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An introduction to theFinite Element Method

Inelastic constitutive models

F. Auricchio

auricchio@unipv.it

http://www.unipv.it/dms/auricchio

Universita degli Studi di PaviaDipartimento di Meccanica Strutturale

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Inelastic constitutive models

Elastic material material for which strain de-

pends only on stress

Inelastic material material for which strain de-pends on stress and possibly on other quantities

Limit our consideration to 1D problems(scalar eqns.)

Elastic material =f()

Inelastic material =f(, )

For an inelastic material the quantityis an extra prob-lem variable (internal variable), hence it requires anew equation (often in evolutive form)

Elastic material =f()

Inelastic material

=f(, )

=g(, )

Internal variable in the sense that it is a variablewhich cannot be measured externally

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Inelastic constitutive models

Standard assumption: strain additive decomposi-tion in elastic and inelastic contribution

=e +i

withe =e() , i = e()

The latter equation is the inelastic strain definition

In the case of linear response for the elastic component

e = E

withEconstant parameter characterizing the materialelastic response

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Standard visco-elastic model

Develop a model which show inelastic strain evolution and

rate-dependency

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Standard visco-elastic model

Internal-variable evolutionary equation:

i =1

1

i (1)

with

: viscosity parameter: internal characteristic time

Full model

=e +i

e =

Ei =

1

1

i

which can be condensed as follows

=E

i

i =1

1

i

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Standard visco-elastic model

Strain control:[input] given=(t)[output] compute(t) andi(t)

Stress control:[input] given=(t)[output] compute(t) andi(t)

For analytical approaches (closed form solutions), wemay consider either strain or stress controlled loadinghistories. Clearly, one of the other can be easier to solve.

For numerical approaches (approximated solutions), ingeneral we consider strain as indipendent variable.

Consider time interval of interest [0, T] and subdi-vide it in sub-intervals

Indicated the generic time sub-interval as [tn, tn+1]

Consider known all the quantities attnas well as theindependent variable at time tn+1. The dependentvariable at timetn+1 should be computed.

For simplicity in a time-discrete numerical framequantities at tn are indicated with the subscript n,while quantities at tn+1 are indicated with no sub-script

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Standard visco-elastic model

Numerical approach (approximated solution)

Strain control: given n, in, n as well as [ =(tn+1)], compute [output] and

i

Stress control: given n,in,n as well as

ext [ext =ext(tn+1)], compute[output] and

i such that

R() =() ext = 0 (2)

Equation 2 can be interpreted as an equilibrium con-dition, solved in general with an iterative Newtonmethod, using the update formula

do while R(k) not equal 0

k+1 =k dR()d

k1

R(k)

k =k+1

end do

dR()/dis the algorithmically consistent tan-gent operatorfundamental to obtain a quadratic

convergenceNote that from Equation 2

dR()

d =

d()

d =ET = tangent modulus

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Standard visco-elastic model

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Visco-elast.model: analytical sols

it is possible to integrate Equation 1

i(t) =

t

exp

t

()

d (3)

where for convenience the initial time is set equal tot=for which we assume i = 0

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Visco-elast.model: analytical sols

Viscosity test orcreep test

Constant load starting at t= 0(t) = 0 for t 0

Possible now to compute the -relation

=e +i con

e

=

1

E

i =

1 exp

t

from which

=

E

+

1 exp t

=

1

E+

exp

t

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Standard visco-elastic model

We can now consider the resulting strain historyFor small time values [t0+], then

i 0

e = 1

E

e

For large time values [t+], then

i 1

e = 1

E

e +i =

1E

+

Att= 0 elastic response with modulus E

Att= +elastic response with relaxed modulus

E =1

1

E+

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Visco-elastic model

Maxwell model:elastic spring and viscous damper in series

Kelvin model:elastic spring and viscous damper in parallel

Standard visco-elastic model:

elastic spring and viscous damper in parallel + elasticspring in series (analyzed model)

More complex spring-dashpot combinations

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Standard visco-elastic model

Numerical treatment

Time-continuous model

=E( i)

i =1

1

i

Time-discrete model Compute stress history from strain history by an

integration technique (strain driven problem)

Thus, knowing strain at tn+1 and the solution attn[i.e. (n, n,

in)], we need to compute the solution

at timetn+1 [i.e. (, i)]

Discrete solution

Time-integration+

solution algorithm

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Standard visco-elastic model

Time-integrationUse implicit Backward Euler formula

=E( i)

i int

=1

1

i

(4)

where t=tn+1

tn

Solution algorithmSince model is linear, no special algorithm is required.

Combining Equation 4, inelastic strain i results as afunction of previous solution and current strain (consis-tent with definition of strain-driven process):

i =

in+

(5)

where:

=

1 +

E

+

1

t

1, =

E

t

Once updated the inelastic strain, possible to use Equa-tion 41 to update the stress.

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Standard visco-elastic model

Tangent modulusd=Ed Edi

anddi =d

Accordingly

ET=E(1 )

Exercise. Develop a matlab code in which: 1) assign a strain history[0, max] in a time interval [0, Tmax]; 2) subdivide time interval [0, Tmax] in

subintervals of length t; 3) for each time istant solve constitutive problem4) plot stress and inelastic strain versus time.

Exercise. Develop a matlab code in which: 1) assign a stress history[0,

max] in a time interval [0, T

max]; 2) subdivide time interval [0, T

max] in

subintervals of length t; 3) for each time istant solve constitutive problem

such to satisfy Equation 2 4) plot stress and inelastic strain versus time.

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Standard visco-plastic model

Develop a model which is rate-dependent and inelastic only

for high stress values (threshold)

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Standard visco-plastic model

Internal-variable evolutionary equation:i =

1

y (6)

with

: viscosity parametery: yielding stress

MacAuley bracket

< x >=

x if x >0

0 if x 0 !!

Full model

=e +i

e =

E

i =1

y

which can be condensed as follows

=E

i

i =1

y

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Standard visco-plastic model

Fundamental difference wrt visco-elastic model is the pres-ence of ayield (limit) functionFdefined as

F = y

Note:

IfF 0, i.e. > y then i = 0

INELASTIC CASE

Fis a function classically defined in stress space

Generalization to remove limitation >0

i =1

|| y

y| y|

for any

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Standard visco-plastic model

One-dimensional case

Extension to three-dimensional case

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Standard visco-plastic model

Numerical treatment

Time-continuous model

=E( i)

i =1

y

Time-discrete model Compute stress history from strain history by an

integration technique (st