A three dimension lattice-spring model with rotational degree of freedom and its application in dynamic crack propagation

https://doi.org/10.1016/j.ijimpeng.2020.103729Get rights and content

Highlights

  • •

    This paper presents a new three dimension lattice-spring model with rotational degrees of freedom.

  • •

    In this paper, a stiffness matrix with rotational degrees of freedom is established.

  • •

    The crack evolution under dynamic tensile loading is studied.

Abstract

Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model (3D-LSM) containing the rotational degree of freedom. Using the finite element method (FEM) and share grid, the spring stiffness coefficient is obtained with a rigorous mathematical logic. This 3D-LSM contains the nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of the system. Wave velocity in elastic solid and dynamic fracture of concrete l-specimen are calculated by both 3D-LSM and elastic theory. Results indicate that this model can present the behavior of the material in the elastic stage. By introducing the energy balance principle, 3D-LSM can simulate dynamic brittle fracture both in mesoscopy and macroscopy. This model is used to analyze the evolution law of crack propagation path and propagation velocity in brittle materials under dynamic tension of shock wave. The results show that the stress concentration area at the crack tip changes during the crack propagation, showing a "butterfly shape" before the crack initiation and a "swept wing shape" after the crack initiation. After the crack initiation, the crack tip velocity jumps instantly and then increases continuously, corresponding to the expansion of a single crack. Then the tip velocity decreases slightly and increases to the maximum crack propagation velocity, corresponding to the crushing stage before bifurcation. After that, the crack growth rate will oscillate significantly, which corresponds to the bifurcation stage of the crack.

Introduction

Originating from typical computational methods, such as frame network method [1], molecular dynamics (MD) [2], [3], [4] and discrete element method (DEM) [5], etc., lattice-spring model (LSM) is one of the earliest simulation methods that was successfully applied to solid mechanics. LSM is gradually developed and matured in the study of brittle fracture of the solid medium [6], [7], [8]. LSM has been widely used in different fields since it has clear physical pictures. For specific cases in different fields, necessary adjustments are applied accordingly for the setting of the model. Because of the different variants of the model, this model has a series of appellations like the lattice model [2,9], the spring network model [10,11], distinct element method [12] and so on.

Many representative works have been done recently on the subject of applying LSM into fracture mechanics. Zapperi et al. [8] studied the phenomena of plasticity and avalanche caused by fracture evolution under the influence of quasi-static tension. Wang et al. [13] simulated the initiation and propagation of wing crack in a pre-cracked brittle sample under quasi-static compression. Based on the two-dimension lattice-spring model, Yu et al. [14] developed a compressive failure model under shock wave, which can effectively present large deformation impact, cavity collapse, crack propagation and medium crush of porous brittle material. They also studied the collapse course and mechanism of a single hole. Under the influence of shock wave, shear stress concentrates around the hole, and shear crack emerges then propagates towards the internal area [15], [16], [17], [18], [19]. Meanwhile, broken medium slips along the shear crack and fills the hole. In the homogeneous materials, the shear crack can extend to a long-distance, while in polycrystalline materials, two kinds of failures exist in the system, namely long cracks and wide cracks, which correspond respectively to the fracture modes of "trans-granular crack to inter-granular crack" and "main crack excites secondary crack". Fairly good results have been obtained by both experiment and simulation [14]. Stress relaxation resulting from cavity collapse is the main influence that microscopic evolution exerts on macroscopic shock response. Pazdniakou [20] utilized a 3D-LSM to study the propagation of the elastic wave in the medium. Zhao [21,22] studied elastic deformation and shock failure of the object through a 3D-LSM. Although the rotational degree of freedom has already been considered in the above-mentioned 3D-LSM, the rotation angle of the model is computed through node displacement, which involves a lot of computing tasks and time [23], [24], [25]. The freedom model without the rotational degree is the “Born model”. However, the Born model may not maintain rotational invariance [26], [27], [28]. The main reason, according to Jagota and Scherer [29], is that shear springs cannot distinguish the differences in the tangential velocities (or displacement) of two nodes owing to a common rotation or shear, as shown in Fig. 1. As a result, global rigid body rotation may incorrectly cause additional strain energy to be generated within the shear spring [30].

In this paper, a new 3D-LSM is proposed which can calculate the rotation angle directly. As shown in Fig. 2, u, v, w denote the translational degree of freedom along x-, y-, z-axis, while θx, θy, θz represent the rotational degree of freedom. All of these six degrees of freedom couple in the lattice. Employing quantitative mapping and shared grids, the stiffness coefficient of 3D-LSM is obtained from stiffness matrix in FEM, the element of which is set as a hexahedron with 8 nodes and 6 degrees of freedom including the rotational degree of freedom.

Section snippets

The theoretical basis of the lattice-spring model

LSM is a network model composed by node(particle) and spring(key, beam, etc.). In this model, parameters such as mass, velocity and displacement are carried by particle, while spring determines interaction force, damping, damage and fracture, etc. Fig. 3 is a cell of the proposed model in this paper, where the particle represents the node and segment represents the spring. Nearest neighbor, next nearest neighbor and third nearest neighbor are also considered in this model. For example, particle

Parameter mapping

When the rotational degree of freedom is considered, force vector on node i can be expressed asFi={FiuMiθ}Twhere Fiu={Fxi,Fyi,Fzi}T represents the principal vector applied on the node and Miθ={Mxi,Myi,Mzi}T represents the principal moment applied on the node. Substituting Eq. (11) and Eq. (12) into Eq. (1), we can obtain{Fiu=KuuΔ+KuθθMiθ=KθuΔ+Kθθθwhere Δj={uj,vj,wj}T, θj={θxj,θyj,θzj}T and j = 1, 2, …, N, in which N indicates the number of total nodes.

The mapping method proposed by Gusev is

The criterion for spring fracture of elastic brittle materials and kinematics integral

In order to use the 3D-LSM model to simulate the fracture process of the material, a failure criterion needs to be defined to identify the key elements to be removed from the model. Most failure criteria used so far in 3D-LSM have been classified into two categories: critical stress (force) criterion and critical strain (displacement) criterion. The critical strain criterion is adopted for calculation in this paper. Because rock and concrete are typical elastic-brittle materials, the

Wave velocity in elastic solid

Both longitudinal wave and shear wave can propagate effectively in solid medium. Wave speed depends only on the characteristics of materials. Longitudinal wave velocity CL and shear wave velocity CR under plane strain can be expressed asCL=1ρE(1−μ)(1+μ)(1−2μ),CR=E2ρ(1+μ)

A comparison of computing results of wave velocity for different materials is shown in Table 1, where parameters of material are also defined. Velocities are calculated from LSM and Eq. (19) respectively.

Computing results from

Crack evolution under dynamic tensile loading

This section mainly studies the application of the lattice-spring model with a rotational degree of freedom in dynamic crack growth and crack growth velocity evolution under shock wave dynamic loading. The calculation model is shown in Fig. 9. Fig. 9(a) is a model with prefabricated cracks. The cracks are transparent edge cracks. The tensile stress in the z-direction is applied on oyz surface, as shown in Fig. 9(b).

In order to verify the validity of the lattice-spring model with a rotational

Conclusion

In this paper, the finite element stiffness matrix is mapped to the spring stiffness coefficient of the lattice-spring model, based on an eight-node hexahedron finite element model containing rotational degrees of freedom, using the model mesh shared by lattice-spring model and finite elements, which makes the choice of spring stiffness coefficient have strict mathematical derivation. The model includes three kinds of neighbor elements: the nearest neighbor, the second neighbor and the third

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 11772090).

References (34)

  • J.G Kirkwood

    The skeletal modes of vibration of long chain molecules

    J Chem Phys

    (1939)
  • P.A. Cundall et al.

    A discrete numerical model for granular assemblies

    Geotechnique

    (1979)
  • G.A. Buxton et al.

    Simulating the morphology and mechanical properties of filled diblock copolymers

    Phys. Rev. E

    (2003)
  • G. Caldarelli et al.

    Self-organization and annealed disorder in fracturing process

    Phys. Rev. Lett.

    (1996)
  • S. Zapperi et al.

    Plasticity and avalanche behaviour in micro-fracturing phenomena

    Nature

    (1997)
  • Y.C. Wang et al.

    Implementation of particle-scale rotation in the 3-d lattice solid model

    Pure and Applied Geophysics

    (2006)
  • A.A Gusev

    Finite Element Mapping for Spring Network Representations of the Mechanics of Solids

    Phys. Rev. Lett.

    (2004)

    • Cited by (1)

    • Micro-holes’ effect on energy absorption ability in three-dimensional brittle materials under impact loading: A preliminary research

      2023, Structures
    View full text
    © 2020 Elsevier Ltd. All rights reserved.