Elastoplastic cup model for cement-based materials

Yan ZHANG*, Jian-fu SHAO

1. College of Mechanics and Materials, Hohai University, Nanjing 210098, P. R. China

2. Laboratory of Mechanics of Lille, UMR 8107 CNRS, Villeneuve d’Ascq 59655, France

1 Introduction

Scientific research of cement-based materials is important for the design of civil engineering and hydroelectric projects. Stress states can be complicated for applications such as underground structures, oil well tubing, and facilities for underground nuclear waste storage.The mechanical behavior of cement-based materials has been widely investigated. However,most previous studies have focused on brittle behavior under tensile and uniaxial compression,and have not dealt with complex stress states. Recent research shows that plastic behavior is also a main property of cement-based materials in complex stress states: there is a transition from brittle to ductile behavior with increasing confining pressure. Fig. 1 shows the stress-strain curves of cement paste in triaxial compression tests under different confining pressures (Yurtdas et al. 2006). We can see, that under high confining pressures, plastic deformation becomes a dominant inelastic mechanism. The mechanical behavior of cement paste under compression is considered to strongly depend on confining pressure.

In general, with plastic pore collapse, the volumetric consolidation of granular materials may be generated when the rearrangement of grains reduces pore space. In porous cohesive materials like cement paste, the volumetric compaction generally results from inelastic pore collapse due to the breaking contact forces between grains. Some research has shown that rock material has high pressure sensitivity and leads to plastic pore collapse in spite of the basic deviatoric shearing behavior. Some elastoplastic cup models have been developed for rocks(Gennaro et al. 2003; Xie and Shao 2006). There has been less research on the constitutive law of cement-based materials over a large range of confining pressure, as opposed to confining pressure on rock materials, even though they are both quasi-brittle materials. Due to the limitations of experimental devices, research on cement-based materials has been conducted only in recent years (Yurtdas et al. 2006). Therefore, two plastic mechanism processes,deviatoric shearing plastic deformation and pore collapse plastic deformation, should be taken into account in the modeling of cement paste over a large range of stress. In this paper, we first propose a cup model for plastic behavior of cement paste under various loading conditions.Then, comparisons between numerical predictions and laboratory tests are presented in order to show the performance of the cup model under compression-dominated stresses. However, the present research is limited to the study of plastic behavior of cement paste, without taking into account effects of strain softening.

Fig. 1 Stress-strain curves of cement paste in triaxial compression test under different confining pressures(ε1 is the axial strain, ε3 is the lateral strain, Pc is the confining pressure, and q is the deviatoric stress)

2 Formulation of constitutive model

In this section, we present the formulation of the constitutive model for the description of mechanical behavior of cement paste by taking into account two plastic mechanisms according to previous review of cement paste.

We assume isothermal conditions for this model. The principal phenomena to be taken into account are elastic deformation, defined by the elastic strain tensor εe, and plastic deformation, defined by the plastic strain tensor εp. Based on the assumption of small strains,the total strain increment dε is composed of an elastic part dεeand a plastic part dεp.Small strains and displacement are assumed. The strain increment partition rule is applied:

As mentioned above, we assume isotropic behavior of cement-based materials. The thermodynamic potential of cement-based materials can be expressed as follows:

where the fourth order tensor C is the elastic stiffness tensor, the function Ψprepresents the locked plastic energy in plastic hardening, and γpdenotes the internal variable of plastic hardening. The standard derivation of the thermodynamic potential yields the state equation:

In the case of isotropic materials, using Hill’s notation (Nemat-Nasser and Hori 1993), the elastic stiffness tensor is defined in the general form:

where k is the bulk modulus of the damaged material, and μ represents the shear modulus.The two isotropic symmetric fourth-order tensors J and K are defined as follows:

The intrinsic mechanical dissipation must meet the following fundamental condition:

The rate form of the constitutive Eq. (3)can be easily written as

The dot denotes the time derivative of variables (or incremental variation of variables in numerical computing procedure).

Under compression-dominated stresses, the mechanical behavior of cement paste is essentially characterized by plastic deformation. Therefore, the mechanical damage due to microcracks is neglected. Thus, the thermodynamic potential for isotropic materials is reduced to the following form:

2.1 Characterization of deviatoric plastic deformation

As mentioned above, the plastic deformation of cement paste, like that in most porous materials, can be characterized by two complementary plastic mechanisms: the deviatoric mechanism and the pore collapse mechanism. The deviatoric mechanism is common for frictional materials and is driven by a shearing phenomenon under deviatoric stress. Classical yield functions are generally based on linear Mohr-Coulomb and Drucker-Prager criteria.Material failure is represented as an ultimate state of plastic yielding. However, we have mentioned that in cement paste material the plastic yielding condition is highly sensitive to confining pressure. Linear yielding criteria are not suitable. In this study, inspired by the previous work of Pietruszczak et al. (1988)and Mohamad-Hussein and Shao (2007), we used the following nonlinear form as the yield function of the deviatoric plastic mechanism:

where q denotes the deviatoric stress; C′ represents the cohesion of material; p corresponds to the mean stress; Pris taken as a reference stress, which is generally fixed at 1 MPa; the parameters n and A define the failure surface; and Sijrepresents the components of the deviatoric stress tensor S. The function g(θ)was introduced to take into account the influence of Lode angle on plastic flow. However, due to the lack of relevant experimental data, the influence of Lode angle was neglected in the present study by assuming that g(θ)=1.

The function α in the yield function defines the plastic hardening law for the deviatoric plastic process. The generalized plastic distortion is generally used as the hardening variable function. For the studied material, the softening behavior is observed only at very low confining pressures. The material hardening is described by an increasing function of plastic distortion as follows:

where α0is the initial yield threshold, and B is a model’s parameter controlling the plastic hardening rate. The variable γsdenotes the generalized plastic distortion, which is defined as follows:

where β is a model’s parameter, taking into account the influence of confining pressure on plastic hardening. Note that the plastic hardening function α increases progressively during plastic flow and closes to the ultimate value defined by α→1, when macroscopic failure is reached.

In most frictional materials, plastic deformation under deviatoric stress exhibits a transition from volumetric compressibility to dilatancy. Non-associated plastic flow rules are generally necessary for present modeling. This is also the case for the cement paste studied here. Inspired by the plastic model proposed by Pietruszczak et al. (1988)for concrete, the following plastic potential is used:

The variable I0is determined by the condition gs=0. The parameter η defines the transition boundary between compressibility and dilatancy on the q-p plane. While the stress points meet the condition ?gs?p= 0, the transition from plastic compressibility to dilatancy occurs. Based on experimental data, it is assumed that the transition boundary can be described as

For the loading path, where only the deviatoric plastic mechanism is activated, the plastic flow rule is written as

where λsis the plastic multiplier of the deviatoric plastic mechanism. The plastic consistency condition can be expressed as

The generalized plastic shear strain ratecan be rewritten as

Therefore, the rate form of constitutive relations is determined to be

Using the plastic flow rule, one obtains the plastic multiplier as follows:

2.2 Plastic pore collapse deformation

According to simplified schematization, the cement paste is idealized as a homogeneous porous material composed of a solid matrix and a connected pore. It is firstly advisable to determine the criterion of plasticity of this material under compressive stresses. With this intention, one took the criterion of plasticity proposed by Gurson (1977)as a starting point,and this criterion was used in this study for metal porous materials subjected to tensile stresses.The deformation and the rupture of material are then determined by the evolution of the pores.This criterion has been widely used and various versions have been proposed for better describing the ductile rupture of porous materials. Recently, some research (Leblond and Perrin 1996; Perrin and Leblond 2000)showed the micromechanical background for the Gurson criterion. Indeed, the Gurson criterion represents the exact solution of the macroscopic criterion of a porous media composed of a solid matrix obeying Von-Mises perfect plasticity and of the spherical pores. According to this result, the macroscopic plasticity criterion depends on the plastic threshold of the solid matrix and macroscopic porosity. Various modifications of the yield surface have been proposed for better correspondence to the experimental data and description of plastic hardening. Inspired by these works, we adopted the general form of the Gurson criterion for the determination of the yield function of the pore collapse mechanism in cement paste. The following function was used:

The parameter q2is introduced to determine the geometrical form of the yield surface. In the initial Gurson criterion, q2=1. The introduction of this parameter is required so that the yield surface correctly reproduces the experimental data. The symbol σ′ denotes the plastic yield stress of the solid matrix. The value of σ′ may vary with plastic deformation. The variation of σ′ can be determined by means of the plastic hardening law. The variable φ is the total connected porosity used as a parameter of the model. In the present study, the cement paste was subjected to mechanical loading. In a general case, when the porosity changes due to mechanical loading, the variation of total porosity can be written as

In order to understand the pore collapse process, the result of the hydrostatic compression test is analyzed. In Fig. 2, a typical stress-strain curve for porous cement paste is shown, where σhis the hydrostatic stress and εvis the volumetric strain. We can find a quasi-linear and reversible phase, corresponding to the elastic compaction of the porous skeleton. The slope of this linear phase provides the elastic compressibility modulus of the material. When the hydrostatic stress exceeds a certain limit, defined as the threshold of pore collapse, progressive irreversible compaction of pores is produced by the progressive destruction of the initial porous structure.

Fig. 2 Experimental data and numerical simulation in hydrostatic compression test of cement paste

Based on the considerations described above and the experimental data from hydrostatic compression tests, the following hardening function is proposed:

Based on the experimental cement paste data, an associated plastic flow rule, rather than the deviatoric plastic mechanism, is used here, giving us the following plastic potential:

For the loading path, where only the pore collapse mechanism is activated, the plastic flow rule is written as

where λcis the plastic multiplier of the pore collapse mechanism. The plastic consistency condition can be expressed as

The rate form of constitutive equations provides the following:

Taking into consideration the flow rule and consistency conditions, λcis written as

2.3 Coupling of two plastic mechanisms

During general loading, the two plastic deformation mechanisms can be activated either separately or simultaneously. Four distinct constitutive domains can be identified:

(1)If fc＜ 0 andfs＜ 0, the applied stress state is fully within the elastic domain or leads to elastic unloading. No plastic flow occurs and we have the following conditions:

3 Calibration of model and numerical simulations

All tests were performed on a pure cement paste with a water-cement ratio of 0.44, using a G-class Portland cement. The parameters of the model were determined from conventional laboratory tests (Yurtdaz et al. 2006), which were composed of triaxial compression tests at different confining pressures and a hydrostatic compression test. Fifteen material and model parameters were considered in this study. Elastic parameters E and v were obtained from the linear part of stress-strain curves in one triaxial test, which were 5 000 MPa and 0.35, respectively.The parameter C′ represents the cohesion of material; in this study, we considered its value one tenth of the failure stress of a uniaxial test, i.e., 3.0 MPa. The parameters n and A were determined by failure stresses obtained in various triaxial compression tests, the values of which were 1.51 and 14, respectively. α0and B are parameters of the plastic hardening law; we drew the β-γpcurve using one triaxial compression test, then plotted a hyperbolic line that fit the experimental points and obtained the representative values of α0and B, which were 0.0 and 8×10-5, respectively. Once B and α0were determined for each test under different confining pressures, the hardening plastic law was obtained through the determination of β,which was 0.23 in this study. The parameter η of -1.8 was obtained from the slope of the transition points from volumetric strain curves in triaxial tests. The parameter σ0′ defines the initial pore collapse yield stress, which was determined to be 70 MPa in this study. The three parameters a, m, and b, equal to 0.9, 0.4, and 180, respectively, determine the rate of pore collapse plastic strain. The influence of q2was analyzed when the value increased and the corresponding plastic threshold in hydrostatic compression decreased, and the value of q2was determined to be 0.9. These values were determined from the hydrostatic compression test. The porosity φ of 0.37 was calculated through a simple comparison between dried samples and samples fully saturated with water. More details about the parameter value determination can be found in Zhang (2008).

(2)The elastic and plastic properties are updated with the current value of chemical damage.

(4)Otherwise, increments of plastic strain are calculated according to loading path using Eqs. (21), (30), or (32).

(5)Stress, strain and internal variables are updated.

Fig. 2 shows the simulations of hydrostatic compression tests. We can see that the present model can well describe the plastic deformation caused by the pore collapse mechanism.

Fig. 3 presents the simulations of triaxial compression tests under different confining pressures. Fig. 3 (a)shows the simulation of a triaxial compression test at a low confining pressure. In this test, the plastic deformation was dominated by the plastic shearing mechanism. At a slightly higher confining pressure, shown in Fig. 3 (b), we obtained a typical ductile mechanism characterized by two plastic phases: the first phase caused by a shearing plastic mechanism, and the second phase caused by the interaction between two plastic mechanisms, which occurred when deviatoric stress reached 40 MPa. In Fig. 3 (c), with the increase of the confining pressure to 17.5 MPa, the deviatoric stress enhancement due to pore collapse was correctly predicted by the proposed model. Overall, there is a good agreement between the simulations and test data. The proposed model correctly predicts the main features of mechanical behaviour of cement paste.

Fig. 3 Experimental data and numerical simulation in triaxial compression test of cement paste under different confining pressures

Obtaining good agreement between the model’s predictions and experimental data in the triaxial compression tests and hydrostatic test for cement paste, the simulations using the proposed model have also been performed for concrete, as reported by Yan and Lin (2007).In Fig. 4, we can see again the performance of this model in predicting concrete behavior under loading conditions, neglecting the pore collapse mechanism because of a lack of the relevant experimental result at a high confining pressure. Under confining pressures ranging from 0 MPa to 12 MPa, the performance of concrete was dominated by the plastic shearing mechanism. Different plastic phases did not occur during the loading process. Therefore, the plastic pore collapse mechanism was not activated in the present modeling.

Fig. 4 Experimental data and numerical simulation in triaxial compression test of concrete under different confining pressures

4 Conclusions

Based on our study, we can draw some conclusions: plastic deformation is one of the main behaviors of cement-based materials when complex stresses are applied, so they cannot simply be considered quasi-brittle materials; and, due to its high porosity, cement-based material such as cement paste and concrete has a pore collapse mechanism, and this mechanism, as opposed to the deviatoric mechanism, is dominant at a high pressure.

We have presented the formulation of a cup model for cement paste under compressiondominated stresses. Based on relevant experimental data, two plastic mechanisms are identified: the deviatoric mechanism and the pore collapse mechanism. The proposed model is implemented in a computer code using the finite element method. We have used the proposed model to simulate typical laboratory tests. The model is able to describe the main features of mechanical behavior of cement paste in different stress states. Also, this cup model can be widely used for other quasi-brittle and high-porosity materials, such as rock and concrete.

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