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A phase-field model of thermo-elastic coupled brittle fracture with explicit time integration

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Abstract

The phase-field method is a very effective way to simulate arbitrary crack nucleation, propagation, bifurcation, and the formation of complex crack networks. The diffusion-based method is suitable for multi-field coupling fracture problems. In this paper, a parallel algorithm of the thermo-elastic coupled phase-field model is implemented in commercial finite element code Abaqus/Explicit. The algorithm is applied to simulate the dynamic and quasi-static brittle fracture of thermo-elastic materials. Further, it is adopted on a structured mesh combined with first-order explicit integrators. Several examples of the quasi-static and dynamic cases of single crack, as well as multi-crack initiation and propagation under thermal shock, are given to demonstrate the robustness of the algorithm. The source code and tutorials provide an effective way to simulate crack nucleation and propagation in multi-field coupling problems.

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Notes

  1. We also wrote Python code to automatically handle the process of modifying input files. One who familiar with Python can just modify the geometric modeling part and the boundary condition imposing part to create their own model.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11532008) the Special Research Grant for Doctor Discipline by Ministry of Education, China (Grant No. 20120002110075) and China Postdoctoral Science Foundation (No. 2019M650699).

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Authors and Affiliations

  1. Applied Mechanics Laboratory, School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Tao Wang, Zhanli Liu, Dongyang Chu & Zhuo Zhuang

  2. LNM, Institude of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Xuan Ye & Xiaoming Liu

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Appendices

Appendix A: Parallel performance study

The phase-field model has high requirements for mesh density and usually requires large-scale calculations. Therefore, parallel computing is particularly important for the wide application of the phase field method. The explicit time integration schemes (including central-difference and forward-difference integration method) are suitable for increasing computational efficiency through parallel computing. In this paper, we implement them using multi-CPU sub-regional calculations. Here we use the example in Sect. 4.5 to study the efficiency of parallel computing. For parallel computing, the entire model is divided into several subdomains according to the number of CPUs that will be used. The information on the common boundary of each subdomain is stored in the public variables, which are synchronized in multiple CPUs via MPI functions in the user subroutine VEXTERNALDB.

Figure 16a shows the increment numbers every two minutes of the model with different number of CPUs. The model has 12,859,435 DOFs and 22,851 incremental steps. We can find that using multiple CPUs can greatly improve the efficiency of calculation, and the wall time is approximately inversely proportional to the number of CPUs used. Figure 16b shows the wall time consumed by the same model with the same number of CPUs (16) and different DOFs. It can be seen that the relationship between the wall time and the number of DOFs is basically linear, which is better than that of the implicit scheme.

Table 6 Solution dependent variables (SDVs) used for visualization
Full size table

Appendix B: One element tutorial

Here we assume that users are familiar with Abaqus/Explicit and its user subroutines. Due to the limitation of page space, only some key modeling details are given.

Each problem has two files for calculation: an Abaqus input file (*.inp) and a FORTRAN source code file (*.for or *.f, depending on the operation system).

Due to the problem of the allocation of the common block (module) for every finite element mesh a new FORTRAN file should be created. Only two variables that should be modified in the provided example file are NodeNum (the number of the nodes in \(\Omega _1\)) and NumEle (the number of the elements in \(\Omega _1\)). Thus, in this example NodeNum = 8, and NumEle=1.

The Abaqus input file is generally written by the software itself. However, we should modify it before initiating the simulation.Footnote 1

In the first section, the parts are created. The nodes are given (*Node) and the elements are generated. After creating all the nodes, a command is given to define the phase-field element type (*User element, nodes=8, type=VU2, properties=3, coordinates=3, variables=16). This command creates an element with eight nodes with five material properties and sixteem status variables. The status variables are used to transport information from one step to the next. It contains the phase-field value and the history variable at each integration point. The meaning of the SDVs is listed in Table 6.

In the next line, we define the concerning DOFs, in this case only the eleventh (11). To create the elements after the command: *Element, type=VU2, the elements are given starting with the serial number then the nodes of the corners in a counterclockwise list: 1, 5, 6, 8, 7, 1, 2, 4, 3. To assign material parameters to the elements a set is created. After which the command *Uel property, elset=Set-part2-ele-1 and the properties are given in the next line (where Set-part2-ele-1 is the name of the set containing all the phase-field elements). The properties are given as follows: viscosity parameter (\(\eta \)), length scale parameter (\(l_c\)) and fracture surface energy (\(g_c\)).

For temperature and displacement elements, the user material subroutine VUMAT is used. We only need to specify the material parameters passed into the user subroutine in *.inp file through the following statement: *User Material, constants=3. The properties are given as follows: Young’s modulus (E) Poisson’s ratio (\(\nu \)) and thermal expansion coefficient (\(\alpha _{\theta }\)). The user subroutine VUSFLD is declared to be used by adding command *User Defined Field to the *.inp file. Command *Field, Variable=1, User declares that the user subroutine VUFIELD is used and that the scope is all nodes (Set-allnode). The loads and boundaries of the displacement and temperature fields (in \(\Omega _1\)) are defined usually as it is done in a normal input file.

See the *.inp file in “Appendix C” for more details.

Appendix C: Supplementary materials

The supplementary materials related to this article include the source code of Abaqus user subroutine and the input file of the one element example.

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Wang, T., Ye, X., Liu, Z. et al. A phase-field model of thermo-elastic coupled brittle fracture with explicit time integration. Comput Mech 65, 1305–1321 (2020). https://doi.org/10.1007/s00466-020-01820-6

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