time marching process for the Boltzmann viscoelastic formu-lation is described in Section 5.5.

5.4. Stress integral equation Boltzmann model

In order to achieve the desired integral representation, Eq. (64) is rewritten for internal points, as follows:

uks Ee Eve

Eve

ZC

ukipi dC

ZC

pkiui dC Ee

Eve

ZX

ekijgijlm _elm dX c

ZC

uki _pi dC

ZX

uki_bi dX

!

Ee Eve

Eve

ZX

ukibi dX 65

For small strain situation, the integral strain equation can be achieved applying the following kinematicrelation:

ekls 1

2

ouk

oxls

oul

oxks 66The differentiation expressed in Eq. (66) is defined with respect to the source point position. Applying Eq. (66)to Eq. (65) and applying the Leibnitz formula to the third integral [17,19] one has

ekls Ee Eve

Eve

ZC

ekilpi dC

ZC

^pkilui dC Ee

Eve

ZX

ekijlgijlm _elm dX ^gkijlgijlm _elms

!

c

ZC

ekil _pi dC

ZX

ekil_bi dX

!

Ee EveEve

ZX

ekilbi dX 67

The differentiation is performed with respect to the source point, therefore only the fundamental valueschange. Terms ^pkil; e

kijl and ^g

kijl are the same previously presented for the Kelvin viscoelastic model, i.e.,

Eqs. (44)(46).

Rewriting the general viscoelastic relation, Eq. (20), as follows:

EveEe

Eve Ee

Cpqklekl rpq

cEe

Eve Ee

~Cpqkl _ekl

cEve

Eve Ee

_rpq 68

and multiplying Eq. (67) by EveEeEveEe

Cpqkl one achieves,

rqqs

ZC

rqiqpi dC Eve

Ee Eve

ZC

pqiqui dC Ee

Ee Eve

ZX

eqijqgijlm _elm dX ~gqijqgijlm _elms

!

cEve

Ee Eve

ZC

rqiq _pi dC

ZX

rqiq_bi dX

!

ZX

rqiqbi dX cEe

Ee EveCqqlm _elms

cEve

Ee Eve_rqqs

69

The terms present in the first bracket of Eq. (69) are an integral following the Cauchy principal sense and gqijqis a free term. Knowing that Cqqlm _e

velm dqidqjCijlm _e

velm, it is possible to add the sixth term, on the right-hand side

of Eq. (69), to the free term ~gqijq, resulting in

rqqs

ZC

rqiqpi dC Eve

Ee Eve

ZC

pqiqui dC Ee

Ee Eve

ZX

eqijqgijlm _elm dX gqijqgijlm _elms

!

cEve

Ee Eve

ZC

rqiq _pi dC

ZX

rqiq_bi dX

!

ZX

rqiqbi dX cEve

Ee Eve_rqqs 70

This is the total stress integral representation for the Boltzmann viscoelastic model. As previously mentioned,the domain integral of strain rate should be performed following the Cauchy Principal Value sense. The new

fundamental values are the same given for the Kelvin viscoelastic model in Eqs. (48)(50).



12/22

The evaluation of the elastic and viscous parts of the stress is performed at the approximation level, but inorder to prepare the necessary equation one can write expression (B1) in a time derivative form, i.e.:

_relij Cijlm _evelm Eve ~Cijlm _e

velm 71

Taking into account Eq. (B7), one transforms Eq. (71) as

rvij gijlmCablm1 _relab cijab _relab or for hl hk cEveCijlm _evelm rvij c _relij 72

From Eqs. (71), (72) and (18) one achieves

gijlmCablm1

_relab relij rij 0 73

This equation is the basis of relaxation functions so widely used in the literature, which are achieved fromthe supposition that the value of rij is constant in a time step. In this paper, we propose a time approximationfor rij and r

elij that allows a more general and direct analysis of the global viscoelastic problem. This approx-

imation is described in details in the numerical time integration section.For each boundary element and internal cell the variables are approached by parametric interpolations, as

follows:

pi /aPai 74ui /aU

ai 75

bi /abai 76

_pi /a _Pai 77

_bi /a _bai 78

gijlm _elm gijlm _ul;m gijlm /a0m _Ual 79

where / and / are, respectively, the adopted shape functions for boundary element and internal cells. Indices aand a are related to the boundary element and internal cell, respectively. Terms Pai ; U

ai ; b

ai ; _P

ai ; _U

ai and

_bai arenodal values of surface forces, displacements, body forces and its time rates, respectively.

Substituting these approximations into integral Eqs. (64) and (70) results in

CkiUis Ee Eve

Eve

Xnee1

ZCe

uki/a dCeP

ai

Xnee1

ZCe

pki/a dCeU

ai

Ee

Eve

Xncc1

ZXc

ekijgijlm/;am dXc _U

al

cXnee1

ZCe

uki/a dCe _P

ai c

Xncc1

ZXc

uki/a dXc _b

ai

Ee EveEve

Xncc1

ZXc

uki/a dXcb

ai 80

rqqs Xnee1

ZCe

rqiq/a dCeP

ai

Eve

Ee Eve

Xnee1

ZCe

pqiq/a dCeU

ai

Ee

Ee Eve

Xncc1

ZXc

eqijqgijlm/;am dXc _U

al

Ee

Ee Eve Xnp

c1

gqijq

npgijlm

/;ams _Ual

cEve

Ee Eve Xne

e1ZCe r

qiq/

a dCe _Pai

cEve

Ee Eve

Xncc1

ZXc

rqiq/a dXc _b

ai

Xncc1

ZXc

rqiq/a dXcb

ai

cEve

Ee Eve_rqqs 81

where np is the number of cells that are connected to the source point s. It means that the free term is calculatedhere as the average of the ones calculated for each common cell. After adopting the same number of sourcepoints as the number of discretization nodes and performing the spatial integration [17,19], one has

HUt Ee Eve

EveGPt

Ee

EveV _Ut cG_Pt cB _bt

Ee EveEve

Bbt 82

rt G0Pt Eve

Ee EveH0Ut

Ee

Ee EveV0 _Ut

cEve

Ee EveG0 _Pt

cEve

Ee EveB0 _bt B0bt

cEve

Ee Eve_rt 83

where t is time.



13/22

5.5. Numerical time integration: (Boltzmann model)

For the Boltzmann model, a detailed explanation about the time integration procedure is necessary. Adopt-ing linear time approximation generates the time marching process

_Us1 Us1 Us

Dt

84

_Ps1 Ps1 Ps

Dt85

_bs1 bs1 bs

Dt86

_rs1 rs1 rs

Dt87

_rels1 rels1 r

els

Dt88

where s + 1 stands for the instant ts+1.Applying the above approximations to Eq. (82) one achieves the following system of algebraic equations:

H Us1 GPs1 Fs 89

where

Gc

Dt

Ee EveEve

G 90

H H Ee

DtEveV 91

Fs Ee

DtEveVUs

c

DtGPs B

c

Dt

Ee EveEve

bs1

c

Dtbs

!92

Boundary conditions are imposed changing columns ofH and G, as usually done in boundary element for-

mulations. As the values at instant ts are known, by solving Eq. (89) one finds Ps+1 and Us+1 and is able tosolve the next time step and so on.In order to calculate the total stress value it is necessary to evaluate _Ps1; _Us1 and _bs1, by means of Eqs.

(84)(86). From these values and Eq. (87) one calculates rs+1 in Eq. (83), as follows:

rs1 G0Ps1

Eve

Ee EveH0Us1

Ee

Ee EveV0 _Us1

cEve

Ee EveG0 _Ps1

cEve

Ee EveB0 _bs1 B

0bs1 c

Dt

Eve

Ee Evers

,

1 c

Dt

Eve

Ee Eve

93

or in a compact form

rs1

G0Ps1

G0 _Ps1

H0Us1

B0bs1

V0 _Us1

94

similarly to Eq. (57) including G0 _Ps1.Substituting approximation (88) into relation (73) it is possible to evaluate the elastic part of stress at the

Kelvin fragment of the adopted Boltzmann viscoelastic model, as follows:

relijs1 cijab1 rijs1 r

elijs

or for hl hk r

els1 rs1

c

Dtrels

.1

c

Dt

95

where

cijab cijab Dt dijab=Dt

relij cijab=Dtrelabs

96

where cijab is defined in Eq. (72). The viscous stress rv

s1 is calculated directly from Eq. (18) using the values of

rs+1 and rels1 determined above.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


14/22

5.6. Time marching processunified approach

In this item the solution procedure is summarized in order to clarify the expressions described in all items ofSection 5. Eqs. (40), (57), (58) and (89), (94) and (95) can be written in a unified form, as:

H Us1 GPs1 Fs 97

rs1 G0Ps1 G

0 _Ps1 H0Us1 B

0bs1 V0 _Us1 98

reg

s1 freg

s1 99

where eg represents e for Kelvin model and el for Boltzmann model and f() is a simple recursive function.As for the initial time t = 0 or s = 0 the restricted displacement, part of vector U1 and the applied surface

force, part of vector P1, are known it is possible to calculate the unknown parts of displacement and surfaceforce vectors, for s = 0, using Eq. (97) as follows:

hkk hkuhuk huu

Uk

Uu

& 's1

guu guk

gku gkk

Pu

Pk

& 's1

F

F

& 's1

100

where k is used for known values and u is used for unknown values. Isolating the unknown values on the left

side of the equation and the known ones on the right (changing columns) one writes

guu hku

gku huu

Pu

Uu

& 's1

hkk guk

huk gkk

Uk

Pk

& 's1

F

F

& 's

F

F

& 's1

101

or simply the resulting linear system of equations

AXs1 Fs1 102

After knowing the complete vectors U1 and P1 one is able to achieve stresses directly from Eqs. (98) and(99). From these values and the prescribed nodal surface forces and displacements for s = 1 the procedureis repeated for any instant s, solving the problem.

6. Examples

In this section three examples are used to check the accuracy and stability of the method. The accuracy istested comparing numerical solutions with analytical ones. The stability is tested varying the adopted time stepfor selected cases. Both Kelvin and Boltzmann model are applied to run the examples. For Kelvin model theinstantaneous Young modulus is not considered.

6.1. Simple stressed bar

The simple bar depicted in Fig. 7 is analysed when subjected to a force at its free end. As its analytical solu-tion is very simple and easily obtained, this example is recommended to verify the accuracy of the proposed

PA

B

x

y

L

h

Physical Properties GeometryEe = 22.5757 kN/mm

2 L = 800.0mm

Eve = 11.0 kN/mm2 h = 100.0mm

= 0.0

= == 45.454545

Analysis a Parameters Loading

t = 1.0 day P = 0.005 kN/mm2

Number of time steps 450

Fig. 7. Geometry, discretizations and other required data.

http://-/?-http://-/?-http://-/?-http://-/?-


15/22

numerical model. The results obtained by analysing this simple bar adopting the proposed two-dimensionalboundary element technique (plane stress conditions were assumed) are compared with the one-dimensionalanalytical solution. The boundary element discretization is given in Fig. 7. The physical constants adopted torun this problem are also given in Fig. 7. For the two-dimensional analysis, the viscous parameters hl and hkare assumed to have the same value, taken here as equal to the one-dimensional case.

The free end displacements at point A, computed by BEM and given by the analytical solution, are depictedin Fig. 8. As it can be seen, no differences are observed when comparing the computed values. The stress valuescomputed at point B are given in Fig. 9. Again, no differences among them are observed.

In Fig. 10 one can see that the present model is quite stable regarding time step length.

0 50 100 150 200 250 300 350 400 450

Time (days)

0.00

0.10

0.20

0.30

0.40

Displacement(mm

)

Analytic

BEM

0 50 100 150 200 250 300 350 400 450

Time (days)

0.10

0.20

0.30

0.40

0.50

0.60

Displacement(mm)

Analytic

BEM

(a) Kelvin Model (b) Boltzmann Model

Fig. 8. Longitudinal displacements at point A.

0 50 100 150 200 250 300 350 400 450

Time (days)

0.000

0.001

0.002

0.003

0.004

0.005

Stress(kN

/mm

2)

Analytic (elastic)

Analytic (viscous)

BEM (total)

BEM (elastic)

BEM (viscous)

0 50 100 150 200 250 300 350 400 450

Time (days)

0.000

0.001

0.002

0.003

0.004

0.005

Stress(kN

/mm

2)

Analytic (elastic)

Analytic (viscous)BEM (total)

BEM (elastic)

BEM (viscous)


Analytic (total) Analytic (total)

Fig. 9. Stresses, re11; rv11; r11, at point B.



16/22

6.2. Internal pressure into a cylinder

A thick cylinder subjected to an internal pressure p is analysed. The double symmetry of the problem istaken into account, therefore reducing the discretization of the body, as shown in Fig. 11. The geometric val-ues and the physical parameters are also given in Fig. 11.

The example is organised in three cases: Case 1 hk = hl = c = 7.14285/day) is used to check the behaviourof numerical solutions for both Kelvin and Boltzmann formulations. Case 2 ( hk = 7.14285/day and hl = 0)

and Case 3 (hl = 7.14285/day and hk = 0) are used to check the extreme material behaviours regardingnon-proportional viscosity (using Kelvin formulation). Case one is compared with the analytical solution inFigs. 12 and 13. The analytical solution is given when plane stress conditions are assumed.

ur r c21 t c1

r1 t

h i 1Ee

1

Eve1 et=c

!

where u is the radial displacement, r is the radius, c = hl = hk. Constants c1 and c2 depend on internal andexternal pressures, pi and pe, respectively and are given by

P

y

x

x

y

R1

R2

Mechanical properties

Eve = 35.0 kgf/cm2

Ee =90.0 kgf/cm2

= 0.4

Geometry

R1 = 25.4cm

R2 = 50.8cm

Other data

t = 1.0 day

number of t's =90

Load

P = 7.031 kgf/cm2

Fig. 11. Geometry, discretization and other required data.

0 50 100 150 200 250 300 350 400 450

Time (days)

0.0

0.1

0.2

0.3

0.4

Displacement(m

m)

dt=0.5day

dt=1.0day

dt=5.0days

dt=10.0days

dt=25.0days

dt=50.0days

0 50 100 150 200 250 300 350 400 450

Time (days)

0.1

0.2

0.3

0.4

0.5

0.6

Displacement(m

m) dt=0.5day

dt=1.0day

dt=5.0days

dt=10.0days

dt=25.0days

dt=50.0days


Fig. 10. Time step dependence.



17/22

c1 r2i r

2e pe pi

r2e r2i

c2 pir

2i per

2e

r2e r2i

Again, for Kelvin formulation, one neglects the terms with Ee, used only with the Boltzmann formulation.The time step length dependence for the proposed scheme is also shown in Figs. 14 and 15. It is worth not-

ing that in those figures plane strain conditions are assumed in order to show the possibilities of the formu-lation. As it can be observed, the obtained results are rather accurate even for large time steps. The finalelastostatic solution is achieved for any chosen time step.

As previously mentioned, in order to describe the generality of the proposed procedure, the cylinder under

consideration is analysed for different values of hl and hk and Kelvin Model. Three situations are depicted in

0 20 40 60 80 100

Time (days)

0

2

4

6

8

10

12

Displacement(cm)

Analytic

BEM

0 10 20 30 40 50 60 70 80 90

Time (days)

0

4

8

12

16

Displacement(c

m)

Analytic

BEM


Fig. 12. Inner wall displacements, comparison with analytical (plane stress).

0 10 20 30 40 50 60 70 80 90

Time (days)

0

2

4

6

8

Displacement(cm)

Analytic

BEM

0 10 20 30 40 50 60 70 80 90

Time (days)

0

2

4

6

8

10

Displacement(cm)

Analytic

BEM


Fig. 13. Outer wall displacements, comparison with analytical values (plane stress).



18/22

Fig. 16. Case 1 means hk = hl = 7.14285, while hk = 7.14285 and hl = 0.0 define Case 2 and hk = 0 andhl = 7.14285 define Case 3. From these figures, one can realise that hl is the most important coefficient forthis problem in comparison with hk. The results are strongly dependent upon hl, while they practically donot vary regarding hk. It means that if one needs to choose just one parameter for an analysis (Part II), itshould be hl.

6.3. Load changing

The same bar of the first example is now analysed when subjected to a more general load. During the first

200 time steps the load is sustained equal to 5 N/mm2

. At this very instant it is changed to zero. The behaviour

0 10 20 30 40 50 60 70 80 90

Time (days)

0

2

4

6

8

10

Displacement(cm)

dt=0.1day

dt=1day

dt=2.5days

dt=5days

dt=7.5days

dt=10days

dt=15days

0 10 20 30 40 50 60 70 80 90

Time (days)

0

2

4

6

8

10

12

14

Displacement(c

m)

dt=0.1day

dt=1day

dt=2.5days

dt=5days

dt=7.5days

dt=10days

dt=15days


Fig. 14. Inner wall radial displacements for several time steps (plane strain).

0 10 20 30 40 50 60 70 80 90

Time (days)

0

2

4

6

Displacement(

cm)

dt=0.1day

dt=1day

dt=2.5days

dt=5days

dt=7.5days

dt=10days

dt=15days

0 10 20 30 40 50 60 70 80 90

Time (days)

2

4

6

8

Displacem

ent(cm)

dt=0.1day

dt=1day

dt=2.5days

dt=5days

dt=7.5days

dt=10days

dt=15days


Fig. 15. Outer wall displacements for several time steps (plane strain).



19/22

of the numerical solution is the same as the analytic one, as seen in Fig. 17. The Dt increment is the same usedin the first example, i.e., one day. It has been assumed that hk = hl.

7. Conclusions and discussion

An alternative methodology for viscoelastic analysis has been presented using boundary element method.This methodology is based on the global movement equation of solids and takes into account the non-localcharacteristics of viscosity by continuity assumptions. The numerical implementation was performed for atwo-dimensional boundary element code using Kelvin and Boltzmann standard viscoelastic models. It wasdemonstrated, by examples that no problems regarding convergence appear using the present formulation.Accurate results were found for selected problems. Another advantage of this technique is that viscoelasticproblems are solved directly by a simple linear time domain scheme and not by residual decaying forces based

on relaxation functions, for which the statement of general boundary conditions are more complex. At the

0

2

4

6

8

10

12

Case 1

Case 2

Case 3Displacement(cm)

Time (day) Time (day)

0 20 40 60 80 100

0

1

2

3

4

5

6

7

Case 1

Case 2

Case 3

Displacement(cm)

(a) Inner wall (b) Outer wall

0 20 40 60 80 100

Fig. 16. Different viscous parameters, Kelvin Model (plane stress).

0 100 200 300 400 500

Time (days)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Disp

lacement(mm)

Analytic

BEM

0 50 100 150 200 250 300 350 400 450

Time (days)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Displacement(mm)

Analytic

BEM


Fig. 17. Displacement of point A for a changing.



20/22

beginning of the developments it was difficult to find a way to generate the Boltzmann model integral equa-tion. A simplification regarding viscous parameters was necessary in order to write the stress rate as surfaceforce rate. This simplification does not invalidate the technique, but after hard considerations it opens the pos-sibility of writing a boundary element formulation considering only boundary integrals. As a consequence theuse of cells can be avoided for both Kelvin and Boltzmann models, as shown in the next part of this study. One

should mention that the use of finite elements for the Boltzmann model considering hk5

hl is recommendedwhen needed [19].

Appendix A. Additional remarks about the use of viscous differential relations

The viscoelastic characteristics hk and hl can be identified from the results of a creep shear test (hl) and auniaxial tensile test (hk) [1]. The same procedure can be followed in order to identify the creep functions J(t)and K(t).

One can relate the creep functions to the viscoelastic characteristics hk and hl, at any instant, by using thefollowing expressions:

hl t

ln1 2lJt KtA1

and

k1 2mhk 2lhl Et

ln1 EJtA2

for t5 0 and, if desired, the values at t = 0, as follows:

hl 1 2lJ0 K0

2ldJt=dt dKt=dtA3

and

k1 2mhk 2lhl 1 EJ0

dJt=dtA4

If one desires hk and hl to be constant, the curves, ln(1 2l(J(t) + K(t))) and ln(1 EJ(t)), should beapproximated by the best straight lines. If one assumes that hk and hl can vary along time, the above expres-sions are directly applied and the proposed formulation does not change.

Appendix B. Boltzmann viscoelastic differential relation

In this appendix some necessary steps to achieve expressions (16) and (17) from Eqs. (13)(15) are pre-sented. It is important to show how to manage constitutive relations to achieve expressions that are moregeneral.

The elastic (instantaneous) part of the Boltzmann viscoelastic model is governed by the Hookes law, i.e.:

reij Cijlme

elm Ee

~

Cijlmeelm B1

where Clmij is the usual elastic constitutive tensor, given by:

Cijlm kdijdlm ldildjm dimdjl B2

in which k and l are the Lame constants, given in terms of Poisson ratio and Young modulus as:

k Eem

1 m1 2m Ee~k B3

l Ee1

21 m Ee~l B4

~Cijlm ~kdijdlm ~ldildjm dimdjl Cijlm=Ee B5

where, for convenience, we defined new dimensionless values~k; ~l and

~Cijlm.



21/22

Using the same idea, one writes the elastic part of the Kelvin viscoelastic stress as follows:

relij Cijlmevelm Eve

~Cijlmevelm B6

The viscous stress is written in a general form as:

rvij gijlm _evelm Eve~gijlm _e

velm Evec~Cijlm _e

velm B7

The viscous stress/strain rate relation is given by the tensor gijlm, written as:

~gijlm hk~kdijdlm hl~ldildjm dimdjl gijlm=Eve B8

where, as for the Kelvin model, hk = hl are the viscous parameters.The inverse representation of Eq. (B5) is:

~Dijlm ~Cijlm1

1

2~l

1

2dildjm dimdjl

m

1 mdijdlm

!B9

Applying Eq. (B9) in Eqs. (B1) and (B6) and taking into account Eq. (13) one writes:

eelm 1

Ee~Dlmijrij B10

evelm 1

Eve~Dlmijr

elij B11

Form Eq. (15) one transforms Eq. (B11) into:

evelm 1

Eve~Dlmijr

veij r

vij

1

Eve~Dlmijrij r

vij B12

Assuming relation (B7) one writes Eq. (B12) as follows:

evelm 1

Eve~Dlmijrij ~Dlmij~gijck_e

veck B13

Substituting expressions (14) and (B10) into Eq. (B13) and rearranging terms, results:

elm ~Dlmij1

Eve

1

Ee

rij ~Dlmij~gijck_e

veck B14

Multiplying Eq. (B14) by Clmqs , given in Eq. (B5), results:

Eve EeEveEe

rqs ~Cqsckeck ~gqsck_e

veck B15

Differentiating Eq. (14) with respect to time, one writes:

_eveck _eck _eeck B16

Substituting relation (B16) into expression (B15) and rearranging indices, results:Eve Ee

EveEe

rqs ~Cqsckeck ~gqsck_eck ~gqsck_e

eck B17

Differentiating, with respect to time, Eq. (B10) and substituting into Eq. (B17) one achieves:

rqs EveEe

Eve Ee

~Cqsckeck

EeEve

Eve Ee

~gqsck_eck

Eve

Eve Ee

~gqsij ~Dijck _rck B18

In order to transform the stress rate into surface force rate it is necessary to impose the simplificationhk = hl = c. In this way expression (B18) turns into:

rij EveEe

Eve Ee ~Cijlmelm cEveEe

Eve Ee ~Cijlm _elm cEve

Eve Ee _rij B19



22/22

This is the general rheological differential relation for the Boltzmann model. It is employed to derive theboundary element procedure proposed here. More complicate rheological models can be introduced followingsimilar procedures as the employed to achieve Eq. (B19).

References

[1] J. Lemaitre, J.L. Chaboche, Mechanics of Solids, Cambridge University Press, Cambridge, 1990.[2] W. Flugge, Viscoelasticity, Blaisdell Publication, USA, 1967.[3] Z. Sobotka, Rheology of Materials and Engineering Structures, Elsevier, Prague, Czechoslovakia, 1984.[4] R.M. Christensen, Theory of Viscoelasticity, Academic Press, New York, 1982.[5] W.H. Chen, C.M. Chang, J.T. Yeh, An incremental relaxation finite element analysis of viscoelastic problems with contact and

friction, Comput. Meth. Appl. Mech. Eng. 109 (1993) 315319.[6] W.C. Carpenter, Viscoelastic stress analysis, Int. J. Numer. Meth. Eng. 4 (1972) 357366.[7] W.H. Chen, T.C. Lin, Dynamic analysis of viscoelastic structures using incremental finite element method, Eng. Struct. 14 (1982) 271

276.[8] S. Yadagiri, R.C. Papi, Viscoelastic analysis of near incompressible solids, Comput. Struct. 20 (1985) 817825.[9] J. Argyris, I.St. Doltsinis, V.D. Silva, Constitutive modelling and computation of non-linear viscoelastic solids. Part I: Rheological

models and integration techniques, Comp. Meth. Appl. Mech. Eng. 88 (1991) 135163.[10] P. Perzyna, The constitutive equations for rate sensitive plastic models, Quart. Appl. Math. 20 (1963) 321332.

[11] D.R.J. Owen, F. Damjanic, Viscoplastic analysis of solids stability considerations, Recent Advances in Non-Linear ComputationalMechanics, Pineridge Press, Swansea, 1982.

[12] J. Munaiar Neto, A study of viscoelastic and viscoplastic constitutive models and the use of Finite Elements algorithms, Doctor inScience Thesis, EESC-USP, Brazil, 1998.

[13] J.H. Argyris, I.S.T. Doltsinis, K.J. William, New developments in the inelastic analysis of quasistatic and dynamic problems, Int. J.Numer. Meth. Eng. 14 (1979) 18131850.

[14] J.C. Simo, T.J.R. Hughes, Computational Inelasticity, Springer, New York, 1998.[15] I. Doltsinis, Elements of Plasticity Theory and Computation, WIT Press, Southampton, 2000.[16] A.D. Mesquita, H.B. Coda, W.S. Venturini, An alternative time marching process for viscoelastic analysis by BEM and FEM, Inter.

J. Numer. Meth. Eng. 51 (2001) 11571173.[17] C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary Element Techniques: Theory and Applications in Engineering, Springer, Berlin,

1984.[18] A.D. Mesquita, H.B. Coda, Alternative Kelvin viscoelastic procedure for finite elements, Appl. Math. Model. 26/4 (2002) 501516.[19] G. Beer, J.O. Watson, Introduction to Finite and Boundary Element Methods for Engineers, Wiley, New York, 1992.


Date post:	05-Apr-2018
Category:	Documents
Upload:	femi-richard-fakoya
View:	216 times
Download:	0 times

Bem Project Must Use

Documents