Hypoplasticity illustrated in terms of Elastoplasticity
J.J. Nader, 1998
One of the main features of soil behaviour, observed in engineering works and in controlled laboratory tests, is the ocurrence of irreversible deformation. In many practical problems it is admissible to neglect the more or less pronounced time dependent character of soil deformation and treat soils as rate-independent materials. To describe soil behaviour as a rate-independent material the most used and known framework is the theory of Elastoplasticity. Since the end of the 50´s, following the pioneer Cam-Clay, new elastoplastic models have been developed with increasing complexity in order to represent soil behaviour. An alternative to elastoplastic models for the mathematical description of soil mechanical behaviour appeared in 1977, created by Kolymbas in his doctoral thesis at the University of Karlsruhe, Germany, and was named Hypoplasticity. With a single constitutive equation, inspired by modern rational mechanics, without recourse to yield or potential surfaces, important phenomena of soil mechanical behaviour can be represented.
We aim in this brief communication at presenting Hypoplasticity taking advantage of the reader's basic knowlegde of Elastoplasticty. A comparison between the two above mentioned theories, restricted by the introductory character of this text, will also be done.
Elastoplasticity
A fundamental ingredient of elastoplastic models is the yield surface in the stress space, which is introduced by a yield function. The yield surface, associated to a particular state of the material, bounds, in the stress space, a region called elastic domain. The most used models are the isotropic-hardening elastoplastic ones where the plastic state is characterized by a single scalar called hardening parameter.
We now introduce two terms extensively used here: elastic and plastic regime.
The regime is said to be elastic when only reversible (elastic) deformation occurs. This reversibility is detected in a stress cycle. In this case the plastic state of the material does not change.
If the deformation is not totally reversible the regime is elastoplastic and the irreversible part of the deformation is called plastic. When the regime, initially elastic, becomes elastoplastic the material yields becoming more deformable.
Since there are more than one definition of stress and deformation tensors, each particular elastoplastic model must define which of them are being employed. For soils we employ, in most modern models, the Cauchy stress T and the stretching D (the symmetric part of the spatial velocity gradient) as a measure of the rate of deformation.
We now proceed giving the definition of elastic and elastoplastic regime in terms of stress. This will provide a relation between stress and rate of deformation.
If the stress to which the material is subject corresponds to a point located within the elastic domain, we specify that, at this instant, the regime is elastic whatever the stress change is. During a certain time interval the stress path remains within the elastic domain and the corresponding regime is elastic. In elastic regime the constitutive equation belongs to Hypoelasticity: T°=h(T,D), wherein T° is the Jaumann derivative of T, also called co-rotated stress rate and the function h(T,D) is isotropic with respect to both arguments and linear and homogeneous of degree 1 in D. To emphasize the linearity in D, we agree to write T°=h(T)[D]. In Elastoplasticity the inverse relation is more usually employed: D=L(T)[T°], with L(T)=h-1(T) (only invertible relations are considered).
We have just considered the case when the stress point is within the elastic domain. Now let us describe what happens if the stress point lies initially on the yield surface. If the stress increment is directed inwards the elastic domain or it is tangent to the yield surface, the regime is elastic, and, so, the same hypoelastic equation relates stretching, stress and stress rate. On the other hand, an elastoplastic regime begins if the stress increment points to the outer region. In its motion outwards the present elastic domain, the stress point enlarges or displaces continuously the yield surface, in such a way that it remains on the yield surface, i.e., at any instant of an elastoplastic regime the stress point is on the yield surface correspondent to that plastic state. Thus the stress point is never outside the region limited by the current yield surface. Well, the constitutive relation for the elastoplastic regime has not yet been mentioned. As an axiom it is stated that the stretching, at any instant of an elastoplastic regime, is the sum of two parts. One is the elastic stretching De, given by the same hypoelastic equation valid for elastic regimes (De=L(T)[T°]); and the other is the plastic stretching Dp.
The constitutive equation relating the plastic stretching, the stress, the stress rate and the state parameters must be specified. Once more a linear relation, this time between plastic stretching and stress rate for each value of the stress, is used. If a continuous material response is to be represented, the plastic stretching for a stress rate tangent to the yield surface must vanish. Thus, it can be shown that the linear relation between plastic stretching and stress rate must have a particular form: the linear function is an inner product of the stress rate and a tensor normal to the yield surface. The tensor n(T), with unitary norm and normal to the yield surface, is employed in the equation. In other words, interpreting it geometrically, it is the scalar projection of the stress rate onto the normal to the yield surface that enters the plastic constitutive equation; if that projection is null, i.e., if the stress rate is tangent to the yield surface, then the plastic stretching is null as well. So the plastic stretching tensor equals that inner product times a tensor function P(T) of the stress and the state parameters, which gives the direction of the plastic stretching: Dp=(n(T):T')P(T), in which T' is the time derivative of the stress tensor T. As n(T):T' is equal to n(T):T°, we may write Dp=(n(T):T°)P(T). The tensor function P(T) is, in many theories, a multiple of the normal to the yield surface n(T) at the corresponding stress point: P(T)=b(T)n(T). This particular form can be either stated a priori or deduced from an axiom regarding the stress power (Drucker's axiom) and, anyway, it is called the normality condition: an allusion to the orthogonality between the plastic stretching and the yield surface at the current stress point. It is important to emphasize that the direction of the plastic stretching does not depend on the direction of the stress rate.
Thus, repeating the main points, we have:
In elastic regime: D=L(T)[T°].
In elastoplastic regime: D=De+Dp=L(T)[T°]+(n(T):T°)P(T)
In this way one can model irreversible deformation. As an example, if the stress T is on the yield surface and a stress rate T' is applied causing elastoplastic regime, the stretching is given by the sum of elastic and plastic ones. On the other hand, if the stress rate is -T', leading to elasticregime, the stretching is the negative of the elastic stretching of the previous case.
Hypoplasticity
The hypoplastic constitutive equation relates Cauchy stress T, the Jaumann derivative of T and the stretching D in the following manner: T°=h(T,D). At this early point we may start the confront with Elastoplasticity. There, at first, yield function and the corresponding surface, loading criterion and two constitutive equations are prescribed. Here the starting point is simpler: one equation.
The function h(T,D) must be isotropic with respect to both arguments to obey the principle of objectivity (material frame-indifference), and so, making use of a representation theorem of tensor algebra, it can be written as:
h(T,D)=f11+f2T+f3D+f4T2+f5D2+f6(TD+DT)+f7(TD2+D2T)+f8(T2D+DT2)+f9(T2D2+D2T2)
in which fi are scalar functions of joint invariants of T and D.
It is clear that for each particular stress T, the relation between T° and D, unlike the hypoelastic case, is non-linear. It is this non-linearity that makes possible the representation of irreversible deformation with a single equation.
When only rate-independent behaviour is to be represented, another restriction is imposed to the above equation: h(T,D) must be positively homogeneous of degree 1 in D.
Since 1977 special cases of the general rate-independent equation have been proposed in order to describe soil behaviour as observed in laboratory tests. All those cases are particular instances of an equation in which T° is given by the sum of two parts; the first one is linear in D (hypoelastic) and the second one is non-linear: T°=L(T)[D]+Q(T,D).
Although in Hypoplasticity there is no previous definition of different regimes, we now show how irreversible deformation is naturally taken into account. For the material to suffer a stretching D, the stress rate T°=L(T)[D]+Q(T,D) is required. If the stretching is -D, the corresponding stress rate is not -T°, but T°=-L(T)[D]+Q(T,-D), with Q(T,-D) different from -Q(T,D) in general, due to the non-linearity (it is strictly positively homogeneous). In Elastoplasticity the difference in behaviour has to be done with two linear functions.
Until today the most used special case of the general hypoplastic non-linear relation is T°=L(T)[D]+N(T)||D||, in which the norm ||D|| is responsible for the non-linearity. Note that the direction of the non-linear part does not depend on D.
For a direct comparison with elastoplastic equations the relation between T° and D can be inverted, giving, restricting the discussion to the just mentioned special case,
D=L-1(T)[T°]-L-1(T)[N(T)]a(T,T°),
where the scalar valued function a(T,T°) equals ||D|| (||D||=a(T,T°), never negative). Defining M(T)=L-1(T)[N(T)] in order to get a more compact equation, it becomes D=L-1(T)[T°]-a(T,T°)M(T). This particular form of hypoplastic equation resembles the one for the stretching in elastoplastic regime: the first part is linear and its direction and magnitude depends on T°; the direction of the second part does not depend on the direction of T°. A difference is that in Elastoplasticity the second part is linear in T°, whereas in hypopasticity it is non-linear.
Note the different material response when T° or -T° is applied. For T° the corresponding D is D=L-1(T)[T°]-a(T,T°)M(T), but, for -T°, it gives D=-L-1(T)[T°]-a(T,T°)M(T).
In Elastoplasticity, yield function and material state parameters define the yield surface, which bounds the elastic domain. As we have already seen, within the elastic domain only elastic deformation occurs, the material is more rigid. The yield surface is a kind of material memory. In Hypoplasticity there is no yield function, no elastic domain. All past information is concentrated in the current stress. Indeed, there are today new versions of hypoplastic equations involving the void ratio and a structural tensor that are more sensible to past deformation history.