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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 739068, 10 pages
A Mathematical Approach to Establishing Constitutive Models for Geomaterials
1Guangdong Provincial Research Institute of Water Conservancy and Hydropower, Guangzhou 510610, China
2State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
Received 1 March 2013; Accepted 25 May 2013
Academic Editor: Pengcheng Fu
Copyright © 2013 Guang-hua Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The mathematical foundation of the traditional elastoplastic constitutive theory for geomaterials is presented from the mathematical point of view, that is, the expression of stress-strain relationship in principal stress/strain space being transformed to the expression in six-dimensional space. A new framework is then established according to the mathematical theory of vectors and tensors, which is applicable to establishing elastoplastic models both in strain space and in stress space. Traditional constitutive theories can be considered as its special cases. The framework also enables modification of traditional constitutive models.
The mechanical properties of geomaterials are complex and essential to make a numerical prediction; thus, many researchers have paid attention to constitutive relations of geomaterials. The simplest constitutive model for geomaterials is the elastic model, among which the common nonlinear models are the Cauchy elastic model, the hyperelastic model, and the hypoelastic model. The Cauchy elastic model assumes that the stress (or strain) in the material depends on the current strain (or stress) only, and not on its history. The constitutive equation for the hyperelastic model is established by the strain energy function or complement energy function. The hypoelastic model assumes that the stress state of an elastic material is associated with both the strain state and the stress path. The typical hypoelastic models are the E-μ and E-B models proposed by Duncan et al. [1, 2] and the K-G model [3–5].
According to the experimental results, most deformations of geomaterials are plastic deformations. Therefore, traditional plasticity theory has often been used to establish constitutive models for soil. For example, Drucker et al.  described the deformation property of soil by traditional elastoplastic theory and proposed a model with conical yield surface affected by hydrostatic pressure. Roscoe et al.  proposed a plastic cap model for normally consolidated clay, which is well known as the Cambridge model. Subsequently, Roscoe and Burland  modified the dilatancy equation in the Cambridge model and proposed a modified Cambridge model with elliptical yield surface. Wroth and Bassett  and Poorooshasb et al.  extended the model to sandy soil, and Yao et al. [11, 12] extended the model to sandy soil and overconsolidated soil. There are plenty of elastoplastic models, such as models with a single yield surface proposed by Desai et al. [13, 14] and Lade et al. [15–18], models with a double yield surface, and three surface models . The concepts of the bounding surface [20–23] and the subloading surface [24, 25], endochronic theory , and disturbed states  have also been applied to establishing constitutive models for geomaterials.
In this paper, a theoretical framework on establishing constitutive models for geomaterials is proposed, the initial thought of which is provided by the first author in 1988 and in 1990s [28–31], and it has been implemented by some researchers to simulate the behavior of jointed rock masses  and soil-structure interface .
2. Classical Elastoplastic Theory of Geomaterials
The incremental form of a stress-strain relationship in traditional geomechanics is generally expressed as Determining is the major topic for constitutive models of geomaterials. Obviously, can be obtained by fitting experimental data, given that experiments on stress and strain tensors are conducted. However, it is extremely difficult to do so. Therefore, experiments on the stress-strain relationship of geomaterials are usually conducted in principal stress/strain space; that is, only the relationship between the principal stress and the principal strain , is obtained. To obtain , the constitutive tensor in general coordinate space should be derived from the stress-strain relationship in principal stress/strain space. From a mathematical point of view, these can be treated as the problems of coordinate transformation [34–36].
The relationship between the plastic strain increment and stress increment in principal stress/strain space is defined as where are the functions of total stress , total strain or stress path.
When the matrix rank of is 1, or , there exists a vector and coefficients to express as Therefore, substituting (3) into (2a) and (2b) gives that is, where According to (4b), is a function of or . When is of a field with potential, there is a potential function such that Substituting (7) into (4b), we have If we assume that and have the same principal directions, the coordinate transformation can be expressed as follows: Substituting (9) into (8) gives
Similarly, from elastic potentials theory, there is a potential function in principal stress space, and is defined as If we assume that and have the same principal direction, the coordinate transformation can be expressed as follows: Substituting (11) into (12),
In conclusion, traditional plastic potential theory corresponds to the case that the matrix in (2a) and (2b) has rank 1, and can be expressed as the gradient vector of a potential function. Based on mathematical principles, a more general potential function-based constitutive framework can be established according to vector field theory and tensor theory as described below.
3. Derivation of Constitutive Framework from Vector Field Theory
Obviously, when the three principal components of the plastic strain increment, , are considered to be components of a vector , the principal components can be expressed as three linearly independent 3D vectors by a vector fitting method. The gradient vectors of three linearly independent potential functions are selected as the linearly independent vectors.
When is expressed in principal stress space, the coordinate orientations of and are the same, and are three linearly independent potential functions in principal stress space, and then the following expression is obtained: where are the coefficients. Suppose that the principal directions of and are the same, and we substitute (14) into (9), giving the tensor expression in general coordinate space as
Similarly, can also be expressed in strain space. Let be three linearly independent potential functions in principal strain space, and the following expression is obtained: Define the plastic stress increment as where is the elastic stiffness matrix, and then the expressions in stress space and strain space are
For the total stress and the total strain, consider the three principal stresses and the three principal strains as vectors in three-dimensional space with the same principal directions, and the following expressions are obtained: where are potential functions with linearly independent gradient vectors in strain space and stress space, respectively.
4. Derivation of Constitutive Framework from Tensor Theory
If and are symmetric second-order tensors with the same principal directions, the following equations are obtained according to tensor theory and vector fitting: where are three independent invariants of , and are three independent invariants of . For example, for the stress tensor , the three independent invariants can be , , ; , , ; or , , , where is the mean stress, is the deviatoric stress, is the Lode’s angle, and , , and are the three principal stresses.
Similarly, if , , then according to (21), we have
It should be noted that the derivations from vector field theory and from tensor theory are actually coincident. The potential functions in the derivation from vector field theory are functions of invariants, which degenerate to tensor form after composite derivation.
5. Elastoplastic Matrix in Stress Space
Without considering the effect of the Lode’s angle and rotation of principal stress, the relations for the plastic strain increment and stress increment are expressed aswhere , , , are parameters.
The following equation is obtained from the potential functions , in (15) without considering the effect of the Lode’s angle: In matrix form, this is It is obvious that According to (27a), (27b), (29), and (30), Since it follows thatwhere is the plastic compliance matrix.
Since , then where is the elastic compliance matrix and is the elastoplastic compliance matrix.
Therefore, the elastoplastic model in stress space can be established once the parameters , , , and are determined. Note that , , , and are not constants, which evolve with stress and strain, as presented in the following sections.
6. Elastoplastic Matrix in Strain Space
According to (17), the plastic stress increment is defined as , where is the elastic matrix and is the plastic strain increment.
, , and are three invariants of the strain tensor, where is the strain Lode’s angle. If we ignore the effect of Lode’s angle and take and as potential functions, that is, in (19), then Written in matrix form, Obviously, From the definition of (17),where , are the elastic bulk modulus and shear modulus.
Consideringsubstituting (39a) and (39b) into (27a) and (27b) yieldsIt can now be calculated thatwhere Substituting (41a) and (41b) into (37), (38a), and (38b) givesIn addition, Substituting (44) into (43a), (43b), and (36) yields where that is, where , and is the elastic Poisson ratio.
It should be noted that it is practically impossible to obtain the total strain in soil, and thus the elastoplastic matrix in stress space is more applicable. However, if we could further extend the framework to the space of strain increment, the practicability becomes promising.
7. Relationship with Traditional Elastoplastic Model
7.1. General Form
The elastoplastic compliance matrix of the traditional elastoplastic model is where and are the yield function and plastic potential function, is the plastic hardening modulus, and is the hardening parameter.
and are usually expressed in terms of the stress invariants, , , . If the effect of the Lode’s angle is not considered, the expression only concerns and , that is,Substituting (48a) and (48b) into (47), we have Comparing this with (33a), (33b), (34a), and (34b), it follows that Equation (33a) and (33b) can be seen as a general formula for traditional constitutive models, and (49) is a special form of (33a) and (33b). For associated models, when ,
Most traditional constitutive models are to determine the relationship between , and , , which can be used to calculate the four model parameters , , , and indirectly. It can be seen from (50a) and (50b) that the traditional nonassociated models actually assume that The associated models still need to satisfy (51a) and also require Rewrite (27a) and (27b) into matrix form
Clearly (51a) requires the determinant rank of the coefficient matrix in (52) to be 1 or requires and to be linearly correlated. Equation (51b) additionally requires the coefficient matrix to be symmetric.
The traditional elastoplastic model in stress space can be translated to strain space by the duality of (45a), (45b), (45c), (33a), and (33b). Substituting (50a) and (50b) into (42), (45a), (45b), and (45c) yields the expression in strain space where
Hence, the stiffness matrix of (53a) and (53b) is obtained using and from the traditional constitutive models, and the transformation from stress space to strain space is thus achieved. Obviously, the transformation only changes the mathematical calculation method of the coefficient and elastoplastic matrices and has no influence on a particular model itself or the loading-unloading criterion of the model. Therefore, the transformation is applicable to all traditional elastoplastic models.
7.2. Modified Cambridge Model
In the modified Cambridge model, and so where , is the hardening parameter (, for the modified Cambridge model), is the stress ratio at critical state, is the initial mean stress, is the initial void ratio, is the slope of the normal compression line (NCL), and is the slope of the unloading line. The elastoplastic compliance matrix in stress space is expressed as The elastoplastic stiffness matrix in strain space is where
A new hardening parameter for the modified Cambridge model was proposed by Yao et al.  as in which where is the potential failure stress ratio.
Yao improved the modified Cambridge model by replacing with (58a) and (58b), which changes in the modified Cambridge model to . The improved constitutive model is a unified hardening model and is suitable for sandy soil, which actually replaces by with (49), or the following expression with (52): where , are the volumetric strain and shear strain, respectively, that are calculated by the modified Cambridge model.
This modification can be further improved. For instance, the volumetric strain and shear strain of triaxial testing are first calculated by the modified Cambridge model, and then the ratio of the volumetric strain and shear strain can be fitted according to the test results, that is Therefore, (52) is modified to
and can be estimated by polynomial fitting or other fitting methods. Yao’s hardening model is obtained when .
The above-mentioned modification method can be extended to other elastoplastic models. The correction coefficients and can be fitted based on the triaxial testing results or other testing results. The obtained matrix can improve the calculation accuracy of existing elastoplastic models or be used to establish new modified models. Note that there is no physical mechanism involved in the framework, and thus the loading-unloading criterion and the evolution of internal state variables of the original model should still be employed in the modified one.
8. Application: A Simple Model
Theoretically, the constitutive model of soil would be established if the parameters , , , and in (27a), (27b), (33a), and (33b) are obtained by experiments such as conventional triaxial test, isotropic compression test, and . test.
For example, the equations for the tangent modulus and tangential Poisson ratio are obtained by fitting the curve from triaxial testing, that is
and can be curve fitted by a polynomial or by the formulae in the Duncan-Chang E-μ model . However, two supplementary equations are needed as there are four unknown parameters in (27a) and (27b). Therefore, an isotropic compression test or . test should be conducted, or the assumptions and are made.
In conventional triaxial test, , , , , and . According to (62a), (62b), (27a), and (27b) and the assumption in (51a) and (51b), we have where is the elastic modulus, and the elastic Poisson ratio is generally taken as 0.3 for soil. It is obvious that is always fulfilled in (63a) and (63b).
By substituting (63a) and (63b) into (33a), (33b), or (35), the elastoplastic compliance matrix in stress space or the stiffness matrix in strain space is obtained. For convenience, we call this model the multiple potential surface model (MPS model).
It should be noted that, contrary to the Duncan-Chang model (DC model), and in the MPS model are not limited by the generalized Hooke’s law; that is, the MPS model is still available when and the stiffness matrix of the model is not singular. Actually, and in the new model are not the traditional modulus and Poisson ratio, but just the slope of the curves.
Figure 1 shows the calculation and test results for triaxial testing of Ottawa silica sand conducted by Wu . The unit weight of the sand is 16.8 kN/m3 (=107 pcf). The test is a conventional consolidated-drained triaxial compression test (CD test). The confining pressures were 68.9, 206.7, and 344.5 kPa (= 10, 30, and 50 psi, resp.). During the CD test, confining pressure was firstly applied and then the specimen was consolidated. Deviation stress was applied in the axial direction after consolidation. Variations of deviation stress and volumetric strain versus axial strain can be acquired in the test.
The calculations were made using the MPS model as well as the Duncan-Chang model, during which , , and were calculated by the method proposed by Duncan and Chang , that is where is the cohesion of the soil, is the friction angle of the soil, is the atmospheric pressure, 100 kPa; , , , , , , and are parameters.
The parameters in the calculation are taken as kPa, , , , , , , and , which are the same for both the DC and MPS models. The value of for the DC model is taken as 0.45 while for the MPS model it is 0.8, which is larger than 0.5. Although the calculation results for deviation stress are identical for the two models, the MPS model can reproduce the dilation of soil. Because of the limitation that in the DC model, the dilatation of soil is not revealed and is not achieved.
Figure 2 shows the calculation and test results of consolidated-drained triaxial compression test (CD test) of a rock-fill material from Hengshan Dam in China. The unit weight of the material is 20.7 kN/m3. The confining pressures were 300, 500, and 800 kPa, respectively.
The parameters for are kPa, .4°, , , , and , which are also the same for both the DC and MPS models. in the DC model is still calculated using (65a) and (65b), and , , and in the calculation, while in the MPS model is calculated using the method proposed by Shen and Zhang ,in which is the deviation stress at failure, ; , and are parameters; , , and in the calculation.
Again, the calculation results for deviation stress are identical for the two models, while the MPS model reproduces the dilatation of soil giving better results than the DC model. Obviously, more appropriate results of volumetric strain can be acquired using the MPS model if we improve the calculation method of . However, this is impossible for the DC model due to the limitation that cannot exceed 0.5 for the nonlinear elastic model.
The main tasks in establishing the constitutive equations for geomaterials are the determination of stress-strain relations in principal stress/strain space and the coordinate transformation of the relationship from principal stress/strain space to general coordinate space. The stress (or strain) and stress increment (or strain increment) in principal stress/strain space is expressed as a vector in a potential field in traditional elastic potential theory and plastic potential theory. However, the vector can be expressed more generally as the gradient vector of linearly independent potential functions. Based on this framework, the traditional models can be easily transformed from stress space to strain space and can be modified in a general way. This framework can also be used to establish new models based on curve fitting. Since it investigates constitutive models from a mathematical point of view independent of the material itself and relevant physical mechanism, the framework can be potentially used in a wider range, not limited to geomaterials. However, the lack of physical insights of materials and constitutive models may also hinder its development, for example, the constitutive model based on the framework may be oversimplified. The loading-unloading criteria as well as the evolutions of internal state variables are not considered in the framework. Therefore, the current framework is mainly useful for modifying the existing models. In the next research, Lode’s angle and noncoaxiality should also be investigated, and more test results are needed to make the verification.
The support of the Natural Science Foundation of China (51279085), National Basic Research Program of China (973 Program 2013CB036402), and the State Key Laboratory of Hydroscience and Engineering (sklhse-2012-D-01) is gratefully acknowledged.
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