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 (

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 [

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 [

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 [

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

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 [

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 that potential the particle feels a force

The forces are used to calculate the velocity of each particle and the new distribution is obtained through Newton’s second law:

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.

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 [

The separation of the cohesive interfaces is calculated from the displacement jump [

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,

Beside the critical separation

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

Taking the material of

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

Multiscale analysis flowchart based on cohesive element.

The cohesive laws, or so-called traction-separation laws (TSL), were introduced into finite element computations for brittle material failure analysis more than 30 years ago [

In order to carry out quantitative evaluation on crack’s impact area at microscale and provide evidence to confirm the reasonable T-S area, two micromechanics models with defects of

The elemental crystal structure and slip plane of

In order to make the comparison of computational results of the two models, the models were set with similar geometrical parameters and mechanics conditions. The length, width, and thickness of the two models were separately 250 Å, 100 Å, and 3 Å, and the lengths of the sharp crack and the blunt crack were both 25 Å (the width of the blunt crack was 25 Å). The tensile displacement load on both upper and lower surfaces of the two models was 0.25 Å/ps. The range of 10 Å from upper and lower surfaces of the crack was taken as the range of T-S area. The T-S data area was divide into 20 judgement areas along the length direction of T-S data area, which means the maximum judgement precision of effective T-S area was 0.5 times crack length, as shown in Figure

Two defected micromechanics models.

Model with blunt crack defect

Model with sharp crack defect

After loading was done, the T-S curves of the two models in all the judgement areas defined in Figure

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Therefore, the T-S curves of the two models in the representative areas such as the 1st, the 3rd, the 12th, the 13th, the 20th, and the 1st~12th were shown in Figures

T-S curve of the model with blunt crack defect.

The 1st judgement area

The 3rd judgement area

The 12th judgement area

The 13th judgement area

The 20th judgement area

The 1st~12th judgement areas

T-S curve of the model with sharp crack defect.

The 1st judgement area

The 3rd judgement area

The 12th judgement area

The 13th judgement area

The 20th judgement area

The 1st~12th judgement areas

Figures

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Through microscopic observation of tensile and fatigue fracture process, void, blunt crack, and sharp crack can be found in the microstructure of

Microstructure of tensile and fatigue fracture surfaces of

Before using cohesive elements to simulate crack propagation, the relationship between the interface cohesive force and the crack opening displacement should be obtained first. Three mechanics models with compound defects of

In order to make comparison among the three models, similar geometrical parameters and mechanics conditions were assigned to the three models. The sizes of the models were 330 Å × 100 Å × 3 Å; the length of the main defect was 25 Å (the width of the main defect of Model I and Model II was 5.2 Å); the distance between the main defect and the secondary defect was 20.15 Å; the diameter of the secondary defect of Model I was 6 Å, and the length of the secondary defect of Model III was 10 Å. Next, the tensile displacement load on both upper and lower surfaces of the three models was all 0.25 Å/ps. The range of 10 Å from upper and lower surfaces of the crack was taken as the range of T-S area. Then the three molecule dynamics models with compound defects could be built as shown in Figure

Three molecule dynamics model with compound defects.

Model I

Model II

Model III

Figures

Traction-separation relationship and deformed mechanisms of Model I under tensile loading.

Traction-separation relationship and deformed mechanisms of Model II under tensile loading.

Traction-separation relationship and deformed mechanisms of Model III under tensile loading.

Figures

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In order to model stable crack growth under static loading and analyze cohesive behavior derived from MD towards greater length scales, we performed a simulation of crack growth for a CT specimen subject to displacement loading via prescribed motion of loading pins. Fracture of a CT specimen could verify whether the cohesive law derived from MD simulations displayed behavior consistent with linear elastic fracture mechanics. The geometry and mesh of our CT specimen were shown in Figure

FEM model geometry (

After the traction-separation results of the three molecule dynamics models with compound defects got in Part 3.2 were made dimensionless, the T-S results could be used in finite element analysis of the specimen showed in Figure

The operation interface of ABAQUAS to simulate the behavior of the cohesive zone model was shown in Figure

The operation interface of ABAQUAS to simulate the behavior of the cohesive zone model.

The crack opening behavior due to displacement of the top and bottom pins could be observed. Before crack propagation begins to occur, the cohesive zone begins to form. Once a critical displacement was reached, crack propagation was seen in Figure

Static crack growth and Mises stress (MPa) contour of the specimen.

Model I

Model II

Model III

Figures

Stress intensity factor of the specimen (three models).

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The authors declare that they have no competing interests.

This work was supported by the National Defense Basic Scientific Research Program of China through the contract of B1520132013-1 and Key Laboratory of Energy Engineering Safety and Disaster Mechanics, Ministry of Education, entitled the multiscale behavior research of interface performance between metal matrix and nanostructured ceramic coatings.