Finite Deformation Analysis on Sandstone Subjected to Thermo-Hydro-Mechanical (T-H-M) Coupling

Abstract

Temperature, water, and confining pressure are crucial environmental factors affecting rock strength and deformability. In this work, temperature–water–confining pressure coupled compression experiments were conducted on sandstone. The effects of each influencing factor on rock strength and deformability are statistically analyzed and discussed. The results suggest that the confining pressure has the most significant effects on the Young’s modulus, Poisson’s ratio, and peak stress, whereas temperature has the least effect on these three parameters under the present test conditions. The evolution of the Young’s modulus, Poisson’s ratio, and peak strength under the influence of the three environmental factors is also studied, from which linear correlations of those parameters with the three factors are obtained. In addition to the parametric study, finite deformation theory and the mean rotation angle are employed to analyze the nonlinear deformation behavior of the test rock, which cannot be comprehensively described by the Young’s modulus and Poisson’s ratio alone. The evolution of the mean rotation angle with respect to stress, strain, and tangent modulus is studied and discussed. A constitutive model based on the mean rotation angle and finite deformation theory is also proposed.

Appendix: Illustration of the Mean Rotation Angle

1.1 Mean Rotation Angle for the Movement (Rigid-Body Rotation) and Deformation of a Body

As shown in Fig. 10, when the configuration $A^{0}$ changes to $A^{1}$ after movement and deformation, the displacement function can be written as

$$ \left. \begin{gathered} u^{1} = \left[ {\left( {1 + \lambda } \right)\cos \varphi - 1} \right]x^{1} - \sin \varphi x^{2} \hfill \\ u^{2} = \left( {1 + \lambda } \right)\sin \varphi x^{1} + \left( {\cos \varphi - 1} \right)x^{2} \hfill \\ \end{gathered} \right\} $$

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where λ and φ are the elongation parameter and rigid-body rotation parameter, respectively.

Fig. 10

Sketch of elongation and movement

• 1. If $\lambda = 0$, $A^{0} \to A^{1}$ is just rigid-body movement without deformation, then based on Cauchy strain theory, the strain tensor $\varepsilon_{j}^{i}$ can be written as

  $$ \left[ {\varepsilon_{j}^{i} } \right] = \left[ {\begin{array}{*{20}c} {\cos \varphi - 1} & 0 \\ 0 & {\cos \varphi - 1} \\ \end{array} } \right]. $$

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  In fact, the components of the strain tensor should be zero since there is no deformation,

  $$ \left[ {\varepsilon_{j}^{i} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ \end{array} } \right]. $$

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  If $\varphi$ = 5°, the principal strain will be $\varepsilon_{1}^{1} = \varepsilon_{2}^{2} = - 0.004$, whereas if $\varphi = 90^{ \circ }$, $\varepsilon_{1}^{1} = \varepsilon_{2}^{2} = - 1$. Results based on Cauchy strain theory are not correct here since $A^{0}$ only has rigid-body rotation.

• 2. If we use finite deformation theory to solve the case above, we can determine the mean rotation angle θ and the strain tensor $S_{j}^{i}$ as follows:

  $$ \theta = \varphi ,\;\;\left[ {S_{j}^{i} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ \end{array} } \right]. $$

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1.2 Mean Rotation Angle of a Deformed Body

As shown in Fig. 11, the rectangle has a configuration change after undergoing simple shear, which causes a shear angle. α is the angle between OA and the horizontal line. After deformation, OA moves to OA′; meanwhile, α changes to β. The displacement function can be written as

$$ \overline{{x^{1} }} = x^{1} + \tan \gamma ,\;\overline{{x^{2} }} = x^{2}, $$

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where $x^{1}$ and $x^{2}$ are the horizontal and vertical coordinates before deformation, and $\overline{{x^{1} }}$ and $\overline{{x^{2} }}$ are the counterparts after deformation. The mean rotation angle can be written as

Fig. 11

Deformation sketch

$$ \theta = \arcsin \left( {\frac{1}{2}\sin \gamma } \right). $$

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Figure 12 shows a plot of β and α when γ = 30° (θ = 14.49°). As shown, the value of β varies with the value of α.

Fig. 12

Plot of β with α