Numerical Manifold Method with Endochronic Theory for Elastoplasticity Analysis

Abstract

Numerical manifold method (NMM) was originally developed based on linear elastic constitutive model. For many problems it is difficult to obtain accurate results without elastoplasticity analysis, and an elastoplasticity version of NMM is needed. In this paper, the incremental endochronic theory is extended into NMM analysis and an endochronic NMM algorithm is proposed for elastoplasticity analysis. It is well known that endochronic theory is one of the widely used elastoplasticity theories which can deal with elastoplasticity problems without a yield surface and loading or unloading judgments. Numerical tests show that the proposed algorithm of endochronic NMM possesses a good accuracy. The proposed algorithm is also applied to analyze a crack problem and a soft clay foundation under traffic loading problem. Results demonstrate the convenience of the endochronic NMM in analyzing elastoplasticity discontinuous problems.

1. Introductions

The numerical manifold method (NMM) was proposed by Shi [1, 2]. It is a numerical analysis method for integrating the discontinuous deformation analysis (DDA) and the finite element method (FEM). The NMM is based on the finite cover approximation theory and approximate displacement by solving the equation on covers. Comparing with FEM, the NMM has the following advantages [3]: (1) the geometry of a model can be discontinuous; (2) it can solve problems with large deformation; (3) for quasistatic problems, the condition of equilibrium is automatically satisfied; (4) there is no need for Gauss integration points by using the simplex integration. Since the original version of NMM proposed, various developments and applications have been achieved during these years [4]. Ma et al. [5] and Gao and Cheng [6] applied NMM to analyze fracture problems. Zhang et al. [7] extended it to predict the growth of complex cracks. Terada and Kurumatani [8] proposed a version of NMM called the finite cover method. Li and Cheng [9] developed meshless NMM based on unit partition. Meshless NMM was extended into crack problems by Li and Cheng [10]. Zhang and Zhou [11] applied NMM to analyze the saturated porous media. Gao and Cheng [12] introduced the complex variable into NMM. Already developed NMM algorithms mainly adopted a linear elastic constitutive model. However, it is difficult to obtain accurate results without elastoplasticity analysis for many problems, and an elastoplasticity version of NMM is needed.

Endochronic theory is an elastoplasticity constitutive theory derived from thermo concepts by Valanis [13, 14] in 1971. It is widely used in metal, sand, concrete, and so on [15, 16]. Comparing with other plasticity theories, endochronic theory has neither conception of a yield surface nor the specification of the loading or unloading rules and can take the effects of the strain history into account by using intrinsic time. Unfortunately, endochronic FEM is not good at dealing with discontinuous problems and mass movement problems which NMM is expert in.

In this paper, we introduce the endochronic theory into NMM and apply the proposed algorithm to analyze a crack problem and a soft clay foundation under traffic loading problem. This paper is organized as follows. In Section 2, the basic conception of NMM and the algorithm of endochronic NMM are described. In Section 3, the accuracy of endochronic NMM is verified by numerical tests. In Section 4, numerical examples are taken to prove the availability of the algorithm. In Section 5, conclusions are given.

2. Endochronic NMM

2.1. The Basic Conceptions of NMM

There are three main components that form the heart of NMM: block kinematics, simplex integration method, and finite covering systems. The block kinematics theory is developed from DDA which can deal with the problems of contact, large deformation, and mass movement. The integration of a function in NMM is evaluated by simplex integration method. This integration method can be implemented on any shape without Gauss integration points. In NMM, there are two sets of covers. One is called physical cover (PC) which contains all the physical features such as the internal face, the domain which needs to be simulated, and the geometries on which the boundary conditions are loaded. Another kind of cover is called mathematical cover (MC) which must cover the physical cover. For example in Figure 1(a), Ω₁ and Ω₂ are the physical domains for analysis and Figure 1(b) shows the mathematical domain. In Figure 1(c), the MC₁ and MC₂ are the mathematical covers and Figure 1(d) shows the physical covers (PC_1-4). The manifold elements (MEs) are the intersection of mathematical covers and physical covers.

(a) Physical domain

(b) Mathematical domain

(c) Mathematical covers

(d) Physical covers

Figure 1 

Mathematical covers and physical covers.

NMM is a numerical method based on the principle of minimum total potential energy. The weak form of governing equation is defined as _Ω𝛿𝜀^𝑇Δ𝜎𝑑𝑉+_Ω𝛿𝑢^𝑇𝜌𝑢̈𝑑𝑉=_Ω𝛿𝑢^𝑇Δ𝑏𝑑𝑉+_Γ𝑡𝛿𝑢^𝑇Δ𝑡𝑑Γ, (1) where 𝜎 is the Cauchy stress tensor, 𝜀 is the small strain tensor, 𝑢 is the displacement vector, 𝑢̈ is the acceleration vector, 𝜌 is the density of the material, 𝑏 is the body force vector, and 𝑡 is the traction vector applied to boundary Γ_𝑡.

2.2. The Incremental Constitutive Equations

Constitutive equations of endochronic theory for homogeneous and isotropic materials could be described as follows: 𝜎_dev=2^𝑧₀𝜌𝑧−𝑧^′𝜕𝜀^𝑃_dev𝜕𝑧^′𝑑𝑧^′,𝑑𝜀^𝑝_dev=𝑑𝜀_dev−𝑑𝜎_dev,2𝜇 (2)(3) where 𝜇 is the elastic shear modulus, 𝜎_dev is the deviatoric stress, and 𝜀^𝑝_dev is the deviatoric plastic strain. In (2), 𝑍 is the intrinsic time measure which can be defined as follows: 𝑑𝑧=𝑑𝜉𝑓(𝜉),𝑑𝜉²=𝑑𝜀^𝑝_dev∶𝑑𝜀^𝑝_dev, (4) where 𝑓(𝜉) is called scale function, and it decides the behavior of material softening or hardening. By this function, the contribution of the plastic strain history can be taken into account. 𝜌(𝑧) is a kernel function and the 𝑚-term Dirichlet series are introduced into this function by Lee [17], which can be written as 𝜌(𝑧)=^𝑛_𝑖=1𝐶_𝑖𝑒^{−𝛼𝑖𝑧} (5) with the requirement that 𝐶_𝑖 and 𝛼_𝑖 are positive for all 𝑖 and ^∞_𝑖=1𝐶_𝑖=∞,^∞_𝑖=1𝐶_𝑖𝛼_𝑖<∞, (6) where 𝐶 and 𝛽 are material parameters.

The incremental form of endochronic NMM which is used in the numerical simulation can be deduced from the integral constitutive equations mentioned above. We assume that the variables at step 𝑛−1 are known, and the increment of deviatoric stress in step 𝑛 can be described as Δ𝜎_dev^𝑛𝑧=𝜎_dev^𝑛𝑧−𝜎_dev^𝑛−1𝑧. (7)

Substituting (2) and (5) to (7), Δ𝜎_dev^𝑛𝑧=2^𝑛Δ𝜀^𝑃_dev𝑛Δ𝑧^𝑚_𝑟=1𝐶_𝑟𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛Δ𝑧}+^𝑚_𝑟=1𝜎^𝑟_dev^𝑛−1𝑧𝑒^{−𝛼𝑟𝑛Δ𝑧}−1. (8)

It can be easy derived that Δ𝜎_dev^𝑛𝑧=2^𝑛Δ𝜀_dev𝑛Δ𝑧^𝑛𝑀^𝑚_𝑟=1𝐶_𝑟𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛}+1Δ𝑧_𝑛𝑀^𝑚_𝑟=1𝜎^𝑟_dev^𝑛−1𝑧𝑒^{−𝛼𝑟𝑛}Δ𝑧−1_𝑛𝑀1=1+_𝑛Δ𝑧𝜇^𝑚_𝑟=1𝐶_𝑟𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛}Δ𝑧. (9)(10)

We assume that the material is plastic incompressibility and the hydrostatic response can be expressed as Δ𝜎_ℎ=3𝐾Δ𝜀_vol, (11) where 𝐾 is the elastic bulk modulus. The total stress Δ𝜎 is that Δ𝜎=Δ𝜎_dev+Δ𝜎_ℎ𝐼. (12)

Substituting (9) and (11) to (12), the incremental stress vector can be represented as follows: ^𝑛Δ𝜎=3𝐾^𝑛Δ𝜀_vol𝐼+2^𝑛Δ𝜀_dev𝑛Δ𝑧^𝑛𝑀^𝑚_𝑟=1𝐶_𝑟𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛Δ𝑧}+_𝑛1𝑀^𝑚_𝑟=1𝜎^𝑟_dev^𝑛−1𝑧𝑒^{−𝛼𝑟𝑛}Δ𝑧−1. (13)

2.3. Discrete Equations for Endochronic NMM

In a manifold element, the relationship of the displacement vector and the strain tensor in (1) can be written as 𝑢=𝑁𝑑,𝜀=𝐿𝑁𝑑=𝐵𝑑, (14) where 𝑁 is the shape function, 𝐿 is the differential operator matrix, and 𝑑 is the unknown vector. From the constitutive equations, the stress tensor can be written as follows: Δ𝜎=𝐷^epΔ𝜀+𝑓^′, (15) where the elastoplasticity matrix 𝐷^ep and vector 𝑓^′ are written as follows: 𝐷^ep=𝐴₁𝐴₂𝐴₂000𝐴₂𝐴₁𝐴₂000𝐴₂𝐴₂𝐴₁000000𝐴₃000000𝐴₃000000𝐴₃,𝐴₁=𝐾+4∑^𝑚_𝑟=1𝐶_𝑟/𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛}Δ𝑧3^𝑛Δ𝑧^𝑛𝑀,𝐴₂=𝐾−2∑^𝑚_𝑟=1𝐶_𝑟/𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛}Δ𝑧3^𝑛Δ𝑧^𝑛𝑀,𝐴₃=∑^𝑚_𝑟=1𝐶_𝑟/𝛼_𝑟1−𝑒^{−𝛼𝑟𝑛}Δ𝑧_𝑛Δ𝑧^𝑛𝑀,𝑓^′=_𝑛1𝑀^𝑚_𝑟=1𝜎^𝑟_dev^𝑛−1𝑧𝑒^{−𝛼𝑟𝑛}Δ𝑧−1. (16)

Substituting (14) and (15) to (1), we finally obtain the discrete equations for endochronic NMM analysis: 𝐾Δ𝑑+𝑀Δ𝑑̈=Δ𝐹, (17) where the stiffness matrix 𝐾, mass matrix 𝑀, and the incremental load Δ𝐹 can be represented as: 𝐾=_Ω𝐵^𝑇𝐷^ep𝐵𝑑Ω,𝑀=_Ω𝜌𝑁^𝑇𝑁𝑑𝑉,Δ𝐹=_Ω𝑁^𝑇Δ𝑏𝑑Ω+_Γ𝑁^𝑇Δ𝑡𝑑Γ+_Ω𝐵^𝑇𝑓^′𝑑Ω. (18)

To consider the dynamic problems, the total time is divided into limited number of periods. By using the Newmark scheme, the acceleration vector can be written as 𝑑_𝑛+1=𝑑_𝑛+Δ𝑡_𝑛𝑑̇_𝑛+Δ𝑡²_𝑛21−𝛽𝑑̈_𝑛+𝛽𝑑̈_𝑛+1𝑑̇_𝑛+1=𝑑̇_𝑛+Δ𝑡_𝑛1−𝛾𝑑̈_𝑛+𝛾𝑑̈_𝑛+1 (19) and then 𝑑̈_𝑛+1=2Δ𝑡²𝛽Δ𝑑−Δ𝑡𝑑̇_𝑛−Δ𝑡²21−𝛽𝑑̈_𝑛. (20)

Here we defined 𝛽 and 𝛾 equal to 1. The acceleration vector is transformed as 𝑑̈_𝑛+1=2Δ𝑡²Δ𝑑−Δ𝑡𝑑̇_𝑛. (21)

The discrete equations for dynamic problem derived from (17) and (21) can be written as 𝐾Δ𝑑=Δ𝐹, (22) where 2𝐾=𝐾+Δ𝑡²𝑀,Δ𝐹=Δ𝐹+2𝑉_𝑛Δ𝑡𝑀. (23)

The details of the algorithm used in the program are described in Table 1.

Table 1 

Description of the algorithm.

3. Performance of Endochronic NMM

3.1. Test A: Behaviors under Cyclic Loading

This example is taken from Lee [17]. He applied the endochronic theory to an oxygen-free high-conductivity copper plate under cyclic loading with FEM and the results coincide with the experimental data [18] very well. In this section, a number of experiments are performed to test the accuracy of the endochronic NMM. The geometry and mathematic mesh of the problem are shown in Figure 2. 𝐹₁ and 𝐹₂ are the forces which are applied on the node of the model and the values of 𝐹₁ and 𝐹₂ are adjusted to ensure that the axial stress is equal to the date from literature [17]. In order to overlook the effect of mathematic covers, we let the mathematic domain have the same area as physical domain. The material parameters, which are provided by Lee, are listed in Table 2 and the material functions: 𝜌(𝑧) and 𝑓(𝜉) are defined as follows: 𝜌(𝑧)=⁷_𝑖=1𝐶_𝑖𝑒^{−𝛼𝑖𝑧},𝑓(𝜉)=1−0.84𝑒^−16.4𝜉,𝜁≤0.035,1−0.637𝑒^−8.48𝜉,𝜁>0.035. (24)

Table 2 

Material properties [17] for the cyclically loaded elements.

Figure 2 

Geometry and mathematic mesh for cyclic loading.

Figure 3 shows that the results from endochronic NMM are identical to the data from Lee [17]. This example can demonstrate that when the mathematic mesh fits with the physical covers, the endochronic NMM will produce exactly the same result as endochronic FEM.

Figure 3 

The axial stress versus strain relation of element 1.

3.2. Test B: The Influence of Mathematical Covers

In FEM, the mesh is identical with the geometric figure, but in NMM the mathematic mesh is arbitrary and the integral form is different from FEM. In this section, as shown in Figure 4, we study the influences of the mathematical mesh by changing the length and position of mathematical cover. We apply a single force on the highest point of the model. Figure 5 is the cyclic loading diagram. The parameters of the endochronic model are shown in Table 3 and 𝑓(𝜉)=1−(𝜉/20).

Table 3 

Material properties for test B.

(a) FEM model

(b) Model 1

(c) Model 2

(d) Model 3

(e) Model 4

Figure 4 

Models for test B (m).

Figure 5 

Cyclic loading diagram.

The values of the intrinsic times, displacement, and residual strain of endochronic NMM are compared with the corresponding values of endochronic FEM in Table 4. It can be easily found that the displacements and the intrinsic times in models 1–4 are exactly the same as those in FEM. The computational results of the stress-strain by endochronic NMM and FEM are shown in Figure 6. The results indicate that the response of a physical cover in a mathematical cover will not be affected by the size, relative position, and shape of mathematical cover in endochronic NMM.

Table 4 

Aggregate of computational results.

Figure 6 

Computation results of axial stress and strain under cyclic loading.

4. Numerical Examples

4.1. Simulation of Crack Problems with Endochronic NMM

This section is devoted to the demonstration of the capabilities of endochronic NMM in simulating the crack problems. A rectangular plate of 40 m wide and 40 m high with a piercing crack of 10 m in the centre is illustrated in Figure 7. Three different meshes, which are shown in Figure 8, have been considered. Tensions are applied on the top and bottom of the model with the maximum being equal to 12.8 (MPa×m). We adopt the same parameters for the endochronic model which are adopted in test A.

Figure 7 

Geometric figure for crack problem.

(a) DOF = 4662

(b) DOF = 1956

(c) DOF = 706

Figure 8 

Three different meshes for crack problem.

The distributions of von-Mises plastic strain with different DOFs are shown in Figure 9, which cannot be obtained by the original NMM. The distributions of von-Mises stress are also shown in Figure 9. It can be observed that the three results are almost the same despite minor discrepancies being found. It seems fair to claim that the endochronic NMM is capable of dealing with the elastoplasticity crack problems.

(a) DOF = 4662

(b) DOF = 1956

(c) DOF = 706

Figure 9 

Distributions of von-Mises stress and von-Mises plastic strain with different DOFs.

4.2. Soft Clay Foundation under Traffic Loading Problem

In this section, a numerical investigation on the mechanical behavior of soft clay under the traffic moving loads is carried out to demonstrate the convenience of the endochronic NMM. The geometry and mesh of the example are shown in Figure 10. Vehicle tire is modeled as a circle of radius 0.5 m. 𝐹 is a concentrated force which is equal to 12.5 kN and applied on the center of the tire. We let the tire move on the foundation five times with 1.00 m/s. In vertical direction, there is one observation point for every 0.5 m. Figure 12 is the stress-strain relations of soil in a particular place under cyclic loading which is tested by Peng et al. [19]. We use the parameters of endochronic model which are listed in Table 5 and 𝑓(𝜉)=1−0.65𝑒^−2.1𝜉. Figure 11 is the comparison of the results from numerical and test. To study the disturbing force generated by the tire, effect of the pore water pressure will be ignored.

Table 5 

Parameters of endochronic model for soft clay foundation under traffic loading problem.

(a) Geometry and boundary conditions of the example

(b) Mesh of the example

Figure 10 

Mesh and geometry of the example.

Figure 11 

Stress-strain relations of soil.

Figure 12 

Displacement of point E.

Figure 12 shows the displacement of point E generated by endochronic NMM and conventional NMM. It could be observed that the endochronic NMM can calculate the land subsidence after the tire rolling through the foundation which cannot be simulated by conventional NMM. Figure 13 shows the additional stress generated by tire at points A–E. It could be found that the curves of additional stress over time are similar to sine curves, which coincide with the conclusion by Hyodo and Yasuhara [20]. Figure 14 is the distribution of equivalent plastic strain when the tire passes the foundation for the first time. The results demonstrate that the endochronic NMM can deal with the elastoplasticity problems conveniently, especially at the problems which the mass movement and contact are in charge of.

Figure 13 

Additional stresses change over time for points A–E.

Figure 14 

Distribution of equivalent plastic strain.

5. Conclusions

The endochronic NMM, as an integration of the numerical manifold method and endochronic theory, is developed especially for elastoplasticity problems. Endochronic theory, which is one of the most typical elastoplasticity theories for modeling material behavior, has been successfully introduced into the presented version of NMM in this paper. Results of tests show that the endochronic NMM is a very useful and accurate method for plasticity analysis. The numerical examples of crack problem and soft clay foundation under traffic loading problem demonstrate the convenience of the endochronic NMM in analyzing elastoplasticity discontinuous problems.

Conflict of Interests

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

Acknowledgments

This research was supported by National Natural Science Foundation of China under Grant no. 51109028 and the State Scholarship Fund of China to pursue study in USA as a visiting scholar under Grant no. 201208210208.