^{1, 2}

^{1, 2}

^{1}

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By analysis of the microscopic damage mechanism of rock, a multiparameter elastoplastic damage constitutive model which considers damage mechanism of tension and shear is established. A revised general form of elastoplastic damage model containing damage internal variable of tensor form is derived by considering the hypothesis that damage strain is induced by the degeneration of elastic modulus. With decomposition of plastic strain introduced, the forms of tension damage variable and shear damage variable are derived, based on which effects of tension and shear damage on material’s stiffness and strength are considered simultaneously. Through the utilizing of Zienkiewicz-Pande criterion with tension limit, the specific form of the multiparameter damage model is derived. Numerical experiments show that the established model can simulate damage behavior of rock effectively.

Rock is a kind of multiphase and inhomogeneous material, with mesoscopic discontinuities randomly distributed. When subjected to loads, discontinuities emerge and develop, leading to the degeneration of material’s strength and stiffness. The exact simulation of the damage behavior of rock requires definition of damage variables based on statistic methods and determination of evolution of those damage variables according to specific physical background so as to establish models like system of microstructures [

Many works have been done on damage property of concrete and geomaterials. Salari et al. [

Many works on damage model are dedicated towards the notion of effective quantities, making the establishment of damage state relatively indirect. In this paper, we follow the approach of taking damage internal variable as a two-order three-dimension tensor but adopt the viewpoint raised by Ying [

Continuum damage models are usually established by introducing particular damage variables and evolution laws, where the damage variables show to what extent damage develops. Nowadays, what is wildly used in a large amount of literature is the concept of effective stress, in which, take the scalar damage variable

As exposed in previous section, rock damage has several basic types, each affecting the strength and stiffness in different way. Similar to plastic internal variables, introduce in constitutive relation a damage internal variable

Assume

From the consistency condition in strain space that

Equations (

Rock’s stiffness matrix degenerates as damage develops. Damage strain is induced by the degeneration of elastic stiffness matrix coefficients; namely,

where

Substitute (

Consider the Clausius-Duhem inequality:

Tension and shear are the main types of damage for rock. Simo and Ju [

Substitute (

To expose the model further, in this section, Zienkiewicz-Pande criterion with tension limit is taken as an example and the specific form of the multiparameter damage model is derived. The criteria currently used very often in research and engineering include Drucker-Prager, Mohr-Coulomb, Hoek-Brown, and unified strength theory suggested by Yu et al. [

In damage judgment, plastic yield surface is often used as the bound for shear damage and stress or strain limit as the bound for tension. Thus, in this section, Zienkiewicz-Pande criterion with tension limit is assumed as the bound for plasticity and damage; namely,

Let elastic flexibility be an isotropic tensor:

Material strength and stiffness degenerate as damage evolves. Here, those strength and stiffness parameters are treated as functions of the reduced form of damage variables

As internal microscopic cracks develop very fast after reaching damage [

The algorithm for solving group of nonlinear equations includes Newton-Raphson method, modified Newton-Raphson method, and quasi-Newton-Raphson method, of which Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity of stiffness matrix. Consider that the model raised in this paper has many factors interacting with each other and thus highly nonlinearized; a framework for nonlinear finite element analysis by using Newton-Raphson method is implemented, which calls specific constitutive model to handle iterative solving.

Typical implementation of material model can be reduced to the following problems:

The establishment of tangent stiffness matrix, which can be implemented using (

State update, namely, update of stress, strain, and material parameters of each quadrature point when strain increment is known. Typical implementation includes explicit method (e.g., Runge-Kutta method) and implicit method (e.g., return map method [

To validate the effectiveness of the model established, consider the cylindrical sample in [

Stress-strain curve under uniaxial compression (MPa).

Consider a cubic finite element model that each side has length 1 m. Let

Stress-strain curve under cyclic uniaxial compression and tension (MPa).

Compression

Tension

In this section, the established model is used for the excavation simulation of Zhen’an underground power station.

Zhen’an pumped storage power station is located in Shanxi Province in China with installed capacity of 1400 MW (4 × 350 MW). The underground powerhouse is made up of the main powerhouse, main transform house, tailrace surge chamber, and several other tunnels connecting each cavern. Mechanical parameters for rock are shown in Table

Mechanical parameters for rock.

Density ^{3}) |
Young’s modulus |
Poisson’s ratio | Cohesion (MPa) | Internal friction angle (°) | Tension limit |
---|---|---|---|---|---|

2.67 | 13.0 | 0.27 | 1.0 | 47.7 | 0.45 |

Computation model (a) and excavation elements (b).

Initial first (a) and third (b) principal stress (MPa).

To simulate the process of excavation, divide the computation into 10 steps, each with a layer shown in Figure

Volume of damage zone (10^{5} m^{3}).

Damage zone of 2# unit section.

After excavation stress, distributions of surrounding rock of each unit section are roughly similar. The first and third principal stress of 2# unit section’s surrounding rocks are shown in Figure

First and third principal stress of 2# unit section (MPa).

What is shown in Figure

Displacement of 2# unit section (mm).

From the analysis above, it is easy to know that the damage zone, stress, and displacement distribution obtained using the model established in this paper are similar to those using Zienkiewicz-Pande or Mohr-Coulomb criterion because of the form of shear damage bound adopted. The introduction of damage mechanism makes it more natural to figure out the shear and tension damage area.

This paper approaches based on the microscopic mechanism of damage for rock material and establishes in an intuitive way a multiparameter elastoplastic damage model for rock that is applicable to engineering. With the assumption that damage comes into existence as the material’s strength and stiffness degenerate and that damage is interconnected with plastic deformation, a revised general form for elastoplastic damage model containing damage variable of tensor form is established. By considering plastic strain separation, the expression of damage variable reflecting the damage mechanism for shear and tension simultaneously is derived. By adopting Zienkiewicz-Pande criterion with tension limit as the bound for plasticity and damage, the specific form for the damage model is derived and implemented.

The model established in this paper is physically intuitive and has relatively well-based theoretical background. Numerical experiments and engineering application show that this model can reflect the damage behavior of rock effectively.

The authors declare that there is no conflict of interests regarding the publication of this paper.