A Constitutive Model for Elastic Visco-plastic Behavior of Weak Sandstones

Squeezing means large time-dependent convergence during tunnel excavation. In the past while tunneling through the weak rock strata in western Taiwan, severe squeezing was encountered and significant remediation costs were required to repair the damage. This phenomenon is closely related to rock mechanical properties, but the mechanism is still not clear. Therefore, it is necessary to explore the mechanical properties of these weak rocks and construct an adequate constitutive model to predict their behaviors. In this paper, we first introduce the deformation characteristics of weak rocks, and then a constitutive model comprising nonlinear elastic and visco-plastic deformation based on the characteristics is proposed. This model has such features as: (1) non-linear elastic deformation; (2) apparent shear dilation, (3) large plastic deformation occurs prior to the failure state, (4) the creep deformation under shear loading. After comparing the actual data with the predictions, it is shown that the proposed model describes well with the elastic, plastic and creep behaviors of weak rock under hydrostatic and triaxial loadings. deformation plays a more dominant role in the volumetric deformation at higher levels of confining stress. Closure of micro-cracks may account for the disappearance of plastic strain at higher levels of confining stress and results in a greater bulk modulus as well. 2.2 Volumetric deformation induced by shear stress The coupling volumetric deformation induced by shear stress is significant. It reveals that the sandstone tends to dilate elastically upon shearing. However, the plastic deformation is comprised of the compression at the earlier stage of shearing and the significant dilation at subsequent stages of shearing, especially when the shear stress is close to the shear strength of the sandstone. Consequently, when the applied shear stress approaches the shear strength of the sandstone, the total deformation shows a significant dilation resulting from both the elastic and the plastic behavior. 2.3 Shear strain induced by shear stress It is indicated that the initial elastic shear modulus increases upon greater levels of hydrostatic stress applied. The loaddeformation curve approximates toward a horizontal asymptote (i.e., failure state) when shear stress approaches the strength of sandstone. Thus, the plastic component reveals that significant plastic shear strain occurs when the shear stress approaches the shear strength and it eventually dominates the total deformation. Besides, analysis data show that associated flow rule is obeyed for the studied rock, i.e. plastic potential surfaces coincide with yield surfaces. 2.4 Creep behavior Under shear loading process, the typical volumetric creep curve is obtained with various constant-loading stresses. The volume first decreases (compressibility); afterwards under higher constant loading stresses, it increases during creep (dilatancy). The details are described in Tsai et al. (2004). It is also found that both plastic flow and visco-plastic flow are orthogonal to yield surface under various hydrostatic stresses, as shown in Fig. 2. Consequently, the associated flow rules can be applied to describe their plastic deformation and visco-plastic deformation. 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Hydrostatic stress (MPa) Sh ea r st re ss (J 2 ) 1 /2 (M Pa ) Plastic flow Viscoplastic flow Irreversible stress work Hypothesized yield surface W = 0.04 W = 0.05 W = 0.06 W = 0.08 W = 0.1 W = 0.16 W = 0.035 Failure envelope Figure 2. Sketch of yield surface, plastic flow and visco-plastic flow. 3 CONSTITUTIVE MODEL OF WEAK ROCK Given the deformational behavior of weak rock observed during the experimental study, an associated constitutive model, based on elastic and visco-plastic concept, can accordingly be developed to describe the aforementioned behavior. 3.1 Elastic Model In view of the non-linear elastic behavior described above, a hyperelastic model (or Green elastic model) is thus adopted. On the basis of such a model, the strain tensor can be determined from an energy density function Ω and the stress tensor σ ij by the following expression: e ij ij e σ ∂Ω = ∂ (1) where e ij e is the elastic strain tensor and Ω the energy density function. From the experimental results, Ω is selected to be: 2 /3 1 1 1 2 1 2 3 2 b I b I J b J − Ω = + + (2) where b1, b2 and b3 e ij e are material constants. Substituting Eq. (2) for Eq. (1), the elastic strain tensor has the following form: 1/ 2 2 1 1 1 2 1 2 2 1 3 2 ( ) 3 e ij ij ij ij b I b I J b I b S e δ σ − − ∂Ω   = = − + +   ∂   (3) According to Eq. (3), the volumetric strain can be obtained as: 11 22 33 2 2 1/ 2 1 1 2 1 9 3 2 e e e e v J b I b I e e e e     = + + = −     (4) The shear strain ( ' 2 2 e J γ = ) can also be determined based on Eq. (3) as: ( ) 1 2 1 3 2 2 e b I b J γ − = + (5) Accordingly, the material constants b1, b2 and b3 can be determined by a curve fitting derived from the experimental curves shown in Figs 3-5 For the tested sandstone, b1, b2 and b3 are found to be 143. 26μ‧(MPa), 1462.8 μ, 30.19 (MPa), respectively. The strains from model prediction are compared to the experimental results, as shown in Figs 3-5. In general, the proposed elastic model is capable of predicting the nonlinear elastic behavior, including shear dilation of sandstone and its prediction has a reasonably good agreement with the experimental curves as well. 0 5 10 15 20 25 30 35 40 45