A Principles of Hyperplasticity part 11 pps

The figure shows the characteristic response of the rate process theory: a linear increase in strength with strain rate at low strain rate appearing as almost a constant strength on the

Figure 11.8 Variation of undrained strength with strain rate: comparison between theory and data from Vaid and Campanella (1977)

not captured accurately Figure 11.8 shows the variation of peak strength with strain rate as observed by Vaid and Campanella, with the theoretical results

(open circles) superimposed The figure shows the characteristic response of the

rate process theory: a linear increase in strength with strain rate at low strain rate (appearing as almost a constant strength on the logarithmic plot) and a linear

Figure 11.9 Undrained stress-strain curves at different strain rates: (a) data from Vaid and ella (1977) and (b) theoretical curves

Campan-increase with the logarithm of the strain rate at high strain rates The strain rate

at which the transition occurs is characterised by the parameter r, and the

inter-section of the two straight line inter-sections indicated on Figure 11.8 occurs at 2

r

H  The slope of the section at high strain rate is approximately Prlog 10e Figure 11.9a shows detail from tests at two strain rates, together with a test in which the strain rate was suddenly changed from the lower to the higher rate at

a strain of about 0.8% Figure 11.9b shows the equivalent theoretical calculation The model is clearly able to capture accurately the transition, which appears as the line connecting the two main curves

Figure 11.10a shows the results of constant stress creep tests, in which strain

is plotted against time for different constant stress values, and Figure 11.10b

Figure 11.10 Comparison of (a) creep data from Vaid and Campanella (1977) and (b) the cal curves

theoreti-Figure 11.11 Creep data in terms of strain rate: (a) data from Vaid and Campanella (1977) and (b) the theoretical model

shows the equivalent calculations Apart from the detail of the curve at

1c 0.5

c

V V , the agreement is very close Figures 11.11a and 11.11b show the

same data presented in terms of strain rate against time Above a certain stress

level, the phenomenon of creep rupture occurs, and Figure 11.12 shows Vaid

and Campanella’s data for the time to rupture against the stress level The

super-imposed open circles show the theoretical calculations No creep rupture is

pre-dicted at stress ratios lower than 0.485

It is remarkable that a model entirely encapsulated by the two potential

func-tions, defined in Equations (11.101)–(11.103) and using only six material

pa-rameters (each of which can be given a clearly defined physical interpretation),

is able to capture the diversity of behaviour shown in Figures 11.7–11.12

11.5.4 Extension of the Model to Three Dimensions

The above model may be extended to three dimensions by generalizing

where K and G are the bulk and shear moduli and the other parameters retain

their original meanings (except that ˆh now bears the same relationship to G as

it previously did to E) A prime is used to indicate the deviator of a tensor

Dif-ferentiation of (11.112) gives

1 0

g ij

ij

kl kl

k w

Further differentiation of (11.113) with respect to time and substitution of

(11.114) then leads to the incremental stress-strain relationship

11.6 Advantages of the Rate-dependent Formulation

The benefits of extending the rate-independent framework to include rate

de-pendency for materials that exhibit any significant rate effects are obvious

There is an additional benefit, however, which is worth a separate discussion It

is well known that when rate-independent behaviour is described as the limiting

case of rate-dependent behaviour, significant simplifications in calculations can

be achieved Finite element calculations of purely plastic behaviour can, for

instance, be efficiently carried out using a viscoplastic algorithm with an

artifi-cial very small viscosity; see, for instance, Owen and Hinton (1980) A

compari-son between the incremental response expressions (11.85) and similar

expres-sions for the rate-independent case developed in Chapter 8 is a good illustration

of the greater simplicity of the rate-dependent equations The source of the

higher complexity of the rate-independent expressions is the necessity to satisfy

the consistency condition for yield In other words, it is necessary to make sure

that the stress state in plastic loading always stays on the yield surface This may

also cause significant numerical difficulties, requiring treatment by special

pro-cedures The rate-dependent framework is free from these limitations

Behaviour of Porous Continua

12.1 Introduction

In previous chapters, we have developed a theory for plastic materials in which the entire constitutive response is determined by specification of two potential functions There are many other areas of continuum mechanics where similar approaches have been made For instance, Ziegler develops theories for viscous materials Many authors treat flow processes within a thermodynamic context and frequently use a dissipation function The special features of rate-indepen-dent materials have been the reason for a slightly different emphasis here, from that in most treatments of the subject

We now explore how the hyperplasticity approach can be generalised and set within the context of a wider variety of types of material behaviour In particu-lar, we shall continue to emphasize the use of two potential functions and the use of Legendre transformations to obtain alternative formulations

When more complex materials are considered, there are two classes of pative behaviour The first is associated with fluxes, for instance flow in a porous medium or the flow of electrical current In these cases, the dissipation is associ-

dissi-ated with the spatial gradient of some variable (e g the hydraulic head for flow

in a porous medium, the voltage for an electrical problem) Constitutive iour is usually described by a linear relationship between the flux and the spatial gradient

behav-The second type of dissipation is associated with the temporal variation of ternal variables The plasticity problems treated in earlier chapters are of this character Viscous behaviour can also be described in this way

in-Most texts that treat the thermodynamics of dissipative continua concentrate either on fluxes or on rates of change of internal variables However, whilst the two problems have much in common, they also have important differences Most obviously, one involves a spatial variation and the other a temporal varia-tion It is tempting to treat both in the same way, and many texts adopt this

approach, using, for instance, “generalised forces” and “generalised fluxes” Here, we adopt a slightly different approach, keeping separate those variables associated with fluxes and those associated with internal variables In this way, the different ways that the two types of process appear in the relevant equations can be made clearer

Rather than considering the possibility of abstract, unspecified fluxes, we find

it more useful to consider a concrete example The case that we consider is

a very important problem in geomechanics and other fields, namely, flow in

a porous medium This is a useful example because the flux itself has mass, which introduces a number of features to the problem that need careful treat-ment The porous medium has to be treated as consisting of two phases, and

there is a partition of the extensive quantities (e g internal energy, entropy)

between the solid skeleton and fluid phases

In previous chapters, we adopted a small strain formulation The problem of coupled fluid and skeleton behaviour cannot be treated rigorously within the small strain framework, because there is a coupling between strains, fluid flow, and density changes In the small strain formulation, the density is treated as

a constant In the following therefore, it is necessary to move to a large strain formulation There is a choice between adopting a Lagrangian approach, in which the problem is formulated in terms of initial coordinates, and an Eulerian approach, in which it is formulated within the current coordinates We adopt the Eulerian approach for much of the following development because this allows

a more direct interpretation of the variables It will prove necessary, however, to transform to Lagrangian variables for part of the analysis

In the small strain approach, for convenience, all extensive quantities were defined per unit volume Since the density was in effect constant, this is equiva-lent to using extensive quantities per unit mass, but avoids a factor of the den-sity appearing throughout the equations In large strain analysis, it is necessary

to use extensive quantities per unit mass, as is usual in thermodynamics, and we adopt this approach below

As mentioned above, we adopt here an Eulerian approach to describe a material

undergoing large strain, i e the description of the material is based on the

cur-rent coordinate system In this, it will be necessary to distinguish between the

time differential of a variable x at a particular point in space, which we shall

denote by w w  , and the material or convective derivative, which represents x t x

the rate of change of an element of the material, which has a current velocity v i

We denote the material derivative by dx dt x x x v  ,i i From this definition,

it follows that the chain rule applies to the convective derivative, e g the rial derivative of xy is xy xy  

mate-12.2.1 Density Definitions, Velocities, and Balance Laws

Consider a volume V fixed in space bounded by a surface S The unit outward

normal to the boundary is n The volume contains porous material with i

a skeleton material of density U and with a porosity n (volume of voids divided s

by total volume) Thus the mass of skeleton per unit total volume is

U  U We should also note that U is the “dry density” in the terminology

of soil mechanics

The velocity of the skeleton at any point is v , so that the mass flux of the i

skeleton per unit area is U , and the outward mass flux per unit area from V is v i

i i

v n

U For conservation of mass, we can write that the rate of increase of mass

within the volume, plus the outward mass flux is zero:

which establishes the link between the material rate of change of dry density and

the dilatation rate

A comment is relevant here about the importance of the assumption that the

volume V is arbitrary This is only justified provided that V is large enough so

that averaged values of stresses, strains, etc., over the volume element are

mean-ingful Such an element is said to be a “representative volume element” In the

context of the mechanics of granular materials, this will typically require that the

element contains many thousands of particles At the same time, the element

must be sufficiently small so that changes of stresses, etc., across the element are

small This requirement conflicts with the first, and there are classes of problems

for which both criteria cannot be satisfied simultaneously Such problems (e g

those involving strong localisation) are not amenable to treatment by

conven-tional continuum mechanics

We now allow for the possibility of fluxes of a pore fluid We shall consider

a pore fluid, the amount of which is specified by the parameter w defined as

mass of fluid per unit mass of skeleton material (i e the water content in the

terminology of soil mechanics) Note that in the study of the mechanics of

granular media, a wide variety of different quantities are used to define the

amount of fluid in the porous medium The flux of the fluid mass is m per unit i

area relative to the skeleton The total flux vector of the fluid is therefore

1

In the above terminology, Gauss’s divergence theorem states that for any variable x that is

con-tinuous and differentiable in V, ³xn dS i ³x dV,i

where w is the Darcy artificial seepage velocity and i v is the average absolute i w

velocity of the fluid

Noting that the mass of the fluid is conserved, there is a balance equation

analogous to (12.1) of the form:

which we can rewrite in local form by using the divergence theorem of Gauss to

obtain the local conservation law:

It is convenient to obtain a combined continuity equation for flow of the

skeleton and pore fluid First, we can note that U U s1n Us n , so that we can

rewrite the mass continuity equation as

If both the soil grains and the pore fluid are incompressible, then this reduces to

the simple form v i i, w i i, Introducing 0 v w U and 1 w v s U , the con-1 s

tinuity equation can also be written v i i, w i i, m v i i,w Uwvw Uv , where s

12.2.2 Tractions, Stresses, Work, and Energy

The tractions (forces per unit area) on the skeleton on the fraction 1 n  of the

boundary S are t , and the pressure in the pore fluid is p which acts on a fraction i

n of the boundary The work done per unit area by the surroundings against the

tractions on S is therefore 1n t vi i, and that done against the pore pressure is

w

i i

npn v

 There are also body forces arising from the gravitational field of

strength g The work done per unit volume by the body forces on the skeleton i

is Uv g i i and on the fluid is Uwv g i w i The heat flux per unit area is q , so that the i

outward heat flux from S per unit area is q n i i

As an extensive quantity, the kinetic energy of all the matter enclosed in

vol-ume V may be written as the sum of the kinetic energies of the skeleton and of

At this stage, we are neglecting tortuosity effects, which are due to the fact

that the pore fluid must take a tortuous path between the skeleton particles, so

that the average speed of the water particles is higher than the magnitude of the

average velocity We shall, however, show how the results can be modified later

to take this into account Now consider the rate of change in kinetic energy in

the volume V, which can be written

The volume integral reflects changes in kinetic energy with time in the

vol-ume, and the surface integrals account for the kinetic energy brought into the

volume due to the skeleton and pore fluid movement through the surface

Ap-plying the theorem of Gauss and grouping the resulting terms,

Recalling the mass balance equations for the skeleton and for the fluid, (12.2)

and (12.6), respectively, we note that the second and fourth integrals vanish We

introduce also the definitions of the accelerations of the skeleton and fluid

12.2.3 The First Law

The First Law of Thermodynamics states that there is a variable, called specific

internal energy, such that the rate of increase of internal energy in the volume

plus the rate of change of the kinetic energy in this volume is equal to the sum of

the rates of energy input at the boundaries plus the rate of work of the body

forces in the volume We attribute a specific internal energy u to the skeleton s

and u to the pore fluid The first law therefore becomes w

We can note that the tractions and pore pressure are related to the stresses by

We can then decompose v i j, into its symmetrical and antisymmetric parts,

identifying the former as the strain rate and the latter as the vorticity tensor:

After substitution of (12.22)–(12.25) in (12.21), we can write

No change in internal energy should, however, be caused by either a ri-

gid body translation or rotation, so that we can conclude that

Vij i,  U 1 w g j Ua j Uwa w j v j and 0 V Z for all ij ji 0 v j and Z These ji

are the virtual work forms of the direct and rotational equilibrium conditions

From the latter, it follows that the antisymmetric part of V must be zero, i e ij

ij

V is symmetrical This condition is usually referred to as that of complementary

shear stresses From the former, it follows that

which can be recognised as the equations of motion (or the static equilibrium

equations in the case of zero acceleration) Equation (12.27) expresses the

mo-mentum balance for the porous medium considered as a whole and has been

derived as part of a formulation rather than postulated However, this equation

is not sufficient to describe of the momentum balance of the pore fluid, which

cannot be derived until some constitutive statement is made about interaction

between the fluid and skeleton The missing fluid balance equation will be

de-rived later as part of the formulation

In view of Equation (12.27), Equation (12.26) reduces to

12.2.5 The Second Law

The Second Law of Thermodynamics can be stated in a number of different

ways We state it here in the form that there exists a function of state, the specific

entropy s, such that the rate of entropy production is non-negative We attribute