On proportional deformation paths in hypoplasticity

The constitutive stress–strain relation for hypoplastic granular materials like cohesionless soil or broken rock is under our consideration. The respective constitutive law is of the rate type, incrementally nonlinear, and it is based on the hypoplastic concept proposed by Kolymbas [29]. Compared to incrementally linear constitutive equations, e.g. for hyperelastic and hypoelastic material laws, the hypoplastic constitutive equations are not differentiable at zero strain rate. This is due to different stiffnesses for loading and unloading typical for inelastic materials. In contrast to the classical elastoplastic concept, the strain in the theory of hypoplasticity is not decomposed into elastic and plastic parts. Detailed discussions about physical aspects of hypoplastic models can be found, for instance, in [20, 31, 36]; their response to cycling loading is studied in [9, 41], shear localization in [7, 10, 16, 23], critical states in [4, 5, 49], behaviour under undrained condition in [37], extension to a micropolar continuum in [23, 24], and thermodynamic aspects in [26, 42, 43].

Other typical representatives for incrementally nonlinear constitutive equations are, for instance, the Armstrong–Frederick model [2], the endochronic model by Valanis [45], the octolinear model by Darve [14, 15], the CLoE model by Chambon et al. [12], and the barodesy model by Kolymbas [30]. For variational approaches to modelling granular and multiphase media, we refer to [1, 27, 33].

Experiments with granular materials show a typical behaviour of the stress path obtained under a monotonic proportional deformation path which was formulated in two rules by Goldscheider [18]. The first rule says that a proportional deformation path starting from a nearly stress-free state results in a proportional stress path. The second rule says that a proportional deformation path starting from an arbitrary stress state asymptotically converges to the same proportional stress path as the one obtained for the initially stress-free state. These asymptotic properties of granular materials can be interpreted by fading memory of the material [21]. This feature is also confirmed by further experiments, e.g. [13, 44], and numerical simulations by the discrete element method, e.g. [25, 34].

As the asymptotic behaviour of the stress path for proportional deformation is an intrinsic property of granular materials, it has also great importance for constitutive models relevant to frictional granular materials. It can be noted that in many constitutive models the asymptotic behaviour is only fulfilled for particular deformation paths. The lack of this property can lead to large deviation of the prediction of stress paths not used for calibration.

In constitutive modelling, the asymptotic property is the starting point for barodesy modelling in [38], and it is also an intrinsic feature in hypoplasticity [20, 22, 28, 35, 36, 40]. With respect to a particular hypoplastic model by Kolymbas [28], stress paths were numerically investigated for initially axisymmetric stress states and axisymmetric proportional deformations. However, no general requirements for the asymptotic behaviour were formulated. A mathematical criterion providing a necessary condition for convergent asymptotic behaviour in incremental models was given first by Niemunis [40]. It requires that the normal distance of the generated stress path to the asymptote decreases with an increase in monotonic deformation, where the asymptote is obtained for the corresponding fixed strain rate starting from the initially stress-free state. The criterion by Niemunis is applicable to all rate-type models; however, it is restricted to proportional deformation paths with contractant volume strain behaviour.

In the present paper, a different strategy is proposed to examine the asymptotic behaviour using the analytical solution for the stress path, which depends on the direction of the proportional strain path, the initial stress state, and the material parameters involved in the constitutive equation. The main challenge in developing proper mathematical tools is the strongly nonlinear character of the underlying differential equations. Following the rate-independent technique developed in [11], we gave a rigorous mathematical proof of the existence of asymptotic states in [8, 32]. This idea is further developed in the present work.

To this end, a simplified version of the hypoplastic model by Gudehus [19] and Bauer [3] is considered. For the sake of simplicity, the pressure- and density-dependent properties of granular materials described in [3, 19] are omitted so that only two material parameters remain in the model. It can be noted that the same simplified version can also be obtained from the hypoplastic model by von Wolffersdorff [46] as shown in [6].

For the simplified hypoplastic model, we construct the solution of the corresponding nonlinear problem in a closed form. In this way, we make also a close link to barodesy models [30]. The exact solution allows us to describe analytically various scenarios of the behaviour of stress paths obtained from monotonic compression, extension, and isochoric deformations. The latter leads to stress limit states or the so-called critical stress states, which can be represented by a conical surface in the space of principal stresses. In particular, we identify the domain of the constitutive parameters which guarantee that for proportional deformation paths the corresponding stress paths starting from an arbitrary initial stress state are asymptotically stable.

We first fix some basic tensor notation. By \({\mathbb {R}}^{3\times 3}_\mathrm{sym}\), we denote the space of symmetric 3-by-3 tensors of second-order and , which are endowed with the usual double dot product, the associated norm, and the trace, respectively:Here, stands for the 3-by-3 identity matrix with the Kronecker symbol \(\delta _{ij}=1\) for \(i=j\), and zero otherwise. In physical interpretation, corresponds to the Cauchy stress tensor, and is the strain rate tensor. For time \(t\ge 0\), we interpret and as time-dependent tensor-valued functions.

With respect to the normalized stress tensor , the general representation of the hypoplastic constitutive equation of the Kolymbas type can be written in the factorized form as the following tensor equation for the objective stress rate:where is a symmetric second-order tensor, is a symmetric fourth-order tensor, and the double dot product is to be interpreted asThe dimensionless parameter \(c<0\) scales the incremental stiffness and can be calibrated, for instance, based on an isotropic compression test. The right-hand side of (1.1) is a homogeneous function of degree one in . Note that dry granular materials are cohesionless, so that only negative principal stresses are relevant to the constitutive equation (1.1). Furthermore, we remark that particular representations of the tensor functions in (1.1) are based on terms from the general representation theorem of isotropic tensor-valued functions [47]. Various explicit versions are proposed in the literature (e.g. [3, 19, 29, 43, 48, 49]). In this paper, we consider a particular version of (1.1) proposed by Bauer in [5] in a simplified manner:with the normalized stress deviator /3, where the symbol stands for the fourth-order identity tensor, the symbol \(\otimes \) denotes the dyadic product of tensors, and the term in (1.1) which is linear in can also be represented as:The constitutive constant \(a>0\) is called limit stress state parameter and characterizes the shape of the conical limit stress surface or the so-called critical stress state surface in the principal stress space [5]. Critical states are defined for a vanishing stress rate under continuous isochoric deformation. For critical stress states, parameter a equals the norm of the normalized stress deviator, i.e. , and it can be related to the so-called critical friction angle [4]. While in the model by Gudehus [19] and Bauer [3] the value of a also depends on the orientation of the stress deviator, parameter a is assumed to be a constant in the present paper. For the granular friction angle \(\phi \in (0,\pi / 2)\) such that \(a =2\sqrt{2/3}\sin \phi / (3 -\sin \phi )\), we get the physical restriction \(a <a_\mathrm{phys}=\sqrt{2/3}\approx 0.8165\) as \(\sin \phi < 1\).

Further we restrict ourselves to strain paths pointing in one fixed direction. Namely, we call a strain path proportional if there exists a time-independent symmetric second-order tensor normalized by and a scalar time-dependent function s(t) with \(s(0) =0\) and \( \dot{s}(t)>0\) such that it holdsIn (2.1), determines a prescribed direction in the space of symmetric second-order tensors, and s represents a monotonic increasing parameter. Moreover, we assume deformations for which the objective time derivative and the material time derivative coincide, i.e. , which, for example, holds for fixed directions of principal stresses. As the material behaviour described by Eq. (1) is isotropic, the strain rate tensor with fixed principal values, i.e. the strain increments are proportional and coaxial, corresponds to a stress rate tensor with zero elements outside the diagonal. Thus, the relevant system of ordinary differential equations (ODE) reduces to three.

In this case, inserting the chain rule \(\mathrm{d}/\mathrm{d}t = \dot{s} \mathrm{d}/\mathrm{d}s\) and according to (2.1) and dividing with \(\dot{s}\not =0\), we can rewrite the rate-independent relations (1) with respect to \(s>0\) in the form:or, equivalently, without factorization:where the formula was used, andIn the principal stress space, the representation of the normalized quantity indicates the direction of the asymptotic stress state for \(s \rightarrow \infty \). This observation will be specified below after the representation formula (5.6). Indeed, formula (5.6.1) shows that the evolution of takes place in the 2D plane generated by constant tensors and . We shall see that the scalar coefficient of is dominant and determines the asymptotic convergence or divergence of the stress path as \(s \rightarrow \infty \).

In system (2.3) of ordinary differential equations (ODE) for , the last term in the right-hand side is nonlinear in . For (2.3), we prescribe the initial condition at \(s=0\):where is a given initial stress tensor.

In the following, we study the behaviour as \(s\rightarrow \infty \) of solutions to the nonlinear initial value problem (2) in dependence of the direction of proportional strain paths, the material parameter a, and the initial stress .

Similarly to the motivation example, we should distinguish contractant (volume-decreasing) from dilatant (volume-increasing) states. For this task, we call by contractant the compression corresponding to , and by dilatant the extension when , while corresponds to the volume-preserving deformation.

To this end, we note that paths with will eventually approach the origin as they are proportional volume-increasing deformation paths. However, there are examples of axial loading paths, e.g. , with . Mašín [34] also observed with discrete element method simulations the so-called asymptotic extension states, that are obtained by with axial loading.

(i) Contractant straining: Let a tensor be prescribed such that . Recalling , we have for all \(a>0\). According to (5.2), it follows and the finite limit (see Fig. 2 the left plot):

From (5.1), we infer \(D^- <0\); hence, the second term in the right-hand side of (5.6) grows exponentially along . The asymptotic behaviour of the first term depends on the sign of \(E^-\): it grows to plus or minus infinity if \(E^-<0\) and decays to a finite number if \(E^- =0\) and to zero if \(E^- >0\). The latter case corresponds to the proportional contractant deformation such that:This condition is equal toand describes the asymptotically stable stress path in (5.6) attracting the direction of as \(s\rightarrow \infty \) by the mean of descent distance:We remark that the decay of the distance in (7.3) holds only for sufficiently large values of s. For instance, the distance may increase in some interval \(s\in (0,s_0)\) with \(s_0>0\) satisfying \(c E^- s_0 +h^-(s_0) =0\), when \(c E^- <0\) and \(h^-(s_0) >0\) for \(C^-<0\) in (7.1).

(ii) Dilatant straining: Let obey . We consider first the case of and similarly to (7.1) (see Fig. 2 the left plot) obtainTherefore, the stable asymptotic behaviour under the proportional dilatant deformation is guaranteed by \(D^+ >0\) and \(E^+ >0\), that is:or, equivalently,then the stress path in (5.6) decays asymptotically to zero:By this, if \(E^+ >D^+\), then the stress path is closer to ; otherwise, it is closer to when the opposite inequality \(E^+ <D^+\) holds.

In the special case of , that is, , from (19) we haveThe asymptotic stability should be considered separately.

If , then and follows the limit in (5.2) (see Fig. 2 the right plot):In this case, \(\exp (c E^+ s +h^+(s))\) and \(\exp (c D^+ s +h^+(s))\) entering the stress path in (5.6) may obey various asymptotic behaviour in dependence of the sign of parameter \(C^+\) in (10.1) and the values of \(E^+\), \(D^+\) in (5.1). The asymptotically stable decay to zero of stress paths is described by the following two cases: either \(C^+ =0\) (hence, \(h^+(s)\equiv 0\)) and \(E^+ >0\), \(D^+ >0\), that is:or \(h^+(s)\rightarrow -\infty \) provided by \(C^+ >0\):In the both cases (10.2) and (10.3), the exponential convergence holds:The non-monotone behaviour of is admissible for dilatant, too.

(iii) Volume-preserving deformation Consider a proportional deformation with ; then, , \(E=a\), and \(D =0\) in (5.1); hence,with , and the limit:We note that the limit states in (11.1) for all admissible directions with and initial stresses form the cone of the limit stress states with the apex in the origin of the space of principal stress axes. From (11.1), we find that ; thus, the interaction of this cone with the deviator plane, i.e. the plane with (see [4]), is represented by the circle of radius a:The analysis of asymptotic behaviour of stress paths under various proportional deformations from Sect. 4 in dependence of the parameters is summarized for convenience in Table 1.

In the following, we omit the superscripts \(-, +\) and study how the parameters D and E defined in (5.1) are connected with the choice of a and . Indeed, from (2.4) we have ; hence, , and we get that D and E are only dependent on a and . We note that the range of is limited by . To prove this, we considered ; therefore, becomes extreme if \(U_{ij}=0, i\ne j\), so only the entries on the diagonal of are distinct from zero. With the arithmetic quadratic mean inequality, we get:and equality holds for \(U_{11}=U_{22}=U_{33}=1/\sqrt{3}\).

As we see in Table 1, the relations of D and E, one to each other and to zero, are crucial for the asymptotic behaviour of as \(s\rightarrow \infty \). Depending on (x-axis) and a (y-axis), these relations are plotted in Fig. 3.

Here, we consider the formal description of the plot, which means the equations of the curves separating the different greyscale areas, starting in the down left corner:

The meaning of the corresponding areas is the following: Further we illustrate the usage of Table 1 for the cases of contractant, dilatant, and volume-preserving deformations.

If , the stress path is asymptotically stable in area 2. We get the lower bound for , which means isotropic compression, byFor \(a>a_{\min }\), we can be sure to have asymptotically stable behaviour for arbitrary proportional deformation (see Fig. 1 from Sect. 3.1).

If , the stress path is asymptotically stable in areas 3 and 4: in area 3, the path tends to zero, while in area 4, it tends to infinity. We get the upper bound for , which means isotropic extension, byFor \(a<a_{\max }\), we have asymptotically stable behaviour for arbitrary proportional extension; in this case, the stress path will also tend to zero.

For the limiting case of , the stress path will tend to a certain stress state at the critical cone. To illustrate the matter, we plot few results of numerical simulation.

In the left plot of Fig. 4, the stress paths are projected in the plane spanned by \(-\sigma _{33}\) and the first median of the \(-\sqrt{2}\sigma _{11}=-\sqrt{2}\sigma _{22}\)-plane. The dot dashed line is the isotropic axis; the outer solid lines are the intersection with the critical cone. In the right plot, the paths are projected onto the deviator stress plane . In the deviator plane, the cutting line with the critical cone forms a circle of radius a according to (11.2). In Fig. 4, we see three example stress paths under volume-preserving deformation with the parameter \(a=0.35\) and three initial stress statesWe can observe that for volume-preserving deformation all stress paths tend to the critical cone independent of the choice of .

For comparison, in the similar coordinate axes as in Fig. 4, in Fig. 5 we see three stress paths starting at the initial stress state with the parameter \(a=0.35\) under three different deformationsWe see that for different deformations we get different behaviour of the stress paths. For (solid curve), we have isotropic compression, and the corresponding stress path tends to the isotropic axis. For (dashed curve), we have uniaxial compression, and we can also observe an asymptotic behaviour. For (dotted curve), we have volume-preserving deformation and the stress path goes to the critical cone.

In this Section, we will discuss the case of isotropic deformation and uniaxial deformation in more detail.

In the paper, for a constitutive model based on the concept of hypoplasticity of the Kolymbas type the stress paths obtained under proportional deformations are investigated. In particular, a simplified hypoplastic constitutive equation is considered where the objective stress rate is a function of the current stress and strain rate. The model is obtained by omitting the influence of the change in the pressure-dependent relative density in the hypoplastic model originally proposed by Bauer and Gudehus. For arbitrary proportional deformations starting from the stress-free state, the corresponding stress paths are linear which is in accordance with the first law by Goldscheider and also an important property for constitutive models relevant to frictional granular materials. From experiments, it is known that significant changes in the density of the granular material can lead to a slightly curved stress path which can also be influenced by grain crushing under higher stresses. Such properties can be taken into account using more enhanced hypoplastic models, which are not considered in the present paper.

The hypoplastic equation considered only includes two constitutive constants: a stiffness parameter and a limit stress state parameter. The former can be related to an isotropic compression test and the latter to the so-called critical friction angle defined in the steady state of a cohesionless granular material under triaxial compression. Although the influence of the direction of the stress deviator on the limit stress state parameter is neglected, it does not mean a restriction of the general results drawn. A relevant relation between the values for the direction of the stress deviator of triaxial compression and the one for another direction can, for instance, be obtained by interpolation.

For the hypoplastic constitutive equation considered, the analytical solution of the stress paths depending on monotonous linear deformation paths is obtained in a closed form. On its basis, the course of the stress paths is examined in dependence of the material parameters and prescribed proportional contractant, dilatant, and volume-preserving deformations. It is shown that stress paths starting from arbitrary initial stress states are usually nonlinear, and their convergence to a proportional stress path (second law by Goldscheider) is asymptotically stable only for a certain domain of the limit stress state parameter, which is related to the critical friction angle.