A hierarchical elasto-plastic constitutive model for rammed earth

Construction and Building Materials 160 (2018) 351–364

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A hierarchical elasto-plastic constitutive model for rammed earth Winarputro Adi Riyono a,⇑, Eric Vincens a, Jean-Patrick Plassiard b a b

Université de Lyon, Ecole Centrale de Lyon, LTDS, Ecully, France Université de Savoie, Polytech Annecy-Chambéry, LOCIE, Le Bourget du Lac, France

h i g h l i g h t s
A hierarchical constitutive model based on elasto-plasticity is developed and presented.
A level is chosen given the quantity of information for model parameter identification.
The first level is able to capture some basic features of rammed earth behaviour.
The second level is able to quantitatively retrieve the non-linear behavior of rammed earth.

a r t i c l e

i n f o

Article history: Received 28 April 2017 Received in revised form 26 October 2017 Accepted 14 November 2017

Keywords: Constitutive model Elasto-plastic Interfaces

a b s t r a c t A hierarchical constitutive model based on the framework of elasto-plasticity is developed to model the mechanical behavior of rammed earth which is a quasi-brittle material. This model holds two mechanisms of plastic deformation, one related to shear failure and another one to a tensile mode of failure. Different hierarchies for the model, which are merely levels of complexity, are proposed according to the amount of information available to identify the model parameters. The model was validated simulating a diagonal compression test and a pushover test on a wallette. The simple elasto-plastic model (level1) was able to capture some basic features of the rammed earth behaviour essentially for a rather small range of deformations, it can be used for a first estimate of the loading capacity of a system. The more sophisticated model (level-2) was able to quantitatively retrieve the non-linear behavior of rammed earth over a larger range of deformations. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Rammed earth structures are obtained by successively compacting layers of moist earth within formworks. The definitive mechanical strength of these structures is obtained after some weeks of drying when the capillary tensions within the pore networks provides a strong bonding effect between the different particles constituting the material. Sometimes, rammed earth is mixed with cementitious materials to obtain impermeable and more durable walls. This construction method is currently becoming more popular because it meets the requirement of sustainable development such as a low embodied energy for the production and the processing of the materials which are locally extracted. Recent studies also showed that the material contributes to the inner comfort of rammed earth houses [24]. In the past, rammed earth was used in areas where stones were not available, for example in large sedimentary basins or large valleys. At present, it is not ⇑ Corresponding author. E-mail address: [email protected] (W.A. Riyono). https://doi.org/10.1016/j.conbuildmat.2017.11.061 0950-0618/Ó 2017 Elsevier Ltd. All rights reserved.

only used for building walls, but also for roofs, foundation, and garden ornaments [32]. In France, there is no regulation yet that rules the design, construction, and also the preservation of rammed earth structures which tend to slow down the development of this constructive technique. This building technique has not benefited from a century of studies, researches and feedback like more conventional technique including reinforced concrete. Nevertheless, there exists a certain amount of scientific works devoted on the basic mechanical of rammed earth including compression, tensile, and other deviatoric stress paths even if not comprehensive. First, on the basis of compression tests, Champiré et al. [11] found that humidity strongly influences the mechanical behaviour of rammed earth. In addition, a decrease of the elastic moduli were observed when unloading is performed from different compression stress levels. Unloadings also evidenced the existence of irreversibilities in the material which rate of creation are different from a given earth to another one. By using compression tests with different layers orientations with respect to the loading compression, Bui and Morel [6] found that rammed earth can be considered as a quasi isotropic material if the layers are adherent one to each

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other. This finding confirmed another study by Bui et al. [8] in which they found that the tensile strength in earthen layers and interface are very similar. Bui et al. [8] performed non homogenous tests, namely Brazilian tests to study the tensile behaviour of rammed earth. They found that the tensile strength of rammed earth can be taken equal to 10% of the corresponding compression strength which is similar to the relationship derived for concrete. More recently, Araki et al. [2] carried out homogenous uniaxial tensile tests and also Brazilian tests on both unstabilised and stabilised earth. The homogenous uniaxial tensile tests showed a pattern typical of quasi-brittle material like concrete with a quasi linear behaviour for the stress-strain curve until a peak, followed by a very sharp softening. They revealed that the tensile strength can vary between 5% to 12.5% of the corresponding compression strength for unstabilised rammed earth and between 15% until 20% of the corresponding compression strength for stabilised one. These extended range of values comes from the natural heterogeneity of the material. Finally, geotechnical shear tests such as the triaxial test and the shear box test can be used to estimate the shear strength of rammed earth. Jaquin et al. [20] used triaxial tests to study the source of shear strength in rammed earth and focused on the contribution of the matrix suction. They found that shear strength will increase as suction increases. Similar results were also obtained by Bui et al. [7]. The shear strength of the material can also be measured at a scale larger than the Representative Volumetric Element by means of a diagonal compression test [30,29,23], though it corresponds to a non homogenous test. To conclude, even if few different homogenous tests are available including unconfined compression tests and confined tests, there is not a comprehensive set of tests for a same material questioning different homogenous stress paths (compression and extension paths with different confining pressures). To model the mechanical behavior of earth, it is not clear whether the available constitutive models for concrete which is like earth (when the water content is as low as 2%) a quasi brittle material can be directly used for rammed earth without any modification. Some authors used simple elasto-plastic models to model the mechanical behavior of rammed earth systems, for example Jaquin [19] or Bui et al. [9] which cannot model the degradation of elastic stiffness observed when unloading. Other authors used models designed within the framework of damage elasticity [8,23] which cannot model the existence of permanent deformations in the material. Phenomena observed throughout experiments incite the use of an elasto-plastic model with damage elasticity. However, this complexity implies the involvement of a numerous set of model parameters that may be hard to identify in the absence of experimental data. In this context, a constitutive model denoted CJS-RE is specifically developed for rammed earth. It is inspired from CJS model which is a hierarchical constitutive model based on the elasto-plasticity theory that was originally designed by Cambou, Jafari, and Sidoroff [10] for granular materials. Different works have improved or extended the usage of this model for complex loadings such as cyclic one-way or two-ways cyclic loadings [16,27,21,4,14]. The interesting idea of this model lies in the hierarchical approach where the appropriated level (of complexity) of the model can be selected according to the information (generally experimental tests) available to identify the model parameters [28] and to the complexity of phenomena to model. This hierarchical framework was also used by Marzec and Tejchman [22] to model the mechanical behaviour of concrete samples in the context of cyclic laboratory tests. Herein, two first levels are given in the framework of elasto-plasticity. This constitutive law can be used for the prediction of the behaviour of rammed earth under mono-

tonous loading. A third level which would involve an elasto-plastic model with damage elasticity is out of the scope of this work. First, some reference experimental tests on rammed earth that were used in this study are exposed. After introducing the two levels of the proposed elasto-plastic model, the process for the identification of the model parameters is given. Finally, the constitutive model CJS-RE is validated by simulating a diagonal compression test and a push-over test which are two different boundary value problems. 2. Reference experiments for the simulations 2.1. Diagonal compression test Experiments are generally required for the modeller to have a better insight onto the mechanical behaviour of a material and onto the variety of phenomena that must be modeled. As important is the necessity of a certain set of experimental data which are required for the identification of the model parameters and the validation of the constitutive model. Herein, the study is based from a set of experiments performed by Silva et al. [29] on unstabilised rammed earth (MAT-1) sourced from Alentejo in Portugal (Fig. 1). The dry density of the material used was around 2100 kg/m3. Fig. 1a depicts shear stress-shear strain curves obtained on different wallettes throughout diagonal compression tests. Fig. 1b gives the typical crack patterns after the test where a vertical crack crosses the whole system and other cracks can also be observed at the edges of the specimen. Eleven wallettes with 550  550  200 mm3 size were loaded under a displacement control with a rate of 2 lm/s. In Fig. 1a, the global behaviour shows a first peak reached at the average level of 0.13 MPa and small deformations followed by a small re-increase of the resistance. One can note that repeatability is not easy to achieve due to the inherent heterogeneity of the system composed of different compacted layers. 2.2. Pushover test The second reference experiment consists of pushover test performed on wallette [15]. Rammed earth materials were sourced from the demolition of an old farmhouse located in Dagneux (Auvergne-Rhône-Alpes region, France). A wallette measuring 1500  1500  250 mm3 (wallette-3) was pushed laterally until failure while a vertical pressure on the wallette equal to 0.3 MPa was used to represent the typical pressure of a two storeys house. The obtained load-displacement curves are depicted in Fig. 2a. In Fig. 2a, the wallette exhibit nonlinear responses until the horizontal resistance of 40 kN at displacement of 9 mm before it reaches yield plateau. Fig. 2b shows cracks pattern where horizontal cracks at the interfaces were found at the bottom left of the wallette and a quasi diagonal cracks at the center part of the wallette. 3. Constitutive equations for CJS-RE 3.1. CJS-RE1: a level-1 model The first level of the model holds the basic features of an elastoplastic model such as Mohr-Coulomb model, adapted to take into account the specificities of quasi brittle materials. The advantage of CJS-RE1, unlike Mohr-Coulomb model, lies in the shape of the shear failure surface which is continuously differentiable. Plasticity is generated whenever the current state of stress reaches the shear failure surface which also acts as a yield surface. There exists two kinds of failures for quasi brittle materials including a failure due to excessive shearing and a failure due to

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Fig. 1. Diagonal compression tests; (a): shear stress vs shear strain and (b): typical crack pattern in the wallette [29].

Fig. 2. Pushover test: (a): load-displacement curve for wallette-3 and (b): crack patterns in the wallette-3 after test [15].

an excessive tensile stress within the material. The associated fails

t

ure surfaces f and f respectively, are given in Fig. 3. The inner envelop of these two surfaces (solid lines) splits the stress space into an inner space where the state of stress is acceptable for the material from an outer space that is not physically reachable for it.

Accordingly, CJS-RE1 model consists of three mechanisms of deformation, an elastic mechanism and two plastic mechanisms. The elastic mechanism is governed by the linear Hooke’s law. The plastic mechanisms are activated when the current state of stress reaches one of the failure surfaces either by shearing or ten-

Fig. 3. Failure surfaces of CJS-RE1; (a): in the meridian plane ðhs ¼ constÞ and (b): in the deviatoric plane ðI1 ¼ constÞ. I1 is the first invariant of the stress tensor, sII is the second invariant of the deviatoric stress tensor, s1 , s2 and s3 are the principal stresses of the deviatoric stress tensor.

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sion. Herein, stresses and strains are positive when of compression type. Due to the existence of three mechanisms, the incremental deformation tensor writes:

e_ total ¼ e_ e þ e_ sp þ e_ tp

ð1Þ

where e_ e is the increment of the elastic deformation tensor, e_ sp is the increment of plastic deformation tensor due to the shear mechanism and e_ tp is the increment of plastic deformation due to the tensile mechanism. 3.1.1. Elastic mechanism The elastic deformations are computed according to Hooke’s law:

e_ e ¼


  m 1þm r trðrÞI d E E

ð2Þ

where E is the Young’s modulus, m is the Poisson ratio, and I d is the identity tensor. 3.1.2. Shear plastic mechanism The shear yield surface is confounded with the shear failure surface which is isotropic and according to the original CJS model writes:

f ðr; Rfail Þ ¼ sII hðhs Þ  Rfail ðI1 þ 3T r max Þ 6 0 s

ð3Þ

with sII is the second invariant of the deviatoric part of the stress tensor r; hðhs Þ is a shape factor that allows to take into account the dissymmetry of the behaviour in compression and extension, I1 the first invariant of the stress tensor. Rfail is the average radius of the shear failure surface and T r max characterizes the maximum tensile strength of the material. They are both model parameters. In Eq. (3), the shape factor hðhs Þ is defined according to the Lode angle as follows:

hðhs Þ ¼ ð1  c cosð3hs ÞÞ1=6

ð4Þ

where hs is a Lode angle in the deviatoric plane such that when hs ¼ 0 , the stress path corresponds to a compression stress path and when hs ¼ 60 to an extension stress path. c is a model parameter which quantifies the dissymmetry of the failure surface. There is some limits in the use of such analytical relationship for the shape factor since it is not warranty the convexity of the yield surface for any value of c. The yield surface is still convex if c is smaller than 0.856. Other shape functions that do not hold such limitation can be found in Bigoni and Piccolroaz [5]. The flow rule of the plastic shear mechanism is non-associated and the direction of the plastic deformation induced by this mechanism is derived from a potential surface g s . The flow rule is defined by the following relationship:

e_ sp ¼ k_ s

@g s @r

ð5Þ

with k_ s the plastic multiplier of the shear mechanism. Instead of defining the potential surface, the direction G of the increment of plastic deformation is directly computed. To allow the creation of phases of contractancy and dilatancy when shearing, G is computed in order to satisfy the following dilatancy law in the original CJS model:

e_ sp v ¼b




sII

1 smvc II



j se_ sp j sII

ð6Þ _ sp

where s is the deviatoric stress tensor, e is the increment of the plastic deviatoric strain tensor. smvc is the value of the second invariII ant of the deviatoric stress tensor at the Maximum Volumetric Contraction (MVC) state for the current value of the mean pressure. For

the sake of simplicity, the MVC surface is isotropic and holds a shape similar to that of the failure surface:

f

mvc

¼ smvc II hðhs Þ  Rmvc ðI1 þ 3T r max Þ

ð7Þ

with Rmvc the average radius of the MVC surface which is a model parameter. Plastic contractive volumetric deformations are generated by the shear mechanism when sII < smvc otherwise plastic dilaII tive volumetric deformations are generated. In Eq. (6), b is a model parameter to ensure that positive volumetric deformations take place for contraction according to the chosen convention. 3.1.3. Tensile plastic mechanism The tensile yield surface is confounded with the tensile failure surface. This yield surface is able to soften and the tensile strength T r is bound to drop to zero when tensile failure is triggered. This yield surface writes: t

f ðr3 Þ ¼ r3  T r 6 0

ð8Þ

where r3 is the minor principal stress and T r is a tensile strength. This criterion is similar to the Rankine criterion for brittle fracture of concrete [12]. Since in experimental unixaxial tensile tests, irreversible deformations due to this mechanism are parallel to the direction of tensile stress, the plastic flow rule of the tensile plastic mechanism is associated. 3.1.4. Tensile softening The softening of the tensile yield surface is defined by an exponential function as follows:


Z  T r ¼ T ini e_ pt dt r exp at

ð9Þ

with T ini r a model parameter which corresponds to the initial value of T r . The tensile softening is characterized by a sharp drop of the tensile strength once the tensile criterion is reached and in this case, a default value of at equal to 1.0 is recommended. Then, for CJSRE1 model, at does not need to be identified. 3.2. Identification of CJS-RE1 model parameters The basic level for the model presented here-before and denoted CJS-RE1 model, involves 7 parameters. A compression test allows the two elastic model parameters E and m to be identified. They are defined as the initial tangential properties of the experimental stress-strain curves. The parameters related to the shear plastic mechanism (Rfail , T r max , and c) can be determined from two compression tests with different confining pressures and an extension test. In the absence of extension test, in this study, c will be set equal to 0.85, a typical value for concrete which warranties the convexity of the shear failure surface [1]. If just an unconfined compression state is available, either Rfail or T r max will have to be stated from usual average values. For the level-1 model, Rmvc coincides with Rfail . It means that dilation takes place at the moment when shear failure is reached. Parameter b can be estimated from the volumetric deformation curve of the compression test. The initial slope of dilation after the maximum volumetric contraction state is taken into account since following this short regime, dilation mainly results from opening cracks in a strong discontinuous medium. The tensile strength T ini r can directly be obtained from a tensile test. If the tensile test is not available, the usual relationship between the maximum uniaxial compression resistance f c and in CJSthe maximum uniaxial tensile resistance f t (equal to T ini r RE model) for rammed earth can be used where f t lies between

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5%f c and 12:5%f c [2]. In this study, T ini is stated to be equal to r 8%f c .

and failure surfaces just shift along the hydrostatic axis. The shear softening law writes:

3.3. CJS-RE2: a level-2 model

res p res T r max ¼ ðT ini r max  T r max Þ expðas ediff Þ þ T r max

ð13Þ

epdiff ¼ keps  eps peak k

ð14Þ

The second level of the model is an elasto-plastic model is encapsulated in CJS-RE1 model. Consequently, it holds the same features as CJS-RE1 but a refinement is introduced with the use of a deviatoric yield surface different from the shear failure surface. It allows the domain of elasticity to be far smaller than in the case of CJS-RE1 (Fig. 4) at the initial stage of the simulation. A simple isotropic hardening of the deviatoric yield surface is added which is suitable when just monotonous loadings are to be simulated. Beside hardening, CJS-RE2 model can also exhibit a shear softening and also a controlled tensile softening which was made possible for the user of CJS-RE1. 3.3.1. Isotropic hardening In CJS-RE2, the size of yield surface can initially be set to a small radius and then can expand isotropically when shearing. The hardening law related to the average yield surface radius R writes:

_ R_ ¼ AðRfail  Rini Þ expðApÞ

ð10Þ

where p_ is the increment of a hardening variable and A is a model parameter which controls the velocity of the isotropic hardening. Two other parameters, Rini and Rfail which are respectively the initial value of the elastic radius and its upper limit must be also identified. The increment of the hardening variable p_ is given by the normality relationship as s

@f ¼ k_ s ðI1 þ 3T r max Þ p_ ¼ k_ s @R

ð11Þ

Eq. (10) can be rewritten so that the value for R can be directly given:

R ¼ Rfail  ðRfail  Rini Þ expðApÞ

ð12Þ

Therefore at the beginning (p ¼ 0), the radius of the yield surface is equal to Rini and when p ! 1 then R value tends to Rfail . 3.3.2. Shear softening In quasi brittle materials and especially in rammed earth, softening results from the development of cracks and decohesion of grains [25]. In this study, shear softening is modelled by decreasing the maximum tensile resistance T r max associated to both the shear yield surface and the shear failure surface. Consequently, the yield

res with T ini r max the initial value of T r max and T r max is the residual value of T r max . Parameter as is a model parameter that controls the rate of the shear softening. epdiff is the norm of the differences between current deviatoric plastic strain tensor eps and deviatoric plastic strain

tensor at peak eps peak . The latter tensor is defined as the state of plastic deformation when the stress path reaches any failure surface. It implies that if the tensile failure surface is first reached, the capacity of the material to resist to shearing automatically decreases according to the model.

3.3.3. Tensile softening In CJS-RE2 model, the tensile yield surface can soften with a velocity that can be controlled through the value of the model parameter at . In CJS-RE1 model, this value was stated to obtain a sudden drop of the tensile resistance towards zero.

3.4. Identification of CJS-RE2 parameters Herein, the identification of the parameters that are considered as new respectively from CJS-RE1 model are exposed. Then, CJSRE2 model needs four additional model parameters compared to CJS-RE1 to be identified. These parameters are the initial elastic radius ðRini Þ of the shear yield surface, the isotropic hardening parameter A or the shear yield surface, as which controls the shear softening velocity and at which controls the tensile softening velocity. Rini can be identified from a compression test and corresponds to the linear zone of behavior in the stress-strain curve. A and as are determined by a trial-and-error method. The other parameters that will be stated are: the residual value for T max denoted T res r max and the average radius of the MVC surface Rmvc which can be different from Rfail in CJS-RE2. Herein, this residual value is stated T res to be equal to 20% T ini r max which is a typical statement for the modelling of quasi-brittle material [25]. Unconfined compression test on different earthen materials proved that Rmvc can be set equal to 0:85Rfail .

Fig. 4. Failure surfaces of CJS-RE2; (a): in the meridian plane ðhs ¼ constÞ and (b): in the deviatoric plane ðI1 ¼ constÞ. I1 is the first invariant of the stress tensor, sII is the second invariant of the deviatoric stress tensor, s1 , s2 and s3 are the principal stresses of the deviatoric stress tensor.

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4. Validation of CJS-RE model The validation of the model is composed of simulations of two different boundary value problems (BVP), namely a diagonal compression test involving material MAT-1 and a pushover test involving material MAT-2. The process starts with the identification of the model parameters for CJS-RE1 and CJS-RE2, then simulations of these two BVP are exposed. 4.1. Identification of the model parameters 4.1.1. Material MAT-1 The identification of CJS-RE1 model parameters for MAT-1 was carried out with just an unconfined compression test (no further test has been carried out by Silva et al. [29]). Then, as mentioned here before, a statement was required on either Rfail or T r max . In fact, these two unknowns can be derived from a set of two equations: first, the value of the maximum compressive strength and secondly a relationship relating the ratio f t =C and the Plasticity Index (PI) of the material [18], where C is the cohesion in MohrCoulomb criterion. The relationship between f t and C was used by Silva et al. [29] in their work. For this purpose, some relationships relating the model parameters of the shear failure surface of CJS-RE1 model and the parameters of Mohr-Coulomb criterion are given in Appendix A. No volumetric deformations were monitored in the work of Silva et al. [29], then for the determination of b, average values from the literature for rammed earth was used [9]. One must remind that volumetric deformations have little impact on the stres-strain curve for quasi brittle materials. Finally, since no data for uniaxial tensile nor Brazilian test were available, at was defined from a previous study involving another earthen material than MAT-1 as a first estimate for its value [2]. The result of the identification process is given in Table 1 and the numerical simulation of the unconfined compression tests for MAT-1 is provided for CJS-RE1 in Fig. 5. As expected, the model fairly succeeded to retrieve the initial tangent of the stress strain curve and the peak of response but the general stress path is excessively stiff. This feature is related to the elasticity which is linear before the shear yield surface is reached. Further model parameters for CJS-RE2 model for material MAT1 considered herein are given in Table 2. The resulting stress-strain curve from the identification process is given in Fig. 6. As expected plasticity generates ductility in the system and allows to fit the average experimental curve. However, one can note that the analytical form of the isotropic hardening function is not complex enough to control both the non linear behavior and the value of deformation at peak. It could be achieved but to the expense of simplicity for the hardening law of the shear yield surface. 4.1.2. Material MAT-2 According to the experiments data from El Nabouch [15], the set of model parameters for CJS-RE1 model was identified and is given in Table 3. First, the parameter of dissymmetry (c) is stated considering the maximum convexity of the shear criterion in the devia-

Table 1 Identified model parameters for CJS-RE1 model for MAT1, related to experiments from Silva et al. [29]. Elastic

Plastic

E = 1036 MPa m = 0.25 MPa

b ¼ 3:0 c ¼ 0:84 Rfail ¼ 0:22 T ini r ¼ 101 kPa T r max ¼ 730 kPa

Fig. 5. Identification of model parameters in CJS-RE1 with a compression test; experiments from Silva et al. [29].

Table 2 Further parameters for CJS-RE2 model for MAT-1, related to experiments from Silva et al. [29]. Elastic

Plastic

E = 1036 MPa

T ini r max ¼ 730 kPa

m = 0.25 MPa

ini T res r max ¼ 0:2T r max A = 0.00013 Rini ¼ 0:08 Rmvc ¼ 0:85Rfail as ¼ 0:0003 at ¼ 0:5

Fig. 6. Identification of model parameters for CJS-RE2 model with a compression test; experiments from Silva et al. [29].

Table 3 Identified CJS-RE1 model parameters for MAT-2, related to experiments from El Nabouch [15]. Elastic

Plastic

E = 760 MPa m = 0.25 MPa

b = 1.0 c = 0.85 Rfail ¼ 0:39 T r ¼ 160 kPa T rmax ¼ 350 kPa at ¼ 1:0

toric plane, therefore c is stated to be equal to 0.85. The plastic shear parameters (Rfail or T r max ) are estimated from known cohesion (C) which is around 135–260 kPa, / between 44 —45 (small shear box tests), and average compression strength on tested

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rammed earth cylinder of 2.0 MPa [15]. Parameter of dilatancy (b) is taken as 1.0 considering that the density of MAT-2 (1878 kg/m3) is lower than MAT-1 (2100 kg/m3). Note that dilatancy in rammed earth is mainly related with structural increases of volume due to opening of cracks and not related to the material itself. Further model parameters for CJS-RE2 model for material MAT-2 are given in Table 4. 4.2. Diagonal compression test: validation of the model A numerical simulation of a diagonal compression test involving MAT-1 was then performed to validate the proposed constitutive model. A diagonal compression test is a non homogeneous test by nature and very different from the test used for the identification of the model parameters (Section 3.2). The simulation is carried out by using FLAC-3D (code ITASCA) where the proposed constitutive model was implemented. The geometry of the model together with the imposed boundary conditions are depicted in Fig. 7. The quantities monitored in the simulation are those indicated by ASTM-E-519 [3] where:

P

s ¼ pffiffiffi

ð15Þ

h 2t

Fig. 7. 3D-Model of the diagonal compression test on a wallette.

s is the shear stress acting on the net area of the specimen (MPa), P is the reaction load (kN), h is the specimen height (mm), and t is the specimen thickness (mm). The shear distortion or shear strain can be calculated as follows:

cd ¼

DV þ DH g

ð16Þ

where cd is the shear strain, DV is the vertical shortening (mm), DH is the horizontal extension (mm) and g is gauge length (mm). The pffiffiffi gauge length is taken to be equal to h 2=3 [29]. The numerical test is displacement controlled with a corresponding velocity of 0.01 lm/s. The global stress-strain response is given in Fig. 8. The initial stiffness of the curve is stiffer than in actual experiments. The presence of interfaces between different compacted layers of material in actual experiments that were not modeling herein may explain such feature [9]. The simulation with CJS-RE1 model shows a local peak of 0.16 MPa before a small drop of the shear resistance. This peak is 25% higher than the average value found in the experiments which is of 0.13 MPa. Afterwards, a re-increase of the shear strength is observed; however this increase is found much higher in the modeling than in the actual experiments. To get more insight onto the observed features through the simulation, Fig. 9 shows the evolution of the plasticity states throughout the simulation of a diagonal compression test. At state 1 which corresponds to a transitory peak of resistance, a tensile failure starts at the center (on the outer surface) of the wallette. At the top and bottom edge of the specimen, some elements experience a shear failure. Between state 1 and state 2 (corresponding to the

Table 4 Further identified CJS-RE2 model parameters for MAT-2, related to experiments from El Nabouch [15]. Elastic

Plastic

E = 760 MPa

T ini r max ¼ 350 kPa T res r max ¼ 70 kPa A = 0.00013 Rini ¼ 0:16 Rmvc ¼ 0:85Rfail as ¼ 0:0003 at ¼ 0:5

v = 0.25 MPa

Fig. 8. Shear stress-shear strain response from a simulated diagonal compression test using CJS-RE1, compared to experiments from Silva et al. [29].

drop end), the tensile failure propagates upwards and downwards respecting the symmetry of the system. While the extent of the pattern is important in the simulation, the phenomena are much more concentrated in the actual experiments (Fig. 1b). This pattern is typical of the continuum approaches as can be seen also in Miccoli et al. [23] and Silva et al. [29]. The propagation splits into two parts close to the top and bottom edges of the specimens in the direction of the previously mentioned zones at failure by shearing. Subsequently, the tensile failure expands in the horizontal direction between state 2 and state 3. Finally, while the zones at failure by tension keep expanding, the elements along the wallette vertical plane of symmetry reach the shear failure surface between state 3 and state 4. The evolution of the number of zones experiencing tensile failure shows that the behavior of the wallette throughout a diagonal compression tests is mainly monitored by the tensile failure criterion. To complement the analysis, Fig. 10 depicts the stress path of the element at the center and outer surface of the wallette. This is the first element that experiences tensile failure and seems to be typical of the ones that monitor the overall behavior of the wal-

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Fig. 9. Evolution of the plasticity states in a simulated diagonal compression tests using CJS-RE1 at different computation stages.

lette throughout the test. While Figs. 10a-c-e-g give information throughout the simulation in the meridian plane, Figs. 10b-d-f-h give other information in a given deviatoric plane indicated in the former figures. Fig. 10a-b give the stress path of the chosen element towards state 1. The stress path is elastic and this element reaches the tensile failure criterion at state 1 (Fig. 10b). From state 1 to state 2 (Fig. 10c-d), the tensile failure surface softens until a T r value equal to zero (state 2). The drop of resistance in Fig. 8 is then related to the softening of the tensile failure surface for the elements at failure by tension. Finally, from state 2 to state 3 (Fig. 10e-f), the stress path follows the tensile failure surface while the mean pressure increases due to the processing compression of the wallette. In Fig. 10g-h, one can see that at state 4, the chosen element reaches the shear failure surface. From now on, the state of stress can only evolve at the junction of the two failure surfaces. The stabilisation of the response in Fig. 9 can be explained by the expansion of the failure by tension which almost spread all over the wallette. As a conclusion, even if the main features of a homogeneous compression test were retrieved by the elastoplastic model CJSRE1, the validation of the model using a non homogeneous test shows a drawback of the constitutive model. Even if the local peak of resistance was obtained with a correct value which may be a sufficient result for a first analysis, the overall resistance at large deformation obtained throughout the diagonal compression test is highly overestimated. To improve the prediction of the mechanical behavior of actual systems along complex stress paths, a refinement of the modeling is required by using CJSRE2 model. The global stress-strain curve obtained by the simulation of the diagonal compression state with CJS-RE2 is given in Fig. 11. Herein, the first local peak is obtained for a shear stress of 0.13 MPa, which is lower than for CJS-RE1 model. The use of CJS-RE2 model has corrected the bad retrieve of the mechanical behavior for larger deformations by CJS-RE1 model. Herein, the shear resistance for larger deformations keeps in the range of the average behavior found throughout experiments for the tested wallettes. As for the case of CJS-RE1 model, the evolution of the plasticity states during the simulation of the diagonal compression test are shown in Fig. 12. The main difference lies in the generation of plas-

ticity (due to shearing) from the very beginning of the test before the tensile yield surface is reached. Moreover, at state 4, the zones at tensile failure (pale blue in Fig. 12) are more visible which is consistent with the crack pattern observed in the actual wallette (Fig. 1b). Fig. 13 shows the evolution of stress path in the central surface element of the wallette. Fig. 13a-c-e-g are view sections of stress path in the meridian plane and Fig. 13b-d-f-h are sections in the deviatoric plane. The stress state reached first the yield surface. Eventually, tensile failure started at the central core of the wallette before the studied element. It led to a relaxation of stresses and then to a drop of the mean pressure before the stress path of the studied element reaches state 1 (Fig. 13a-b). This deviation of the mean stress towards smaller values explains why the tensile failure surface is reached by the studied element for a smaller value of sII which is revealed in Fig. 11 than in the case of CJS-RE1. From state 1 to state 2 (Fig. 13c-d), the tensile failure surface softens (same feature for the yield surface) but during this period, the average radius of the yield surface keeps constant (softening is obtained by a reduction of the tensile resistance T r and of the maximum tensile resistance T rmax ). From state 2 to state 3 (Fig. 13e-f), the stress path keeps on the tensile failure surface and just a small increase of the mean pressure is required to reach the shear yield surface. From now on, the stress path keeps at the junction of the tensile failure surface and of the shear yield surface. The softening of T r max (that started previously when the tensile failure surface was reached) and the hardening of the mean radius R (the mean pressure increases due to the reserve of resistance of the wallette) are then again activated. Note that between state 2 to state 3, the rate of hardening (rotation of yield surface) is greater than the shear softening (shifting of the yield surface). From state 3 to state 4 (Fig. 13g-h), the rate of hardening is getting slower (as plastic deformation are generated) than throughout the previous stage, while the shear softening process keep moving. This condition force the stress path to move from point 3 to point 4. From state 3 to 4, the softening of T r max monitored the behavior of the element which can also be noticed at the scale of the wallette (many of the elements in the wallette hold this behavior).

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Fig. 10. Stress path of an element in a simulated diagonal compression test (CJS-RE1).

The drop is supposed to go on until the residual value for T r max is reached. To conclude, shear softening in CJS-RE2 model has an important contribution to limit the re-increase of the shearing resistance which is excessive when using CJS-RE1 model. In CJS-RE2, the

shear plastic mechanism (together with the shear failure surface) is coupled with the tensile plastic mechanism through the softening law driving the loss of the maximum resistance T r max . CJS-RE2 model was able to retrieve the average stress strain curve obtained throughout experiments within a wide range of deformations.

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Fig. 11. Shear stress-shear strain response from a simulated diagonal compression test using CJS-RE2, compared to experiments from Silva et al. [29].

However, one can note that the general ductility of the wallette is not perfectly modeled with CJS-RE2 model and too high initial stiffness and a peak of the curve that is obtained for smaller deformations than in actual experiments. This effect may be attributed to the statement of a homogenous system where no interfaces between different compacted layers have been herein modeled [9]. However, this refinement of the modeling seems oversized for such a BVP where mechanical properties of interfaces are generally unavailable and dispersion of results is so significant. 4.3. Pushover test on a wallette: validation of the model A second boundary value problem involving a pushover test on a wallette with material MAT-2 (see Tables 3 and 4) is simulated for the validation of CJS-RE model. The geometry of the model together with the imposed boundary conditions are depicted in Fig. 14. First, vertical pressure of 0.3 MPa is imposed on the upperpart of the wallette. Then lateral displacement is applied at each point in the upperpart of wallette with rate of 0.01 lm/s. The horizontal displacement (d) was measured at the upperleft of the wallette. Fig. 15a gives both the load-displacement curve obtained throughout simulation using CJS-RE1 and the response recorded throughout the experiments by El Nabouch [15]. The simulation with CJS-RE1 reaches a peak for 50 kN with a corresponding dis-

placement of 6 mm. The curve is generally stiffer than in the experiments which was expected and the ultimate loading capacity is over-estimated by 25% in the simulation. Nevertheless, CJS-RE1 may give a first estimate of the loading capacity of the system with few effort to identify the model parameters. Fig. 15b depicts the plastic states at three different computation stages (reference points given in Fig. 15a). At state 1, a detachment at the wallette bottom right is noticeable together with a zone at shear failure in the bottom left part. At state 2, a tensile failure is found to propagate from the bottom left to top left of the wallette. This pattern holds true at state 3. The general failure mode is similar to the experiments (shear failure at the bottom left of the wallette). Nevertheless, the pattern of plastic points is different from the experimental evidences where a quasi diagonal crack was observed on wall-3 [15]. Another simulation is carried out using CJS-RE2 model. Fig. 16a gives the result of the simulation for the pushover test performed together with the corresponding variables monitored throughout the experiments. A better result for the simulation is obtained qualitatively and quantitatively. The response is generally softer than with CJS-RE1 and the maximum resistance is closer to the experimental data. However, the pattern of plastic points is similar with what was obtained with CJS-RE1 (Fig. 16b). To conclude, CJS-RE2 model was able to retrieve a better loaddisplacement curve obtained throughout experiments within a larger range of deformations. Nevertheless, one can note that the general ductility of the wallette is not perfectly modeled with CJS-RE2 model. In the simulation, the initial stiffness and the loading capacity are overestimated while failure is obtained for a too large deformation. This departure may be due to the absence of the interfaces which probably improves the result though it has been proven experimentally that the mechanical properties of the interfaces are close to those of the layers of compacted earth. The role of local defaults in the actual wallettes can also alter its loading capacity but this aspect is difficult to monitor throughout experiments. 4.4. Parametric study A parametric study was performed to investigate the role of some model parameters in the response of the wallette throughout the diagonal compression test and the pushover test. The parametric study has focused of some main model parameters involved in the plastic shear mechanism and plastic tensile mechanisms. Fig. 17 depicts the non-dimensional variables related to the resistance obtained at the first local peak (state 1 of Fig. 12) in

Fig. 12. Evolution of the plasticity states in a simulated diagonal compression tests using CJS-RE2 at different computation stages.

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361

Fig. 13. Stress path of an element in a simulated diagonal compression test (CJS-RE2).

the diagonal compression test. The normalisation is processed using the values obtained in Section 4.2 for the shear resistance ðs=sref Þ and the value of the model parameter used in Table 5 ðx=xref Þ. The steeper the evolution of the curve for a given parameter, the more influence its holds in the simulation. According to the Fig. 17, parameters Rfail ; T r , and at are the parameters which

much influence the value at state 1 in the diagonal compression test. The influence of parameters A and as are of second order but may be more important to explain the second part of the curve in Fig. 12. Fig. 18 provides a similar analysis in the case of the pushover. The ratio H=Href corresponds to the normalised maximum loading

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Fig. 14. 3D-Model of the pushover test on a wallette.

capacity (normalised with the value found in the simulation performed in Section 4.3). The ratio x=xref is similar to the one exhibited in the parametric study performed for the diagonal compression test. According to Fig. 18, the parameters Rfail and T r are the parameters with a major role to explain the maximum response in the pushover test. On the other hand A; as , and at hold a secondary role. This last results emphasis the fact that the model calibration requires adequate experimental characterization experiments in order to seize the relevant behaviour of the material.

5. Conclusion In this paper, a hierarchical elasto-plastic constitutive model for studying the mechanical behaviour of rammed earth structures has been presented. There are two different levels for the constitutive model CJS-RE be used for monotonously loaded structures related to the number of different experiments tests available for the identification of the model parameters. CJS-RE1 model requires the identification of seven parameters while there are twelve parameters to identify for the more complex version CJS-RE2. The model strictly requires two compression tests with two different confining pressures, an extension test, and a tensile test. The measurements of the volumetric deformations may be required. If some

tests are not available, relationships with Mohr-Coulomb model can be used but some typical values can also be recommenced. The models have been validated on an actual diagonal compression test and pushover test whose stress path is different from the stress path involved in the identification of the model parameters. The simple elasto-plastic CJS-RE1 model is able to capture some basic features observed in a compression test which was used in the identification of the model parameters. In the simulation of diagonal compression test, the model gives a good estimate of the first local peak related to the tensile failure of the wallette. But, subsequently a large reincrease of the shear resistance is observed contrary to the experiment which is due to the too large extent of the elastic domain. In the pushover test, simulation with CJS-RE1 produces an overestimate results. Nevertheless, it might be used as a first estimation of the loading capacity of the system. A definite improvement was obtained with CJS-RE2 thanks to the reduced elastic domain and due to the shear softening law associated to the shear yield surface. The shear softening law involves a coupling between the tensile failure surface and both the shear yield surface and the shear failure surface. In the case of CJS-RE2, the prediction of wallette behaviour in a diagonal compression test and pushover test was fairly good. There is a close relationship between the overall behavior of the wallette and the stress path followed by the element at its center and at its surface throughout the diagonal compression test. The study of the stress path followed by this element complemented and explained why CJS-RE1 model is enough precise to predict the phenomena measured at the global scale and for small deformations and why CJS-RE2 model is required to better retrieve the phenomena observed for a much larger range of deformations. Finally, a parametric study involving the two boundary value problems solved herein showed that the radius at failure Rfail , the tensile strength T r , and the tensile softening at are the parameters with the major influence on the first local peak in the diagonal compression test. In the case of the pushover test, Rfail and T r play a major role to explain the value of the maximum load capacity of the wallette. Therefore, experimental investigations of the mechanical behaviour of rammed earth should determine carefully these peculiar model parameters. In a context where the seismic behavior of rammed earth construction is addressed, two approaches for the seismic design may be used, after the European Standard [17]: pushover tests or cyclic tests. The first one correspond to a monotonic loading (Section 4.3) for which the CJS-RE2 model is suitable. The second approach requires further developments including the degradation of the stiffness of the material when cycling together with a kinematic hardening in order to model irreversibilities in the material

Fig. 15. (a): Load-displacement response from a simulated pushover test on a wallette using CJS-RE1 vs experiments, (b): evolution of plastic points.

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363

Fig. 16. (a): Load-displacement response from a simulated pushover test on wallette using CJS-RE2 vs experiments, (b): evolution of plastic points.

when unloading. It implies to encapsulate a new hierarchy in the present model. Moreover, as the construction of a wallette requires the compaction of successive layers of moist earth, the modeling of the wallette as a heterogeneous system is questionable. Further studies are required to investigate the relevance and the limitations of this approach. Acknowledgements

Fig. 17. Non-dimensional relationship between the maximum shear stress and the parameters assessed (x) for the homogeneous system in the diagonal compression test.

Table 5 Parameters considered in the parametric study. Parameters

Reference value ðxref Þ

Lower value

Upper value

A Rfail y Rfail à

0.00013 0.22 0.39 0.0003 8%f c 0.5

0.00010 0.18 0.38 0.0001 5%f c 0.02

0.00016 0.26 0.41 0.0005 12%f c 1.0

as Tr

at à y

Rfail in the pushover test. Rfail in the diagonal compression test.

This work was supported by the French Research program ANR12-VBDU-0001-06 funded by the National Agency of Research (ANR). The author expresses their gratitude for the financial support from ANR. Appendix A. Relationship between CJS-RE and Mohr Coulomb model Some parameters in CJS-RE model are closely related to the parameters of Mohr-Coulomb model. In fact, four CJS-RE parameters can be estimated if the parameters of Mohr-Coulomb model are known. These parameters are T r max ; c; Rfail , and b. Maximum tensile strength (T r max ) which is the apex of CJS-RE model are related to Mohr-Coulomb model parameters by the following relationship:

T r max ¼

pffiffiffi 3ðCÞ cotð/Þ

ðA:1Þ

where cohesion (C) is estimated based on the Plasticity Index (PI) [18]:

ft ¼ 0:34 þ 0:01ðPIÞ C

ðA:2Þ

Therefore, by set the tensile strength around 5%f c and 12:5%f c [2] and also Plasticity Index from the experiments references (PI = 0.7), we will get acceptable cohesion between 182–454 kPa. Finally, the value of C = 318 kPa was taken according to the average value of the acceptable cohesion. By using this value, we get T r max equal to 730 kPa. Dissymmetry of the shear failure surface (c) can be estimated by comparing the ratio of the radius at failure in the tensile and the compression meridian ðRt =Rc Þ between the model (Eq. (A.3)). So, for / ¼ 37 , it is associated with c ¼ 0:84 in CJS-RE model. This value of c still satisfy the convexity which assures a stable material behaviour according to the postulate of Drucker [13]. Fig. 18. Non-dimensional relationship between the maximum horizontal load and the parameters assessed (x) for the homogeneous system in the pushover test.

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CJSRE
1 Rt 1c 6 ¼ Rc 1þc
MC Rt ð3  sin /Þ ¼ ð3 þ sin /Þ Rc

ðA:3Þ

Radius at failure can be deduced as follows. From MohrCoulomb model, we can write:

pffiffiffiffiffiffi

r1 ¼ r3 ðN/ Þ  2C N/ with N/ ¼

ð1 þ sin /Þ ð1  sin /Þ

In an axisymmetry triaxial condition, sII ¼

ðA:4Þ qffiffi 2 3

j r1  r3 j,

I1 ¼ r1 þ 2r3 , and the radius at failure Rfail in CJS-RE model can be deduced from Eq. (3) using Eqs. (A.1) and (A.4):

Rfail ¼

rffiffiffi 2 3

! pffiffiffiffiffiffi 1 ðr3 ðN/  1Þ  2C N/ Þð1  cÞ6 pffiffiffi pffiffiffiffiffiffi r3 ðN/ þ 2Þ  2C N/ þ 3 3ðCÞ cotð/Þ

ðA:5Þ

If c and T r max are known, Rfail can be identified through:

Rfail

! rffiffiffi 1 2 qfail ð1  cÞ6 ¼ 3 Ifail 1 þ 3T r max

ðA:6Þ

where qfail is a deviatoric stress at failure and Ifail 1 is the first invariant of stress tensor. In the unconfined compression test qfail ¼ f c and Ifail 1 ¼ f c. Parameter of dilatancy (b) can be determined from the dilatancy angle (w) according to Purwodihardjo [26] as

pffiffiffi 2 6 sin w b¼ ðr  1Þð3  sin wÞ

with r ¼

Rfail Rmvc

ðA:7Þ

Stating that the Maximum Volumetric Contraction (MVC) state is reached for Rmvc ¼ 0:85Rfail , then r = 1.18. If the dilatancy angle (w) ranges between 0 and 20 whether dealing with soils, concrete, or rocks [31], then the range for b is found between 0.0 and 3.5. References [1] M. Allouani, Identification des lois de comportement de sols definitions de la strategie et de la qualite de l’identification (Ph.D. thesis), Ecole Centrale de Lyon, 1993. [2] H. Araki, J. Koseki, T. Sato, Tensile strength of compacted rammed earth materials, Soils Found. 56 (2016) 189–204. [3] ASTM-E-519, Standard test method for diagonal tension (shear) in masonry assemblages, West Conshohocken, 2002. [4] Y. Bagagli, Modélisation du comportement cyclique des sols et des interfaces sol-structure (Ph.D. thesis), Ecole Centrale de Lyon, 2011. [5] D. Bigoni, A. Piccolroaz, Yield criteria for quasibrittle and frictional materials, Int. J. Solids Struct. 41 (2004) 2855–2878. [6] Q.-B. Bui, J.-C. Morel, Assessing the anisotropy of rammed earth, Constr. Build. Mater. 23 (2009) 3005–3011.

[7] Q.-B. Bui, J.-C. Morel, S. Hans, P. Walker, Effect of moisture content on the mechanical characteristics of rammed earth, Constr. Build. Mater. 54 (2014) 163–169. [8] T.-T. Bui, Q.-B. Bui, A. Limam, S. Maximilien, Failure of rammed earth walls: from observations to quantifications, Constr. Build. Mater. 51 (2014) 295–302. [9] T.-T. Bui, Q.-B. Bui, A. Limam, J.-C. Morel, Modeling rammed earth wall using discrete element method, Continuum Mech. Thermodyn. 28 (2016) 523–538. [10] B. Cambou, K. Jafari, Modèle de comportement des sols non cohérents, Revue française de géotechnique (1988) 43–55. [11] F. Champiré, A. Fabbri, J.-C. Morel, H. Wong, F. McGregor, Impact of relative humidity on the mechanical behavior of compacted earth as a building material, Constr. Build. Mater. 110 (2016) 70–78. [12] W.-F. Chen, Plasticity in Reinforced Concrete, J. Ross Publishing, 2007. [13] D.C. Drucker, A definition of stable inelastic material. Technical Report DTIC Document, 1957. [14] J. Duriez, E. Vincens, Constitutive modelling of cohesionless soils and interfaces with various internal states: an elasto-plastic approach, Comput. Geotech. 63 (2015) 33–45. [15] R. El-Nabouch, Q.-B. Bui, O. Plé, P. Perrotin, Assessing the in-plane seismic performance of rammed earth walls by using horizontal loading tests, Eng. Struct. 145 (2017) 153–161. [16] K. Elamrani, Contribution à la validation du modèle C.J.S. pour les matériaux granulaires (Ph.D. thesis), Ecole Centrale de Lyon, 1992. [17] Eurocode-8, Design of structures for earthquake resistance-part 1: general rules, seismic actions and rules for buildings, Ref. No. 1998-1:2004: E, 2004. [18] H. Fang, T. Hirst, A method for determining the strength parameters of soils, Highway Res. Rec. 463 (1973) 45–50. [19] P. Jaquin, Analysis of Historic Rammed Earth Construction (Ph.D. thesis), Durham University, 2008. [20] P. Jaquin, C. Augarde, D. Gallipoli, D. Toll, The strength of unstabilised rammed earth materials, Géotechnique 59 (2009) 487–490. [21] M. Maleki, B. Cambou, P. Dubujet, Development in modeling cyclic loading of sands based on kinematic hardening, Int. J. Numer. Anal. Methods Geomech. 33 (2009) 1641–1658. [22] I. Marzec, J. Tejchman, Enhanced coupled elasto-plastic-damage models to describe concrete behaviour in cyclic laboratory tests: comparison and improvement, Arch. Mech. 64 (2012) 227–259. [23] L. Miccoli, D.V. Oliveira, R.A. Silva, U. Müller, L. Schueremans, Static behaviour of rammed earth: experimental testing and finite element modelling, Mater. Struct. (2014) 1–14. [24] J. Morel, A. Mesbah, M. Oggero, P. Walker, Building houses with local materials: means to drastically reduce the environmental impact of construction, Build. Environ. 36 (2001) 1119–1126. [25] T. Namikawa, S. Mihira, Elasto-plastic model for cement-treated sand, Int. J. Numer. Anal. Methods Geomech. 31 (2007) 71–107. [26] A. Purwodihardjo, Modélisation des déformations différées lors du creusement des tunnels (Ph.D. thesis), Ecole Centrale de Lyon, 2006. [27] A. Purwodihardjo, B. Cambou, Time-dependent modelling for soils and its application in tunnelling, Int. J. Numer. Anal. Methods Geomech. 29 (2005) 49– 71. [28] W.A. Riyono, E. Vincens, J.P. Plassiard, A hierarchical constitutive model for rammed earth, in: EMI International Conference, 25–27 October, Metz France, 2016, pp. 219/404. [29] R. Silva, D. Oliveira, L. Schueremans, P. Lourenço, T. Miranda, Modelling the structural behaviour of rammed earth components, in: Twelfth International (Ed.), Conference on Computational Structures Technology, Civil-Comp Press, Stirlingshire, Scotland, 2014. [30] R.A. Silva, D.V. Oliveira, T. Miranda, N. Cristelo, M.C. Escobar, E. Soares, Rammed earth construction with granitic residual soils: the case study of northern Portugal, Constr. Build. Mater. 47 (2013) 181–191. [31] P.A. Vermeer, R. De Borst, Non-associated plasticity for soils, concrete and rock, Heron 29 (1984). [32] P. Walker, R. Keable, J. Martin, V. Maniatidis, Rammed Earth: Design and Construction Guidelines, BRE Bookshop Watford, 2005.