The geometric description of incompatible deformation of deep rock mass and its evolution

Rock mass has a complex internal structure. Under the action of external loads, incompatible deformation develops in rock mass that makes the concept of elementary volume and the Saint Venant’s condition of compatible deformation problematic. The incompatibility of the deformation indicates the non-coincidence of the internal and the external metrics and the breaking of the Euclidean structure of the space. Therefore, the use of differential geometry to describe the incompatible deformation is natural. At present, the expression of non-Euclidean models is not concise and compact; the constitutive relation is not complete and the evolution of incompatibility parameters is absent. In this study, with the help of the orthogonal frame method, we used the generalized distortion tensor, torsion tensor, and Riemann tensor to describe the incompatible deformations. The used geometric parameters have clear physical meaning and can be used as thermodynamic variables. By constructing Helmholtz free energy and using irreversible thermodynamics, we obtained the constitutive equations. For completing the constitutive equations, the evolution equations of the used geometric parameters are derived. In this manner, the non-Euclidean description of the incompatible deformation extends the classical models of deformable bodies.


Introduction
Rock mass has complex internal structure formed by geological structural fracture zones, fissures, joints, and cracks (commonly known as weakened planes), ranging from 106 to 10−6 m (from continental scale to mesoscopic and microscopic scales) [1][2][3]. The deformation of the rock mass is primarily concentrated on the weakened surfaces and the weakened zones of the structural blocks. When rock mass is subjected to the external force, rock mass deforms and fails at the largest structural level first because the strength at this level with the largest crack width is the least. When the external force continues to increase, the rock mass blocks at the smaller structural levels begin to deform and fail. That is, the structural levels of activated deformation and failure of rock mass gradually transform to the smaller levels. Therefore, the failure of the rock mass can be considered as a continuous process of symmetric breaking and the progressive symmetry localization. In this case, the displacement cannot be determined from the distortion tensors uniquely, that is, defects appear, and the compatibility condition cannot be satisfied: where ij H is the strain tensor, that is, the additional parameter hijk R for a description of the deformation of medium appears, which is nothing else, but the Riemann tensor.
In differential geometry, a manifold is a basic concept. A manifold is the extension of Euclidean space and is the result of sticking together small pieces of Euclidean space [4]. This situation determines that the concept of the manifold can be used to describe various defects in the medium. In this case, the description of deformation and failure of the medium requires the introduction of additional parameters to describe the internal defect structure of the medium and the usual parameters of continuum mechanics. Thus, there is a requirement to transform the Euclidean model to the non-Euclidean model of the medium. In physical literature, Kondo and Bilby [5][6][7][8][9][10] reported the necessity of introducing non-Euclidean parameters. In fact, for the first time, they suggested using affine metrics to describe defect structures. Several theories deal with elastic continua with internal defects inside [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. The common features of the different types of non-Euclidean models are that the torsion tensor is usually identified with dislocation, the Riemann tensor with disclination, and the sectional curvature with point defect. These tensors are then used to characterize the geometry of interactions between particles within a material. Any small perturbation of the structure, for example, the introduction of the above geometric tensors, leads to a non-Euclidean model. In general, this structure is stable to small changes of curvature tensor, torsion tensor, and non-metric tensor. At present, the expression of non-Euclidean models is not concise and compact; moreover, in the constitutive relation, the evolution of incompatibility parameters is not directly related to the measurable parameters. Therefore, this study will focus on these problems. In theories of gravity, the orthogonal frame method is very effective [29] because all gravity theories can be expressed in orthogonal frames in the form independent of the choice of the coordinates, i.e., in a general covariant form. In addition to purely geometric considerations, the idea of mapping all quantities to geometric quantities in a tangent space has physical causes, because only in a flat space can a geometric quantity be physically explained. In this method, all other geometric quantities can be associated with the primary vector frames in the tangent space. The abovementioned characteristics of the orthogonal frame method determine that it can be effectively used in the establishing non-Euclidean models of media. Therefore, this study will discuss the application of the orthogonal frame method in the establishment of a non-Euclidean model of the medium. By incorporating geometric parameters into Helmholtz free energy and using irreversible thermodynamics, we obtained the constitutive equations. For completing the constitutive equations, we derived the evolution equations of the used geometric parameters. In this manner, the non-Euclidean description of the incompatible deformation extends the classical models of deformable bodies.

Description of non-compatible deformation of media by orthogonal frame
The modification of the Euclidean models describing the construction of defects is related to the change of the intrinsic geometric properties. In this study, the image of a continuous medium is assumed to be a point set M, and the mapping from such a point set to the Euclidean space R3 with coordinates 1, 2,3 P ] P is given. In geometry, the set endowed with differential homeomorphic coordinates is called a differentiable manifold, whereas  (2) and the velocity is The trajectories of particles in Euclidean space are the integral curves of the following equation: Along each such trajectory, it is clear that the initial coordinates are constant: where Dt D is the material derivative.
Taking the derivative of Eq. (5) is called the distortion tensor [30].
which has the following form in components: The metric tensor can then be expressed in terms of distortion tensor in the following form: The evolution equation of G can then be obtained from Eq.
The following tensors Thus, the necessary and sufficient conditions for the deformation compatibility of the medium is that Burgs tensor B equals to 0 (B = 0). The existence of defects in the medium ensures the Burgs tensor is no longer 0, and the deformation compatibility condition of the medium is no longer satisfied. Therefore, space is non-uniform. In differential geometry, orthogonal frames (quaternions) are often used to describe non-uniform spaces. In other words, a point of a manifold is considered as the common origin of two coordinate frames (affine frame and orthogonal frame): The affine frame is associated with an arbitrary coordinate system, whereas the orthogonal frame is associated with local orthogonal coordinates. The mixed scalar product of vectors belonging to different frames is the Lamé coefficients: be a non-degenerate transformation; therefore, the base vector of the observer space is as follows: Because of the existence of defects in the medium, this basis cannot be obtained using the Jacobian formula; thus, the basis is not coordinated. Therefore, the derived symmetric tensor from Eq. (17) (18) can be used as the internal metric tensor of medium with an internal defect structure. Note is the external metric tensor of the medium. The Algebraic structure of the internal metric tensor g is the same as that of the external metric tensor G .  (19) Only when P ij C 0 and P P G i i h , the internal metric tensor g can be consistent with the external metric tensor G . The above quantities constitute non-holonomic quantities, which constitute general covariant tensors for indices Q P, .
The comparison between Eqs. (19) and (13) shows that P ij C is similar to the Burgs tensor However, unlike Eq. (8), the source term appears on the right side of the equation.
The appearance of the source term is because of the appearance of defects in the process of deformation, which leads to the incompatibility of deformation and the destruction of continuity. The breakdown of continuity results in either overlap or void within the medium. Similarly, the evolution equation for the internal metric tensor is as follows:  The irreversible process of the medium can be expressed in terms of the second law of thermodynamics, which can be expressed as the Clausius-Duhem inequality [31]:

Constitutive relations
where 0 U is the medium density, T is the absolute temperature scale, and i S is the irreversible part of the entropy generation. It can be expressed as follows: where \ is the Helmholtz free energy, S is the entropy per unit mass, T ' is the temperature is the heat flux, and ij H is the deformation rate.
Generally, it is assumed that the process of heat conduction does not depend on the local thermodynamic process; thus, we can split Eq. (26) into two independent inequalities: (i) local entropy production inequality The strain rate is then divided into the sum of reversible and irreversible parts: (29) where e H is the reversible part of the deformation and S is the irreversible part of the deformation. The Helmholtz free energy is assumed to be a state function dependent on the reversible deformation, the internal variables i D , and the temperature T, that is, Now, it is an established physical fact that the glide of dislocations is responsible for the critical plastic deformations that correspond to the production of Riemann-Christoffel curvature, but not necessarily of torsion tensor. Therefore, the criterion of yielding of the matter is mostly expressed in terms of Riemann-Christoffel curvature tensor [7]. Here, we considered the internal variable R 1 D and the following Helmholtz free energy function: