A Principles of Hyperplasticity part 10 doc

This feature of the material behaviour is due to the fact that the dissipation function is chosen to be a homogeneous first-order function of the internal variable rate.. Mathemati-cally

Rate Effects

11.1 Theoretical Background

11.1.1 Preliminaries

So far we have considered only materials that are rate-independent, that is to

say, the response is the same irrespective of the strain rate This feature of the

material behaviour is due to the fact that the dissipation function is chosen to be

a homogeneous first-order function of the internal variable rate

Mathemati-cally, this can be expressed through Euler’s equation as

ij ij

d d

w

D

Several features of the behaviour of rate-independent materials follow from

this special form of the dissipation function In particular, the special case of the

Legendre transform of a homogeneous first-order dissipation function (see

Appendix C, Section C.6) gives rise to the existence of a yield surface

Many materials, whilst primarily rate-independent, do show some small

de-pendence on the strain rate Typically, the yield stress may be observed as

in-creasing marginally with the strain rate Creep under sustained stresses and

relaxation of stress at fixed strain are processes that are also related to strain rate

effects Most geotechnical materials, for instance, exhibit these types of

behav-iour to a certain extent

This type of response is often modelled semi-empirically, frequently with

dif-ferent (and not always consistent) theories used to model the rate-dependence

of strength and the processes of creep and relaxation Properly, however, all

these phenomena should be encompassed within a single approach that

ex-plains all rate-dependent processes This can be achieved within the frame-

work described in this book by considering dissipation functions that are not

homogeneous, first-order functions of the internal variable rate, so that

Equa-tion (11.1) does not apply

As stated before, the formalism adopted in this book is based on the method

used by Ziegler (1977) He describes principally materials that are

rate-depen-dent (concentrating mainly on linear viscous materials, for which the dissipation

function is quadratic), and devotes relatively little attention to the special case of

rate-independence Here, we have taken the inverse approach Having started

with the rate-independent case, we shall now examine briefly the implications of

departures from it This approach offers a different, and we believe useful,

in-sight into the treatment of rate-dependent materials

Following the concepts behind Ziegler’s method, but adopting a slightly

dif-ferent terminology and sequence to the argument, first we define:

ij ij

d d

Q wD

wD 

(11.2)

Comparing with Equation (11.1), we see that Q for a rate-independent 1

material For any dissipation function d which is a homogeneous function of

degree n in D , it follows from Euler’s theorem that ij Q is simply a constant

w

wD

this case, it follows once again that, by comparison with Equation (4.8),

F  Fij ijD ij 0 Then, following exactly the same argument as before, we

adopt the constitutive hypothesis (equivalent to Ziegler’s orthogonality

condi-tion) that F F This form of the equations corresponds exactly to Ziegler’s ij ij

original approach (although expressed slightly differently)

The presence of the factor Q in the above formalism is not too much of an

in-convenience when the dissipation is a homogenous function of the internal

vari-able rate, because in this case Q is simply a constant However, if the dissipation

is not a homogeneous function of the rates, as proves useful to describe

materi-als with a weak rate-dependence, then the presence of the factor Q in Equation

(11.3) is a significant inconvenience Specifically in this case, d is said to be

a pseudo-potential rather than a true potential, and it is not possible to take the

Legendre transform of d to interchange F and ij D as dependent and inde-ij

pendent variables

11.1.2 The Force Potential and the Flow Potential

If, however, d can be written in the form ij

d is homogeneous and first order in D , then z d ij { Next, it is clear from

Euler’s equation that if d is homogeneous and of order n, then z d n

Rather than (11.3), we prefer to adopt the following definition of the

dissipa-tive generalised stress:

ij ij

z

w

F

From Equation (11.4), it follows that F D ij ij d as before, so that once again

F  Fij ijD ij 0, and we make the constitutive assumption F F ij ij

The principal advantage of using the function z is that, unlike the dissipation

function d (which is a pseudo-potential), the function z serves as a true potential

for F The function z could properly be defined as the generalised stress poten- ij

tial, but for brevity, we shall simply refer to it as the force potential Because it

serves as a potential for the generalised stresses, a simple Legendre

transforma-tion can be made:

 ij, ij, ij ij ij

such that

ij ij

w

w

D wF

The function w has a clear analogy with the yield function in the

rate-independent case, but because z is not homogeneous and first order in the rates,

the Legendre transform is no longer the degenerate special case; so although

0

y for the rate-independent case, in general, the condition w does not 0

apply The function w could properly be called the internal variable rate

poten-tial, but again for brevity, we shall simply call it the flow potential Note that the

sum of the force and flow potentials is equal to the dissipation function

z w d The potentials z and w have been defined elsewhere [see, for example,

Maugin (1999)], but previous authors seem to have concentrated almost

exclu-sively on the linear viscous case where both are quadratic Confusingly, both z

and w have been referred to in the literature as dissipation potentials, a

termi-nology we deliberately avoid here

We shall now explore how the potential z can be defined if the dissipation

function d is known Since z is obtained by integration from d (see below), it is

determined only to within an additive constant To make the definition of z

precise, we shall specify that z when the internal variable rates are all zero 0

Since d in this case, and w = d – z, it also follows that w = 0 when the rates are 0

zero

As noted above, if d is a homogeneous function of order n in the internal

vari-able rates, we can choose z d n , which is also of order n, so that

where each of the N functions d is itself homogeneous and of order k n in the k

internal variable rates, then z can be chosen as

1

N k k k

d z n

application of the definition (11.10) leads to a potential with the required

prop-erty that F w wD satisfies ij z ij F D ij ij d Thus by setting W in (11.10), we o 1

obtain

1  

0

ij ij

A simple change of variable to x lnW gives an alternative expression to

(11.11) which may be more convenient in some cases:

Therefore, we have demonstrated that, if the dissipation exists as a function

of the internal variable rates, then it is possible to derive the force potential by

integration

11.1.3 Incremental Response

In the rate-independent case, we found that the incremental response could

always be derived if the model was specified by one of the energy functions (u, f,

g , or h) and the yield function y Alternatively, if the dissipation function is

de-fined then, although the incremental response can be derived by applying

vari-ous ad hoc procedures for particular models, it has not proved possible to do

this by a completely general and automated procedure

We find that there is a parallel position for the rate-dependent materials If w

is specified, then the incremental response can be obtained automatically,

whereas if z is specified, then ad hoc procedures again have to be used for each

model

Assuming that the model is specified by the Gibbs free energy g and the flow

potential w, we can define the following differential relationships:

w

w

D wF

Note that for a rate-dependent material, the time increment in a calculation

has a real physical meaning, whilst for a rate-independent material, the time

increment is artificial For a real increment of time dt, (11.15) can be rewritten

As well as the stresses and strains, it is necessary to update D and ij F be-ij

cause the differentials in Equation (11.15) are functions of these variables The

updating is achieved by using Equations (11.6) and (11.14)

11.2 Examples

11.2.1 One-dimensional Model with Additive Viscous Term

The simplest of one-dimensional elastic-plastic models is defined by the

func-tions:

2

2

g E

V

where E is the elastic stiffness and k is the strength

On the other hand, an equivalent viscoelastic model would be defined by

(11.17) together with

2

where P is the viscosity

A simple elastic-viscoplastic model is obtained by combining the two

dissipa-tion funcdissipa-tions:

2

A schematic representation of the model represented by Equations (11.17)

and (11.20) is given in Figure 11.1, in which it can be seen that the plastic and

viscous elements act in parallel

Figure 11.1 Schematic representation of elastic-viscoplastic model

Noting that the dissipation function is now non-homogeneous and using any

of (11.8), (11.9), or (11.11), one can derive

z k

so that V ! , and the signs of V and k D must always be the same A corollary is

that if V d , then k D  0 and H V  E, i e incrementally elastic behaviour

oc-curs Thus the elastic region is bounded by a “yield surface” in stress space

where are Macaulay brackets, x 0,x , 0 x x x, t The above result 0

can be obtained in a mathematically more rigorous and concise way by using the

terminology of convex analysis; see Chapter 13 and Appendix D

It now follows that, differentiating (11.22) and substituting (11.26),

S

k E

V V

P



Considering tests at a constant strain rate H , the viscoplastic part of the

re-sponse described by (11.27) can be rearranged as

S

Ek

E d

where A is a constant of integration For the first loading, D ! 0 and V at k

Figure 11.2 shows the normalised stress-strain curves for PH k 0.0,0.1, 0.2,

and 0.3 (the first is equivalent to the model without viscosity) It can be seen that

this model provides a satisfactory starting point for describing rate-dependent

plastic behaviour, in which there is a linear increase in strength with strain rate

We have obtained the above response from the specification of the force

po-tential Alternatively, we can subtract (11.21) from (11.20) to obtain the flow

potential w  PDd z 2 2 Substituting (11.26), but writing F instead of V, we

can express the flow potential as a function of F:

P k 

0 0

1 0

2 0

k

V

k EH

Figure 11.2 Normalised stress-strain curves for different constant strain rates

We could have taken Equation (11.32) as our starting point rather than (11.21)

Differentiating (11.32) gives

S

which [using (11.23)], immediately leads to (11.26), so that the derivation of

a response based on the flow potential w is briefer than the derivation using the

force potential z

11.2.2 A Non-linear Viscosity Model

An alternative approach to modelling viscous effects is to modify (11.18) to

1

where the constant r, with the dimensions of strain rate, has been introduced to

maintain the dimension of stress for the constant k It immediately follows that

There is now no purely elastic region, and non-linear viscous behaviour

oc-curs whenever the stress is non-zero Noting that Equation (11.36) implies that

the signs of V and D must be the same, it can be rearranged to

1 1

r k

Equation (11.39) cannot be integrated analytically without recourse to special

functions, but Figure 11.3 shows stress-strain curves for n 1.1 and different

strain rates obtained by numerical integration (using forward differences with

an interval of E kH 0.1) The curves shown are for H  r 1,2, 3, and 4

A test at infinitesimal strain rate would simply give V 0

Figure 11.4 shows the stress-strain curves (again produced by numerical

inte-gration) for H  r 1 for n values of 2, 1.5, 1.2, and 1.1 It shows how the

elastic-plastic response is approached asymptotically, for a given strain rate, as no 1

Although the curves in Figures 11.2 and 11.3 are at first sight remarkably

similar, there are a number of important differences in the character of the

re-sponse Firstly, the curves shown are not necessarily for comparable strain rates

Also, for the second case, infinitesimally slow straining results in zero stress,

whilst in the first case, it gives the elastic-plastic response Once straining is

stopped, the stress in the first model relaxes to V , but in the second model, it k

relaxes to V 0

Clearly, the models represented by Equations (11.21) and (11.35) could both

be expressed within a more general model defined by

1 2

in which the first model is obtained by setting n and 2 k2 P , and the sec-r

ond model simply by setting k1 This serves as a good example of the way 0

this approach to the formulation of constitutive models allows to be set them

within a hierarchy, where simpler models are subsets of more complex models

The model defined by Equation (11.40) is used as the basis for a continuum

P k 

1 2 3

k

V

k EH

Figure 11.3 Normalised stress-strain curves for non-linear viscosity model

P k 

k

V

k EH

2 1 H

P k 

1 1 H

P k 

5 1 H

P k 

Figure 11.4 Stress-strain curves for different powers of n

11.2.3 Rate Process Theory

We now consider a more sophisticated example of dependence Many

rate-dependent processes can be regarded as thermally activated processes The

re-sulting approach is known as rate process theory We omit the details here, but

the final result is that the rate x of the process depends on the driving force q in

the following way:

sinh

where A and B are functions of the temperature We shall consider fixed

tem-perature here, and treat A and B as constants Mitchell (1976) gives a useful

dis-cussion of the theory in the context of the mechanics of soils

Identifying x with D and q with F, we are expecting a relationship of the

we obtain a dissipation function of the form,

Note that this form of d cannot be decomposed as pseudo-homogeneous in the

form of Equation (11.7) (at least without recourse to an infinite series) However,

for a one-dimensional model, we can apply (11.9): effectively, we rearrange

d FD as F d D and then integrate the equation dz dD F Then we obtain

Using (11.43) in place of (11.24), we follow exactly the same procedure as in

the example in Section 11.2.1 and derive the following in place of Equation

H Results for this case are plotted in Figure 11.5 for H r 0,



0.1, 1, and 10 when Pr k 0.04 Rate process theory results in a shear strength

which, for low values of H , increases linearly with strain rate, whilst at high r

values of H , it increases linearly with the logarithm of the strain rate This r

represents quite realistically the behaviour of a number of materials, particularly

geotechnical materials

Alternatively, if a formulation based on the flow potential is required, then we

can note that w  Pd z r D 2 r2r, which can be expressed as

Note that if this form is used, then the expression for D , and hence for the

strain rate follow very simply by differentiation

represents quite realistically the behaviour of a number of materials, particularly

geotechnical materials... that for a rate-dependent material, the time increment in a calculation

has a real physical meaning, whilst for a rate-independent material, the time

increment is artificial... obtained by numerical integration (using forward differences with