# A Unified Nonlinear Elastic Model for Rock Material
## 1 Introduction  
Engineering rock masses are discontinuous and often heterogeneous and anisotropic. Jointed rock masses are the basic units that are used to study the properties of engineering rock masses, and their deformation characteristics under different stress conditions are crucial to the performance of geotechnical engineering structures. Since a large number of projects are located in the shallow layer of Earth, which experiences low crustal stress, it is of great significance to study the deformation characteristics of jointed rock masses under conditions of low and medium stress smaller than 10 MPa and 20 MPa, respectively [1]. Under such circumstances, both intact rocks and joints (as two components of the jointed rock mass) will undergo nonlinear elastic deformation. Many scholars have performed extensive research on the nonlinear deformation of joints and different models have been proposed. [PERSON] [2] first used semilogarithmic functions in order to describe the normal deformation characteristics of joints. [PERSON] [3] used the hyperbolic model in order to describe the relationship between normal stress and the normal closure of joints. On this basis, [PERSON] and [PERSON] [4, 5] proposed a relatively simple model by introducing two parameters: initial joint stiffness and maximum closure. [PERSON] and [PERSON] [6] proposed a simple exponential model and a modified exponential model. [PERSON] and [PERSON] [7, 8] proposed a power function model based on Hertzian contact theory. Other scholars [9, 10, 11] have also developed and improved the above models and the nonlinear elastic deformation of joints has had a great influence on their shear strength [12, 13]. Since it is difficult to directly measure the deformation of joints, it is indirectly evaluated by subtracting the deformation of the intact rock specimen from the jointed rock mass. However, during such an operation the nonlinear elastic deformation property of the intact rock is unconcernedly ignored (although not always). The nonlinear elastic deformation in the initial compaction stage of the intact rock was customarily considered to be caused by sampling disturbance [14]. Without considering the initial compaction stage, hyperbolic [15], logarithmic [16], double exponential [17] and exponential [18] stress-strain models for different rock types have been proposed. However, there is a significant difference in the nonlinear part of stress-strain relationships between the observed and estimated values [19]. The widely used stress-strain curves for intact rocks with cracks [20] and weak rocks [21] have validated that the nonlinear elastic deformation in the initial compaction stage is remarkable. It has been shown that the porosity of the rock and the fractures in the rock can not only reduce its strength and elastic modulus [22; 23], but also cause its nonlinear characteristics [24]. [PERSON] [25] and [PERSON] [26] proposed different improved models in order to describe the rock compression stage based on the Duncan model. [PERSON] [27] introduced a compression factor in order to characterize the change in porosity and established a nonlinear damage constitutive model reflecting the rock compression stage. Based on particle contact theory, [PERSON] [28] established a constitutive model by using spring elements and G-W contact elements in order to express the linear and nonlinear deformation of contacts between adjacent particles, respectively. [PERSON] [29; 30] used engineering strains and natural strains based on Hooke's law to describe the deformation of the rock matrix and the porosity and fractures respectively, and proposed the two-part Hooke's model. This model was used to quantitatively calculate the crack closure stress of the rock [31; 32] and was introduced into the statistical damage constitutive model of the rock [33; 34].