Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 2156936, 13 pages

http://dx.doi.org/10.1155/2016/2156936

## The Deformation Mechanisms in Process of Crack Propagation for Alpha Titanium with Compounding Microdefects

College of Architecture and Environment, Sichuan University, Chengdu 610065, China

Received 26 February 2016; Revised 22 May 2016; Accepted 23 May 2016

Academic Editor: Hiroshi Noguchi

Copyright © 2016 Ying Sheng and Xiang-guo Zeng. 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.

#### Abstract

The multiscale analysis method based on traction-separation law (TSL) and cohesive zone law was used to describe the cross-scale defective process of alpha titanium (*α*-Ti) material with compounding microdefects in this paper. First, the properties of T-S curve and the reasonable range of T-S area relative to the length of defects were discussed. Next, based on the conclusions above, the molecule dynamics analysis of three models of *α*-Ti with compounding microdefects was conducted and cross-scaly simulated. The phenomenon, principles, and mechanisms of different compound microscale defects propagation of *α*-Ti were observed and explained at atomic scale, and the effects of different microdefects on macrofracture parameters of materials were studied.

#### 1. Introduction

Microdefects commonly occur on materials. Microdefects of materials not only occur during the manufacture and fabrication process, but also form new microdefects under different loading circumstances. For example, micro tip cracks will occur under fatigue loading, and voids or micro blunt cracks will occur under impact loading [1]. And the material properties strongly depend on the atom structure and microstructure of itself [2]. Hence, in order to improve the security of materials during practical application, it is significant to study the effect of microdefects on macrocharacteristics of materials.

How to build proper models and find methods to carry out cross-scale analysis on the micro and macroproperties of materials becomes central attention of many scholars. Currently, there are two classic research methods: the serial multiscale method and the parallel multiscale method. For the serial method, the numerical model of mesoscale elements needs to be built first, then based on the micro numerical results, the macrostructure parameters can be estimated according to the multiscale theories, and finally the obtained parameters can be applied to macrostructure simulation by some methods, such as homogenization method [3]. For the parallel method, the micro and macro numerical models can be built simultaneously, and then different cross-scale methods can be used to connect and exchange data between macro and micro models, such as Macroscopic/Atomistic/Ab initio Dynamics (MAAD) method [4], Quasi Continuum (QC) method [5], and coarse-grained molecular dynamics (CGMD) method [6].

There are mainly two types of models to study crack propagation and fracture problems: one is based on classical fracture mechanics models and the other is based on damage mechanics models. Among them, the cohesive method based on damage mechanics is one of the widely applied methods. It applies to both macrocrack propagation and microcrack propagation [7, 8]. The basic ideas of the cohesive method are applying traction-separation law (TSL) to simulate cohesive forces among atomic lattice and therefore avoid singularity of crack tip and adopting cohesive separation principles to define the properties of cohesive interface elements. The cohesive separation principles include cohesive plastic property, cohesive elastic property, fracture, fiber damage, mechanics defect, circular loading defect, and other behaviors. The cohesive method is appropriate to apply to concrete, metals, compound materials, and many other materials. Houachine et al. [9] applied the cohesive method to successfully predict the interface integrate stress of FRP and concrete beam; Kebriaei et al. [10] used the cohesive method to build multiscale model of AA1050 and AA5754 alloy connection and described integrate properties of connection interface. Grogan et al. [11] used the cohesive method to successfully simulate the formation and propagation of microcracks of composite laminates materials.

However, what is the optimum range of T-S region related to the length of the initial crack is also an open issue. If T-S area exceeds the affected range of the crack too much, it will not only increase meaningless computation because of the unreasonable model, but also make the traction value obtained through computation less than real value. If the value of T-S area is too small, then it is unable to accurately evaluate the impact of microcrack propagation on macromechanics performance of the material. In order to carry out quantitative evaluation on crack’s impact area at microscale and provide evidence to confirm the reasonable T-S area, this paper discussed the properties of T-S curve as well as reasonable range of T-S area relative to the defect length first by taking *α*-Ti material as an example.

Ti alloy has several advantages such as high stress, small density, corrosion resistance, and good deform property under low temperature. That is why it is widely used in aviation, shipping, mechanic production, and weapon industry and also frequently endures high-speed impact loading during its application [12]. Ti has allotropism phenomenon. Under the temperature of 882°C, it becomes hexagonal crystal packed (HCP) structure, and it is also called *α*-Ti. *α*-Ti is an anisotropic material and its mechanical properties of different crystal orientation are different. In this paper, taking *α*-Ti material as an example, the traction-separation law (TSL) method was adopted to discuss the properties of T-S curve as well as the reasonable range of T-S area relative to the defect length. Then based on the conclusions, the open source code of LAMMPS [13] developed by Sandia laboratory was adopted to analyze the molecule dynamics of the three models of *α*-Ti with compound microdefects, and AtomEye software [14] was also used to carry out the visualization of the atom configuration during the process of deformation. In addition, the different compound microdefects propagation phenomenon, principles, and mechanisms of *α*-Ti were also observed and explained at atomic scale. Finally, the cohesive zone law multiscale analysis method was adopted to study the effects of different microdefects on macrofracture parameters of materials.

#### 2. Multiscale Simulation Methods

##### 2.1. Brief Review of Molecular Dynamics

Molecular Dynamics (MD) method is one of simulation methods widely used for molecular systems. In the MD method the initial distribution is random, and every new distribution is derived from the previous one by using the interactions between the particles. Consider one particle in the system. Every other particle attracts or repels it. The interaction depends on the positions of the particles and and contributes to the total potential energy of particle:

In that potential the particle feels a forcewhich accelerates the particle in a certain direction.

The forces are used to calculate the velocity of each particle and the new distribution is obtained through Newton’s second law:where is the velocity of the particle. When the velocity of each particle at a given time is calculated by solving the equation of motion (3), one allows every particle to move with that velocity a short period of time and then reevaluates the potential energy, force, and thus the velocity.

The MD method follows classical mechanics of motions and is therefore purely deterministic. It has a real time coordinate and the trajectory therefore follows the changes of the system in time. It is not limited to systems at equilibrium but can be used to study systems under external perturbations.

##### 2.2. Traction-Separation Laws in Cohesive Zone Models

The idea for the cohesive model is based on the consideration that infinite stresses at the crack tip are not realistic. Models to overcome this drawback have been introduced by Dugdale [15] and by Barenblatt [16] for the first time. The material separation and thus damage of the structure are classically described by interface elements, and no continuum elements are damaged in the cohesive model. The behavior of the material is spitted in two parts, the damage-free continuum with an arbitrary material law and the cohesive interfaces between the continuum elements, which specify only the damage of the material. The interface elements open when damage occurs and lose their stiffness at failure so that the continuum elements are disconnected. For this reason the crack can propagate only along the element boundaries.

The separation of the cohesive interfaces is calculated from the displacement jump [], that is, the difference of the displacements of the adjacent continuum elements,

The separation depends on the normal and the shear stress, respectively, acting on the surface of the interface. When the normal or tangential component of the separation reaches a critical value, or , respectively, the continuum elements initially connected by this cohesive element are disconnected, which means that the material at this point has failed.

Beside the critical separation , the maximum traction (stress at the surface of the continuum element) is used as a fracture parameter, also denoted as cohesive stress. The value of only describes the maximum value of a traction-separation curve , in the following denoted as cohesive law. Like the separations, the stresses can also act in normal or in tangential direction, leading to normal or shear fracture, respectively. The shape of the curve , which is assumed to be a material independent cohesive law, is defined differently by various authors [15, 17, 18].

The integration of the traction over separation, either in normal or in tangential direction, gives the energy dissipated by the cohesive elements, :

Beside the form of the T-S curve, which was assumed to be a model quantity, there are two material parameters, that is, the maximum separation stress , which has to be overcome for final fracture, and the separation at failure . These quantities define the height and the width of the curve and give (together with the function of the curve) the dissipated energy per cohesive element as a result.

##### 2.3. Multiscale Simulation Process

Taking the material of *α*-Ti as an example, our approach studying crack propagation includes the use of MD simulation to obtain T-S relations in cohesive elements. In our present work, the crack propagation of compact tension (CT) model containing cohesive elements, whose T-S relation was derived from MD simulations, is simulated. The method may be called the atomic based cohesive zone models (CZM) method. These results enable us to make several recommendations to improve the methodology to obtain cohesive laws and better comprehend the crack propagation behavior of *α*-Ti.

The multiscale analysis flow based on cohesive elements is displayed in Figure 1.