Modelling of Rammed earth
Numerous studies have been carried out related to the experimental characterization of rammed earth at diﬀerent scales, including the material scale and wall scale. On the other hand, there are few studies carried out on the coupled hydro-mechanical modelling of rammed earth. These studies have used elasto-plastic constitutive models and damage models for prediction of the mechanical behavior of rammed earth.
Rammed earth in dry state behaves as a quasi-brittle material. Damage models are more suitable for the constitutive modeling of quasi-brittle materials such as concrete. When earthen material is subjected to mechanical loading or environmental conditions, microscopic defects and cracks can develop. The distributed defects in the material are responsible not only for crack initiation and final fracture but also induced deterioration or damage such as a reduction of strength and stiﬀness (Zhang et al. 2010 ). This behavior can be studied by the use of a damage model which can represent the change in material properties and its failure due to initiation of damage, its growth and propagation. The damage models require the definition of an appropriate damage variable to represent the macroscopic eﬀects of microscopic cracks. If the damage is assumed isotropic, a scalar damage variable can be used, and for anisotropic damage, a tensor damage variable is required. Also, a constitutive equation, including damage variable to describe the mechanical behavior is required. The diﬀerent damage models can be isotropic elastic damage model, elastoplastic damage model, and elasto-viscoplastic damage model.
The most commonly used isotropic damage model is Mazars model , generally used for concrete. The constitutive relationship between stress and strain in Mazars model is given by the following relationship, which is based on generalized Hooke’s law: = (1 D)E e (2.2)
where, E is Hooke matrix, e is the elastic strain, and D is the scalar damage variable which varies between 0 and 1. D is defined as a combination of two damaging modes defined by Dc and Dt which also varies between 0 and 1 depending on the state of damage in compression and tension respectively.
where, c and t are weight coeﬃcients depending on the principal strains.
Bui et al. 2014  used damage model to simulated the mechanical behavior of rammed earth wallettes under compression loading by taking into account the complex behavior of rammed earth, i.e. non-linearity, cracking and damage. This study used Mazar’s model which is an isotropic nonlinear damage model. It can identify the gradual degradation in stiﬀness caused by micro-cracks. The wallettes analyzed were considered homogeneous and isotropic. The numerical model could reproduce the initial stiﬀness in compression and the strength, but could not predict the behavior once the cracking begins, and stiﬀness decreased (figure 2.30). It is a common limitation of Mazar’s model as it can reproduce the maximal load but not the behavior curve. On the other hand, the Mazar’s model cannot reproduce the behavior cycles, which is not suitable in the case of drying/humidification cycles as in the case of earthquakes.
Another approach for constitutive modeling of rammed earth is the elastoplasticity frame-work. This approach is more adapted to represent the behavior of the post-peak phase as rammed earth is not perfectly brittle. Rammed earth can show significant ductility, especially at higher moisture conditions. The assumption of elasto-plasticity is that for small strain conditions, the total strain can be decomposed into elastic (d e) and plastic (d e) strain (equation 2.4).
Apart from the linear elastic Hooke’s law, other models have been used to model elasticity in rammed earth. Francois et al. 2017  used a non-linear hypoelastic law to determine the elastic component of strain. Hypoelasticity is generally used to model ma-terials that exhibit non-linear, but reversible stress-strain behavior even at small strains. The strain in the material depends only on the stress applied and not on the rate or history of loading. The stress is a non-linear function of strain even when the strains are small. The Young’s modulus (E) was taken as a function of mean eﬀective stress through a hyperbolic function:
where, p’ is the mean eﬀective stress, Eref is the reference Young’s modulus at the reference mean eﬀective stress p0ref , and ne is a material parameter. By using mean eﬀective stress in the expression of elastic modulus, the eﬀect of suction on the stiﬀness was taken into account.
Concerning plasticity, there are numerous models used for geomaterials such as Mohr-Coulomb, Drucker-Prager, and Cam-Clay model. Mohr-Coulomb model is an elastic-perfectly plastic model used to model soil behavior. It uses two parameters which define the failure criterion, i.e. cohesion (c) and friction angle ( ). In addition, it uses a parameter to describe the flow rule, i.e. dilatancy angle ( ) coming from the non-associative flow rule to model the irreversible change in volume due to shearing. Since it is a perfectly plastic model, it does not include strain hardening or softening eﬀects. The Mohr-Coulomb yield surface in the principal stress space is shown in figure 2.31. The simplification of Mohr-Coulomb model where the hexagonal shape of the failure surface is changed with a circle is the Drucker-Prager model (figure 2.31). In simple 3D space, the hexagonal failure cone is replaced by a simple cone in Drucker-Prager failure model. It uses the same parameters to define the failure criterion.
Figure 2.31: Mohr-Coulomb and Drucker-Prager yield surface in the isometric principal stress space
Figure 2.32 shows the results of the simulation for the unconfined compressive strength tests at diﬀerent initial suction states. The non-linear elasto-perfectly plastic model used was able to predict the eﬀect of suction on the strength and stiﬀness. How-ever, the sudden transition from elastic to perfectly plastic is not consistent with the experimental observations. It is a known drawback of classic elasto-plastic models.
Figure 2.32: Experimental and modelling results of unconfined compressive strength at diﬀerent suction states from Francois et al. 2017 .
In addition to these models, there is Cam-Clay model which uses the strain hardening theory of plasticity to formulate the stress-strain model. It is based on critical state and generally used for normally consolidated or lightly overconsolidated soils.
Finally, for the constitutive modelling of unsaturated soil, it is essential to in-corporate the eﬀect of suction on the failure criterion. Thus the approach which has been used in literature to do a coupled analysis is discussed in the next part.
The link between the mechanical behavior and the hydraulic conditions can be considered by an approach that takes into account the hydro-mechanical coupling. As rammed earth is present in an unsaturated state, soil suction has a significant eﬀect on the stress state and thus the mechanical behavior. In unsaturated soil mechanics, the stress state of a porous medium can be represented by two independent stress variables such as net vertical stress ( ua) and matric suction (ua uw) which are measurable and have an experimental significance. On the other hand, an approach which uses single eﬀective stress to define the stress state can also be used. Eﬀective stress is the stress which is being transferred by grain to grain contact and responsible for the mobilization of shear strength in the soil. In order to represent the results of tests at diﬀerent suction conditions in a single stress framework, Bishop’s generalized eﬀective stress can be used:
where, 0 is the eﬀective stress tensor, is the net stress tensor, s is the suction, ij is the Kronecker delta ( ij=0 if i 6= j, else = 1) and is the eﬀective stress parameter which is a function of degree of saturation.
It helped to obtain a unique failure criterion by including the eﬀect of water retention on strength directly in the stress definition. To define the unified failure criterion, the intrinsic strength parameters: eﬀective cohesion (c’) and eﬀective friction angle ( 0) were determined by Gerard et al. 2015 . Consolidated and undrained (CU) triaxial tests at diﬀerent confining pressures were performed on saturated samples. The value of cohesion and friction angle obtained was 6.2 kPa and 36.5 respectively. Using these values, a Mohr-Coulomb failure criterion was drawn. This unified failure criterion and the various unconfined compressive strength and indirect tensile strength tests represented as their Mohr circle at failure state are shown in figure 2.33. The test which were carried out in saturated conditions did not fit well with the failure envelope. Otherwise, a decent fitting of the experimental data with the proposed unified failure criterion was observed.
Figure 2.33: Mohr circles at failure in terms of eﬀective stress for UCS (left) and indirect tensile strength test (right) at diﬀerent initial suction conditions .
In this chapter, the bibliographic review of the various experimental and numerical studies carried out on rammed earth was presented. Firstly various key mechanical parameters from an engineering point of view were discussed. Compressive strength was the most widely used indicator of strength for the construction of rammed earth structure. The dif-ferent standards and guidelines indicated that the minimum compressive strength should be in the range of 1-2 MPa.
Table of contents :
1.1 Earth as a construction material
1.2 Rammed earth
1.2.1 Advantages and limitation of earthen structures
1.3 Thesis objective and outline
2 Literature review
2.1 Key mechanical parameters
2.2 Factors influencing the mechanical characteristics
2.2.1 Sample geometry and scale
2.2.3 Clay content and nature
2.2.4 Dry density and method of compaction
2.3 Suction as a variable for describing water state
2.4 Modelling of Rammed earth
2.4.1 Damage models
2.4.2 Elasto-plasticity models
2.4.3 Coupled analysis
3 Hydro-mechanical behavior at material scale
3.2 Geotechnical characterization of the material
3.2.1 Particle size distribution
126.96.36.199 Sieve Analysis
188.8.131.52 Sedimentation Analysis
3.2.2 Atterberg limits
3.2.3 Other characterization tests
3.3 Sample Preparation
3.3.1 Proctor compaction test
3.3.2 Double Compaction
3.4 Hydric conditions
3.4.2 Soil water retention curve (SWRC)
3.4.3 Control of suction
184.108.40.206 Liquid-vapor equilibrium method
220.127.116.11 Axis translation technique
3.4.4 Hydric properties of rammed earth
18.104.22.168 Soil water retention curve
22.214.171.124 Hydraulic conductivity
3.4.5 Conditioning of samples
3.5 Influence of suction on mechanical parameters
3.5.1 Unconfined compressive strength test
3.5.2 Direct shear tests: Influence of suction on shear parameters
3.5.3 Unsaturated triaxial tests
3.5.4 Intrinsic shear parameters
3.6 Towards constitutive modeling
4 Experiments and simulation at structural scale on columns
4.2 Experimental study
4.2.2 Experimental Protocol
126.96.36.199 Sample Preparation
188.8.131.52 Sensor calibration and layout
4.2.3 Results of the drying phase
4.2.4 Unconfined compression test on columns
184.108.40.206 Experimental procedure
220.127.116.11 Results of the compression test
4.3 Numerical analysis
4.3.1 Theoretical aspects of CODE_BRIGHT
18.104.22.168 Balance equations
22.214.171.124 Constitutive equations and equilibrium restrictions
4.3.2 Material parameters
126.96.36.199 Hydro-thermal parameters
188.8.131.52 Mechanical parameters
4.3.3 Numerical simulations of drying phase
184.108.40.206 Initial conditions and boundary conditions
220.127.116.11 Drying phase simulation results
4.3.4 Numerical simulations of compression phase
18.104.22.168 Initial state and boundary conditions
22.214.171.124 Compression phase simulation results
4.4 Conclusions and Perspective
5 Case study: THM coupled simulations of rammed earth walls
5.2 General considerations for the simulations
5.2.1 Failure envelope
5.2.2 Environmental conditions
5.3 Single Wall
5.3.1 Compression of wall at compaction hydric state
5.3.2 Drying in warm conditions
5.3.3 Drying in cold conditions
5.4 Two walls joined at right angle
5.5 Conclusion and Perspective
6 Conclusions and perspectives
7.1 Etude expérimentale à l’échelle du matériau
7.1.1 Test de résistance à la compression non confiné (UCS)
7.1.2 Essai de cisaillement direct (DST)
7.1.3 Test triaxial non saturé
7.1.4 Test triaxial saturé
7.1.5 Vers une modélisation constitutive
7.2 Etude expérimentale à l’échelle structurelle sur colonnes
7.2.1 Comportement au séchage
7.2.2 Comportement de compression
7.3 Simulation du comportement de séchage et de compression des colonnes de pisé
7.3.1 Simulation de phase de séchage
7.3.2 Simulation de phase de compression
7.4 Étude de cas: simulations couplées THM de murs en pisé