# Modelling the poroplastic damageable behaviour of earthen materials

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## Abstract

This paper presents a new constitutive model on the poroplastic behaviour of earthen materials accounting for stiffness degradation, using the approach of continuum damage mechanics. The poroplastic behaviour is modelled based on the bounding surface plasticity (BSP) theory and the concept of effective stress while isotropic damage is modelled using a scalar variable. Plastic flow and damage evolution occur simultaneously in a coupled process which take into account the impact of suction. The model was successfully validated against results of triaxial compression tests performed at different relative humidities and confining pressures. Despite the relatively small number of material parameters, this model can reproduce the essential features of earthen materials behaviour observed experimentally: suction-induced hardening and stiffening, post-peak softening, as well as the progressive transition from contractant to dilatant volumetric behaviour. Use of the BSP theory allows to reproduce a smooth stress–strain relation as experimentally observed, instead of an abrupt change upon plastic yielding predicted by classic elastoplastic models. Furthermore, the present model also furnishes a quantitative description on the degradation of elastic properties hitherto not accounted for, thanks to the additional scalar damage variable.

## Keywords

Unsaturated rammed earth Bounding surface plasticity Suction Damage Degradation## 1 Introduction

Unfired crude (untreated) earth is a hygroscopic porous material that contains generally a small quantity of active clay minerals. Currently, considering the context of sustainable construction, reduction of energy consumption and greenhouse gas emission, it is regarded as a promising non-industrial construction material. Several construction technics using earthen materials have been invented, leading to monolithic walls, like rammed earth and cob, or to masonry units, like adobe or compressed earth blocks [1]. After constructions, earthen buildings being always subject to atmospheric conditions possess relatively low water contents and high suction which impact significantly on their mechanical behaviour.

In order to study the behaviour of unsaturated earthen materials, a series of large-scale laboratory tests were carried out by several researchers [2, 3, 4, 5]. The main conclusion drawn from all these experiments is that the mechanical properties of earthen materials, whatever their origin and their implementation technic, are strongly influenced by their water content: even a small increase of the latter leads to a significant reduction of both the failure strength and Young’s modulus *E*, while the irreversible plastic strain is increased. Apart from plastic deformations and suction effects, a progressive but non-negligible degradation of Young’s modulus *E* with increasing stress level were observed in recent study [6, 7]. This stiffness reduction is conjectured to be a consequence of damage induced by initiation and development of micro-cracks. The above experimental observations motivate the development of a new constitutive model which can simulate the poroplastic behaviour of earthen materials while accounting for damage effects.

To begin with, the main drawback of classic elastoplastic models is their erroneous prediction of a sharp transition at yield from elastic to elastic–plastic behaviour (jump of apparent stiffness), as well as their difficulty to reproduce realistic volumetric behavior (typically a smooth transition from contractancy to dilatancy). Models developed using the Bounding Surface Plasticity (BSP) theory can overcome these limitations. This theory, firstly proposed by Mroz [8] and Dafalias et al. [9, 10] to simulate metal behaviors, was successively applied by Bardet [11] on saturated sands and later on followed by Gajo and Muirwood [12] as well as by Yu and Khong [13]. BSP theory was later extended to partially saturated materials in order to describe suction effects [14, 15, 16, 17, 18, 19]. In addition, BSP theory is also applied on geomaterials other than soils due to its versatility, such as rockfills and clay-fouled ballast [20, 21].

In parallel, there have been extensive studies on the development of coupled elastoplastic damage models [22, 23, 24, 25, 26]. However, most of them are devoted to modeling dry materials without considering coupled hydromechanical interactions, not to mention the capillarity effects due to partial saturation. In recent years, several researchers have attempted to couple damage with hydromechanical interactions. For instance, a coupled poro-elastoplastic damage model is proposed by Shao et al. [27] and extended to partially saturated conditions in order to study coupled hydromechanical damageable behaviours in drying–wetting processes. Zhang et al. [28] developed a unified elastic–viscoplastic damage model to describe long-term hydromechanical behavior of argillites under unsaturated condition. Bui et al. presented a new constitutive model accounting simultaneously for the impact of damage on hydraulic and mechanical properties of unsaturated poroplastic geomaterials, by means of the thermodynamic framework for partially saturated media [29]. The present work follows a similar path of this last reference.

After a summary of experimental investigations showing the main trends on the mechanical behaviour of earthen materials, a new constitutive model is presented for unsaturated earthen materials. It is based on the BSP theory and the effective stress concept. It adopts a non-associative flow rule and a simple radial mapping rule, and takes into account an isotropic damage variable. Plastic straining and damage evolution occur as two coupled processes. The influence of suction on plastic straining and damage evolution is taken into account. Finally, the performance of this model is investigated by comparing the numerical simulation with a series of triaxial tests results on compacted earth samples at different hydraulic conditions and confining pressures.

## 2 Summary of experimental investigations

^{3}) is also applied to Mix1 and Mix2. Cylindrical samples (3,5 cm diameter and 7 cm high) were fabricated using a home-made mold which allows a double compaction (compression applied simultaneously at both ends of the sample) of the raw earth at controlled displacement rate. After fabrication, samples were cured at three different relative humidity conditions (23, 75, and 97%) at 23 °C until reaching equilibrium. The total suctions, denoted by \(s_{\text{t}}\), corresponding to the three values of relative humidity (RH) can be calculated by Kelvin’s law:

*s*. In the rest of this paper, the term ‘suction’ will therefore refer to ‘matric suction’. The matric suctions corresponding to the three values of RH (23, 75, and 97%) imposed at 23 °C are 201, 39 and 4 MPa respectively.

*f*

_{c}) for one particular specimen using this type-1 loading, the second loading type consisting of loading–unloading cycles at prescribed stress levels, respectively 20, 40, 60 and finally 80% of

*f*

_{c}was conducted on two other specimens prepared under identical conditions. The second loading–unloading cycle was used to estimate the evolution of Young’s modulus

*E*with the level of axial stress applied, assuming unloading is reversible. Young’s modulus

*E*is therefore computed from the following equation:

*f*

_{c}. All the remarks mentioned above are also observed for Lim and Mix2.

*E*measured during the unloading–reloading cycle at 20%

*f*

_{c}and the evolution with relative humidity. Young’s modulus decreases drastically with the rise of relative humidity. Results presented in Fig. 2b on the evolution of Young’s modulus

*E*versus stress level indicate that there is a progressive and significant degradation of the material stiffness when the stress increases. This degradation of stiffness is faster at low stress and slower at high stress. Consistently with the stiffness degradation, macro-cracks oriented in the direction of major principal stress appeared during the compression test.

To summarize, results of our experimental investigation indicate that the basic mechanical behaviour of unsaturated earthen materials can be characterized by irreversible plastic straining and stiffness degradation. The first behaviour is commonly modelled using classic elastoplasticity theory, despite its unrealistically prediction of a stiffness-discontinuity on the stress–strain curve at yield. To improve this, it is proposed to adopt the approach of Bounding Surface Plasticity (BSP). The stiffness degradation due to microcracks has not been considered in the literaure on earthen materials constructions. To address this issue, it is proposed to follow the approach of continuum damage mechanics.

## 3 A poro-elastoplastoplastic damageable constitutive model

As mentionned in the previous section, the construction of the new constitutive model requires to address two mechanisms: a poroplastic mechanism capable of simulating the suction-dependent plastic straining and a damage mechanism to capture the stiffness degradation. The poroplastic mechanism in this new model follows the same approach as the CASMNS model developed by Lai et al. [19] for unsaturated soils. The formulation is based on the concepts of effective stress and bounding surface plasticity theory. To accurately simulate volume changes (e.g. transition from compressibility to dilatancy), a state-dependent non-associative flow rule is used, by adopting the plastic potential of Yu [30]. The starting point is the definition of an effective stress to account for effects of partial saturation.

### 3.1 Partial saturation and effective stress

There are two main classical modelling approaches on partially saturated soils. All of them need two independent stress variables. The classic BBM model is based on net stress and suction whereas a few others use an effective stress and suction [14, 15, 31]. Some recently appeared models proposed to use the degree of saturation instead of suction as a generalised stress variable [32, 33, 34]. This new approach seems to be promising, but the feedback is still limited. In this paper, we follow the approach based on an effective stress and suction.

*, s*is the difference between pore air and pore water pressure (\(u_{\text{a}} - u_{\text{w}}\)), referred as suction, \(\delta_{ij}\) is the second order identity tensor, and

*χ*is defined as the effective stress parameter and generally is state dependent. In all the following, a prime (‘) above a stress variable will denote the effective stress counterpart. In many practical cases, air pressure can be identified as the atmospheric pressure, taken as the reference datum for stresses and fluid pressures, hence, Eq. (3) can be simplified to:

*χ*have been proposed in the existing literature: some researchers identified the effective stress parameter with degree of saturation \(S_{\text{r}}\) [5, 36]; while others argued that \(\chi\) should be a function of suction [37]. In this study, the functional form (9) below firstly proposed by Khalili and Khabbaz [38] has been adopted:

*α*is a material constant and \(s_{\text{e}}\) represents the air-entry suction which can be determined from water retention measurements.

*e*) and a plastic component (superscripts

*p*):

For the damage mechanism, a damage variable *D* is introduced as an internal variable to simulate the effects of microcracks. For the sake of simplicity, only isotropic damage is considered here and \(D \in \left( {0,1} \right)\) is a scalar variable.

### 3.2 General concept of bounding surface plasticity

The concept of bounding surface plasticity (BSP), also known as two-surface plasticity, was first originated by Mroz [8] and Dafalias et al. [9, 10], and applied by Bardet [11] and Yu et al. [13] to soils. Russell and Khalili [39] appear to be the first who applied the BSP theory to model unsaturated soils. Similar to the bounding surface models, sub-loading surface models are also widely used for modelling complicated plasticity behaviour for soils. Based on Hashiguchi’s initial framework [40], Yao et al. proposed a simple but very useful unified hardening equation [41] and corresponding models to consider complicated soil behaviour like hardening behaviour related to thermal effects [42] and hardening behaviour related to unsaturation [43].

### 3.3 Elastic mechanism accounting for damage

*K*and

*G*. To account for the effects of damage on the stress-dependent elastic properties, the following expressions are adopted based on the previous results of [23, 27, 29]:

*D*is the isotropic damage variable, \(\nu\) the Poisson ratio, \(\kappa\) a material constant, and \(e_{0}\) the initial void ratio.

### 3.4 Bounding surface, loading surface and hardening mechanism

*M*represents the slope of critical state line (CSL) in \(p' - q\) plane, while \(n\) and \(r\) are two other model constants. The first constant \(n\) specifies the shape of the bounding surface and the second constant \(r\) is a space ratio used to control the intersection point of the CSL with the BS. \(p_{\pi }\) is a variable analogue to the preconsolidation pressure in the classic Cam Clay model that determines the current size and position of the bounding surface. For simplicity, the loading surface (LS) is defined simply by homotheticity relative to the BS, as illustrated in Fig. 4:

### 3.5 Plastic potential, non-associative flow rule and plastic modulus

*m*is another model constant which governs the plastic potential, whereas constants

*M*and

*n*as well as the stress ratio \(\eta\) have already been defined. This equation implies the dilatancy depends only on the stress ratio and becomes zero at the critical stress ratio \(\eta = M\). A plastic flow rule which satisfies the above dilatance ratio can be written as:

### 3.6 Damage mechanism

*D*and/or the conjugate damage force \(Y_{D}\). Although the latter can be derived directly from the framework of thermodynamics, this usually leads to overly complex expressions. An alternative simpler family of damage evolution laws, which has been successfully applied to concrete and rocks under compressive deviatoric stresses is based on the tensile part of principal strains. Following the approach of Mazars [23], the tensile part of the principal strain, noted \(\varepsilon_{i}^{ + }\), is defined by:

Note that the above definition is consistent with the sign convention of positive compressive strains. Moreover, in conventional triaxial compression tests, the axial compressive strain \(\varepsilon_{1}\) is positive while the radial tensile strain \(\varepsilon_{3}\) is negative.

### 3.7 Elastoplastic compliance matrix accounting for damage

## 4 Model validation against experimental data and discussion

### 4.1 Determination of model parameters

The proposed bounding surface plastic damage model requires 14 parameters for its complete definition. They characterize particular aspects of material behaviour: (1) \(\nu\) and \(\kappa\) to describe the elastic behaviour. (2) Seven constants on the plastic behaviour: \(\lambda_{0}\) and \(\varGamma\) define the position of critical state line (CSL) in the \(e - \ln p'\) plane while *M* defines the slope of CSL in \(p^{\prime} - q\) plane; \(n\) and \(r\) specify the shape of bounding surface; \(h\) controls the hardening modulus; and *m* controls the stress-dilatancy. (3) Another four model parameters are needed to account for suction effects: \(s_{e}\) and \(\alpha\) define the effective stress used in this model, \(k_{1}\) and \(k_{2}\) account for suction effects on hardening and volumetric deformability respectively. (4) Finally, the constant \(k_{3}\) controls damage evolution.

The above model parameters can be determined from a few classic tests in the laboratory. \(\nu\), \(\kappa\) and \(\lambda_{0}\) can be obtained from classic oedometric and triaxial consolidation tests at full saturation under drained conditions: \(\kappa\) is based on the slope of elastic unloading–reloading line on \(e - \ln p^{\prime}\) plane, \(\nu\) is the Poisson’s ratio (can be determined from axial vs radial strain rates during elastic unloading) and \(\lambda_{0}\) is detemined from the loading part of the isotropic triaxial consolidation tests expressed in \(e - \ln p^{\prime}\) plane. \(\varGamma\) and *M* associated with the definition of critical state line can be deduced using triaxial tests at different controlled suctions. The parameter \(k_{2}\) defining the reduction of volumetric deformability \(\lambda \left( s \right)\) with suction can be assessed by carrying out virgin oedometric compression tests at different controlled suctions. \(s_{\text{e}}\) is identified as air-entry suction determined from the classical water retention curve and a unique value \(\alpha = 0.85\) was adopted for the three different soils as mentioned in the Sect. 3.1. The remaining parameters: \(h, m, n, r,\) \(k_{1}\) and \(k_{3}\) have to be determined by trial and error using experimental results.

### 4.2 Model implementation

Model parameters adopted in simulation for Lim and Mix1

Parameter symbols | Role or/and physical meanings | Lim | Mix1 |
---|---|---|---|

\(\nu\) | Poisson’s ratio | 0.05 | 0.05 |

\(\kappa\) | Slope of unloading line in the | 0.005 | 0.002 |

\(\lambda_{0}\) | Volumetric compressibility constant | 0.025 | 0.024 |

\(e_{0}\) | Initial void ratio | 0.37 | 0.37 |

| Critical state line in | 1.65 | 1.85 |

| Constant specify the shape of bounding surface | 4 | 3.5 |

| Space ratio | 1.4 | 1.4 |

| Constant to calibrate the hardening modulus | 1.6 | 1.6 |

| Constant to define plastic potential | 1.8 | 2.2 |

\(s_{\text{e}}\)(MPa) | Air-entry suction | 0.77 | 0.29 |

\(\alpha\) | Constant to define effective stress parameter | 0.85 | 0.85 |

\(k_{1}\) | Constant to control suction effects on the hardening parameter | 0.039 | 0.015 |

\(k_{2}\) | Constant to control suction effects on volumetric compressibility | 0.005 | 0.005 |

\(k_{3}\) | Constant to describes the damage evolution | 0.05 | 0.08 |

### 4.3 Stress–strain behaviour and volumetric evolution

As shown in Fig. 5b, the volumetric strain versus axial strain \(\varepsilon_{v}\)–\(\varepsilon_{1}\) as predicted by this model is also in good agreement with the experimental data. This confirms that the model can indeed reproduce the complex volumetric behaviour: transition from contraction to dilation state. The model also captures the suppressed dilation induced by the increasing relative humidity (suction decreasing).

At lower hydraulic state RH = 23%, some deviations are observed both in stress strain curve and volumetric evolution, especially for the post-peaking stage. One possible explanations is that some form of localization may appear, which cannot be simulated by a purely continuous modelling approach. Note that this kind of deviation is also faced with in previous modelling work [27, 29].

By making a comparison between Figs. 5 and 6, it is worth noting that the model also captures the increased deviator stress and overall reduced dilation with an increase in confining pressure. As confining pressure increases, particle crushing begins to dominate plastic deformation, resulting in a reducing tendency for a peak in the shear resistance and for a volumetric expansion.

### 4.4 Degradation of elastic property

In addition, it is worth noticing the degradation rate of Young’s modulus at RH = 75% (higher suction, lower water content) is higher than that at RH = 97% (lower suction, higher water content), which is conform to the physical intuition that higher water content should result in higher ductility.

## 5 Conclusions

A new constitutive model for unsaturated earthen materials is proposed accounting simultaneously for the impact of suction and confining pressure on mechanical properties. This model is formulated using the concept of effective stress and bounding surface plasticity (BSP) theory under a critical state framework. It adopts an whale-head shaped loading and bounding surfaces, a simple radial mapping rule, a non-associative flow rule which generally gives a better description on volumetric behaviour, and suction-dependent hardening controlled by plastic volumetric strains. At the present stage of development, only isotropic damage is considered. Damage evolution rate is assumed to be driven by tensile strain and restrained by suction. Plastic flow and damage evolution occur as coupled processes.

The model has been used to simulate the triaxial compression tests subjected to different relative humidities and confining pressures. Good agreement is generally obtained between model predictions and experimental data. Of fundamental importance in practice, this model only requires 14 independent soil parameters for its definition and makes it ideally suited for engineering applications. Despite the minimal number of parameters, the model is able to reproduce the essential trends in the behaviour of partially saturated earthen materials: suction-induced hardening and stiffening, smooth stress–strain behaviour, post-peak softening ae well as contractancy-dilatancy transition.

Furthermore, the present model qualitatively describes a degradation of elastic properties observed in experimental data. The model in its current form is intended for monotonic loading, an extension of this model to cyclic loading will be the subject of future work.

## Notes

### Acknowledgements

The present work has been supported by the French Research National Agency (ANR) through the ‘‘Villes et Bâtiments Durables” program (Project Primaterre no. ANR-12-VBDU-0001). The first author is supported by the China Scholarship Council (CSC) with a PhD Scholarship (File No.201406300070) for his research work.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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