TIME-DEPENDENT AND NON-LINEAR PROBLEMS 165

In the same way as before we can obtain the integral equation in

terms of our transformed variables.

cUj+ f l/,PJi dr = [ PtUJt άΓ + f

pB

t

U%

άΩ (6.66)

Jr Jr JQ

Note that the integral over Ω only involves the transform of the

body force appearing in the original equation (6.55). As before we

solve (6.66) for a sufficient number of values of ω and numerically

invert the transformed variable Uj(x

9

ω) to obtain the time-

dependent displacements u, (x, t). A direct consideration of the

magnitudes of Uj (x, ω) for different ω will give the natural modes of

vibration of the elastic body.

6.7 VISCOELASTIC PROBLEMS

Viscoelastic laws can be applied to materials to represent creep if the

rate at which the creep strain develops depends not only on the

current state of stress and strain but in general, on the full history of

the applied loading.

For a linearly viscoelastic material the total stress at time t due to

the initial strain and the incremental changes in strain is

15

σ

υ

(ή =

λ(ήδ

υ

ηΙ

κ

+ μ(ί) (^ + ^) +

+ λ(ί-τ)δ^ύ

ΚΛ

{τ)άτ +

+ J μ(ί-τ) {ujjT) + u

itJ

(x))dx (6.67)

where t is the time since loading, u° are the initial components of

strain and λ, μ are the time-dependent Lame parameters (they are now

time dependent and can be interpreted as relaxation moduli).

Note that equation (6.67) can be written as a Laplace convolution,

i.e.

σ

υ

(ί) = ^Χ, +

μ(ιι?,

ι

+ ιι^) +

+ K K

k^ij

+ μ*

(ii;.

i

+

li,·,

j) (6.68)

Consider for simplicity only one component. We can do this by

assuming that the bulk modulus (£/3(l - 2v)) is constant or that the

166 TIME-DEPENDENT AND NON-LINEAR PROBLEMS

Poisson ratio is constant. Let us use the second assumption with

v = 0. This gives λ = 0 and reduces μ to E/2. Hence equation (6.68)

becomes,

*ij(t) = ß7v +

ß*7ü

(6.69)

where y

tj

= (u

h t

+ u

u}

) = y

jt

.

In order to represent the time-dependent behaviour of the material

McHenry

17

proposed the following equation to approximate the

relaxation modulus:

μ(ί) =

$

0

+Σ

S^l-e-"")

(6.70)

/ = i

This equation can be fitted to experimental data by reducing it to a

set of linear equations in S

k

and giving y

l

n values of time over the

transition range of the experimental data.

Assuming y° = 0, equation (6.69) can be reduced to the following

by taking the Laplace transform,

L(a

l}

) =

K

^)L(y%) (6.71)

where from (6.70)

LOi(ie)) = - \s

0

+ X (-^— ) 1 (6.72)

The stress-strain relationship of the viscoelastic material has thus

been reduced to the normal static form (<r

i;

= μγ^) in the transform

plane (equation (6.71)). That is, we use a 'quasi-static' analysis.

To obtain a quasi-static analysis in boundary element terms we can

set up an integral equation relationship for the displacements similar

to equation (5.23), but using the constitutive relations (6.71) to

construct the fundamental traction tensor components Pfj. κ will

appear as a parameter of our problem and we must solve for a

sufficient number of values in order that the inversion of our Laplace

transforms can be accurately performed.

For the fully transient viscoelastic problem it is possible to follow

Cruse

1

but instead of the constitutive equations (6.42) we have to use

(6.68),

which becomes equivalent to the normal static form under the

Laplace transformation. The fundamental influence tensor U* would

remain the same but the traction influence tensor P£ would become

altered by the new constitutive relations. The normal integral

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