^{1}

This paper discusses the Joule heating effect and residual compressive stress near the crack tip under the electro-thermo-structural coupling state. For the crack tip field, the compressive condition is important for retarding or stopping the crack growth.

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 [

Based on the past references, the Joule heating phenomenon near the crack tip is well known. Figure

Electric current concentration and hot spot at crack tip.

Under the Joule heating, the compressive stresses can be produced around the crack tip [

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.

Figure

Geometric and loading conditions.

To consider practical conditions, the temperature-dependent material properties in Table ^{2}-°C.

Temperature-dependent physical properties of mild steel [

Temperature (°C) | Young’s modulus |
Yielding strength |
Coefficient of thermal expansion |
Thermal conductivity |
Specific heat |
Resistivity |
---|---|---|---|---|---|---|

21 | 206.8 | 248 | 10.98 × 10^{−6} |
64.60 | 444 | 0.14224 × 10^{−6} |

93 | 196.5 | 238 | 11.52 × 10^{−6} |
63.15 | 452.38 | 0.18644 × 10^{−6} |

204 | 194.4 | 224 | 12.24 × 10^{−6} |
55.24 | 511.02 | 0.26670 × 10^{−6} |

315.5 | 186 | 200 | 12.96 × 10^{−6} |
49.87 | 561.29 | 0.37592 × 10^{−6} |

426.7 | 169 | 173 | 13.50 × 10^{−6} |
44.79 | 611.55 | 0.49530 × 10^{−6} |

537.8 | 117 | 145 | 14.04 × 10^{−6} |
39.71 | 661.81 | 0.64770 × 10^{−6} |

648.9 | 55 | 76 | 14.58 × 10^{−6} |
34.86 | 762.34 | 0.81788 × 10^{−6} |

760 | 6.9 | 14 | 14.05 × 10^{−6} |
30.46 | 1005.3 | 1.0109 × 10^{−6} |

871 | — | — | 13.05 × 10^{−6} |
28.37 | 1005.3 | 1.1151 × 10^{−6} |

982 | — | — | — | 27.62 | 1005.3 | 1.1582 × 10^{−6} |

1093 | — | — | — | 28.52 | 1189.6 | 1.1786 × 10^{−6} |

1204 | — | — | — | — | 1189.6 | 1.2090 × 10^{−6} |

Poisson’s ratio ^{3}, melting point = 1521°C.

Bilinear elastoplastic stress-strain properties (SIG: stress, units: Pa; EPS: strain, dimensionless;

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.

In this paper, the analysis is the electro-thermo-structural coupled-field problem. First, for the electric current field, it obeys the following equations [^{2}), electric potential (V), and resistivity (Ω-m), respectively.

For the transient thermal analysis, the rules are as follows [^{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 thermoelastic analysis couples the thermal and elastic stress fields as follows [^{2}), strain (dimensionless), body force (N/m^{3}), displacement (m), acceleration (m/s^{2}), Young’s modulus (Pa), Poisson’s ratio (dimensionless), stress invariant (

In this study, the elastoplastic stress-strain behavior is considered. The von Mises yield criterion [

The finite element equations of the electro-thermo-structural coupled-field analysis are as follows [

Referring to Figure ^{2}-°C and

The software ANSYS is adopted to perform the finite element modelling and calculation. In Figure

Finite element model. (a) Entire view of plate, (b) local region near crack, and (c) three-dimensional view.

The electro-thermo-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 [^{2}), ^{−5} Ωm, and ^{-5 }m (0.001 inch) [

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 ^{2} is used to replace the concentrated current

Using the limited electric potential extrapolation technique (LEPET) [^{-3/2}. According to the analogy and analytical methods [^{-3/2}. By comparing both values of

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

(a) Electric current density field (units: A/m^{2}). (b) Temperature field (units: °C).

In this section, the geometric and loading conditions of the steel plate are the same as the above section. As shown in Figure

Electric loading, unloading, and cooling history.

In Figure

Time-history of crack tip temperature (

Figures

Time-history of stress near crack tip (

Time-history of stress near crack tip (

Time-history of stress near crack tip (

In Figures

Figure

Contour of stress

Stress distribution in front to crack tip (

Figure

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

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

Concept for stopping crack growth.

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.

Electric load (direct current, DC) (A)

Time (s)

Electric loading time (s)

Plate dimensions (m)

Crack length (m)

Specific heat (J/kg-°C)

Temperature (°C)

Mass density (kg/m^{3})

Electric field (V/m)

Electric current density (A/m^{2})

Electric potential (V)

Resistivity (Ω-m)

Heat flux (W/m^{2})

Thermal conductivity (W/m-°C)

Heat generation of Joule heating (W/m^{3})

Stress (Pa)

Strain (dimensionless)

Displacement (m)

Young’s modulus (Pa)

Poisson’s ratio (dimensionless)

Coefficient of thermal expansion (1/°C)

Reference temperature (°C)

Yielding strength (Pa)

von Mises equivalent stress (Pa)

Force (N)

Heat flow rate (W)

Electric current (A)

Coefficient of convection (W/m^{2}-°C)

Ambient temperature (°C)

Electric conductance of contact surface (m^{−2}Ω^{−1})

Thermal conductance of contact surface (W/m^{2}-°C)

Contact resistivity of contact surface (Ω-m).

The author declares that there are no conflicts of interest.

The author would like to acknowledge the Ministry of Science and Technology in Taiwan for the financial support under Contract nos. MOST 104-2221-E-131-030 and MOST 105-2221-E-131-014.