A Principles of Hyperplasticity Part 7 pdf

8.3.3 Dissipation Functional From the First Law of Thermodynamics, the definition of the mechanical dissi-pation d Chapter 4, it follows that In Chapter 4, the dissipation is defined as

8.3.3 Dissipation Functional

From the First Law of Thermodynamics, the definition of the mechanical

dissi-pation d (Chapter 4), it follows that

In Chapter 4, the dissipation is defined as a function of state and rate of change

of internal variable Similary, we can write a dissipation functional in the form:

which can be compared with Equation (4.7) We use the superscript g simply to

indicate that the arguments of the function are those used for the Gibbs free

energy: numerically d d g

8.3.4 Dissipative Generalised Stress Function

By analogy with the definition

g ij ij

d

w

F wD , we define:

ˆˆˆ

g ij ij

d

w

F

Since ˆd will be first order in ˆ g D for rate-independent materials, it follows ij

from Euler’s theorem that

Equations (8.8) and (8.12) can be compared with Equations (4.6) and (4.8) It

follows from (8.8) and (8.12) that

8.4 Legendre Transformations of the Functionals 137

This condition is entirely consistent with the Second Law of Thermodynamics

but is a more restrictive statement In a similar way to Chapter 4, we shall adopt

it here simply as a constitutive hypothesis: it defines a class of materials that

satisfy thermodynamics Wider classes of materials that satisfy thermodynamics

but violate our constitutive hypothesis could exist However, the class defined by

this hypothesis proves very wide, encompassing realistic descriptions of many

materials Furthermore, these descriptions are very compact in that only two

scalar functionals need to be defined We have adopted the ˆF and ˆF notation to

indicate the fact that these quantities are always equal in our formulation (by

hypothesis), but are separately defined [Equations (8.6) and (8.10)]

8.4 Legendre Transformations of the Functionals

8.4.1 Legendre Transformations of the Energy Functional

In the original approach, a variety of Legendre transformations between energy

functions were used, e g fH D T ij, ij, g V D T  V H Transformations ij, ij,  ij ij

that involve variables (as opposed to functions of internal coordinates) are

simi-lar to the original form, e g fª¬H D Tij,ˆij, º¼ gª¬V D T  V Hij,ˆij, º¼ ij ij Those involving

the internal function and the generalised stress function are slightly more

com-plex Thus instead of the original gV D T ij, ij, g V F T  F D we have ij, ij,  ij ij

ˆ

ij ij

8.4.2 Legendre Transformation of the Dissipation Functional

The only relevant transformation is the singular transformation from the

dissi-pation functional to the yield functional The original transformation was, for

instance, of the form Oy gV D T Fij, ij, , ij F D ij ij d gV D T Dij, ij, ,ij 0 together

with the result

g ij ij

ˆ

g ij

which is the analogy of the normality condition in conventional “associated”

plasticity Note, however, that in the conventional approach, the plastic strain

rate (the internal variable rate) is given by the differential of the yield function

with respect to the stress; here it is given by the differential with respect to the

generalised stress This allows the current formulation to encompass

Chapter 4 demonstrates how, given knowledge of the energy function and the

yield function, it is possible to derive the entire incremental response for an

elastic-plastic material within the adopted formalism This is of particular

im-portance because non-linear material models are frequently implemented in

finite element codes for which an incremental response is required

The derivation of the incremental response begins with differentiation of the

energy function, giving the results summarised in the sixth row of Table 8.1

Further differentiation gives the rates of the variables This is set out for the

single internal variable in general form as Equation (4.15), which, for the

par-ticular case of the Gibbs free energy, takes the following form, easily obtained by

double differentiation of the Gibbs free energy:

Table 8.1 Examples of comparisons between different formulations

Single internal variable Internal function

Equations (8.20)–(8.22) are used together with the flow rule, Equation (8.18), to

The multiplier function ˆO is obtained by substituting the above equations in the

consistency condition, which is obtained by differentiation of the yield function

Equation (4.17) results in the condition,

8.5 Incremental Response 141

Note that Equations (8.29)–(8.32) are analogous to Equations (4.18)(4.21)

Finally, Equation (8.29) is substituted in Equations (8.23)–(8.25) to obtain the

complete incremental relationships, which can be expressed In a similar way to

Equation (4.23):

Thus we can see that the entire constitutive response of the material pressed through the incremental stress-strain relationships and the evolution equations for internal variables) can be derived from the original two thermo-dynamic functionals

(ex-In Chapter 4 we discuss a number of cases in which constraints are imposed (for example, on the rates of the internal variables) Constraints may also be necessary within this new formulation, but have not been addressed here The purpose here has been to set out the basic theory of a new approach to plasticity theory with an infinite number of yield surfaces The following chapters will pursue examples in detail It is useful, however, to set out a simple example to demonstrate how the formalism can be used In section 8.6 we develop the gen-eral equations for a kinematic hardening plasticity model, and in section 8.7 describe a particular model for the one-dimensional case

8.6 Kinematic Hardening with Infinitely Many Yield Surfaces

The advantage of the multiple surface models is clearly that they are able to fit the non-linear behaviour of certain materials more accurately across a wide range of strain amplitudes This is important, for instance, in modelling geo-technical materials (Houlsby, 1999) The disadvantage is that a large number of material parameters (associated with each yield surface) are necessary In this section, we take the modelling of non-linearity to its logical conclusion by intro-ducing an infinite number of yield surfaces Paradoxically, this reduces the number of material parameters required to specify the models, although at the expense that certain functions also have to be chosen

8.6.1 Potential Functionals

The hyperplastic formulation for multiple yield surfaces (Section 7.3) can be further extended to describe a continuous field of kinematic hardening yield surfaces of the type originally suggested by Mroz and Norris (1982) The general formulation of continuous hyperplastic models is given in Sections 8.2–8.5 Be-low we show how the continuous hyperplastic models are capable of reproduc-ing decoupled associated kinematic hardening plasticity with a continuous field

of yield surfaces For this case, the specific Gibbs free energy is a functional (rather than function) of the stress and an internal variable function Dˆij K :

8.6 Kinematic Hardening with Infinitely Many Yield Surfaces 143

where Y is the domain of K The function * K is a weighting function, such

that * K K is the fraction of the total number of the yield surfaces having d

a dimensionless size parameter between K and K  K d

The dissipation functional, which is a functional of internal variable function

and its rate Dˆij K , is also required:

Furthermore, in the following, only dissipation functionals with no dependence

on stress are considered This automatically leads to models in which the flow

rule (in the conventional sense in plasticity theory) is associated

8.6.2 Link to Conventional Plasticity

The field of yield functions is related to the function ˆg ,ˆ ,ˆ , 

ij ij ij

the dissipation functional (8.43) by the Legendre transform (see Appendix C),

where the rate of internal variable function Dˆij K is interchanged with the

dissipative generalised stress function F K Noting that here we are consider-ˆij

ing only cases where the dissipation does not depend on the stress, the

dissipa-tive generalised stress function Fˆij K is defined by

The transformation from the dissipation to the yield function is a degenerate

special case of the Legendre transformation due to the fact that the dissipation is

homogeneous and first order in the rates Therefore, this transformation results

in the following identity:

As seen from Equation (8.45), a complete field of yield functions is contained

in the equation of the dissipation functional (8.43) in a compact form

The Gibbs free energy functional (8.42) allows the definition of the strain

ten-sor:

1

ˆ,

Here ˆij d

8

plastic strain Hij p It is convenient also to define the elastic strain

The flow rule for the field of yield surfaces is obtained from the properties of

the degenerate special case of the Legendre transformation (8.45) relating yield

and dissipation functions (see Appendix C):

ˆ ˆˆ

ˆ

g ij

We restricted the dissipation function to exhibit no explicit dependence on the

true stresses, so again it follows that the normality presented by Equation (8.47)

in the generalised stress space also holds in the true stress space

The dependence of the dissipation functional on the internal variable

func-tion Dˆij K is transferred to the field of yield functions by the Legendre

trans-formation (8.45) where Dˆij K plays the role of a passive variable Therefore,

the strain hardening rule is obtained automatically through the functional

de-pendence of the yield function on the internal variable (or plastic strain)

func-tion Dˆij K

The generalised stress function is defined by Frechet differentiation of the

Gibbs free energy functional (8.42) with respect to the internal variable function,

U K associated with the internal variable function and defined as the

differ-ence between the true stress and generalised stress function By applying

function can be expressed as

Equation (8.50) is interpreted as the translation rule for the field of yield

sur-faces when the dissipation function (and hence also the yield function) exhibits

no explicit dependence on the true stresses

8.6 Kinematic Hardening with Infinitely Many Yield Surfaces 145

8.6.3 Incremental Response

Two possibilities exist for each value of K If the material state is within the yield

surface, yˆgDˆij K F K K  , no dissipation occurs and ,ˆij ,  0 O K If the ˆ 0

material point lies on the yield surface, ˆy gDˆij K F K K , then plastic ,ˆij ,  0

deformation can occur provided that O K t In the latter case, the incremental ˆ 0

response is obtained by invoking the consistency condition of the field of yield

g ij ij

Differentiation of Equation (8.46) and substitution of (8.47) in both the result

and in (8.50) gives the incremental stress-strain response,

d

8

ww

ˆ

g ij

y D K F K K  or when (8.52) gives a negative value of O K ] ˆ

Description of the constitutive behaviour during any loading requires a

pro-cedure for keeping track of F Kˆij and Dˆij K K8 , and this is achieved by ,

using Equations (8.54) and (8.55)

8.7 Example: One-dimensional Continuous Hyperplastic

Model

We now give a simple example of a continuous hyperplasticity model to

intro-duce the techniques used to manipulate the functionals to derive a constitutive

E g

ˆ

g

d E

a

 Kw

Derivation of the stress-strain law involves noting that, on first yield from an

initially unstressed state, combination of (8.61) and (8.62) gives (together with

ˆF F ), for the elements that are yielding,

ˆ  1  3ˆ

S

E k

8.7 Example: One-dimensional Continuous Hyperplastic Model 147

ˆS

 K

where K specifies the size of the largest yield surface which has become active *

and V  Kk * S D ˆ 0 For simplicity, now consider monotonic loading for which

*

3 0

1

1

a d

(8.70)

This may be readily integrated, noting that appropriate initial conditions can be

obtained from Equations (8.66) and (8.68), to give the hyperbolic stress-strain

The asymptotic strength is V , the initial stiffness E, and the secant stiffness k

at V k 2 is E a

It can be shown that on stress reversal the model gives pure kinematic

hard-ening behaviour, and the stress-strain curve obeys the “Masing” rules

8.8 Calibration of Continuous Kinematic Hardening Models

In the above example, we showed how the choice of a particular pair of

func-tionals leads to a material model with a specific stress-strain curve In certain

circumstances, it may be of value to reverse this process We may wish to

cali-brate the model by specifying the shape of the stress-strain curve and from this,

deduce the form of the functionals

In defining a model using the above approach, there is considerable freedom

in the way the functionals can be expressed For instance, in generalizing the

Iwan model with a finite number of yield functions to a model with an infinite

number, there is a choice of

x the way the hardening stiffnesses (plastic moduli) hˆ K are distributed,

x the way the strength parameters kˆ K are distributed,

x the use of the weighting function * K

In this chapter, we explore two alternative ways of calibrating models First, in

Section 8.9, we show how the weighting function * K can be calibrated,

us-ing a one-dimensional model example Next, in Section 8.10, calibration of the

plastic moduli function is demonstrated in a multidimensional example

8.9 Example: Calibration of the Weighting Function

8.9.1 Formulation of the One-dimensional Model

In this one-dimensional example, we consider that hˆ K H is constant, and

ˆ

k K K The function k * K is left undetermined at first, and it will be seen that

the form of this function will be determined from the shape of the stress-strain

curve The model constitutive behaviour is defined by two potential functionals:

8.9 Example: Calibration of the Weighting Function 149

The back stress is defined from (8.49):

ˆ2

ˆ

g H

w

Differentiation of (8.74) yields wˆy g wF D Kˆ Sˆ  and wyˆg wD , so that ˆ 0

the incremental Equations (8.52)–(8.55) reduce to:

1 0

For each yield function, one out of two cases takes place If O K d , no dis-ˆ 0

sipation related to the Kth yield function occurs, so that D K ˆ 0 and ˆF K V 

Alternatively, O K ! , in which case dissipation occurs (“plastic” behaviour), ˆ 0

so that F K ˆ 0 and for monotonic loading, substitution of (8.78) in (8.77) and

(8.76) gives

* 0

where *K is the largest K for which F K  K ˆ k 0

8.9.2 Analogy with the Extended Iwan’s Model

This incremental behaviour is identical to the constitutive behaviour of the

con-tinuous Iwan (1967) model, defined as an extension of the Iwan model described

in Section 7.4 In this extension, the number of slip elements with slip stresses

k K Kk is continuous and described by the distribution function * K , so

that * K K d is the fraction of the total number of slip elements having slip

stress between kK and kK  Kd  This model simulates one-dimensional

elas-tic-non-linear plastic stress-strain behaviour Elongation of the E spring gives

elastic strain He , and elongation of the distribution of the H K springs

con-tributes the plastic strain D Kˆ to the total plastic strain; their sum gives the

total strain H It is assumed here that the elastic coefficients of all H K springs

are the same and equal to H

func-tionals leads to a material model with a specific stress-strain curve In certain

circumstances, it may be of value to reverse this process We may