## Abstract

For the long-term behavior and safety assessment of rockfill dams, not only the shape of the dam body, the loading history, the geological condition of the dam foundation and abutments, the assessment of possible seismic hazards and seepage events caused by defects of the sealing are important, but also the time dependent mechanical behavior of the dam materials used can be of significant influence. In this paper a novel hypoplastic constitutive model for moisture sensitive, coarse-grained rockfill materials is presented. In the constitutive equations, the so-called solid hardness is a key parameter to reflect the influence of the state of weathering on the mechanical response. With respect to the evolution equation for the solid hardness, creep and stress relaxation can be modeled for dry and wet states of the material in a unified manner. The performance of the model is demonstrated by comparing the numerical simulation with experimental data.

### Keywords

- rockfill material
- wetting behavior
- creep
- stress relaxation
- hypoplasticity

## 1. Introduction

Rockfill dams have become very popular among dam engineers due to of their simple construction sequence, short construction period and low costs compared to concrete dams. An economic aspect also lies in the fact that rockfill materials of different types are usually available on site can be used in appropriate zones of the dam. The frequent use of weathered and moisture-sensitive rockfill materials requires the precise recognition of the mechanical behavior of these materials under the expected load and environmental conditions. The prediction of the post-construction settlements of rockfill dams is a challenging task because of uncertainties of environmental events, which can occur during the entire lifetime of the dam [1, 2]. Changes of the moisture content of weathered rockfill material can lead to sudden settlements that could have relatively large values, without any changes in the applied load [3, 4, 5, 6, 7, 8]. For the long-term behavior of rockfill dams, not only the shape of the dam body, the loading history, the geological condition of the dam foundation and abutments, the assessment of possible seismic hazards and seepage events caused by defects of the sealing are important, but also the time dependent behavior of the dam materials used can be of significant influence [9, 10, 11]. Thus, the proper modeling of the time dependent behavior of rockfill material under different loading and environmental condition plays an important role for the design, construction, operation and safety assessment of rockfill dams. For instance, post construction settlements may affect the amount of the bending and the crack propagation in concrete slaps of concrete face rockfill dams [12, 13, 14, 15, 16].

In engineering disciplines time dependent deformations under constant stress are termed creep, however, this term does not reflect the individual mechanisms that different types of materials can exhibit. Rheological properties of weathered rockfill materials are strongly influenced by the state of weathering and the mechanical and environmental boundary conditions to which the material is subjected. Thus, the concept for the experimental investigations requires an appropriate adaptation to the conditions relevant at the construction site. In this context also scale effects resulting from the differences between the grain size distribution and pre-compaction in laboratory tests and in the filed must be taken into account [17, 18, 19].

Long-term deformations are usually an accumulation of deformations related to various events and the proper interpretation of the relevant physical and hydro-chemical mechanisms is important for the evaluation of the data obtained from laboratory experiments and field measurements as well as for the numerical modeling. A general distinction can be made between time independent deformations, which are the instantaneous part of the deformation due to applied load changes, and time dependent deformations, which can also take place under constant load. The former, for instance, can be initiated by rapid changes of the water level in the reservoir, the change of the effective stresses caused by a change of suction of fine grained materials, hydraulic fracturing and piping as a result of the seepage-driven internal erosion of solid particles. Time dependent deformations are influenced by the mineralogical composition of the solid material, the frequency and the orientation of micro-cracks, the grain size distribution, the pre-compaction, the moisture content, the stress state and the evolution of weathering [20, 21, 22, 23, 24]. Progressive weathering caused by mechanical and hydro-chemical weathering has a significant influence on the time dependent process of the degradation of the solid hardness and as a consequence on the resistance of the material against compaction and shearing. In dam engineering it is common to differentiate between long-term creep and so-called collapse settlements. While the former is related to the rheological properties of the rockfill material and controlled by gravity load and the effect of water impounding, the latter can usually be observed immediately after a change of the moisture content in the stressed rockfill material. Collapse settlements are also called instantaneous wetting deformation and characterized by a spontaneous increase of the deformation velocity.

Changes of the moisture content can be caused by different events like climate changes, leakage as an effect of defects of the dam sealing and the dam foundation, as well as by rainwater infiltration into the dam body. According to Terzaghi [25] collapse phenomenon are also time dependent and related to the decrease in the grain crushing strength, especially at the contact points. Plastification of grain contacts and grain crushing bring about local instabilities in the grain skeleton. Rheological properties of weathered rockfill materials are more pronounced in the wet state of the material than in the dry state [4, 9, 11, 26]. Depending on the state of weathering of the rockfill material a change of the moisture content can initiate an acceleration of the crack propagation of stressed rockfill grains, which leads to a reduction of the solid hardness and consequently to a sudden increase of the settlement velocity. Figure 1a shows an example for an increase of the settlement velocity after flooding under a constant vertical stress of −0.8 MPa in an oedometer device [3]. The course of the deformations is qualitatively similar for greywacke and sandstone and can be divided into three parts. After applying the vertical stress on the initially dry material, an instantaneous settlement can be detected, which is larger for sandstone. Under constant stress, creep can be observed in the dry state and also following a sudden flooding of the specimen. Flooding leads to a sudden jump of the settlement rate but there is no clear sharp jump in the settlement which indicates that so-called collapse settlements can also assumed to be time dependent. While for the dry material the creep velocity decreases slowly, the high settlement rate immediately after flooding fades out very fast. It is experimentally evident that the compressibility of weathered rockfill material strongly depends on the pre-compaction and moisture content of the material, i.e. the compressibility is higher for a less compacted material and higher for the wet than for the dry material. This is also clearly visible for example from the results of oedoemeter compression tests with weathered broken granite in Figure 1b [27, 28]. Great efforts have been made to investigate and to model the complex mechanisms of wetting deformations at the micro- and macro-level, e.g. [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

The focus of this paper is on the time dependent process of degradation of the stiffness of weathered, coarse grained and moisture sensitive rockfill materials and its constitutive modeling under dry and wet states. A particular version of a hypoplastic model developed within the past decade is presented and its performance is verified by comparison of numerical simulations with experimental data. It is an aim of the present paper to propose simple calibration procedures for the material parameters based on standard laboratory tests. As a measure of the state of weathering the so-called “solid hardness of the grain assembly” is a key parameter in the material model proposed [47, 48, 49, 50, 51, 52]. Particular attention is paid on a refined modeling of the influence of the coupled behavior between the state of weathering, the stress state and the packing density of the rockfill material on the calibration of those material parameters relevant for collapse settlements, long-time creep and stress relaxation. In this context the modeling of the change of the rheological material parameters during the lifetime of the dam, the concept of the solid hardness is put forward to simulate a repeated acceleration of the degradation of the solid hardness. Such events are relevant for repeated changes in the moisture content of the rockfill material caused by local defects of the sealing and heavy rain water infiltration into the rockfill material of the dam body. While the focus of the present paper is on the theory of constitutive modeling and its calibration, the application of the proposed material model to different types of rockfill dams can be found for instance in [53, 54, 55, 56].

The present paper is organized as follows:

In Section 2 resent developments of modeling the compression behavior of weathered, creep and moisture sensitive rockfill materials under dry and wet condition are summarized. To this end the so-called “solid hardness” is defined in the sense of a continuum description and is a state parameter in Bauer’s compression law [47, 57, 58]. As an example of how the concept of the solid hardness can be incorporated into a general three dimensional material model the adaptation of the compression law to the bulk modulus of a nonlinear elastic material model is outlined. In incrementally non-linear material models, for instance, the concept of the solid hardness was embedded into non-polar constitutive models, e.g. [58, 59, 60] and into micro-polar models, e.g. [61, 62, 63, 64, 65, 66, 67, 68]. In the present paper the proposed enhanced constitutive model for the solid hardness is a time dependent quantity and a measure of the state of weathering of the rockfill material [47]. With respect to the evolution equation of the time dependent process of degradation of the solid hardness creep and stress relaxation are modeled in a unified manner. Particular attention is paid to the adaptation of the velocity parameter in the evolution equation of the solid hardness to experimental creep and stress relaxation curves.

Section 3 gives a brief introduction to the hypoplastic constitutive model by Gudehus [69] and Bauer [58] which was originally developed for a constant solid hardness and a time independent material behavior. The incrementally nonlinear constitutive equation describes the stress rate as a function of the current void ratio, the Cauchy stress and the strain rate. In contrast to the concept of elasto-plasticity, the framework of hypoplasticity does not need to decompose the deformation into elastic and plastic parts, which allows a relatively easy calibration of the material parameters involved. The hypoplastic constitutive model captures the extended theory of “critical state soil mechanics” and describes the influence of pressure and density on the incremental stiffness, peak friction angle and dilatancy angle using only eight material parameters. It is also outlined in detail how the compression law by Bauer can be embedded into the hypoplastic model with help of a consistency condition.

In Section 4 the hypoplastic constitutive model shown in Section 3 is extended to describe also time dependent material properties which are relevant for weathered and moisture sensitive rockfill materials. In addition to the current void ratio and stress, the solid hardness and its rate are also state quantities of the extended model. As a consequence creep and stress relaxation properties are usually coupled and the incremental stiffness, peak friction angle, dilatancy angle are also influenced by the evolution of the degradation of the solid hardness.

In Section 5 the capability of the proposed hypoplastic model to model the mechanical behavior of rockfill materials under dry and wet conditions are verified by comparison of numerical simulations with experimental results. In order to simulation sudden changes of the creep velocity initiated by repeated acceleration of the degradation of the solid hardness during the lifetime of the dam an extended version for the evolution equation of the solid hardness is proposed. The enhanced version permits the simulation of multistep degradation of the solid hardness and a refined simulation of collapse settlements and long-time creep.

Throughout the paper the sign convention of rational solid mechanics is adopted, i.e. compressive stresses and strains, and their rates are negative. Indices on vector and tensor components refer to an orthonormal Cartesian basis and the symbol

## 2. Compression law by Bauer

This section deals with the modeling of essential mechanical properties of weathered and moisture sensitive rockfill materials under isotropic and oedometric compression. To this end the so-called “solid hardness” is defined in the sense of a continuum description and it is a state parameter in the compression law by Bauer. An evolution equation for the degradation of the solid hardness of the rockfill material is used to model the influence of the reduction of the incremental stiffness caused by progressive weathering, instantaneous wetting deformation and long-time creep. Examples for the adaptation of the velocity parameter in the evolution equation of the solid hardness to experimental creep and stress relaxation curves are outlined in detail.

### 2.1 Introduction of the solid hardness in the sense of a continuum description

Isotropic or oedometric compression tests carried out with various granular materials show qualitatively similar compression curves. At lower stresses the reduction of the void ratio is explained by sliding of neighboring grains against each other and a reorientation of the grain skeleton into a denser state. Under higher pressures the additional compaction is mainly related to progressive grain crushing. In a semi-logarithmic representation, the compression curves show an S-shape which is also clearly visible from the experimental data for two different sand materials in Figure 2 [70]. For different initial void ratios, the distance between the compression curves becomes smaller with an increase in the mean pressure and for higher pressures the curves merge together. This means that the memory of the material on the initial density fades out due to both grain crushing and reorientation of grains. It is known from experiments that the point of inflection is related to the pressure level where grain crushing becomes dominant. Such a compression behavior was also observed for arbitrary granular materials and it is also verified by numerical simulations with the discrete element method, e.g. [71, 72]. Experimental work has shown that the pressure at the point of inflection depends mainly on the mineral composition and the state of weathering of the solid material. As the point of inflection shows no noticeable influence on the initial density, it is a well-defined state of the material under compression. The pressure where the point of inflection appears is in the following called “solid hardness” and it is a material parameter in the compression law by Bauer [57, 58]. In this context it is worth noting that the solid hardness is defined for the compression behavior of an assembly of grains and does not mean the hardness of a single grain.

In the case of frequently used constitutive models the compression behavior represented in a semi-logarithmic representation is approximated by a straight line as illustrated in Figure 3a. However, the approximation is only applicable to a limited pressure range. For higher pressures the compression line, NCL, leads to the non-physical area of negative void ratios. A behavior of this kind cannot occur when using the compression law proposed by Bauer [57, 58]. The exponential function captures the whole pressure range in a consistent manner as depict in Figure 3b. In particular, Bauer’s isotropic compression law describes the reduction of the void ratio

Eq.(1) represents the upper bound of the pressure dependent maximum void ratio

For rockfill materials the value of the solid hardness is rather high, and usually it cannot be achieved with standard isotropic compression apparatus available in a soil mechanics laboratory. On the other hand, in an oedometer device higher pressures are easier to carry out. Investigations show that for practical application the solid hardness can also be calibrated with sufficient accuracy if data are used which are obtained from an oedometer test instead of an isotropic test. Standard oedoemeter devices usually only allow the measuring of the vertical stress

The embedding of the compression law (1) into general 3-D constitutive models can be accomplished with the help of a consistency condition. As a heuristic example the embedding into an elastic material model is demonstrated in Appendix A. The implementation of Eq.(1) into an enhanced constitutive model for rockfill materials is discussed in Section 3 and outlined in detail in Appendix B.

### 2.2 Time dependent process of degradation of the solid hardness

The degradation of the solid hardness caused by progressive weathering of the rockfill material is a time dependent process and causes a reduction of resistance of the material against shearing and compaction. A chemical reaction of the weathered rockfill material with water can accelerate the process of weathering leading to grain breakage and as a consequence to collapse settlements and creep deformations. Experimental investigations show that the solid hardness of the dry material is higher than the solid hardness of the wet material, and that the transition from the dry to the wet state is a time dependent process as illustrated in Figure 4.

In order to take the current state of weathering into account, the constant solid hardness

The change of the void ratio with time

Substituting the identity

The evolution Eqs. (3) and (4) describe creep and stress relaxation in a unified manner. In particular, for the special case that during degradation of the solid hardness the pressure is kept constant, i.e.

and

respectively. With respect to the state quantities at the reference time

The integration of the identity

The comparison of Eq.(7) with Eq.(8) yields the creep strain

On the other hand for the special case that the volume is kept constant, i.e.

With respect to

For the irreversible degradation of the solid hardness with time, i.e.

where parameter *t*:

where parameter

Weathered rockfill materials can also undergo creep deformation and stress relaxation under dry condition and it can be distinguished between instantaneous deformation and time dependent deformation. The time dependent compressibility of the dry material can be explained by delayed grain crushing and it can also be modeled by a degradation of the solid hardness. In this case

The value of parameter

From a pure stress relaxation test the value of

where

## 3. Embedding the solid hardness into hypoplasticity

To demonstrate the embedding of the solid hardness

The quantities in Eq.(16) denote:

Because of the nonlinearity of

The peak friction angle and dilatancy are predictions of the constitutive Eq. (16) and not material constants. The influence of pressure and density on the peak friction angle and dilatancy angle is modeled using the pressure dependent relative density factor

where

In Eq.(18) the pressure dependent quantities

where

The influence of the pressure and density on the incremental stiffness is modeled using the stiffness factor

The first term on the right hand side of Eq.(21) is the density dependent part and a relation between the pressure dependent maximum void ratio

where

Factor

The hypoplastic constitutive model also captures properties according to the theory of the “critical state soil mechanics” [75]. In particular, in critical states the pressure dependent void ratio

The hypoplastic constitutive Eq. (16) contains eight material parameters, i.e.

## 4. Extension of the hypoplastic model to weathered and moisture sensitive rockfill materials

Although the hypoplastic constitutive Eq. (16) is of the rate type, the material behavior described is rate independent. In order to take into account rheological properties of weathered and moisture sensitive rockfill materials in the hypoplastic model outlined in the previous section the constant solid hardness

In Eq.(25) the degradation of the solid hardness is coupled with the evolution of stress and strain. As the pressure dependent limit void ratios defined in Eq.(20) are also functions of the current state of the solid hardness a degradation of the solid hardness also results in a reduction of the limit void ratios. For a lower solid hardness the stiffness factor

For general boundary conditions the evolution of creep is obtained for

and for pure stress relaxation, i.e. stress changes under

For the special case of isotropic stress states under

## 5. Numerical simulations

In the following the performance of the proposed constitutive concept for modeling the degradation of the solid hardness is demonstrated by comparing experimental data with the results of numerical simulations of a multistep, one dimensional creep test, and triaxial creep tests under different deviatoric stresses. For numerical modeling of wetting induced settlements a smooth transition from an almost sudden settlement, i.e. collapse settlement, to the creep deformation is assumed. A smooth transition of the kind is also observed in laboratory experiments and can be modeled with an extended evolution equation of the solid hardness as demonstrated in the following.

### 5.1 Combined modeling of collapse deformation and creep

In this subsection the results of wetting experiments with broken sandstone carried out by Fu et al. [77] are considered for the simulation with a simplified version of the hypoplastic constitutive model proposed in [60], which differs from the hypoplastic model (25) in that for the density factor the simplified relation

Numerical investigation with respect to the constitutive relation (13) show [60], that the prediction of the time dependent deformations immediately after wetting deviates from the experimental data. In particular, at initiation of creep the axial creep strain

Herein

The results of numerical simulations using the extended evolution Eq. (28) are in good agreement with the experimental results as shown for the axial strain

### 5.2 Multistep creep

In order to simulation multistep creep initiated by repeated acceleration in the degradation of the solid hardness, e.g. caused by repeated rain water infiltration events into the rockfill body of the dam, two different concepts can be suggested for practical application. Either each event is modeled independent of the previous one, i.e. the values for parameters

In Eq.(29) the terms on the right hand side consist of the contribution

### 5.3 Influence of the stress deviator on the evolution of the creep deformation

The results of numerical simulations of the influence of the deviatoric stress on the stress strain relationship and on the creep behavior for a moisture sensitive broken sandstone are shown together with the experimental data in Figure 9. The 11 material parameter of the extended hypoplastic constitutive model were calibrated by Bauer et al. [50] using the experimental data obtained by Li [78]. As no creep behavior of the dry material in the triaxial compression test was reported, the calibration of the hypoplastic material parameters for the dry material were carried out for a constant solid hardness. In particular, the following set of material parameters were obtained for the initially dry material:

For monotonic triaxial compression under a lateral stress of

The behavior after water saturation at different deviatoric stresses is shown in Figure 9b. It is clearly visible that the axial creep strain

## 6. Conclusions

In this paper the long-term behavior of weathered and moisture sensitive coarse-grained rockfill material is reviewed and the main mechanical properties are modeled in a phenomenological manner using a novel constitutive model based on the frame work of hypoplasticity. In order to take into account of the influence of grain crushing and the time dependent process of degradation of the strength of the solid material on the incremental stiffness, a so-called “solid hardness” is introduced as a state parameter into the constitutive model. The solid hardness is defined for monotonic isotropic compression of a grain assembly and it is a key parameter in the compression law by Bauer. It is shown that creep and stress relaxation are usually coupled. The calibration procedure of the material parameter relevant for the velocity of degradation of the solid hardness is outlined for different states at the creep curve and stress relaxation curve obtained in isotropic compression experiments. It is demonstrated that the strategy used for the embedding of the compression law into an extended 3-D hypoplastic continuum model can also be applied to other classes of constitutive models. With an enhanced hypoplastic constitutive model the influence of the state of weathering, the evolution of the degradation of the solid hardness, the packing density of the rockfill materials and the stress state on the incremental stiffness, the peak friction angle and the dilatancy angle can be modeled in a unified manner using a single set of material parameters. These properties of the model are confirmed by the comparison of numerical simulations with experimental data, in particular for triaxial compression tests under dry and wet conditions as well as for creep tests under different deviatoric stresses. It is shown that the evolution of the volume strain curve under creep strongly depends on the amount of the stress deviator where the rockfill material is saturated. The extended version of the evolution equation for the degradation of the solid hardness permits a refined modeling of collapse settlements and long-time creep caused by repeated changes of environmental conditions.

## Acknowledgments

The author wishes to thank Dr. Z.Z. Fu, Dr. L. Li, Mr. M. Khosravi and Mr. S. Safikhani during their stay at the Institute of Applied Mechanics at Graz University of Technology for preparing parts of the numerical simulations shown in this paper.

As a heuristic example for the adaptation of the compression law (1) to a nonlinear elastic material description the following isotropic elastic material law is considered

Herein the bulk modulus,

With respect to the relationship

Substitution of Eq.(A3) into Eq.(A2) leads to:

The compression law (1), i.e.

can be transformed to:

The comparison of relation (A4) with relation (A5) yields for the bulk modulus

In the initial stress free state, i.e.

For the shear modulus,

where

The material parameter **Figure B1**. In **Case I** parameter **Case II** the inclination of the volume creep curve at **Case III** and **Case IV** deals with the calibration of parameter

## B.1 Case I

Substituting the evolution Eq. (12) for the solid hardness into Eq.(5) one obtains for the rate of the volume strain under creep:

Herein the degraded solid hardness

In **Case I** the inclination

one obtains for parameter

## B.2 Case II

In **Case II** the inclination

to:

## B.3 Case III

For stress relaxation under constant volume staring from

Substituting the evolution relation (12) for the solid hardness into Eq.(B6) one obtains:

The inclination

and with respect of Eq.(15), i.e.

where

## B.4 Case IV

As in experiments the inclination

with:

Parameter

to:

In order to embed the compression law (1) in a consistent manner into the hypoplastic constitutive Eq. (16) the rate of the mean pressure, i.e.

With respect to the relations in (C1) the rate of the mean pressure calculated from the hypoplastic constitutive Eq. (16) reads:

From the compression law (1) one obtains:

From the consistency condition