Numerical Analysis of Joule Heating Behavior and Residual Compressive Stress around Crack Tip under High Electric Load

The Joule heating behavior and electro-thermo-structural coupled-field around the crack tip were firstly investigated and discussed by Russian scholars in the 1980s [1, 2]. They used mathematical methods to analyze the crack-tip field under the electric current. Following their pioneer research, many typical studies were done for solving this problem in the last decades [3–12]. Based on the past references, the Joule heating phenomenon near the crack tip is well known. Figure 1 illustrates the concept of this problem. Due to the Joule heating effect, the thermoelectric phenomenon occurs in the conductive material under the electric load. In other words, the material gets hotter when it is subjected to the electric current. If the material has cracks or fractures, the electric current density concentration occurs around the crack tip. Then this electric concentration causes a local hot region at the crack tip due to the Joule heating. In addition, the electric current density has r−1/2 singularity at the crack tip [6]. This is similar to the elastic stress field. Under the Joule heating, the compressive stresses can be produced around the crack tip [1, 2]. This crack-tip compressive stress field is important to reduce or stop the potential crack growth. However, the past references did not show the time-history of the crack-tip stress. During the electric loading, unloading, and cooling process, the cracktip stress may present the tensile or compressive state. If the plastic strain or deformation occurs at high temperature, the residual stress around the crack tip will be an important topic for the fracture problem. In this paper, the Joule heating behavior and residual stress around the crack tip will be investigated using the electro-thermo-structural coupling finite element analysis. The temperature and electric current density fieldswill be also obtained for estimating the crack tip behavior. In particular, this primary studywill discuss the residual compressive stress and its importance for stopping the crack growth.


Introduction
The Joule heating behavior and electro-thermo-structural coupled-field around the crack tip were firstly investigated and discussed by Russian scholars in the 1980s [1,2].They used mathematical methods to analyze the crack-tip field under the electric current.Following their pioneer research, many typical studies were done for solving this problem in the last decades [3][4][5][6][7][8][9][10][11][12].
Based on the past references, the Joule heating phenomenon near the crack tip is well known.Figure 1 illustrates the concept of this problem.Due to the Joule heating effect, the thermoelectric phenomenon occurs in the conductive material under the electric load.In other words, the material gets hotter when it is subjected to the electric current.If the material has cracks or fractures, the electric current density concentration occurs around the crack tip.Then this electric concentration causes a local hot region at the crack tip due to the Joule heating.In addition, the electric current density has  −1/2 singularity at the crack tip [6].This is similar to the elastic stress field.
Under the Joule heating, the compressive stresses can be produced around the crack tip [1,2].This crack-tip compressive stress field is important to reduce or stop the potential crack growth.However, the past references did not show the time-history of the crack-tip stress.During the electric loading, unloading, and cooling process, the cracktip stress may present the tensile or compressive state.If the plastic strain or deformation occurs at high temperature, the residual stress around the crack tip will be an important topic for the fracture problem.
In this paper, the Joule heating behavior and residual stress around the crack tip will be investigated using the electro-thermo-structural coupling finite element analysis.The temperature and electric current density fields will be also obtained for estimating the crack tip behavior.In particular, this primary study will discuss the residual compressive stress and its importance for stopping the crack growth.

Problem Statement
Figure 2 shows the geometric and loading conditions of the steel plate in this study.The plate with an edge crack is subjected to the constant direct current (DC)  0 .The electric loading time is   .The plate is made of the mild steel with the dimension  ×  × .The crack length is .This problem will be simulated by the three-dimensional finite element analysis with the solid model.
To consider practical conditions, the temperaturedependent material properties in Table 1 [9,13] are adopted in the finite element analysis.Also, the elastoplastic material properties as shown in Figure 3    Joule heating effect, the electric-current-induced thermostructural problem is transient.The initial temperature is 21 ∘ C. The convection coefficients on all solid/air interfaces are set as 10 W/m 2 -∘ C. The contact condition between crack surfaces is considered as the coupled-field problem.The electric current and heat flow can pass through the crack surfaces when the crack contact occurs.The detailed information of the contact condition will be described in the next section.

Principles and Finite Element Modelling
3.1.Basic Principles.In this paper, the analysis is the electrothermo-structural coupled-field problem.First, for the electric current field, it obeys the following equations [14]: where E, J, , and  are the electric field (V/m), electric current density (A/m 2 ), electric potential (V), and resistivity (Ω-m), respectively.For the transient thermal analysis, the rules are as follows [14,15]: Modelling and Simulation in Engineering where q  , , , q , ,   , and  are the heat flux (W/m 2 ), thermal conductivity (W/m-∘ C), temperature ( ∘ C), heat generation (W/m 3 ) of Joule heating, mass density (kg/m 3 ), specific heat (J/kg-∘ C), and time (s), respectively.
The finite element equations of the electro-thermostructural coupled-field analysis are as follows [18]: where U, T, V, F, Q, and I are the vector forms of the displacement, temperature, electric potential, force, heat flow rate, and electric current, respectively.The material constant matrices M, C, C t , C tu , K, K t , K ut , and K v are the structural mass, structural damping, thermal specific heat, thermostructural damping, structural stiffness, thermal conductivity, thermo-structural stiffness, and electric conductivity, respectively.The coupled heat flow matrix Q contains the effects of the thermal loading and Joule heating.C tu and K ut are thermo-structural coupled terms.Equation ( 5) is a directly coupled nonlinear equation which is solved using the Newton-Raphson iterative method [18].

Boundary and Initial Conditions.
Referring to Figure 2, the boundary and initial conditions are listed as follows: The convective heat transfer on all structure/air interfaces is considered.The convection condition ℎ = 10 W/m 2 -∘ C and  ∞ = 21 ∘ C is applied.

Finite Element Modelling.
The software ANSYS is adopted to perform the finite element modelling and calculation.In Figure 4, it shows the finite element model of the steel plate with the typical dimensions:  = 240 mm,  =10 mm,  = 1 mm, and  = 5 mm.The mesh is constructed by the 20-node solid element SOLID226 which has the capability of the electro-thermo-structural coupled-field analysis.Due to the symmetry, only half thickness is analyzed.In addition, the quarter-point elements are used to simulate the  −1/2 singularity at the crack tip [9][10][11][12]19].

Contact Condition on Crack Surfaces.
The electrothermo-structural contact condition on both crack surfaces is considered in the numerical analysis.In ANSYS, two contact element types, TARGE170 and CONTA174, are adopted.Aside from the contact stress, the electric and thermal contacts are considered as the following equations [18,20]: where  el and  th are, respectively, the electric conductance and thermal conductance of the contact surfaces.The terms ( 1 −  2 ) and ( 1 −  2 ) are, respectively, the electric potential difference and temperature difference between both contact surfaces.In this study,  th is assumed to be very large so that the thermal contact resistance on crack surfaces can be ignored.However,  el is defined as follows [20]: where   and   are the electric contact resistivity and characteristic length, respectively.For mild steel, the typical values  el = 6.29 × 10 8 1/(Ωm 2 ),   = 6.2586 × 10 −5 Ωm, and   = 2.54 × 10 −5 m (0.001 inch) [13] are used.In addition, the coefficient of friction on crack surfaces is neglected.

Validation of Finite Element Model.
The accuracy of the finite element model must be valid.For the validation, the coupled-field analysis is simplified to the pure electric problem.Furthermore, the SOLID226 elements are degenerated to SOLID231 elements for the ANSYS model in Figure 4.
In this case, the resistivity is 7.2 * 10 −7 Ω-m.Also, the distributed current  0 = 6.25 * 10 6 A/m 2 is used to replace the concentrated current  0 .Using the limited electric potential extrapolation technique (LEPET) [9], the electric current density factor   at the crack tip can be obtained from the finite element results.In this case, the numerical result of   is 8.857 * 10 5 Am −3/2 .According to the analogy and analytical methods [8,9,21], the analytical solution of the edge crack problem under electric load  0 is where  = 1.128 for the case of / = 0.5.Substituting all values to (9),   is 8.836 * 10 5 Am −3/2 .By comparing both values of   , the finite element result has good agreement with the analytical solution with the small numerical error 0.24%.

Electric Concentration and Hot
Spot at Crack Tip.The Joule heating effect makes high temperature field in the steel plate.Due to the local electric concentration at the crack tip, a hot spot exists.In Figure 5, it shows the electric current density and temperature fields around the crack under the following conditions:  = 240 mm,  = 10 mm,  = 1 mm,  = 5 mm, and  0 = 950 A. The electric loading time is   = 0.3333 s.The numerical results obviously prove the phenomena of the electric concentration and hot spot at the crack tip.

Residual Compressive Stress at Crack Tip.
In this section, the geometric and loading conditions of the steel plate are the same as the above section.As shown in Figure 6, the unloading and cooling processes are added in the analysis to obtain the residual stress around the crack tip.The loading process is during  = 0∼0.3333s.Then the electric current is removed and it remains the cooling process till  = 600 s.In Figure 7, the time-history of the crack tip temperature is shown.It can be seen that the crack tip temperature reaches the maximum value (969 ∘ C) at the end of the loading time (  = 0.3333 s).After the unloading process, the temperature decays quickly and approaches the steady state.At  = 600 s, the crack tip temperature decreases to 25 ∘ C.
Figures 8-10 show the time-history of the normal stress   at the point B near the crack tip.The distance from B to the crack tip is 0.2 mm.In Figure 8, it shows the negative values of   during the cooling process till  = 600 s.Also, it approaches the steady state.At  = 600 s, the value of   is −301 MPa.The negative value implies the compressive normal stress.In other words, the residual compressive stress exists near the crack tip after the electric load is removed.
In   Figure 11 shows the contour of the stress   near the crack tip at the end of the cooling process ( = 600 s).There are negative stress values around the crack tip.It means that the residual compressive stresses occur in this area.According to the results of Figure 11, Figure 12 shows the stress distribution in front to the crack tip.The compressive stress field is also investigated near the crack tip.

Effects of Electric Load.
Figure 13 shows the time-history of the stress at point B under  0 = 950 A and  0 = 400 A. Comparing two curves in the figure, it is found that there is no residual stress under  0 = 400 A. It implies that the magnitude of the electric load is an important parameter for producing the residual compressive stress near the crack tip.If the electric load is not sufficient, the temperature and stress will be too small to make the yielding (plastic) strain and residual stress.

Conclusions
From the finite element results, the electric current density concentrates at the crack tip.Due to the Joule heating, it causes a hot spot at the crack tip.The residual compressive stress appears near the crack tip due to the high temperature and plastic deformation.Furthermore, the compressive condition can retard or stop the crack growth.The concept for stopping crack growth is shown in Figure 14.
This paper provides a primary study and conclusion of the residual compressive stress near the crack tip under the electro-thermo-structural coupling state.The compressive condition is practically important to the fracture mechanics problem.
It is important to study the fatigue crack growth due to the Joule heating caused stresses.The extending work will be considered in the future research.

Figure 1 :
Figure2shows the geometric and loading conditions of the steel plate in this study.The plate with an edge crack is subjected to the constant direct current (DC)  0 .The electric loading time is   .The plate is made of the mild steel with the dimension  ×  × .The crack length is .This problem will be simulated by the three-dimensional finite element analysis with the solid model.To consider practical conditions, the temperaturedependent material properties in Table 1[9,13] are adopted in the finite element analysis.Also, the elastoplastic material properties as shown in Figure3are considered.Under the

Figure 4 :
Figure 4: Finite element model.(a) Entire view of plate, (b) local region near crack, and (c) three-dimensional view.
Figures 9 and 10, the detailed stress values during shorter time intervals are shown.The stress causes complicated fluctuation during the electric loading and unloading processes.Then it decays and approaches the steady state with sufficient cooling time.

Figure 13 :
Figure 13: Time-history of stress near crack tip under different electric loads.