Unified elastoplastic finite difference and its application

Appl. Math. Mech. -Engl. Ed., 344, 457 474 013 DOI 10.1007/s10483-013-1683-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 013 Applied Mathematics and Mechanics English Edition Unified elastoplastic finite difference and its application Zong-yuan MA 1, Hong-jian LIAO, Fa-ning DANG 1 1. School of Civil Engineering and Architecture, Xi an University of Technology, 5 South Jinhua Road, Xi an 710048, P. R. China;. Department of Civil Engineering, Xi an Jiaotong University, 8 Xianning West Road, Xi an 710049, P. R. China Abstract Two elastoplastic constitutive models based on the unified strength theory UST are established and implemented in an explicit finite difference code, fast Lagrangian analysis of continua FLAC/FLAC3D, which includes an associated/nonassociated flow rule, strain-hardening/softening, and solutions of singularities. Those two constitutive models are appropriate for metallic and strength-different SD materials, respectively. Two verification examples are used to compare the computation results and test data using the two-dimensional finite difference code FLAC and the finite element code ANSYS, and the two constitutive models proposed in this paper are verified. Two application examples, the large deformation of a prismatic bar and the strain-softening behavior of soft rock under a complex stress state, are analyzed using the three-dimensional code FLAC3D. The two new elastoplastic constitutive models proposed in this paper can be used in bearing capacity evaluation or stability analysis of structures built of metallic or SD materials. The effect of the intermediate principal stress on metallic or SD material structures under complex stress states, including large deformation, three-dimensional and non-association problems, can be analyzed easily using the two constitutive models proposed in this paper. Key words elastoplastic constitutive model, unified strength theory, explicit finite difference, effect of intermediate principal stress Chinese Library Classification O344 010 Mathematics Subject Classification 74C05 1 Introduction As one of the most important constitutive relations, strength theory yield or strength criterion was implemented in various nonlinear computer codes based on the finite element method FEM and the finite difference method FDM. Strength theory also plays an important role in material nonlinearity analysis, e.g., bearing capacity evaluation for structures. Metallic materials e.g., steel, copper, and aluminum have equal strength both in tension and compression, and hydrostatic stress has little influence on the strength of metallic materials. Nonmetallic Received Sept. 14, 01 / Revised Nov. 0, 01 Project supported by the National Natural Science Foundation of China No. 411776 and the Central Financial Funds for the Development of Characteristic Key Disciplines in Local Universities Nos. 106-00X101 and 106-5X105 Corresponding author Zong-yuan MA, Ph.D., E-mail: mzy gogo@hotmail.com

458 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG materials e.g., concrete, rock, and soil have different strengths in tension and compression i.e., the strength-different SD effect, and hydrostatic stress has a marked influence on the strength of SD materials. Plasticity theory can be divided into two components for strength theory, classical plasticity for metallic materials and generalized plasticity for SD materials. Three forms of work-hardening/softening or strain-hardening/softening law are used in plasticity theory, i.e., the isotropic hardening/softening case, the kinematic hardening/softening case, and the anisotropic hardening/softening case. The anisotropic hardening/softening law is basically discussed in generalized plasticity for SD materials. Many complex stress tests, such as the combined tension-torsion test and the true triaxial test, have verified that the intermediate principal stress has big influence on the mechanical behavior of metallic materials and SD materials. Lode [1] performed combined-stress experiments on thin-walled tubes made of iron, copper, and nickel. Taylor and Quinney [] did experiments on mild steel, copper, and aluminum thin-walled tubes that were subjected to tension and torsion. The effect of the intermediate principal stress was verified for all of the tested materials. Ivey [3] and Mair and Pugh [4] performed experiments with combined tension and torsion stress experiments of thin-walled tubes made of aluminum and pure copper and demonstrated the effect of the intermediate principal stress. Michelis [5] and Mogi [6] obtained experimental results from true triaxial testing of many rock samples and showed that the effect of the intermediate principal stress must be considered for rock. The results of complex stress tests of concrete indicate that the intermediate principal stress also has a marked effect on the strength of concrete [7 8]. The results from true triaxial tests show that the intermediate principal stress has a marked effect on the characteristics of soil under three different principal stresses [9 13]. Tresca and von Mises are two widely used yield criteria for metallic materials with the same yield stress in both tension and compression. The Mohr-Coulomb criterion is a widely used strength criterion for nonmetallic materials concrete or geomaterials, and it considers the SD and hydrostatic stress effects. However, the effect of the intermediate principal stress is not taken into account by the Tresca and the Mohr-Coulomb criterion, and some metals strengths exceed the yield surface of the von Mises criterion under a complex stress state [3 4]. Several strength criteria that take the intermediate principal stress and nonlinear yield surface into account, such as the William and Warnke [7], Lade and Duncan [14], and Matsuoka and Sun [15] criteria, which are proposed for geomaterials. Moreover, a smooth model for the Mohr-Coulomb criterion was established by Zienkiewicz and Pande [16], using the shape function suggested by Gudehus and Argyris, which used in many computer programs. However, those nonlinear yield surface criteria are appropriated for particular materials only. For example, the William-Warnke criterion is appropriate for concrete, and the Lade-Duncan and Matsuoka-Nakai criteria are both appropriate for sand. In addition, nonlinear yield surface criteria are difficult to implement in computer programming or to apply in theoretical analysis. The advances in strength theories of materials under complex stress states in the 0th century have been summarized by Yu [17]. Theories and methods.1 Unified strength theory UST The twin-shear stress element and the multi-shear stresses element, proposed by Yu [18], are shown in Fig. 1g and Fig.1h. The stress elements shown in Fig.1a, Fig.1b, and Fig. 1c are principal stress element, maximum shear stress element, and octahedral shear stress element, respectively. The multi-shear element see Fig. 1d is a rhombic dodecahedral multiple slip element differing from that of the three dimensional principal stress used in common continuum mechanics. There are three sets of principal shear stresses τ 13, τ 1, and τ 3 and normal stresses σ 13, σ 1, and σ 3 acting on the same sections. The maximum principal shear stress τ 13 equals the sum of the other two, i.e., τ 13 = τ 1 +τ 3. Thus, there are only two independent components among the three principal shear stresses. If the states of the three principal stresses change,

Unified elastoplastic finite difference and its application 459 e.g., if the intermediate principal stress σ increases from the value of the minimum principal stress σ 3 to that of the maximum principal stress σ 1, the intermediate principal shear stress τ 1 will be decreased, and τ 3 will be the intermediate principal shear stress. The Tresca and Mohr-Coulomb criteria consider only the maximum principal shear stress τ 13. Thus, the Tresca and Mohr-Coulomb criteria are both single-shear strength criteria and do not take into account the effect of the intermediate principal stress. The UST which takes the influence of the intermediate principal shear stress and the intermediate principal stress into account with a bilinear function, was proposed by Yu [19]. The expression of the UST for metallic materials equal strength in tension and compression can be formulated as follows: { f = τ13 + bτ 1 C, τ 1 bτ 3, f = τ 13 + bτ 3 C, τ 1 bτ 3. 1 Fig. 1 Various polyhedral element action by different stresses The expression of the UST for SD materials different strength in tension and compression can be formulated as follows: { f = τ13 + bτ 1 + βσ 13 + bσ 1 C, τ 1 + βσ 1 bτ 3 + βσ 3, f = τ 13 + bτ 3 + βσ 13 + bσ 3 C, τ 1 + βσ 1 bτ 3 + βσ 3, where β is a coefficient reflecting the effect of the normal stress on the strength of materials, b is a coefficient reflecting the effect of the intermediate principal shear stress τ 1 or τ 3 on the strength of materials, and σ 13, σ 1, and σ 3 are the normal stresses corresponding to τ 13, τ 1, and τ 3 that act on the sides of a multi-shear stress element. The stresses τ 13, τ 1, τ 3, σ 13, σ 1, and σ 3 can be obtained from the principal stresses σ 1, σ, and σ 3 as follows: τ 13 = σ 1 σ 3 σ 13 = σ 1+σ 3, τ 1 = σ 1 σ, σ 1 = σ 1 + σ, τ 3 = σ σ 3,, σ 3 = σ + σ 3. 3

460 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG Therefore, the expressions of the UST for metallic materials and SD materials can be written in the forms of principal stresses as follows: f = σ 1 1 1 + b bσ + σ 3 σ s when σ 1 σ 1 + σ 3, f = 1 1 + b σ 1 + bσ σ 3 σ s when σ 1 σ 1 + σ 3, f = bσ + σ 3 1 + bn ϕ σ 1 + c f = σ 3 σ 1 + bσ + c N ϕ 1 + b when σ 1 + sin ϕ Nϕ when σ 1 + sinϕ Nϕ σ 1 + 1 sin ϕ σ 3, σ 1 + 1 sinϕ σ 3, where σ s is the tensile yield stress for metal, c and ϕ are the cohesion and friction angle for SD material, and N ϕ = 1 + sinϕ/1 sinϕ. The order of the three principal stresses follows σ 1 σ σ 3. For the non-associated flow rule [0], the plastic potential function g can be written as follows: g = bσ + σ 3 σ 1 when σ 1 + sin ϕ σ 1 + 1 sin ϕ σ 3, 1 + bn ψ g = σ 3 σ 1 + bσ N ψ 1 + b when σ 1 + sin ϕ σ 1 + 1 sin ϕ σ 3, where N ψ = 1+sin ψ 1 sin ψ, and ψ is the dilation angle for SD material. If ψ < ϕ, then the plastic flow rule is non-associated, while for the associated flow rule, ψ = ϕ. Direct comparisons with the analytical method can be made when the associated flow rule is used in numerical methods. We can see that the effect of the intermediate principal stress is an inherent property for materials. The different effects of the intermediate principal stress for different materials can be reflected by the coefficient b in the UST. The effect of the intermediate principal stress on the strength of materials is increased with increasing values of the coefficient b. A series of convex strength criteria can be obtained when the value of the coefficient b varies in the range of 0 b 1, and more materials can be described. The UST offers the possibility to choose a reasonable strength criterion for studies or applications. In addition, the UST can also introduce a family of non-convex strength criteria when b < 0 or b > 1. The limit loci of the UST on the deviatoric plane or principal stress space are shown in Fig., where θ b is the stress angle for the junction of two yield surfaces and depends only on the friction angle of the material. The yield surfaces of the UST cover the entire region of convex theory from the lower bound UST b = 0.0 to the upper bound UST b = 1.0. The Mohr-Coulomb strength criterion is the special case of the UST when the coefficient b is equal to zero. Other nonlinear yield surface criteria can be encompassed or linearly approximated by the UST when 0.0 < b < 1.0. Comparisons between the yield surfaces of the UST and test data for metallic materials [1 4], granite [1], and sand [10] are plotted in Fig. 3. The results shown in Fig. 3 indicate that the intermediate principal stress has certain influences on the strengths of different metallic materials or geomaterials under complex stress states, and the potential strengths of different materials can be predicted by the UST with various values of the parameter b. Yu et al. [1] proposed an elastoplastic associated and non-associated constitutive model based on the UST for SD materials and implemented it in a two-dimensional finite element program. The objective of this study is to establish two elastoplastic constitutive models for metal and SD materials based on the UST and implement those two models in the finite difference code fast Lagrangian analysis of continua FLAC/FLAC3D. Those two new elastoplastic constitutive models for 4 5 6

Unified elastoplastic finite difference and its application 461 FLAC/FLAC3D are programed easily in computer code and can be widely used in material nonlinearity analysis under complex stress states for both metallic and SD materials. Fig. Limit loci of UST on deviatoric plane Fig. 3 Effects of intermediate principal stress for different materials. Finite difference and its elastoplastic constitutive model FLAC/FLAC3D is a two/three-dimensional finite difference code with an explicit computation scheme. The behavior of structures built of metallic materials or SD materials that may undergo plastic flow when their yield limits are reached can be simulated easily by

46 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG FLAC/FLAC3D [ 3]. An approach of mixed discretization is applied in FLAC/FLAC3D, as described by Marti and Cundall [4]. The components of the strain-rate tensor for a tetrahedral element are expressed as follows: ε ij = 1 6V 4 l=1 v l i nl j + v l j nl i S l. 7 The critical time step corresponding to the explicit finite difference scheme is given by the following expression [5] : t = T π = m k, 8 where T is the period of the system, k is the stiffness between two nodes, and m is the point mass. An important factor for the convergence of FLAC/FLAC3D is the out-of-balance force of the whole system. The out-of-balance force approaches zero when the medium reaches equilibrium or steady flow, and the calculation process of FLAC/FLAC3D is converged. Two elastoplastic constitutive models based on the UST, which are appropriate for metallic and SD materials, respectively, are written in C++ as user-written constitutive models and compiled as dynamic link library DLL files that can be loaded into the FLAC or FLAC3D code. The direction of the plastic strain increment is not uniquely on the additional corners of bilinear yield surfaces in the UST. Thus, corner singularity treatment must be processed. The method of dividing the stress space is used to address the problem of corner singularity in FLAC/FLAC3D. Figure 4 shows the stress space of the bilinear yield surfaces divided by a function σ = σ1+σ3 or σ = 1+sin ϕ σ 1 + 1 sin ϕ σ 3 on the deviatoric stress plane. In this paper, the vector summation method is used to eliminate the corner singularity on the corner of two yield surfaces in the UST [6 8]. Fig. 4 Stress space dividing and corner singularity treatment for UST on deviatoric plane The elastic relations between elastic principal strain increments and principal stress increments in the constitutive model for FLAC/FLAC3D are as follows: σ i = S i ε e n, i = 1, n, 9 where S i is a linear function of the elastic principal strain increments ε e n. The plastic principal strain increments can be written as follows: ε p i = λ g σ i, 10

Unified elastoplastic finite difference and its application 463 where λ is a non-negative multiplier if plastic loading occurs. The expression of the elastoplastic constitutive model for FLAC/FLAC3D can be formulated as follows: g σi N = σi I λ S i, 11 σ n where λ S i g σ n is the plastic principal strain component in the matrix of the constitutive model, g is the plastic potential function, σi I are the stress components obtained from the incremental elastic law, and σi N are the new stress components obtained from the plastic flow rule. The plastic factor λ can be written as follows: λ = fσ I n f S n g σ n f0. 1 After substituting the expressions 4, 5, and 6 into 10 and 11, the elastoplastic constitutive model based on the UST can be obtained. For the metallic materials, the term λs i f σ n is expressed as follows: When σ 1 σ 1 + σ 3, S 1 λ f, λ f σ, λ f = λα α 1, S λ f, λ f σ, λ f = λb 1 + b α 1 α, S 3 λ f, λ f σ, λ f = λ 1 + b α 1 α. 13a When σ 1 σ 1 + σ 3, S 1 λ f, λ f σ, λ f = λ 1 + b α α 1, S λ f, λ f σ, λ f = λb 1 + b α α 1, S 3 λ f, λ f σ, λ f = λα 1 α, 13b where α 1 = K+ 4G 3, α = K G 3, K is the elastic bulk modulus, G is the elastic shear modulus, and the plastic factor λ is expressed as follows: λ = fσ1 I, σi, σi 3 f S g 1 σ n, g S σ n, g when σ 1 S3 σ n f0 σ 1 + σ 3, f σ1, I σ, I σ3 I f S g 1 σ n, g S σ n, g S3 σ n f 0 when σ 1 σ 1 + σ 3. For SD materials, the term λs i f σ n is expressed as follows: 14

464 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG When σ < 1+sin ϕ σ 1 + 1 sin ϕ σ 3, S 1 λ g, λ g, λ g α = λ α 1, σ N ψ S λ g, λ g, λ g bα 1 1 = λ + α 1, σ 1 + bn ψ 1 + bn ψ S 3 λ g, λ g, λ g α 1 b = λ + α 1. σ 1 + bn ψ 1 + bn ψ 15a When σ > 1+sin ϕ σ 1 + 1 sin ϕ σ 3, S 1 λ g, λ g, λ g = λ σ α 1 N ψ b α 1, 1 + b 1 + b S λ g, λ g, λ g 1 = λ α 1 bα 1, σ N ψ 1 + b 1 + b S 3 λ g, λ g, λ g 1 = λ α 1 α, σ N ψ where the plastic factor λ is expressed as follows: fσ1 I, σi, σi 3 λ = f S 1 g, g σ, g, S g, g σ, g, S3 g, g σ, g f0 when σ < 1 + sin ϕ σ 1 + 1 sin ϕ σ 3, f σ I 1, σi, σi 3 f S 1 g, g σ, g, S g, g σ, g, S3 g, g σ, g f 0 when σ > 1 + sin ϕ σ 1 + 1 sin ϕ σ 3. 15b In the stage of strain-hardening/softening, the shear hardening parameter e ps is used to characterize the plastic shear strain, whose incremental form is defined as follows: e ps = 1 εps 1 εps m + ε ps εps m + ε ps 3 εps m 1 16, 17 where ε ps 1, εps, and εps 3 are the three plastic principal strains of the shear strength envelope, and ε ps m = ε ps 1 + εps + εps 3 /3. It is observed from 17 that the hardening parameter eps equals the square root of the second invariant of the strain. For the plastic hardening/softening stage, the yield function f for metallic materials can be written as follows: fσ n, σ s e ps = 0, 18 18 belongs to the isotropic hardening/softening case of classical plasticity theory. The yield function f for SD materials can be written as follows: fσ n, ce ps, ϕe ps = 0, 19 19 belongs to the anisotropic hardening/softening case, if the value of the strength parameter ϕ is varied with the hardening parameter e ps. 19 changes to the isotropic hardening/softening case, if only the value of the strength parameter c is varied with the hardening parameter e ps.

3 Verification problems Unified elastoplastic finite difference and its application 465 3.1 Perforated plate in tension An experiment on a thin perforated plate in tension was produced by Theocaris and Marketos [9]. The material they tested was aluminum alloy with linear strain-hardening characteristics. The yield function f for linear strain-hardening can be written as follows: fσ n, σ s + He ps = 0, 0 where H is the plastic modulus for linear strain-hardening material, and the isotropic hardening law is used to analyze this problem. The material properties for the perforated plate were reported by Theocaris and Marketos as follows: Young s modulus E = 70 000 MPa, Poisson s ratio ν = 0., the yield stress σ s = 43 MPa, and the linear hardening modulus H= 40 MPa. This problem is analyzed by Zienkiewicz [30] using FEM with the variable stiffness iterative method, assuming plane stress conditions and applying the von Mises criterion. In this paper, the same problem is analyzed using FLAC with the UST and ANSYS with the von Mises criterion. The FD or FE meshes, with dimensions and boundary conditions are given in Fig. 5. A quadrilateral element with linear interpolation Plane4 and the variable stiffness iterative method are used in ANSYS. A tensile load σ is applied to the top of the perforated plate, and the tensile load σ is divided into nine load increments a single load increment is 0.1 σ s. The relationship of average tensile stress σ mean along a section AB versus maximum tensile strain ε yy at the first yield point is shown in Fig. 6. The results from FLAC based on the UST b = 0.5 and b = 0.366 and from FEM using the von Mises criterion show excellent agreement. Comparison of the tensile stress σ yy along AB at the load σ/σ s = 0.455 obtained by numerical simulation and experiment is shown in Fig. 7. A satisfactory agreement is achieved among the tensile stress distributions obtained from FDM and FEM with the von Mises criterion. A slight discrepancy exists between the experimental data and the model predictions in the plastic region. Fig. 5 FD and FE meshes with boundary conditions for perforated plate plane stress

466 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG Fig. 6 Stress-strain relationship for perforated plate Fig. 7 Stress distribution at root of plate with load σ/σ s = 0.455 3. Bearing capacity of strip footing An example was given by Humpheson and Naylor [31] to analyze the influence of different forms of strength criteria i.e., Mohr-Coulomb, Drucker-Prager, William-Warnke, and Gudehus- Argyris on the bearing capacity of a strip footing using FEM with 3 parabolic elements. This example was examined again by Zienkiewicz and Pande [16]. Some differences in the values of the ultimate bearing capacity obtained using different strength criteria were detected for a plane strain and flexible footing on a weightless SD material based on the associated flow rule ψ = ϕ. The same problem is analyzed using the two-dimensional explicit finite difference code FLAC with the UST. The foundation soil is considered to be a linearly elasticperfectly plastic and homogeneous material with the following properties: Young s modulus E= 40.0 MPa, Poisson s ratio ν = 0., the cohesion c = 10.0 kpa, the friction angle ϕ = 0, and the dilation angle ψ = ϕ. The uniform vertical pressure is gradually increased the incremental pressure is 0. MPa until the algorithm could not converge. The ultimate bearing capacity is reached when the numerical nonconvergence occurres. Figure 8 shows the limit loci of the UST, Drucker-Prager, and nonlinear yield surface criteria William-Warnke and Gudehus- Argyris on the deviatoric plane. Figure 9 shows the FD meshes with the dimensions and boundary conditions used in analysis. The values of the ultimate bearing capacity P u /c calculated by Humpheson and Naylor and obtained using FLAC are presented in Table 1. Figure 10 shows a comparison of the values of ultimate bearing capacity P u /c obtained from FLAC using the UST with those calculated by Humpheson and Naylor. The values of P u /c calculated by Humpheson and Naylor with the William-Warnke and Gudehus-Argyris criteria lie between the values obtained using the UST for b = 0.5 and b = 0.5, and the values of P u /c calculated by Humpheson and Naylor with the Gudehus-Argyris criterion are close to the UST solution for b = 0.5. 4 Application problems 4.1 Large deformation and non-association problem Large deformation is a common phenomenon for metallic or nonmetallic structures under static or dynamic loading. There are two ways to describe the movement in continuum mechanics, the Lagrangian description and the Eulerian description [30]. In the Lagrangian formulation, which is often applied in solid mechanics, the nodes of the Lagrangian mesh move together with the material. Large deformations may lead to problematic mesh and large element distortions

Unified elastoplastic finite difference and its application 467 in Lagrangian simulations. In the Eulerian formulation, an Eulerian reference mesh is needed to trace the motion of the material in the Eulerian domain. This is a field description that is often applied in fluid mechanics. Fig. 8 Limit loci of UST and other criteria on deviatoric plane Fig. 9 FD meshes with boundary conditions for bearing capacity of strip footing plane strain Table 1 Ultimate bearing capacity P u/c values from FLAC and Humpheson & Naylor P u/c Yield criteria Humpheson & Naylor FEM FLAC FDM Drucker-Prager 14.8 14.9 Mohr-Coulomb/UST b = 0.0 15.1 15.0 Drucker-Prager extension cone 16.9 17. UST b = 0.5 19.1 Williams-Warnke 1.1 Gudehus-Argyris.6 UST b = 0.5 3.0 UST b = 1.0 30.5 Drucker-Prager compression cone 34.0 35.5

468 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG Fig. 10 Comparison bearing capacity P u/c for flexible strip footing values for different criteria The updated Lagrangian method is used for large deformation analysis in FLAC/FLAC- 3D [ 3]. Incremental displacements are added to the coordinates so that the grid moves and deforms with the material it represents. Considering the large displacements, displacement gradients, and rotations, the node coordinates and element stresses are updated after each step calculation, and the rotation components must be taken into account in calculation of the element stresses. The updates of node coordinates and element stresses are expressed as follows: x l i t + t = x l i t + tv l i t + t, 1 σ ij = σ ij + σ C ij, σ C ij = ω ik σ kj σ ik ω kj t, where x l i and v l i are the components of coordinate and velocity for nodes in large deformation analysis and σij C is the stress correction for displacement gradients and rotations and not taken into consideration in small deformation mode. The fundamental hypothesis of associated plasticity, which is applicable in plasticity theory for metallic materials, does not hold for some SD materials, e.g., concrete, rock, or soil [0]. The plastic volume dilation of SD materials is much higher than that predicted by the associative flow rule, and the plastic volume dilation can be reduced by the non-associative flow rule. Considering the influence of large deformations and the non-associative flow rule, an example of a prismatic bar with a square cross section subject to axial compression load is simulated in FLAC3D. The updated Lagrangian formulation is used to analyze the large deformation of the prismatic bar in compression. The material of the prismatic bar has the following properties, Young s modulus E = 50.0 MPa, Poisson s ratio ν = 0., the cohesion c = 10.0 kpa, the friction angle ϕ = 30, the dilation angle ψ = ϕ, ψ = ϕ, and ψ = 0. The boundary conditions and dimensions used in the analysis are shown in Fig. 11. The relationship between the axial compression load and the axial displacement of the prismatic bar is shown in Fig. 1. For small deformations, the uniaxial stress state is closely approximated by the prismatic bar being subject to pure axial compression, and the intermediate principal stress has little influence on the strength and deformation of the prismatic bar. However, the uniaxial stress state is changed to a complex stress state when the deformation is large. As Fig.1 shows, a difference induced by the effect of the intermediate principal stress is detected from the results of the Mohr-Coulomb criterion and UST b = 1.0. The influence of the intermediate principal stress is decreased with decreasing values of the dilation angle for the non-association problem. However, little effect of the intermediate principal stress is indicated for small deformations.

Unified elastoplastic finite difference and its application 469 Fig. 11 Dimensions and boundary conditions for large deformation analysis three-dimensional Fig. 1 Relationship between axial compression load and axial displacement 4. Strength of soft rock under complex stress states The hump curve function for the friction angle ϕ is used to simulate the strain-softening behavior of soft rock in this study. The hump curve function is shown as follows: ϕe ps = eps H + Re ps H + Pe ps, 3 where the slope of the hump curve is controlled by the parameter H, and the peak and residual values of the hump curve are controlled by the parameters P and R, and the anisotropic hardening/softening law is used to analyze this problem. A conventional triaxial shear test on diatomaceous soft rock was conducted in Refs. [3 33]. The strength parameter c for diatomaceous soft rock varies slightly with increasing value of ε 1, but the strength parameter ϕ has a peak and residual value. Figure 13 shows the variation in friction angle for diatomaceous soft rock with the axial strain increased. Based on the variation in friction angle determined with the conventional triaxial test, the values of the parameters H, P, and R in 3 can be determined. A hexahedral element with unite length is used to simulate a cubical specimen of diatomaceous soft rock, and the conventional triaxial and true triaxial tests are simulated using the

470 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG three-dimensional finite difference code FLAC3D. The confining pressure σ 3 is applied by the stress boundary condition on all sides of the cubical specimen element to generate the consolidated stress. For the conventional triaxial test simulation simple stress state, the three principal stresses follow σ 1 σ = σ 3, a vertical velocity load 1 10 6 m per step is applied to the top of the cubical specimen element to simulate the maximum principal stress σ 1, and the confining pressure σ 3 is also applied on both sides of the cubical specimen element. The bottom of the cubical specimen element is fixed in the vertical direction, so that both ends of the cubical specimen element are rigid and smooth. For the true triaxial test simulation complex stress state, the three principal stresses follow σ 1 σ σ 3 and the maximum principal stress σ 1 and confining pressure σ 3 also depend on the velocity and the stress boundary condition. The intermediate principal stress σ depends on the stress boundary condition according to the principal stress ratio of σ σ 3 /σ 1 σ 3. The magnitude of σ is controlled by the program to observe the principal stress ratio of σ σ 3 /σ 1 σ 3 during the true triaxial test simulation. The technology of the true triaxial test for a cubical specimen based on the principal stress ratio of σ σ 3 /σ 1 σ 3 is widely used in complex stress tests for geomaterials [9 10]. The values of the material property parameters used in the numerical analysis are presented in Table. Because the cubical specimen of diatomaceous soft rock has little volume dilation after the conventional triaxial test, the dilation angle ψ is set to be zero non-associated flow rule throughout the whole calculation process for the triaxial and true triaxial test simulations. Figure 14 shows the relationships of stress σ 1 σ 3 versus axial strain ε 1 for diatomaceous soft rock as measured in the triaxial test and as predicted by FLAC3D. The comparison between the model prediction and the measured results shows that the stress-strain relationship of diatomaceous soft rock can be simulated by the strain-hardening/softening model with the hump curve function. Fig. 13 Variation of friction angle ϕ with axial strain ε 1 for diatomaceous soft rock Fig. 14 Relationship of stress σ 1 σ 3 to axial strain ε 1 for diatomaceous soft rock measured by conventional triaxial test or computed by FLAC3D Table Material parameters used in true triaxial tests simulation of diatomaceous soft rock Name Diatomaceous soft rock Linearly elastic-perfectly plastic Strain softening Elastic modulus Poisson s ratio Cohesion E/MPa ν c/kpa P H R 500 0.5 40 0.01 0.04 0.004 5

Unified elastoplastic finite difference and its application 471 The strain-hardening/softening constitutive model based on the UST is then used to simulate a true triaxial test for diatomaceous soft rock. The hardening/softening rule for the friction angle ϕ follows. The stress-strain relationships for diatomaceous soft rock predicted by the true triaxial test simulation for various values of the principal stress ratio σ σ 3 /σ 1 σ 3 are shown in Fig.15. The results shown in Fig.15 indicate that the peak strength q p and the residual strength q s for diatomaceous soft rock under a complex stress state are higher than under a simple stress state. The stress-strain curve obtained using the Mohr-Coulomb criterion is insensitive to the variations in the intermediate principal stress. The peak strength q p and residual strength q s values obtained from the true triaxial test simulation for various values of the principal stress ratio σ σ 3 /σ 1 σ 3 are shown in Fig.16. The peak strength q p and residual strength q s first increase and then decrease with increasing values of the principal stress ratio σ σ 3 /σ 1 σ 3. The values of q p and q s under the stress state of σ = σ 1 σ σ 3 /σ 1 σ 3 = 1.0 are lower than those under the stress state of σ = σ 3 σ σ 3 /σ 1 σ 3 = 0.0. Fig. 15 Stress-strain curves yield by true triaxial test simulation under different principal stress ratios

47 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG Fig. 16 Variation of peak and residual strength from true triaxial test simulation with principal stress ratio of σ σ 3/σ 1 σ 3 5 Conclusions Two new elastoplastic constitutive models, appropriate for metallic materials and SD materials, are presented and implemented in the explicit finite difference code FLAC/FLAC3D. Two verification examples are used to compare test data and computation results. Good agreement is achieved between the test results and model predictions, and the two constitutive models proposed in this paper are verified. Two application examples, a prismatic bar subject to axial compression and soft rock under a complex stress state, are analyzed. Based on the results of this study, the following conclusions can be drawn: i The two elastoplastic constitutive models proposed in this paper, based on the unified strength theory with bilinear expression and yield surfaces, can be easily implemented and used in the explicit finite difference code FLAC/FLAC3D. ii These two elastoplastic constitutive models are verified using two classical examples and may be widely used in bearing capacity evaluation or stability analysis of structures built of metallic or SD materials in situations, where the effect of the intermediate principal stress must be taken into account. iii The results from the analysis of the large deformation and non-association problem indicate that the effect of the intermediate principal stress on large deformation problems is related to the plastic volume dilation of SD materials. The true triaxial test simulation results for diatomaceous soft rock suggest that the intermediate principal stress has a significant influence on the strength and strain-softening behavior of diatomaceous soft rock under a complex stress state. iv Considering the entire region between the lower bound using either the Tresca or the Mohr-Coulomb criterion and the upper bound UST b = 1.0 of the convex theory, the effect of the intermediate principal stress on metallic or SD materials or structures under complex stress states, including large deformation, three-dimensional or non-association problems, can be easily analyzed using the two new elastoplastic constitutive models proposed in this paper. Acknowledgements The first author expresses gratitude to Prof. Mao-hong YU of Xi an Jiaotong University, People s Republic of China. Discussion with Prof. YU is of great help and value.

Unified elastoplastic finite difference and its application 473 References [1] Lode, W. Versuche ueber den Einlfuss der mittleren Hauptspannung auf dsa fliessen der metals eisen kupfer und nickel. Z. Physik., 36, 913 939 196 [] Taylor, G. I. and Quinney, H. The plastic distortion of metals. Phil. Trans. Roy. Soc., 30, 33 36 1931 [3] Ivey, H. J. Plastic stress-strain relations and yield surfaces for aluminium alloys. J. Mech. Eng. Sci., 3, 15 31 1961 [4] Mair, W. M. and Pugh, H. L. D. Effect of pre-strain on yield surfaces in copper. J. Mech. Eng. Sci., 6, 150 163 1964 [5] Michelis, P. Polyaxial yielding of granular rock. J. Eng. Mech., 1118, 1049 1066 1985 [6] Mogi, K. Effect of the intermediate principal stress on rock failure. J. Geophys. Res., 70, 5117 5131 1967 [7] William, K. J. and Warnke, E. P. Constitutive model for the triaxial behavior of concrete. Proceedings of Seminar on Concrete Structures Subjected to Triaxial Stresses, 1 30 1975 [8] Du, X. L., Lu, D. C., Gong, Q. M., and Zhao, M. Nonlinear unified strength criterion for concrete under three-dimensional stress states. J. Eng. Mech., 1361, 51 59 010 [9] Green, G. E. and Bishop, A. W. A note on the drained strength of sand under generalized strain conditions. Geotechnique, 191, 144 149 1969 [10] Lade, P. V. and Duncan, J. M. Cubical triaxial tests on cohesionless soil. J. Soil Mech. Found. Div., 9910, 793 81 1973 [11] Ochiai, H. and Lade, P. V. Three-dimensional behavior of sand with anisotropic fabric. Journal of Geotechnical Engineering, 10910, 1313 138 1983 [1] Matsuoka, H. and Nakai, T. Stress-deformation and strength characteristics of soil under three difference principal stress. Proceeding of Japan Society of Civil Engineers, 3, 59 70 1974 [13] Xing, Y. C., Liu, Z. D., and Zheng, Y. R. A failure criterion of loess in Chinese. Journal of Hydraulic Engineering, 1, 1 19 199 [14] Lade, P. V. and Duncan, J. M. Elastoplastic stress-strain theory for cohesionless soil. J. Geo. Eng. Div., 10110, 1037 1053 1975 [15] Matsuoka, H. and Sun, D. A. Extension of spatially mobilized plane SMP to frictional and cohesive materials and its application to cemented sands. Soils and Foundations, 354, 63 7 1995 [16] Zienkiewicz, O. C. and Pande, G. N. Some Useful Forms of Isotropic Yield Surfaces for Soil and Rock Mechanics, Wiley, New York, 179 190 1977 [17] Yu, M. H. Advances in strength theories for materials under complex stress state in the 0th Century. Appl. Mech. Rev., 553, 169 18 00 [18] Yu, M. H. Twin shear stress yield criterion. Int. J. Mech. Sci., 51, 71 74 1983 [19] Yu, M. H. Unified Strength Theory and Its Applications, Springer, Berlin 004 [0] Vermeer, P. A. and de Borst, R. Non-associated plasticity for soils concrete and rock. Heron, 93, 3 64 1984 [1] Yu, M. H., Yang, S. Y., Fan, S. C., and Ma, G. W. Unified elasto-plastic associated and nonassociated constitutive model and its engineering applications. Comput. Struct., 716, 67 636 1999 [] Itasca Consulting Group. FLAC-Fast Lagrangian Analysis of Continua, User s manual, Minneapolis 005 [3] Itasca Consulting Group. FLAC 3D-Fast Lagrangian Analysis of Continua in 3 Dimensions, User s manual, Minneapolis 005 [4] Marti, J. and Cundall, P. A. Mixed discretization procedure for accurate modelling of plastic collapse. Int. J. Numer. Anal. Meth. Geomech., 6, 19 139 198 [5] Bathe, K. J. and Wilson, E. L. Numerical Methods in Finite Element Analysis, Prentice-Hall, New Jersey 1976

474 Zong-yuan MA, Hong-jian LIAO, and Fa-ning DANG [6] Koiter, W. T. Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with singular yield surface. Quart. Appl. Math., 11, 350 354 1953 [7] Sloan, S. W. and Booker, J. R. Removal of singularities in Tresca and Mohr-Coulomb yield functions. Commun. Appl. Numer. Mech.,, 173 179 1986 [8] Yu, H. S. A close-form solution of stiffness matrix for Tresca and Mohr-Coulomb plasticity models. Comput. Struct., 533, 755 757 1994 [9] Theocaris, P. S. and Marketos, E. Elastic-plastic analysis of perforated thin strips of a strainhardening material. J. Mech. Phys. Solids, 1, 377 390 1964 [30] Zienkiewicz, O. C. The Finite Element Method in Engineering Science, McGraw-Hill, New York 1971 [31] Humpheson, C. and Naylor, D. J. The Importance of the Form of the Failure Criterion, University of Wales, Swansea 1975 [3] Liao, H. J., Su, L. J., Pu, W. C., and Yin, J. H. Test and numerical analysis of the constitutive relation of a diatomaceous soft rock. Marine Georesources & Geotechnology, 1, 183 00 003 [33] Liao, H. J., Pu, W. C., and Yin, J. H. Numerical modeling of the effect of strain rate on the stress-strain relation for soft rock using a 3-D elastic visco-plastic model. Int. J. Rock. Mech. Min., 413, 43 43 004