Effects of interfacial discontinuity on the fracture behavior in the superconductor-substrate system

Lang Jiang, Xiaoqiang Ren, Zhiwen Gao,*, Youhe Zhou

a Key Laboratory of Mechanics on Disaster and Environment in Western China attached to the Ministry of Education of China, Lanzhou University, Lanzhou 730000, China

b Department of Mechanics and Engineering Science, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000,China

Keywords:Superconductor-substrate system Interfacial discontinuity Fracture behavior Activation processes

ABSTRACT This study concerns a two-dimensional model and the corresponding virtual crack closure technique (VCCT) implemented to solve the general boundary value problems that may explain why interface discontinuity has effects on the fracture behavior in the superconductor-substrate system. The interfacial discontinuity can be classified according to the material properties'continuity and their derivatives' continuity at the interface. For nonhomogeneous superconductor and substrate specimens with various material properties, a VCCT method is developed to calculate their fracture behavior. Furthermore, the effects of applied magnetic field amplitude and nonhomogeneous parameters are extensively and parametrically studied in two activation processes (zero-field cooling and field cooling). The integrative and computational study presented here provide a fundamental mechanistic understanding of the fracture mechanism in the superconductor-substrate system and sheds light on the rational design of interfacial continuity.

1Introduction

RE-123 coated superconductors can simultaneously motivate a variety of current of both long length and high critical current, therefore, in contrast to engineering structures, they have been applied to power cables, fault current limiters and current leads [1]. The fundamental analysis of interfacial transition of material property between superconductor and substrate has been established in a continuum mechanics framework [2]. Interfacial discontinuity not only affects the strength properties but also the fracture and failure behavior of these systems. To test the interfacial transition response, we presented an interface model to account for the effects of interfacial transition between superconductor and substrate on the giant magnetostriction in a high temperature superconductor (HTS) [2].

The fracture analysis of HTS has become an important field in the superconductor mechanics research. Zhou and Yong [3]firstly studied the crack problem for a long rectangular slab of superconductor under electromagnetic force. Yong et al. invest-igated the crack problem for a thin superconducting strip in a perpendicular magnetic field [4]. Gao et al. studied the dynamic fracture problem of the superconductor under an alternating magnetic field [5]. These fracture analyses were conducted based on the assumption that superconductors were isotropic homogeneous material. In fact, superconductor materials are inhomogeneous [6, 7]. Based on the heterogeneity of superconductors, Gao et al. [8] analyzed the fracture problem of an inhomogeneous high temperature superconductor (HTS) slab under electromagnetic force by using real fundamental solutions.Li proposed real fundamental solutions to study the mode-I crack problem [9-11]. For superconductor-substrate systems, using numerical methods to study the effects of interfacial discontinuity on the fracture behavior is more convenient. The virtual crack closure technique (VCCT) is a powerful tool for the energy release rate calculation which is basic for fracture mechanics.VCCT was firstly introduced by Rybicki and Kanninen [12] for two-dimensional (2D) crack problems and then applied to three-dimensional (3D) crack problems by Shivakumar et al.[13]. Xie proposed an interface element for VCCT which was used to study the dynamic crack propagation under mixed mode loading and dynamic loading [14-16]. Xie then studied the strain energy release rate for a moving delamination front of arbitrary shape based on VCCT [17, 18]. For inhomogeneous superconductor materials, we can easily use VCCT to obtain the stress intensity factors from the energy release rate.

The present work develops an interfacial discontinuity model based on the material properties' continuity and their derivatives' continuity at the interface. This model is founded on the fact that the fracture behavior of the superconductor-substrate systems depend on interfacial discontinuity. The goal of this work is to provide a framework that considers interfacial discontinuity, applied magnetic field and critical current density as physical parameters to accounts for fracture behavior in inhomogeneous superconductor. Then, an extensive parametric study is conducted on superconductor-substrate systems under electromagnetic loading. Results show that interfacial discontinuity, applied magnetic field, and critical current density all can cause significant effects on the fracture behavior of inhomogeneous superconductors.

The paper is organized as follows. The basic framework of VCCT and the physical governing equations of inhomogeneous superconductor-substrate systems will be reviewed in Sect. 2.The qualitative features of the obtained results will be discussed in Sect. 3. The summary and conclusions of this research will be presented in Sect. 4.

2Mechanical model

To characterize the effects of interfacial discontinuity on the fracture behavior of superconductor-substrate systems, the superconductor and substrate materials are modeled as isotropic inhomogeneous material. Illustrated in Fig. 1 is an inhomogeneous superconductor-substrate system with an interfacial crack placed in a magnetic field Baoriented parallel to the z direction,where the crack lies in the x-y plane. The crack of length 2a is parallel to the upper and lower boundaries. A rectangular coordinate system is established with the rightward x axis parallel to the crack line and the upward y axis. The superconductor bulk is assumed to be sufficiently long in the z direction. Therefore,the demagnetization effects can be neglected [19]. The Young's moduli of the substrate are denoted by, where E0is the Young's moduli of interface. The Young's modulus of the superconductor is supposed to vary with the interfacial discontinuity:

From Fig. 1(b), it can be seen that, in case (A), the Young's modulus is discontinuous at interfaces, at which all of the mechanical properties and their derivatives are discontinuous. In case(B), the Young's modulus first-order derivatives are continuous at interfaces. In case (C), the Young's modulus first-order and second-order derivatives are continuous at interfaces. In case(D), the Young's modulus mechanical properties and their derivatives are all continuous.

The electromagnetic body force arising from flux pinning can be expressed as

where J is current density and B is magnetic induction intensity.

In the model, the current density J only has x direction component Jx_sc, and the magnetic induction intensity B only has z direction component Bz, and

Fig. 1.a Schematic illustration of a superconductor-substrate system. b Geometry of the interfacial discontinuity transition with varying Young's moduli.

The Lorenz force can be simplified as

Therefore, the electromagnetic body forces in the superconductor of y direction can be expressed as

We can calculate the stresses induced by the interaction between the magnetic field and the persistent current inside superconductors. For critical state Bean model [20], the full penetration field is. To simplify notation, we use the following notation:

3Numerical results and discussion

A model is presented to investigate the effects of interfacial discontinuity on the fracture behavior in superconductor-substrate systems. The presented model is applied to calculate the stress intensity factors in two activation processes. On account of lacking corresponding experimental data, numerical results have been obtained for the interface crack in a superconductorsubstrate system under electromagnetic force. The following material properties are used in the analysis (see Table 1):

3.1Fracture behavior in zero-field cooling (ZFC) activation processes

For the zero-field cooling activation processes, as the applied field bais reduced from 2 to 0, the flux density is piecewise linear and can be expressed as

where e0=1-(ba，max-ba).

As the applied field bais reduced from 4 to 2, the flux density can be expressed as

where e0=1-(ba，max-ba)/2.

Figure 2 shows the variation of the stress intensity factor as the applied field is decreased from ba=2 to ba=0. From the results, we can see the stress intensity factors are zero in the initial stage. When applied field decreases from ba=1.5 to ba=0, the stress intensity factors become positive and increases as as badecreases, which means that the interface crack is on a growthtrend. It is indicated that the total forces are compressive for 1.5≤ba≤2, and become expansive for 0≤ba＜1.5. For ba,max=4, when badecreased from 4 to 2, the stress intensity factors deceased linearly as shown in Fig. 3. Figures 2 and 3 show the variation of stress intensity factors at four distinct discontinuity conditions.When the discontinuity increases, the stress intensity factors increase.

Table 1The parameters in present model

Figures 4 and 5 show the effects of superconductor material's nonhomogeneous parameter on the stress intensity factors for different applied magnetic fields. From the results, it can be found that the stress intensity factors increase as the nonhomogeneous parameter increases. In addition, the value of stress intensity factors for positive nonhomogeneous parameter is larger than the negative nonhomogeneous parameter.

Figures 6-8 show the profiles of the crack opening displacement (COD) for different interfacial discontinuity. It is obvious from Fig. 6 that with the increase of the interfacial discontinuity,the COD decreases. The COD is plotted in Fig. 7 as the applied magnetic field is reduced from 4 to 2. As we can observe from the plots, when the applied field increases, the COD increases.Figure 8 shows the profiles of the COD when the values of the nonhomogeneous parameters increase from -0.5 to 0.5. It is obvious that with the increase of the nonhomogeneous parameters, the COD decreases.

Fig. 2.Normalized stress intensity factors (SIFs) vary with magnetic field for different interfacial discontinuous conditions (,-α=β=0.5).

Fig. 3.Normalized SIFs vary with magnetic field for different interfacial discontinuous conditions (, -α=β=0.5).

3.2Fracture behavior in the field cooling activation processes

Fig. 4.Normalized SIFs vary with magnetic field for different nonhomogeneous coefficients (,).

Fig. 5.Normalized SIFs vary with magnetic field for different nonhomogeneous coefficients (,)

Fig. 6.The profiles of the crack opening displacement for different interfacial discontinuous conditions (, ba=4,)

Field cooling as a method to activate trapped-field magnets requires a much weaker field source to accomplish full activation. The method is to cool an HTS in a fixed magnetic field Bfcthen the applied field is removed, and a large part of the field remains trapped inside the superconductor. Bfcdenotes the external field applied during the cooling process. It is assumed that Bfcis also the flux density frozen in the superconductor when the subsequent field descent starts. To simplify notation, we use the following notation

For the magnetic field bfc＞ 1, as the applied field bfcis reduced to 1, the flux density can be expressed as

When magnetic field is reduced from 1 to 0, the flux density can be expressed as

For magnetic field bfc≤1, as the applied field bfcis reduced to 0, the flux density can be expressed as

Fig. 7.The profiles of the crack opening displacement for different magnetic fields (,,)

Fig. 8.The profiles of the crack opening displacement for different nonhomogeneous coefficients (,, ba=4,)

where e0=(1-bfc-ba)/2.

The stress intensity factors are plotted in Figs. 9 and 10 for different interfacial discontinuity. Shown in Fig. 9 are the stress intensity factors as bfc=1 and badecrease from 1 to 0. It is clear that the stress intensity factors increase as the applied field reduces, and the maximum value is reached at ba=0. For bfc=1.5,when the applied magnetic field descent starts at bfc＞1, the maximum stress intensity factor is reached at bfc-1. When the field descent starts at bfc≤1, the stress intensity factors increase as the applied field reduces, and the maximum value is reached at bfc=0.

Figures 11 and 12 show plots of the stress intensity factors for different interfacial discontinuity with different nonhomogeneous parameters. As seen from the figures, as the nonhomogeneous parameters increase, the stress intensity factors increase.

Fig. 9.Normalized SIFs vary with magnetic field for different interfacial discontinuity (bfc=1, -α=β=0.5)

Fig. 10.Normalized SIFs vary with magnetic field for different interfacial discontinuous conditions (bfc=1.5, -α=β=0.5).

Figures 13-15 present the profiles of COD for different discontinuity conditions during the field cooling process. Again,when the applied field increases, the COD increase, as shown in Fig. 13. Figure 14 present the COD profile for case C at bfc=1.5 and=0.5. It can be found that the COD has the maximum value as the applied magnetic field baequals to bfc-1. When ba＞bfc-1,with the applied magnetic field decrease, the COD increase.When ba＜bfc-1, with the applied magnetic field decrease, the COD decrease. The plots of the effect of nonhomogeneous parameters on COD are shown in Fig. 15. The larger the nonhomogeneous parameters, the smaller the COD are.

4Concluding remarks

A numerical model for the interfacial continuity of the superconductor-substrate system has been proposed, developed and implemented to predict the fracture process for different interfacial continuity under electromagnetic force. The interface model is formulated in terms of coupled theory of magnetic field and mechanics. It also combines the VCCT method in the numerical code. Two activation processes are presented to provide an insight into the effects of interface discontinuity and nonhomogeneous parameters on the fracture process of the interface in the superconductor-substrate system. The numerical results have shown that the macroscopic mechanical response of the fracture parameters is very sensitive to the interfacial discontinuity and nonhomogeneous parameters. Another example is considered to understand the COD response to the superconductor-substrate system.

Fig. 11.Normalized SIFs vary with magnetic field for different nonhomogeneous coefficients (bfc=1, E2=E(C)).

Fig. 12.Normalized SIFs vary with magnetic field for different nonhomogeneous coefficients (bfc=1.5, E2=E(C)).

Fig. 13.The profiles of the crack opening displacement for different interfacial discontinuous conditions (bfc=1.5, ba=0.5,-α=).

Fig. 14.The profiles of the crack opening displacement for different magnetic fields (, bfc=1.5,).

Fig. 15.The profiles of the crack opening displacement for different nonhomogeneous coefficients (, bfc=1.5, ba=0.5,).

Acknowledgments

The authors gratefully acknowledge the financial supports provided by the National Natural Science Foundation of China(11772142, 11272140, and 10902046) and the Fundamental Research Funds for the Central Universities (lzujbky-2015-176).