A Principles of Hyperplasticity Part 5 ppt

5.3 Frictional Plasticity and Non-associated Flow In the previous section, we demonstrated that hyperplasticity can reproduce sim-ple elastic perfectly plastic models where the size of

5.2.3 Rigid-plastic Models

Rigid-plastic materials are special cases in which energy is dissipated only, and

not stored They can be described without use of internal variables because there

is no elastic strain The total strain is simply equal to the plastic strain To

scribe such materials without internal variables, it would be necessary to

de-velop a theory in which the dissipation could depend on the strain rate, e g

 ij, ij

already adopted, but simply introduce a set of constraints c ij H  D ij ij 0

Using the f formulation, we start from f 0 and, because of the constraints,

Then, as before, we use the dissipation function d k 2 D D ij ij together with

the constraint c Dkk 0, so that F ijc k 2 Sij Dij Combining this with

Equations (5.25), (5.26) and the constraint equation gives V ijc k 2 Sij Hij ,

from which both the yield surface and the flow rule follow Note that the mean

stress Vkk /kk Fkk is undetermined by the constitutive law because of the

0

Dkk

The same result could also have been obtained by specifying the yield surface

as y F F ij ijc c k 2 and deriving the expression for 0 Dij and hence (from

F w wD V The rest of the formulation follows as above Note that if

the two starting functions are g and y, then no additional constraints need to be

introduced This is an example of a more general observation that the g and y

formulation often offers the simplest route for deriving constitutive behaviour

5.3 Frictional Plasticity and Non-associated Flow

In the previous section, we demonstrated that hyperplasticity can reproduce

sim-ple elastic perfectly plastic models where the size of the yield surface in the

devia-toric plane does not depend on the stress state This class of models describes

a wide range of materials – from metals to saturated clays However, most

granu-lar materials exhibit a property called friction, characterized by dependence of

shear strength on normal stress As will be demonstrated below, the hyperplastic

formulation can easily reproduce this behaviour with an interesting and

impor-tant restriction – the flow rule for these models cannot be associated

5.3.1 A Two-dimensional Model

We shall start by introducing a simple frictional model defined in a

stresses The corresponding normal and shear strains are H and J, respectively

Consider the model specified by

V H

J

cw

wDcw

which, by eliminating /, can be combined to produce the yield surface in the

generalised stress space (see Figure 5.4):

H V

model (2.41) in true stress space (with P* = P + E) and the flow rule (2.43) with the plastic potential (2.44), as shown in true stress space in Figure 2.5

5.3.2 Dilation

As seen from Figure 5.4, within the framework of the above model, when E > 0, plastic shearing leads to positive increments of normal plastic strains which cause an increase in volume This is a well-known phenomenon in the behaviour

of dense granular materials, called dilation Experimental data, however, show

that the associated flow rule would grossly overestimate the amount of dilation

An associated flow rule is obtained in this model only if P = 0 The ability of this hyperplastic model to accommodate non-associated flow by allowing E > 0 is important for realistic modelling of the behaviour of frictional materials Fur-thermore, in this model, the purely empirical observation of P* = P + E is ex-plained for dense materials that exhibit dilation As demonstrated in Chapter 4, the Second Law of Thermodynamics demands that mechanical dissipation be non-negative Applied to Equation (5.28), this requirement yields P t 0 In the case P = 0, then P* = E, the dissipation is zero, and the flow is “associated” in the conventional sense Clearly such a model of frictional behaviour is unrealistic because of the implication of zero dissipation

Figure 5.4 Yield surface and plastic potential for a frictional model

There are no restrictions on the sign of E Clearly, when E = 0, we observe

in-compressible behaviour, which is achieved in the critical state Choosing E < 0

allows for contraction, exhibited by some loose granular materials, at least for

a limited range of strains

5.3.3 The Drucker-Prager Model with Non-associated Flow

Extension of the two-dimensional hyperplastic model into the general

six-di-mensional stress-strain space is formulated through the following potential

mn mn

MV  FB Dcc

which can be rearranged to eliminate the rates of the plastic strains and give the

yield surface in generalised stress space, expressed as

tion 5.3.1

5.4 Strain Hardening

5.4.1 Theory of Strain-hardening Hyperplasticity

The general hyperplastic framework was presented in Chapter 4 Here we revisit

briefly some of its highlights, relevant for generation of strain-hardening

hyper-plastic models

Potential Functions

The constitutive behaviour of a dissipative material can be completely defined

by two potential functions The first function is either the Gibbs free energy

ı Į șij, ij, 

dependence on temperature in the following because we are not considering

thermal effects If g is specified, then the strain is obtained by

of arguments where a function appears within a differential, so that we would

write, for instance, simply F w wDij g ij

By a suitable choice of Dij, it is possible to write the Gibbs free energy so that

the only term which involves both Vij and Dij is linear in Dij:

1 ij 2 ij 3 ij ij

Furthermore, if g3 is also linear in the stresses, then it is always possible (again

by a suitable choice of Dij) to choose simply g3 Vij V : ij

 ij, ij 1 ij 2 ij ij ij

The second function required to define the constitutive behaviour is (assuming

that g is the specified energy function) either a dissipation function

g

ij ij ij

d d V D D t or a yield function y y gV D Fij, ij, ij For a rate-0

independent material, dissipation is a homogeneous function of order one of the

plastic strain rate tensor In this case, dissipation and yield functions are related

by a degenerate Legendre transformation The dissipative generalised stress is

defined as F wij d gV D Dij, ij,ij wD Then from a property of the transforma-ij

tion, it follows that D Owij y gV D Fij, ij, ij wFij, where O is an arbitrary

non-negative multiplier The formulation is completed by the orthogonality

assump-tion of Ziegler (1977), which is equivalent to the assumpassump-tion F Fij ij

Link to Conventional Plasticity

The strain hardening hyperplastic formulation, although based entirely on

specification of two potential functions, intrinsically contains all the

compo-nents of conventional plasticity theory As mentioned above, the yield function,

although in generalised stress space, is obtained as a degenerate Legendre

trans-form of the dissipation function When the Gibbs free energy can be written in

the form of (5.43), no elastic-plastic coupling occurs, and it follows that

The interpretation of the above is that D plays exactly the same role as the ij

conventionally defined plastic strain p

H H  D w wV ) The generalised stress in this case also acquires

a clear physical meaning: F w V Dij g ij, ij wD V  wij ij g2 Dij wD The ij

generalised stress simply differs from the stress by the term

ij ij g ij ij

U D w D wD , known as the “back stress” In conventional kinematic

hardening plasticity, the “back stress” would normally be associated with the

stress coordinates of the centre of the yield surface

The flow rule is given by the expression

 , , 

g

ij ij ij ij

where O is an arbitrary non-negative multiplier When the yield (and

dissipa-tion) function does not depend on stress explicitly, this flow rule is associated in

both true and generalised stress spaces

Finally, the hardening rule is also specified by the two potential functions

Isotropic hardening is defined by dependency of the dissipation

g

ij ij ij

d d V D D t or yield function y gV D Fij, ij, ij on the internal 0

variable D Kinematic hardening is defined by dependency of the “back stress” ij

ij ij g ij ij

U D w D wD on internal variable D , as defined by the second term ij

of the Gibbs free energy function The translation rule for the yield surface can

be recognized in the expression

Dissipation and Plastic Work

It is important to note the difference between the dissipation d F D , which ij ij

we require to be non-negative on thermodynamic grounds, and the rate of

ij ij

W V H Consider, for instance, uncoupled plasticity for which

we can write gV Dij, ij g1 Vij g2 Dij  V D In this case, as discussed ij ij

above, the internal variable plays the role of the plastic strain p

W  V Hd   F D U D  Using (5.46), we see that dissipation and plastic

work are identical only if

form of the energy function

Incremental Stress-strain Response

In a way similar to conventional plasticity, two possibilities exist in the

descrip-tion of the incremental response of a strain-hardening hyperplastic material

Consider a non-frictional material, either its state is within the yield surface

(y gD Fij, ij0), in which case no dissipation occurs, and O = 0 If the material

point lies on the yield surface (y gD Fij, ij ), then plastic deformation can 0

occur, provided that O t In the latter case, the incremental response is ob-0

tained by invoking the consistency condition of the yield surface:

g ij ij

Differentiation of Equation (5.44) and substitution of (5.45) in the result and in

(5.46) give the incremental stress-strain response:

and the update equations for internal variable and generalised stress:

g ij ij

y

w

D OwF

The multiplier O defined from Equation (5.48) is derived from the consistency

condition ygD Fij, ij 0 Therefore, it applies only wheny gD Fij, ij 0 and

0

O ! For all other cases (i e. either when y gD Fij, ij0 or when (5.48) results

in a negative O value), then O 0

5.4.2 Isotropic Hardening

One-dimensional Example

Consider a model with constitutive behaviour completely defined by the

follow-ing specific Gibbs free energy potential function gV D, :

,2

which defines the linear elastic range ª¬ Dk ;k D º¼ for the generalised stress

F Differentiation of the yield surface gives

The incremental response of the hyperplastic model is found from Equations

2 2

, when 02

occurs (“elastic” behaviour), D  0:

SS

The constitutive behaviour described by these incremental equations is

shown in Figure 5.5 for a first loading with positive (OAB) or negative (OCD)

Figure 5.5 Behaviour of model with linear isotropic hardening: (a) in true stress space; (b) in

gener-alised stress space

plastic strain, followed by an unloading From Figure 5.5a, it is clearly seen

that the initial linear elastic range >k k; @ of the model undergoes expansion

For plastic loading, we observe strain hardening with the tangent modulus

1

E EH E H

In generalised stress space, where the effect of the elastic strain is eliminated,

the behaviour appears as rigid-plastic with linear strain hardening/softening, as

in Figure 5.5b

A further modification is required if this isotropic hardening model is to

pro-vide realistic modelling on subsequent cycles of unloading and reloading The

expression k D k H D would imply softening if (after first having positive

plastic straining) D were later to be reduced The hardening should be a

func-tion of the accumulated plastic strain This can be achieved by introducing

a second kinematic variable E, which is defined through the constraint equation

0

c E  D   The hardening is then expressed in the form k E  Ek H For

a first loading, this modification makes no difference to the model, but it gives

realistic behaviour on unloading, as illustrated in Figure 5.6

Multidimensional Example (the von Mises Yield Surface)

The following model is an example of isotropic hardening hyperplasticity in

six-dimensional stress space The constitutive behaviour of the model is again

de-fined by two potential functions In this case, these are supplemented by the

plastic incompressibility condition, Dkk 0, which is introduced as a constraint

(see Chapter 4) The first function is the specific Gibbs free energy:

ij ij ll kk ij ij ij ij g

Figure 5.6 Behaviour of unmodified and modified models with linear isotropic hardening on

reversal of plastic strain direction

where a prime notation is used to denote the deviator of a tensor It follows that

ij ij ij G

13

where k Dijc ! is the strength in simple shear 0

The plastic incompressibility condition is introduced as a side constraint,

0

kk

aug-mented dissipation function,

g

ij ij ij ij ij kk

The deviatoric and hydrostatic parts of the dissipative generalised stress tensor

are obtained by differentiating the augmented dissipation function:

As expected, the mean dissipative generalised stress is undetermined by the

constitutive equation The plastic strain rates are eliminated from Equation

(5.73) generating the von Mises yield function:

The incremental stress-strain response during plastic flow is easily derived using

(5.48)–(5.51):

12

ij ij ij G

13

This response is completely consistent with the stress-strain behaviour of the

following isotropic hardening plasticity model The linear elastic region is

bounded by a von Mises yield surface, which can be expressed in true stress

space as V V ij ijc c 2k Dij 2 (Figure 5.7) Within the yield surface, the stress-0

strain behaviour is governed by Hooke’s law:

1312

ij ij

K G

During plastic flow, these expressions give the elastic component of the total

strain An associated flow rule is implied by the model, together with the plastic

incompressibility condition, so that D OVijc 2 ijc andDkk 0 In linear

harden-ing, when k Dijc k H D D , the stress-strain response of the model to one-ij ijc c

dimensional loading is similar to that presented in Figure 5.5 To obtain realistic

unloading behaviour, it is again necessary to introduce a variable representing

the accumulated plastic strain, e g through the constraint c E  D D   ij ijc c 0

and then defining k E  E The response will then be as for the modified k H

model in Figure 5.6

5.4.3 Kinematic Hardening

One-dimensional Example

Consider a model with constitutive behaviour completely defined by the

follow-ing specific Gibbs free energy potential function gV D, :

,

H g

which defines the linear elastic range >k k; @ for the generalised stress F

Differ-entiation of the yield surface gives

where we can derive

2 2

S, when 02

k kH

The constitutive behaviour described by these incremental equations is shown

in Figure 5.8 From this figure, it is clearly seen that the linear elastic range

>k k; @ of the model does not undergo any expansion, just being translated

along the stress axis following the current stress state In generalised stress

space, where the effects of “back stress” and elastic strain are eliminated, the

behaviour appears as rigid-perfectly plastic (Figure 5.8b)

This behaviour is consistent with the stress-strain behaviour of the St-Venant

model with linear kinematic hardening The original St-Venant model

(Figure 5.2) is a spring with elastic coefficient E in series with a sliding element

with slip stress k This model simulates one-dimensional linear elastic-perfectly

plastic stress-strain behaviour The linear hardening is introduced by

incorporat-ing another sprincorporat-ing with elastic coefficient H (Figure 5.9) in parallel with the

slid-ing element An elongation of the E sprslid-ing gives elastic strain H(e)

, whereas an

elongation of the H spring gives plastic strain D, their sum gives the total strain H

During initial loading OA, before the stress reaches the value of the slip

stress k, the behaviour is linear elastic and is governed by the elongation of the E

Figure 5.8 Cyclic stress-strain behaviour of the St-Venant model with linear kinematic hardening:

(a) in true stress space; (b) in generalised stress space

a wide range of materials – from metals to saturated clays However, most

granu-lar materials exhibit a property called friction, characterized... empirical observation of P* = P + E is ex-plained for dense materials that exhibit dilation As demonstrated in Chapter 4, the Second Law of Thermodynamics demands that mechanical dissipation be... class="page_container" data-page= "5" >

5. 4 Strain Hardening

5. 4.1 Theory of Strain-hardening Hyperplasticity

The general hyperplastic framework was