Thermomechanical formulation of a small-strain model for ... - CiteSeerX

10.1098/rspa.2000.0673

Thermomechanical formulation of a small-strain model for overconsolidated clays By A. M. P u z r i n1 , G. T. H o u l s b y2 a n d J. B. B u r l a n d3 1

Faculty of Civil Engineering, Technion|Israel Institute of Technology, Haifa 3200, Israel 2 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK 3 Department of Civil and Environmental Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BU, UK Received 8 September 1999; revised 13 April 2000; accepted 24 July 2000

A constitutive model for the stress{strain behaviour of overconsolidated clays at small and intermediate strains is presented. The model had previously been formulated by Puzrin & Burland in terms of classical plasticity theory with certain additional assumptions. It is presented here within a rigorous formalism based on thermomechanical considerations, described by Puzrin & Houlsby and termed the `continuous hyperplastic’ method. The entire constitutive behaviour is derived from the speci cation of two scalar functionals. The model serves as the rst example of the derivation of a non-trivial constitutive model within this approach. Keywords: clay; functional; hyperplasticity; plasticity; small strain; soil

1. Introduction Puzrin & Burland (1998) presented a plasticity model for the small and intermediate strain behaviour of overconsolidated clays. The purpose of the model was to reproduce the characteristic S-shaped curve in a plot of normalized secant shear sti¬ness G=G0 against shear strain log(® ), and to generalize this variation of sti¬ness to any stress path. The model was based on the concepts of plasticity theory. It employed an inner yield surface which encloses a region of stress space in which the response is purely elastic, and an outer surface representing the outer boundary of nonlinear behaviour. Once the outer surface is reached, further monotonic loading is at a constant sti¬ness. The plastic strains which occur if the stress point lies between the inner and outer surfaces were calculated using an empirically determined relationship, logarithmic in form (Puzrin & Burland 1996), that was found to t the behaviour of clays well. The model was rigorously calibrated against a number of tests on overconsolidated clay. The model does not incorporate any features related to gross plastic strains or to failure, and these would need to be added for a comprehensive description of soil behaviour. A concern that arises in the formulation of complex plasticity models is that the resulting stress{strain law may be such that the model material violates thermodynamic principles. One way of addressing this problem is to base plasticity theory on a ® c 2001 The Royal Society

Proc. R. Soc. Lond. A (2001) 457, 425{440

425

426

A. M. Puzrin, G. T. Houlsby and J. B. Burland

formulation which automatically obeys the laws of thermodynamics. Houlsby (1981) developed such an approach, based on the work of Ziegler (1983). This approach has been further developed by Collins & Houlsby (1997) and by Houlsby & Puzrin (2000). In this approach, the material behaviour is entirely determined by the speci cation of two scalar functions, conveniently taken as the Gibbs free energy and the dissipation function. The entire response, including the incremental stress{strain relationships, is determined by standard manipulation of these functions, as set out by Houlsby & Puzrin (2000). By analogy with elasticity theory, in which elastic behaviour derived from potential functions is termed `hyperelastic’ (Fung 1965), we shall use the term `hyperplastic’ to describe the models using the formulation of Houlsby & Puzrin (2000). The work also has points in common with the work of Reddy & Martin (1994), although a stronger emphasis is placed here on the role of the scalar potentials. A limitation of the hyperplastic models is that they result in `hard’ yield surfaces, in which there is a sudden change from elastic to elastic{plastic behaviour, and the theory cannot accommodate models, such as that developed by Puzrin & Burland (1998), in which there is a smooth change of behaviour. This problem was addressed by Puzrin & Houlsby (2001). They modi ed the hyperplastic formulation so that the material was speci ed in terms of two functionals instead of functions. (A functional is loosely de ned as a function of a function.) In e¬ect, this approach replaces the concept of a single plastic strain with that of a eld of an in nite number of plastic strain components. Single yield surfaces are correspondingly replaced by a eld of an in nite number of yield surfaces. We shall term such models `continuous hyperplastic’ models. The concept may be thought of as an extension of the use of multiple yield surfaces (see, for example, Mroz & Norris 1982). The èndochronic’ theories proposed by Valanis (1975) and Bazant (1978) also set out to derive smooth transitions between elastic and plastic behaviour. The work presented here, however, has little in common with the endochronic approach in which (a) it is attempted to capture the e¬ects of history in a single ìntrinsic time’ and (b) yield surfaces are abandoned (although later re-introduced in some variants of the theories). In contrast, we capture the e¬ects of history using a tensorial function, and a eld of yield surfaces is central to our development. The purpose of this paper is to reformulate the model presented by Puzrin & Burland (1998) within the continuous hyperplastic method. This is an important development for two reasons. Firstly, it demonstrates that the Puzrin & Burland (1998) model, does, in fact, obey thermodynamic principles, in spite of the fact that these were not embodied in its original empirical basis. Secondly, this is the rst time the continuous hyperplastic approach has been used to de ne a non-trivial model for soil behaviour, and so the way in which the model is developed is important. Because the Puzrin & Burland (1998) model has already been demonstrated to reproduce the behaviour of overconsolidated clays well, further comparisons between the model and test data are not given here. Instead we simply demonstrate that the new formulation results in the same material behaviour as the previous one. For proportional loading paths from the initial stress point, the continuous hyperplastic model produces identical behaviour to the earlier model. For other paths, including those with multiple stress reversals, the continuous hyperplastic model behaviour is identical to that of a classical plasticity model with an in nite number of kinematic Proc. R. Soc. Lond. A (2001)

A small-strain model for clays

427

hardening elliptical yield surfaces, each one with the associated ®ow rule and Ziegler’s translation rule. In the following, all stresses are è¬ective stresses’ in the sense used in soil mechanics; in other words, they are the total stresses minus the pore ®uid pressures. It is these stresses which control the soil skeleton strains, and are indeed work-conjugate to them (Houlsby 1979). For brevity, we omit, however, the prime notation for e¬ective stress. Following the convention usual in soil mechanics, compressive stresses and strains are taken to be positive.

2. Formulation of kinematic hardening continuous hyperplastic models (a) Potential functionals The general formulation of continuous hyperplastic models is given by Puzrin & Houlsby (2001). It is not possible to reproduce here all the stages in the development of this approach, and reference to Houlsby & Puzrin (2000) and Puzrin & Houlsby (2001) is necessary for de nitions. The continuous hyperplastic models are capable of reproducing decoupled associated kinematic hardening plasticity with a continuous eld of yield surfaces. For this case, the speci c Gibbs free energy is a functional of stress and the internal variable function ¬ ^ ij (² ), g[¼

ij ; ¬ ^ ij ]

= g1 (¼

ij )

¡ ¼

¬ ^ ij (² )¡ (² ) d² +

ij Y

g^2 (¬ ^ ij (² ); ² )¡ (² ) d² ;

(2.1)

Y

where Y is the domain of ² . The function ¡ (² ) is a distribution function, such that ¡ (² ) d² is the fraction of the total number of the yield surfaces having a dimensionless size parameter between ² and ² + d² . The second functional, required for de ning the constitutive behaviour, is the dissipation functional, which is a functional of the internal variable function and its rate ¬ ^_ ij (² ), dg [¼

_

ij ; ¬ ^ ij ; ¬ ^ ij ]

d^g (¼

= Y

ij ; ¬ ^ ij (²

); ¬ ^_ ij (² ); ² )¡ (² ) d² >0:

(2.2)

Furthermore, in the following, only dissipation functionals with no dependence on stress are considered. This automatically leads to models in which the ®ow rule (in the conventional sense in plasticity theory) is associated. (b) Field of yield functions The eld of yield functions is related to the function d^g (¼ ij ; ¬ ^ ij ; ¬ ^_ ij ; ² ) of the dissipation functional (2.2) by the Legendre transform (see appendices to Collins & Houlsby 1997; Puzrin & Houlsby 2001), where the rate of internal variable function ¬ ^_ ij (² ) is interchanged with the dissipative generalized stress function À îj (² ). Noting that here we are considering only cases where the dissipation does not depend on the stress, the dissipative generalized stress function À îj (² ) is de ned by

À îj (² ) = Proc. R. Soc. Lond. A (2001)

@ d^g (^ ¬ ij (² ); ¬ ^_ ij (² ); ² ) : @ ¬ ^_ ij (² )

(2.3)

428


For brevity in the following we shall omit the list of arguments where a function appears within a di¬erential, so that the above is written simply as

À _ ij (² ) =

@ d^g : @ ¬ ^_ ij

It is necessary, however, to be rather precise about the arguments of the functions. The transformation from the dissipation to yield function is a degenerate special case of the Legendre transformation due to the fact that the dissipation is homogeneous and rst order in the rates. Therefore, this transformation results in the following identity: ¶^ (² )^ y g (¬ ^ ij ; À îj ; ² ) = À îj (² )¬ ^_ ij (² ) ¡

d^g (¬ ^ ij ; ¬ ^_ ij ; ² ) = 0;

(2.4)

where y^g (À îj (² ); ¬ ^ ij (² ); ² ) = 0 is the eld of yield functions and ¶ ^ (² ) is a non-negative multiplier. As is seen from (2.4), a complete eld of yield functions is contained in the equation of the dissipation functional (2.2) in a compact form. (c) Flow rule The ®ow rule for the eld of yield surfaces is obtained from the properties of the degenerate special case of the Legendre transformation (2.4) relating yield and dissipation functions (see appendices to Collins & Houlsby 1997; Puzrin & Houlsby 2001), ¬ ^_ ij (² ) = ¶ ^ (² )

@ y^g : @ À îj

(2.5)

Since we restricted the dissipation function to exhibit no explicit dependence on the true stresses, it follows that normality presented by (2.5) in the generalized stress space also holds in the true stress space. The Gibbs free energy functional (2.1) allows for de nition of the strain tensor, "ij = ¡

@g[¼

ij ; ¬ ^ ij ]

@¼

ij

=¡

@g1 + @¼ ij

¬ ^ ij (² )¡ (² ) d² :

(2.6)

Y

The interpretation of the above is that the quantity ¬ ^ ij (² )¡ (² ) d² Y

plays exactly the same role as the conventionally de ned plastic strain "pij . It is convenient also to de ne the elastic strain "eij = ¡ @g1 =@¼ ij . (d ) Strain hardening rule The dependence of the dissipation functional on the internal variable function ¬ ^ ij (² ) is transferred to the eld of yield functions by the Legendre transformation (2.4), where ¬ ^ ij (² ) plays the role of a passive variable. Therefore, the strain hardening rule is obtained automatically through the functional dependence of the yield function on the internal variable (or plastic strain) function ¬ ^ ij (² ). Proc. R. Soc. Lond. A (2001)


429

(e) Translation rule The generalized stress function is de ned by Frechet di¬erentiation of the Gibbs free energy functional (2.1) with respect to the internal variable function, resulting in @^ g2 : (2.7) À ·îj (² ) = ¼ ij ¡ @ ¬ ^ ij It is convenient to introduce the `back stress’ function associated with the internal variable function and simply de ned as a di¬erence between the true stress and generalized stress function. By applying Ziegler’s orthogonality principle in the form À îj (² ) = À ·îj (² ), the back stress function can be expressed as @g2 » îj (² ) = ¼ ij ¡ À îj (² ) = ; (2.8) @ ¬ ^ ij which, after di¬erentiation, yields » ^_ij (² ) = ¼ _ ij ¡

À ^_ ij (² ) =

@ 2 g^2 _ ¬ ^ kl (² ): @ ¬ ^ ij @ ¬ ^ kl

(2.9)

Equation (2.9) can be interpreted as a translation rule for the eld of yield surfaces in the case when the dissipation function (and hence also the yield function) exhibits no explicit dependence on the true stresses. (f ) Incremental response For each value of ² two possibilities exist. Either the material state is within the yield surface (^ yg (^ ¬ ij (² ); À îj (² ); ² ) < 0), in which case no dissipation occurs and ^ ¶ (² ) = 0. If the material point lies on the yield surface (^ y g (¬ ^ ij (² ); À ^ ij (² ); ² ) = 0), ^ then plastic deformation can occur provided that ¶ (² ) >0. In the latter case, the incremental response is obtained by invoking the consistency condition of the eld of yield surfaces, @ y^g _ @ y^g _ y^_ g (À îj (² ); ¬ ^ ij (² ); ² ) = ¬ ^ ij (² ) + (2.10) À ^ ij (² ) = 0: @ ¬ ^ ij @ À îj Substitution of (2.5) and (2.9) into (2.10) leads to the solution for the multiplier ¶ ^ (² ), ¶ ^ (² ) =

@ y^g ¼ _ ij @ À îj

@ y^g @ 2 g^2 @ y^g ¡ @ À îj @ ¬ ^ ij @ ¬ ^ kl @ À ^kl

@ y^g @ y^g : @ ¬ ^ ij @ À îj

(2.11)

Di¬erentiation of (2.6) and substitution of (2.5) into both the di¬erential and (2.9) give the incremental stress{strain response @g1 @ y^g "_ij = ¡ ¼ _ kl + ¡ (² ) d² (2.12) h¶ ^ (² )i @¼ ij @¼ kl @ À ^kl Y and the update equations for the size and position of the eld of yield surfaces involved in plastic loading @ y^g ¬ ^_ ij (² ) = h¶ ^ (² )i ; (2.13) @ À ^kl @ 2 g^2 @ y^g » ^_ ij (² ) = h¶ ^ (² )i ; (2.14) @ ¬ ^ ij @ ¬ ^ kl @ À ^kl Proc. R. Soc. Lond. A (2001)

430


where h i are Macauley brackets (i.e. hxi = x, x > 0, hxi = 0, x 60), ¶ ^ (² ) is de ned by (2.11), when y^g (¬ ^ ij (² ); À îj (² ); ² ) = 0 and ¶ ^ (² ) = 0 when y^g (^ ¬ ij (² ); À îj (² ); ² ) < 0. Description of the constitutive behaviour during any loading requires a procedure for keeping a track of » îj (² ) and ¬ ^ ij (² ), ² 2 Y .

3. Continuous hyperplastic formulation of the Puzrin & Burland small-strain model The Puzrin & Burland (1998) model was expressed using the e¬ective stress and strain parameters used to describe triaxial tests in soil mechanics. The e¬ective stresses are de ned by the vector ¾ = fp0 qgT , and the corresponding strains by " = fv; "gT . The initial stresses are ¾0 = fp00 q0 gT . Consider a model with constitutive behaviour de ned by the following two potential functionals. For simplicity, the weighting function ¡ (² ) has been taken as unity and ² is a dimensionless measure of the size of the yield surfaces. (I) The speci c Gibbs free energy functional, g[¾; ®] ^ =¡

1 (¾ 2

¡

¾0 )T D ¡ 1 (¾ ¡ ¡

1

¾0 )T

(¾ ¡

¾0 ) 1 ®(² ^ ) d² +

2 0

0

1

h(² )®(² ^ )T B ¡ 1 ®(² ^ ) d² : (3.1)

(II) The dissipation functional, 1

_ = dg [®]

(ae + (aL ¡

ae )² )

0

^_ )T B ¡ 1 ®(² ^_ ) d² >0; ®(²

(3.2)

^ = f^ where ® ¬ v ¬ ^ s gT is the internal variable vector function and D is an elastic sti¬ness matrix de ned by D=

K¤ J

J 3G¤

(using the anisotropic elastic form described by Graham & Houlsby (1983)). The matrix B is de ned by B=

jDj ¡ 1 1 ¡ m D = ; ¡ m n2 3G¤

where n2 = K ¤ =3G¤ and m = J=3G¤ . This particular form of B simpli es the relationship between the elastic and plastic behaviour. The parameter ae de nes the size of the inner yield surface (within which behaviour is purely elastic), and aL the size of an outer yield surface (de ning the outer limit of the variable sti¬ness region). On further monotonic straining once the outer surface is reached, the sti¬ness is constant. Finally, h(² ) is a function which de nes the shape of the stress{strain curve in the nonlinear region. The speci c form of this function will be addressed later. Proc. R. Soc. Lond. A (2001)


431

4. Derivation of the model from potential functions ^ ) is de ned by applying (2.3) The dissipative generalized stress vector function Â(² to the dissipation functional (3.2), Â(² ^ )=

À ^p À ^q

=

^_ ) a(² )B ¡ 1 ®(² ^_ )T B ¡ 1 ®(² ^_ ) ®(²

;

(4.1)

where a(² ) = ae + ² (aL ¡ ae ). We can observe that the internal variable rates can be eliminated from (4.1), leading to the following continuous eld of elliptical yield functions which are, in fact, independent of ®(² ^ ) y^g (Â(² ^ ); ² ) = Â(² ^ )T B Â(² ^ )¡

a(² )2 = 0;

² 2 [0; 1];

(4.2)

where a(² ) = ae + ² (aL ¡ ae ) is the semi-diameter of each elliptical yield surface in the p direction. In the true stress space, this eld of elliptical yield surfaces is given by (¾ ¡

¾0 ¡

^ ))T B(¾ ¡ ½(²

¾0 ¡

^ )) ¡ ½(²

a(² )2 = 0;

a(² ) 2 [ae ; aL ];

(4.3)

^ ) is the `back stress’ vector function, and the quantity (¾0 + ½(² ^ )) de nes where ½(² the stress coordinates of the centre of each elliptical yield surface. As is seen, the yield functions constituting the eld do not depend explicitly on the internal function ®(² ^ ), therefore the yield surfaces do not undergo any size or shape changes or rotation during plastic loading. The hardening of the model is purely due to kinematic translation. The two yield surfaces bounding this eld, with semi-diameters ae and aL , correspond to the boundaries of the `linear elastic region’ (LER) and `small strain region’ (SSR), respectively, as de ned by Puzrin & Burland (1998). Applying (2.8) to the speci c Gibbs free energy functional (3.1), we obtain the `back stress’ function ^ )= ½(²

» ^p » ^q

=¾¡

¾0 ¡

^ ) = h(² )B ¡ 1 ®(² ^ ): Â(²

(4.4)

The ®ow rule for the eld of yield surfaces is obtained applying (2.5) to the yield functions (4.2), ^_ ) = 2¶ ^ (² )B Â(² ^ ) = 2¶ ^ (² )B(¾ ¡ ®(²

¾0 ¡

^ )); ½(²

(4.5)

where ¶ ^ (² ) is a non-negative multiplier de ned from the consistency condition. As is seen from (4.5), the associated ®ow rule holds separately for each yield surface, both in the generalized and the true stress space. An expression for the strain vector, and its decomposition into elastic and plastic components, is obtained from (2.6) and (3.1), 1

" = "e + "p = D ¡ 1 (¾ ¡

^ ) d² : ®(²

¾0 ) +

(4.6)

0

The translation rule for the eld of the yield surfaces is given by (4.4) and (4.5), ½(² ^_ ) = h(² )B ¡ 1 ®(² ^_ ) = 2h(² )¶ ^ (² )(¾ ¡ Proc. R. Soc. Lond. A (2001)

¾0 ¡

½(² _ )):

(4.7)

432


This rule, known as the Ziegler translation rule (Ziegler 1959), states that during plastic loading the incremental displacement of the centre of each yield surface occurs along the radius-vector connecting this centre with the current stress state. The incremental stress{strain response of the model follows from (2.12), 1

"_ = D ¡ 1 ¾_ +

h¶ ^ (² )iB(¾ ¡

¾0 ¡

^ )) d² ; ½(²

(4.8)

0

where h i are the Macauley brackets and ¶ ^ (² ) is de ned by equation (2.11), when ^ ); ² ) = 0, y^g (Â(² ¶^ (² ) =

(¾ ¡

^ ))B ¾_ ¾0 ¡ ½(² : 2h(² )a2 (² )

(4.9)

The plastic strains are updated using (4.5) and (4.9), ^_ ) = 2h¶ ^ (² )iB(¾ ¡ ®(²

¾0 ¡

^ )): ½(²

(4.10)

Then the `back stresses’ are updated using (4.4).

5. Behaviour of the model during initial proportional loading (a) Elasticity with cross-coupling During initial loading emanating from the initial stress state ¾0 , before the stress state reaches a yield surface with semi-diameter a(² ), the plastic strain component ^ ) associated with this yield surface is equal to zero. In this case, from (4.4) it ®(² ^ ) = 0, so that the eld of yield surfaces is initially follows that the back stress ½(² centred around the initial stress state ¾0 , (¾ ¡

¾0 )T B(¾ ¡

¾0 ) ¡

a(² )2 = 0;

² 2 [0; 1]:

(5.1)

The initial proportional loading emanating from the initial stress state ¾0 is described by the following expression ¾_ =

a_ a (¾ ¡ aa

¾0 );

(5.2)

where aa =

(¾ ¡

¾0 )T B(¾ ¡

¾0 ):

(5.3)

^ ); ² ) < 0, ² 2 [0; 1], so that, When aa < ae , from (4.2) it follows that y^g (¾ ¡ ¾0 ¡ ½(² from (4.9), ¶ ^ (² ) = 0 and the stress{strain behaviour is elastic, with cross-coupling described by the rst term in (4.8), _ "_ = D ¡ 1 ¾:

(5.4)

This form of constitutive behaviour was proposed by Graham & Houlsby (1983) and used by Puzrin & Burland (1998) to describe the stress{strain behaviour within the LER. Proc. R. Soc. Lond. A (2001)


433

(b) Logarithmic normalized stress{strain curve When aa 2 [ae ; aL ], from (4.2) it follows that y^g (¾ ¡ y^g (¾ ¡

¾0 ¡ ¾0 ¡

² 2 [0; ² a ]; ² 2 [² a ; 1];

^ ); ² ) = 0; ½(² ½(² ^ ); ² ) < 0;

where ²

a

=

(5.5)

aa ¡ ae ; aL ¡ ae

so that aa = a(² a ). Then, substituting (5.2) into (4.9) and the result into (4.7), and solving the di¬er^ ), ² 2 [0; ² a ], we obtain ential equation (4.7) for ½(² 1¡

½(² ^ )=

a(² ) (¾ ¡ aa

¾0 ):

(5.6)

Substitution of (4.4) and (5.6) into expression (4.6) yields ²

"= 1+ 0

a

jDj 1¡ 3G¤ h(² )

a(² ) aa

d²

D ¡ 1 (¾ ¡

¾0 ):

(5.7)

De ning ¾L as the stress point at which the stress path reaches the SSR boundary, we then de ne a normalized stress y such that (¾ ¡ ¾0 ) = y(¾L ¡ ¾0 ). It also follows that y = aa =aL . We de ne also a normalized strain x such that " = xD ¡ 1 (¾L ¡

¾0 ):

(5.8)

The normalization is chosen so that, within the linear elastic region, x = y. At the boundary of the LER, y = x = xe = ae =aL . Equation (5.7) may be rewritten in normalized form as ²

jDj y¡ 3G¤ h(² )

a(² ) aL

aa ¡ ae y¡ = aL ¡ ae 1¡

xe xe

a

x=y+ 0

It should be noted that ²

a

=

d² :

(5.9)

is itself a function of y. Double di¬erentiation of (5.9) with respect to y yields, after some manipulation, h(² ) = h = =¡

y¡ 1¡ 1

1¡

jDj xe 3G¤ 1

1¡

xe xe

jDj xe 3G¤

d2 x dy 2

¡1

d2 y dx2

¡1

dy dx

3

:

(5.10)

Equation (5.10) establishes the relationship between the shape of the normalized stress{strain (x; y) curve and the function h(² ). Proc. R. Soc. Lond. A (2001)

434

A. M. Puzrin, G. T. Houlsby and J. B. Burland 1 1

1

1

b

y xe 1 x

xe

0

xL

Figure 1. Normalized stress{strain curve.

Any plasticity model requires a speci c form of the hardening law to be chosen, usually on the grounds of tting experimental data. Puzrin & Burland (1996, 1998) found that the following logarithmic function tted experimental data extremely well. It is expressed as a normalized stress{strain curve describing both deviatoric and volumetric behaviour during any initial proportional loading ( gure 1). Note, however, that no particular importance is attached to the speci c mathematical form adopted here, and other expressions may provide a comparable quality of t to experiments. The normalized equation is y = xe + (x ¡

xe )(1 ¡

¬ (ln(1 + x ¡

xe ))R );

(5.11)

where R= ¬ =

1 ¡ xe ¡ xL ¡ xe (xL ¡

b

(1 + xL ¡

xL ¡ 1 xe )[ln(1 + xL ¡

xe ) ln(1 + xL ¡ (xL ¡ 1)

xe )]R

xe )

;

;

(5.12) (5.13)

where xL is the normalized limiting strain at the SSR boundary and b the ratio between the tangent sti¬ness at the SSR and LER boundaries. Double di¬erentiating (5.11) and substituting it into (5.10), we obtain the following expression for the sti¬ness function h(² ) =

1 1¡

jDj [X (1 ¡ ¬ (ln X )R ) ¡ ¬ R(X ¡ 1)(ln X )R¡ 1 ]3 ; xe 3G¤ ¬ RX(ln X )R¡ 2 [(1 + X) ln X + (R ¡ 1)(X ¡ 1)]

(5.14)

where X (² ) is de ned by the relationship ² =

(X ¡ 1) (1 ¡ 1 ¡ xe

¬ (ln X )R ):

The small-strain constitutive model is completely de ned by (3.1), (3.2) and (5.14). It can be shown that substitution of (5.14) into incremental equations (4.7){(4.9) and subsequent integration of these equations along any straight e¬ective stress path result in a normalized stress{strain curve identical to that given by (5.11) and used by Puzrin & Burland (1998) in their model. We have thus achieved the principal objective here: the model formulated by Puzrin & Burland on a largely empirical basis has been formulated within the thermodynamic context. Proc. R. Soc. Lond. A (2001)


435

The model requires seven independent parameters: ae , aL , b, xL , m, n and G¤ . The whole set of parameters can be derived from a single consolidated undrained triaxial test with local deformation measurements (Puzrin & Burland 1998). It is freely admitted that (5.14) is a rather complex, and perhaps unlikely, form of the expression for h(² ). However, this form was only chosen in order to result in the form of the nonlinear stress{strain curve already used by Puzrin & Burland (1998) and given in (5.11). They chose this particular form because it models extremely well (much better than other proposed curves) the rapid loss of sti¬ness of soils at very small strains. Other, simpler, forms of h(² ) could, however, be chosen; and the corresponding stress{strain curve readily obtained. One purpose of expressing models within a thermodynamic context is that the expressions employed for the storage and dissipation of energy can be interpreted in physical terms, and sometimes be identi ed with particular physical processes. This approach has met with some success, for instance, in the plasticity of metals. Such progress is of course a long-term aim of the work presented here, as this o¬ers the opportunity to unify theoretical and experimental observations. If, for instance, simpler forms of h(² ) can be identi ed which result in good ts to experimental results, then it may be possible to provide physical explanations for the form of the function. This is a long-term aim, and no such explanation is at present o¬ered here. (c) Ìncremental strain energy’ criterion Burland (1989) proposed that the boundary of the SSR could be de ned by a contour of ìncremental strain energy’ U , which he de ned for radial stress paths by dU = (¾ ¡ ¾0 )T d". Burland & Georgiannou (1991) showed experimentally that such contours were elliptical in shape for certain soils. The fact that both energy potentials (3.1) and (3.2) contain quadratic forms with the same shape matrix B, and its inverse, lead to important consequences described below. Di¬erentiation of expression (5.8) yields "_ = xD _ ¡ 1 (¾L ¡

¾0 );

(5.15)

so that, for a proportional loading path, U_ = y x(¾ _ L ¡

¾0 )T D ¡ 1 (¾L ¡

¾0 ) = y x_

3G¤ a2L ) : jDj

Since y is a function of x (equation (5.9)), it follows that U is a unique function of the normalized strain x, or alternatively of the normalized stress. Since each yield surface also corresponds to a single value of the normalized stress, it follows that the yield surfaces in the model are contours of ìncremental strain energy’ for initial proportional loading. Therefore, for any xed yield surface (5.1), including the LER and SSR boundaries, the same incremental strain energy is required to reach this surface during an initial proportional loading in any direction. This observation is consistent with the incremental strain energy criterion proposed by Burland (1989) and used for de nition of these boundaries by Puzrin & Burland (1998). We introduce this result here simply as an observation about the model: there is no fundamental theoretical reason why yield surfaces should correspond to contours of incremental strain energy. Proc. R. Soc. Lond. A (2001)

436

A. M. Puzrin, G. T. Houlsby and J. B. Burland (d ) Ùndrained’ loading

An important case in soil mechanics is the so-called ùndrained’ loading, in which the volume is constant. Because of the form of (5.7), it can be seen that a proportional loading path produces a straight path in strain space. Conversely, straight strain paths will result in straight stress paths. One such example is undrained loading (v_ = 0). For this case, it follows that (¼ ¡ ¼

0)

p0 ¡ p00 q ¡ q0

=

y 0 D " x

=

=

y x

J" ; 3G¤ "

so that the undrained stress path is inclined at a constant slope 3G¤ =J in (p0 ; q) space. Indeed, Puzrin & Burland (1998) suggest that the value of the parameter m = J=3G¤ should be inferred directly from the slope of the e¬ective stress path in an undrained test.

6. Behaviour of the model during proportional cyclic loading Consider proportional unloading, taking place after stress reversal at a stress state ¾r reached during initial loading emanating from the initial stress state ¾0 , and described by the following equation (¾ ¡

2aa (¾r ¡ ar

1¡

¾0 ) =

¾0 );

(6.1)

where aa =

(¾ ¡

½(² a ))T B(¾ ¡

¾0 ¡

¾0 ¡

½(² a ))

is the semi-diameter of the largest of the yield surfaces currently involved in reverse plastic loading and ar =

(¾r ¡

¾0 )T B(¾r ¡

¾0 )

is the semi-diameter of the largest of the yield surfaces involved in the initial monotonic loading. It follows from the initial loading conditions that (¾r ¡ Recalling the de nition (¾ ¡

¾0 ) =

ar (¾L ¡ aL

¾0 ) = y(¾L ¡ y=

¾0 ):

¾0 ), it follows that, during the unloading,

ar ¡

2aa aL

:

(6.2)

When aa < ae , from (4.2) it follows that y^g (¾ ¡ ¾0 ¡ ½(² ^ ); ² ) < 0, ² 2 [0; 1], so that, from (4.9), ¶ ^ (² ) = 0, and the stress{strain behaviour is elastic with crosscoupling described by (5.4). When aa 2 [ae ; aL ], we divide the eld (see equation (5.1)) of the yield surfaces y^g (¾ ¡ ¾0 ¡ ½(² ^ ); ² ) into the following three regions. De ning ² Proc. R. Soc. Lond. A (2001)

r

=

ar ¡ ae ; aL ¡ ae


437

so that ar = a(² r ), the rst region corresponds to the range ² 2 [² r ; 1], and the yield ^ ); ² ) in this region were involved neither in initial loading, surfaces y^g (¾ ¡ ¾0 ¡ ½(² nor in subsequent reverse loading, therefore their back stresses are zero. ^ ); ² ) In the second region, where ² 2 [² a ; ² r ], the yield surfaces y^g (¾ ¡ ¾0 ¡ ½(² were involved in initial loading, but not in subsequent reverse loading, therefore their back stresses can still be de ned using expression (5.6), ^ )= ½(²

1¡

a(² ) (¾r ¡ ar

ar ¡

¾0 ) =

a(² ) aL

(¾L ¡

¾0 ):

(6.3)

^ ); ² ) Lastly, in the third region, where ² 2 [0; ² a ], the yield surfaces y^g (¾ ¡ ¾0 ¡ ½(² were involved both in initial loading and subsequent reverse loading, so that their back stresses are de ned by substituting expression (6.1) into the following equation: ½(² ^ ) = (¾ ¡

¾0 ) +

a(² ) (¾r ¡ ar

ar ¡

¾0 ) =

2aa + a(² ) (¾L ¡ aL

¾0 ):

(6.4)

Substitution of (6.1){(6.4) into expression (4.6) yields the same expression as (5.8), but in this case where ²

x=y+ 0

a

²

a(² ) jDj y+ ¤ 3G h(² ) aL

r

d² + ²

a

ar ¡ a(² ) jDj ¤ 3G h(² ) aL

d² :

(6.5)

At the point of stress reversal, ¾r , the normalized strain is de ned using (5.9), ²

jDj yr ¡ 3G¤ h(² )

r

xr = y r + 0

where ²

r

=

a(² ) aL

d² ;

(6.6)

yr aL ¡ ae aL ¡ ae

is itself a function of yr = ar =aL , the normalized stress at the stress reversal. Then, using (6.2) and (6.6), equation (6.5) can be rewritten as ²·

x · = y· + 0

where ²· =

y·aL ¡ ae ; aL ¡ ae

jDj y· ¡ 3G¤ h(² )

x · = 12 (xr ¡

x)

a(² ) aL and

d² ;

y· = 12 (yr ¡

(6.7)

y):

As is seen, equation (6.7) produces the same logarithmic normalized stress{strain curve as given in (5.9) and (5.11), subject to transformations described by the above expressions. The initial curve is enlarged in both coordinate directions by a factor of two, rotated by 180¯ and originates from the stress reversal point on the initial loading normalized stress{strain curve ( gure 2). This kind of unloading behaviour is consistent with Masing rules, and con rms the pure kinematic hardening nature of the model presented here. It can be further shown that as soon as the normalized stress y reaches the value ¡ yr , the third region vanishes, and the normalized stress{strain behaviour is again described by the initial logarithmic normalized stress{strain curve as given in (5.9) and (5.11). This curve is rotated by 180¯ and initiates from the origin. Generally, it Proc. R. Soc. Lond. A (2001)

438

A. M. Puzrin, G. T. Houlsby and J. B. Burland y - yr - xr

xr

0

x

- yr Figure 2. Normalized stress{strain curves for uniform proportional cyclic loading.

can be shown that, for a proportional cyclic loading, a cycle of a larger amplitude wipes out any memory of all preceding events of a smaller magnitude. If another stress reversal takes place, with subsequent reloading, the initial portion of the reloading normalized stress{strain curve is again given by (6.7), but with x · = 12 (x ¡ xr ) and y· = 12 (y ¡ yr ), which is again consistent with Masing rules and produces closed hysteretic loops for uniform proportional cyclic loading ( gure 2). Again, if the loading is continued beyond the largest previous stress reversal, the reloading produces the initial normalized stress{strain curve (5.9). It can be shown that for transient and cyclic loading the continuous hyperplastic formulation of the model produces the stress{strain behaviour identical to that of a classical plasticity model with an in nite number of kinematic hardening elliptical yield surfaces, each one with the associated ®ow rule and Ziegler’s translation rule.

7. Concluding remarks The purpose of this paper has been to demonstrate that a model for the smallstrain behaviour of soils, previously presented by Puzrin & Burland (1998) within classical plasticity concepts, together with some additional rules for handling stress reversals, can be expressed within a rigorous thermomechanical formulation. This serves two purposes. Firstly, it is demonstrated that the Puzrin & Burland model does not violate thermodynamic principles. Secondly, the utility of the continuous hyperplasticity approach suggested by Puzrin & Houlsby (2001) is demonstrated. The modelling of the small-strain nonlinearity of soils is one of the key challenges of current theoretical soil mechanics. The establishment of a theoretical framework within which such models can be achieved is regarded as an important step towards a fuller understanding of soil behaviour. This paper was completed while A.M.P. and G.T.H. were academic visitors at the Centre for O® shore Foundation Studies at the University of Western Australia, and they are grateful for support from COFS and UWA to enable this collaboration.

Appendix A. Nomenclature a(² ) = ae + ² (aL ¡ aa

ae )

ae Proc. R. Soc. Lond. A (2001)

semi-diameter of elliptical yield surfaces semi-diameter of the largest yield surface in plastic loading semi-diameter of the inner yield surface bounding


the LER semi-diameter of the outer yield surface bounding the SSR semi-diameter of the largest yield surface in initial monotonic loading ratio between the tangent sti¬ness at the SSR and LER boundaries

aL ar b jDj ¡ 1 1 ¡ m D = ¡ m n2 3G¤ dg [¼ ij ; ¬ ^ ij ; ¬ ^_ ij ] B=

K¤ J

D=

J 3G¤

g[¼ ij ; ¬ ^ ij ] h(² ) m = J=3G¤ n2 = K ¤ =3G¤ x xe = ae =aL xL xr y yr = ar =aL y^g (À îj (² ); ¬ ^ ij (² ); ² ) = 0 ¬ ^ ij (² ) ^ = f^ ® ¬ v ¬ ^ s gT ¡ (² ) "ij "eij "pij " = fv "gT ² ²

a

=

aa ¡ ae aL ¡ ae

²

r

=

ar ¡ ae aL ¡ ae

Y ¶ ^ (² ) » îj (² ) ^ ) ½(² ¼

439

ij

¾ = fp0 qgT ¾0 = fp00 q0 gT ¾r Proc. R. Soc. Lond. A (2001)

dimensionless compliance matrix dissipation functional elastic sti¬ness matrix speci c Gibbs free energy functional function de ning the nonlinear shape of the stress{strain curve material constant in model material constant in model normalized strain normalized strain at the LER boundary normalized limiting strain at the SSR boundary normalized strain at stress reversal normalized stress normalized stress at stress reversal eld of yield functions internal variable function internal variable vector function distribution function strain tensor elastic strain plastic strain strain vector in triaxial space dimensionless measure of the size of the yield surfaces dimensionless size of largest yield surface currently in plastic loading dimensionless size of largest yield surface in initial monotonic loading the domain of ² a non-negative multiplier back stress function back stress vector function stress tensor e¬ective stress vector in triaxial space initial e¬ective stress vector in triaxial space stress state at stress reversal

440


À îj (² ) À ·îj (² ) ^ ) Â(²

dissipative generalized stress function generalized stress function dissipative generalized stress vector function

References Bazant, Z. P. 1978 Endochronic inelasticity and incremental plasticity. Int. J. Solids Structures 14, 691. Burland, J. B. 1989 Small is beautiful|the sti® ness of soil at small strains. Ninth Laurits Bjerrum memorial lecture. Can. Geotech. J. 16, 499{516. Burland, J. B. & Georgiannou, V. N. 1991 Small strain sti® ness under generalised stress changes. In Proc. 10th Eur. Conf. Soil Mech. Found. Eng., Florence, vol. 1, pp. 41{44. Collins, I. F. & Houlsby, G. T. 1997 Application of thermomechanical principles to the modelling of geotechnical materials. Proc. R. Soc. Lond. A 53, 1975{2001. Fung, Y. C. 1965 Foundations of solid mechanics. Englewood Cli® s, NJ: Prentice-Hall. Graham, J. & Houlsby, G. T. 1983 Elastic anisotropy of a natural clay. G¶eotechnique 33, 165{ 180. Corrigendum. G¶eotechnique 33, 354. Houlsby, G. T. 1979 The work input to a granular material. G¶eotechnique 29, 354{358. Houlsby, G. T. 1981 A study of plasticity theories and their applicability to soils. PhD thesis, Cambridge University. Houlsby, G. T. & Puzrin, A. M. 2000 A thermomechanical framework for constitutive models for rate-independent dissipative materials. Int. J. Plasticity 16, 1017{1047. Mroz, Z. & Norris, V. A. 1982 Elastoplastic and viscoplastic constitutive models for soils with application to cyclic loading. In Soil mechanics|transient and cyclic loads (ed. G. N. Pande & O. C. Zienkiewicz), pp. 173{218. Wiley. Puzrin, A. M. & Burland, J. B. 1996 A logarithmic stress{strain function for rocks and soils. G¶eotechnique 46, 157{164. Puzrin, A. M. & Burland, J. B. 1998 Non-linear model of small-strain behaviour of soils. G¶eotechnique 48, 217{233. Puzrin, A. M. & Houlsby, G. T. 2001 A thermomechanical framework for rate-independent dissipative materials with internal functions. Int. J. Plasticity. (In the press.) Reddy, B. D. & Martin, J. B. 1994 Internal variable formulations of problems in elastoplasticity: constitutive and algorithmic aspects. Appl. Mech. Rev. 47, 429{456. Valanis, K. C. 1975 On the foundations of endochronic theory of viscoplasticity. Arch. Mech. 27, 857. Ziegler, H. 1959 A modi¯cation of Prager’s hardening rule. Q. Appl. Math. 17, 55{65. Ziegler, H. 1983 An introduction to thermomechanics, 2nd edn. Amsterdam: North-Holland.

Proc. R. Soc. Lond. A (2001)