BEM of Viscoelatic Model

8/2/2019 BEM of Viscoelatic Model

1/8

A GENERAL VISCOELASTIC ANALYSIS BY THE BOUNDARY ELEMENT

METHOD

Arthur Dias Mesquita1,

and Humberto Breves Coda2

ABSTRACT

From basic assumptions of viscoelastic constitutive relations and weight residual techniques aBoundary Element procedure is achieved for both Kelvin and Boltzmann models. Imposing spatial

approximations and adopting convenient kinematical relations for strain velocities, a system of time

differential equations is achieved. This system is solved adopting linear approximations for displacements,

resulting in a time marching methodology. This approach avoids the use of relaxation functions and makes

easier changes in boundary conditions along time, natural or essential. An important feature of the resulting

technique is the absence of domain discretizations, which simplifies the treatment of problems involving

infinite domains (tunnels and cavities inside the soil). Some examples are shown in order to demonstrate

the accuracy and stability of the technique when compared to analytical solutions.

Keywords: Viscoelasticity, Boundary Elements, Time integration.

INTRODUCTION

In some recent works (see e.g. Mesquita et al (2001), Mesquita and Coda (2001), Mesquitaand Coda (2001)) the authors developed a new time marching process for both Finite ElementMethod (FEM) and Boundary Element Method (BEM) to solve viscoelastic problems. These

formulations are based on differential constitutive relation for Kelvin and Boltzmann viscoelasticmodels.The novelty and main objective of this paper is avoiding internal cells, resulting in aviscoelastic Boundary Element formulation performing discretizations only at the boundary of

the analysed body. Another feature of the developed formulation is that, for Boltzmann model,the total time dependent displacements and stresses are achieved directly from the time marchingprocess. The formulation proposed here is quite different from the ones usually found in literature.

The most of the works developed so far follows, basically, three main procedures. All of them arebased on relaxation functions or Laplace-Carson transforms (see e.g. Lemaitre and Chaboche

(1990), Flgge (1967), Sobotka (1984), Chris tensen (1982)). Various examples are shown inorder to demonstrate the accuracy and stability of the formulation.

1 Dept. of Structural Eng., University of So Paulo, So Carlos, SP/Br 13566960. E-mail: [email protected] Dept. of Structural Eng., University of So Paulo, So Carlos, SP/Br 13566960. E-mail: hbcod [email protected]


2/8

2

BASIC RELATIONS FOR VISCOELASTICITY

This section is divided into two main parts related to the Kelvin and Boltzmann relations.

Kelvins model

Using rheological models defined in the uniaxia l space is the usual way adopted to describethe viscoelastic behaviour of solids. A simple representation, very often adopted to describe thiskind of behaviour, is the Kelvin-Voigt viscoelastic, figure 1.

E

Fig. 1. KelvinVoigt viscoelastic model (uniaxial representation).

From figure 1, the following relations are stated:

v

ij

e

ijij == (1)

v

ij

e

ijij += (2)

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

lm

lm

ij

e

lm

lm

ij

e

ij CC == (3)

Similarly, the following relation gives the viscous stress components:

lm

lm

ij

v

lm

lm

ij

v

ij

&& == (4)

In this work, a further simplification is assumed, i.e., == , in order to obtain only

boundary values at the integral equations. From the previous equations one writes:

lmlmijlm

lmijij CC &+= (5)

The actual infinitesimal equilibrium equation is written as follows:

jjj

v

i,ij

e

i,ij ucub &&& +=++ (6)

Expression (6) is rewritten for the absence of inertial and frictional forces, as:

0bjv

i,ij

e

i,ij =++ (7)


3/8

3

Boltzmann model

Another viscous representation is the so-called standard Boltzmann model. This model canbe described in a uniaxial representation, as illustrated in figure 2.

Fig. 2: Boltzmann viscoelastic model(uniaxial representation)This model is represented by a serial arrangement of Kelvin model and an elastic relation. It

reproduces both instantaneous and viscous behaviour of a material. From figure 2 one writes:

ve

ij

e

ijij == (8)

velm

elmlm += (9)

vij

eij

veij +=

l (10)

From previous equations, one writes (Mesquita & Coda (2001)):

ij

eve

velm

lmij

eve

evelm

lmij

eve

eveij

EE

EC~

EE

EEC~

EE

EE

&&

+

+

+

= (11)

INTEGRAL AND NUMERICAL EQUATIONS

In order to save space, the usual passages are skipped, for more details one can see Mesquitaet al (2001), Mesquita and Coda (2001), Mesquita and Coda (2001).

Kelvins model

From equations (5) and (7), following weighting residual technique it is possible to find:

dBbdupdupdpu)p(uC)p(uC kiiikiikii*kiikiiki +=+ && (12)

The term kiC is the same obtained in elastostatic formulations. Writing equation (12) for

internal points, differentiating it concerning the source point position and using the viscoelasticstress strain relation, results:

dBbdupdupdp)p()p( iqiiiqiiqi*

iq

v

q

e

q

+=+ & (13)

Using equation (2), equation (13) turns into:

dBbdupdupdp)p( iqiiiqiiqi*iqq += & (14)

Ee

Eve

e ve


4/8

4

In this scheme equation (3) is written in the following form (Mesquita et al (2001)):

elij

vij

vij

velm

lmij

velm

lmij

elij

1C

1C

&&&& ==== (15)

Substituting the above relation into equation (2) results:

0ijeij

eij =+ & (16)

All variables are approximated in space and time. In space isoparametric quadratic boundaryelements are adopted and all integrals are made as in any BEM formulation, resulting:

)t(Bb)t(GP)t(UH)t(HU +=+ & (17)

)t(b'B)t(U'H)t(U'H)t(P'G)t( += & (18)

To solve equation (17) linear time approximation for velocity is adopted, resulting:

s1s1s FGPUH += ++ (19)

where

Ht

1H

+=

(20)

1sss BbHUt

F ++=

(21)

The system (19) is solved for actual time, i.e., ts+1. The boundary conditions along time areprescribed by interchanging columns of H and G .To calculate the total stress the same timeapproximation is applied on equation(18), i.e.:

1s1s1s1s1s b'BU'HU'HP'G +++++ += & (22)

The elastic stress is calculated applying linear time approximation on equation (16).

+

+= ++

t1

t

es1s

e1s

(23)

The viscous part comes from equation (2).

Boltzmann model

From equations (11) and (7), it is possible to write:


5/8

5

[ ]

dBbE

EEdBbdpu

)p(uCdupdupdpuE

EE)p(uC

kiive

veekiiiki

ikiikiikii*ki

ve

veeiki

+++

++

=

&&

&&

(24)

The stress integral representation for total stress is given by:

[ ] )p(EE

EdBbdBbdp

EE

E

dupEE

Edup

EE

Edp)p(

q

vee

ve*

iqi

*

iqii

*

iq

vee

ve

iiq

vee

veiiq

vee

vei

*iqq

&&&

&

+++

+

++

+

=

(25)

The elastic instantaneous stress and the viscoelastic stress are equal to the total stress. It isnecessary to separate the viscous and elastic parts of the viscoelastic stress, as:

el

ij

v

ij

v

ij

ve

lm

lm

ij

ve

lm

lm

ij

el

ij

1C

1C

&&&& ==== (26)

Substituting equation (26) into equation (10) results the following differential equation.

0=+ ijelij

elij & (27)

All variables are approximated in space and time, resulting:

)t(BbE

EE)t(bB)t(PG)t(UH)t(GP

E

EE)t(HU

ve

vee

ve

vee +++++

= &&& (28)

)t(EE

E)t(b'B)t(b'B

EE

E

)t(P'GEE

E)t(U'HEE

E)t(U'HEE

E)t(P'G)t(

vee

ve

vee

ve

vee

ve

vee

ve

vee

ve

&&

&&

++

++

++

+

+=

(29)

To solve equations (28) and (29) and (27) linear time approximation is used for all variables,resulting:

s1s

ve

vee1s FGP

E

EE

tUH +

++= ++

(30)

where

Ht

1H

+=

(31)

+++= + s1sve

veesss b

tb

E

EE

tBGP

tHU

tF

(32)


6/8

6

The time dependent boundary conditions are prescribed by interchanging columns of H and

G . Using the results, 1sU + and 1sP + , one calculates 1sP +& , 1sU +& and 1sb +& . From these values, it is

easy to calculate the total stress level at 1st + as follows:

+ +

++

++

+

+= +++++++ 1s1s

vee

ve1s

vee

ve1s

vee

ve1s

vee

ve1s1s b'Bb'B

EE

EP'G

EE

EU'H

EE

EU'H

EE

EP'G &&&

+

+

+

+vee

ves

vee

ve

EE

E

t1

EE

E

t

(33)

The elastic stress at the viscoelastic part of the Boltzmann model is:

+

+= ++

tt

elss

els

111 (34)

From expression (10) and equation (34), one achieves the viscous stress

EXAMPLES

Kelvin viscoelastic model

Thick cylinder subjected to a sudden internal pressure P

The geometry, discretization and physical properties are depicted in figure 3.

y

R1

R2

P

x

x

y

Physical properties Geometry

E1=350N/cm2

R1=25.4cm=0.4 R2=50.8cm=7.14285days

Time parameters Loading

t=1.0day P=70.31 N/cm2

Fig. 3. Geometry, discretization and physical properties.

The inner and outer wall radial displacements obtained applying this numerical formulationare compared with the analytical ones in figures 4(a) and (b), respectively. The numerical resultswere obtained adopting a time step of one day.

0 15 30 45 60 75 90Time(days)

0

2

4

6

8

10

12

Displacement(cm)

Analytic

BEM

0 15 30 45 60 75 90

Time(days)

0

1

2

3

4

5

6

Displace

ment(cm)

Analytic

BEM

Fig. 4. (a) inner wall displacement, (b) outer wall displacement


7/8

7

The numerical behaviour is almost the same as the analytical one.

Boltzmann viscoelastic model

Simple stressed bar

The structure is modelled by quadratic boundary elements, a quarter of the body is discretized.The geometry, discretization and physical properties are given in figure 5.

B

AL

h P

x

y Physical properties Geometry

E1=22.5757kN/mm2

L=800mmE2=350N/cm

2h=100mm

=0.0=45.454545days


t=1.0day P=5 N/mm2

Fig. 5. Geometry, discretization and physical properties.

In figure 6(a) the displacement of point A is given regarding time steps length. One can see thatthe results are very stable and converges to the analytical solution. The time step length is varied

from one day to fifth days.

0 5 0 1 00 1 50 2 00 2 50 30 0 3 50 40 0 4 50

Time(days)

0.1

0.2

0.3

0.4

0.5

0.6

Displacement(cm)

1day

5days

10days

25days

50days

0 5 0 1 00 1 50 2 00 2 50 3 00 3 50 40 0 4 50

Time(days)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Displacement(mm)

Analytic

BEM

Fig. 6. (a) Time step sensibility, (b) Varying loading

The results for stresses are as good as the obtained for displacement, but are omitted here tosave space. In figure 6(b) the displacement of point A is shown when the load is removed at day200. The adopted time step is one day.

Tunnel subjected to a sudden internal pressure P

y

r

xP

soil

Physical properties Geometry

E1=900N/cm2

r=25.4cmE2=350N/cm

2

=0.4=7.14285days


t=0.5days P=70.31N/cm2

0 1 5 30 45 60 75 9 0Time(days)

0

1

2

3

4

5

6

7

8

9

10

Displacem

ent(cm

)

Analytic

B EM

Fig. 7. (a) Discretization, (b) Physical properties, (c) Radial displacement


8/8

8

This example exhibits that using the proposed formulation it is possible solving an exteriorBoltzmann (or general) viscoelastic problem using only a boundary discretization. The cavity is

modeled by adopting quadratic boundary elements, as depicted in figure 7(a). The physicalparameters are shown in figure 7(b). The tunnel radial displacement behavior is shown in figure7(c), it is hard distinguishing the numerical and the analytical results (plane stress).

CONCLUSIONS

It has been shown, along the paper, a stable and accurate way to perform viscoelasticanalysis by the Boundary Element Method. It consists in considering the viscous elastic relationas a non-local property of the continuum. From this assumption, the time integration should bedone after spatial approximation. The main advantage of the presented technique is that theintegral representation posses only boundary values. It has been imposed a spatial approximationfor boundary values achieving a system of time differential equations. Adopting linear timeapproximation for velocity, surface force and stress rates one easily solves this system. A very

elegant treatment is given for the stress determination. The proposed formulation has beendeveloped and implemented for both Kelvin and Boltzmann viscoelastic models. Any desiredviscous model could be added to the formulation following similar steps.

REFERECES

Christensen, R.M., (1982) Theory of Viscoelasticity. Academic Press, New York.Flgge, W., (1967) Viscoelasticity. Blaisdell Pub., USA.Lemaitre, J. and Chaboche, J.L.(1990)Mechanics of Solids, Cambridge University Press.

Mesquita, A.D.; Coda, H.B. and Venturini, W.S. (2001) An alternative Time marching processfor viscoelastic analysis by BEM and FEM, International Journal for Numerical Methods in

Engineering,, 51: 1157-1173.

Mesquita, A.D. & Coda, H.B. (2001) Alternative Kelvin viscoelastic Model for Finite Elements,

Applied Mathematical Modelling. (In revision)Mesquita, A.D. & Coda, H.B.(2001) An Alternative time integration for Boltzmann viscoelastic

model: a BEM approach,, Computer and Structures, Accepted.Sobotka, Z.(1984) Rheology of materials and engineering structures. Elsevier Science

Publishers, Prague, Czechoslovakia.

APPENDIX I. NOTATION

The following symbols were used (strain), (stress), (viscous compliances), C(elastic

compliances), (viscous parameters), (unified viscous parameter), b (domain forces),

u (displacements), p (surface forces), E (Young modulus), (domain), (boundary),

e (elastic), ve (viscoelastic), v (viscous), le (elastic Boltzmann), t(time), s (discrete time).

BEM of Viscoelatic Model

Documents

Transcript of BEM of Viscoelatic Model