PSI - Issue 52

Jan Sladek et al. / Procedia Structural Integrity 52 (2024) 133–142 Author name / Structural Integrity Procedia 00 (2019) 000–000

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2. Gradient theories for the Joule heating The Joule heating is a multi-physical problem, where a high temperature is induced by the electric current density focused at the crack tip vicinity. Electric changes are much faster than the response of elastic fields and the response of thermal fields in real materials. Since, the thermal response is much slower than the response of elastic fields, the quasi-static approximation is justified for both the electric and elastic fields. The steady-state governing equation for the electric current is given as , 0 i i J  . (1) The Latin subscripts denote the Cartesian components and take values 1,2, in this paper. Furthermore, a subscript following the comma denotes the partial derivative with respect to the corresponding spatial coordinate. In electric conductive materials the constitutive equation for the electric current i J is given by the Ohm`s law where j E is the electric intensity vector, ij s is the tensor of electrical conductivity and  is the scalar electric potential. The Joule heating generated in an electrical conductor is given by (3) The Joule heating is mainly generated at the crack tip vicinity due to the singularity of the electric current at the crack tip in the linear theory. Then, large gradients of temperature and strains should be considered in real continuum models. Gradient theories of elasticity and heat transfer appear to be most suited for a study of this problem. The uncoupled thermoelasticity is considered here, where mechanical fields are influenced by thermal ones, however the thermal fields are not influenced by mechanical ones. Next, the heat transfer problem is solved in the first step. The heat flux vector i  is proportional to temperature gradients where ij  is the tensor of thermal conductivity coefficients. If higher-order derivatives of temperature are included into the rate of thermal potential in the higher-grade theory (Sladek et al. 2021) one can write (5) where  and c are the mass density, specific heat, and ik m is the higher-grade flux. This higher-order flux is canonically conjugated field with , ik  in linear theory (6) and ikjl  is the tensor of higher-order thermal conductivity coefficients. To simplify theory these material coefficients are considered to be proportional to the thermal conductivity with one microstructural length scale parameter T l (micro-thermal length scale parameter), 2 ikjl T lk ij l     , with lk  being the Kronecker delta. Then, one can write 2 , ik T ij jk m l    . (7) Applying the variation principle one can derive the governing equation for the heat conduction (Sladek et al. 2021) , , j ij i j Q E J s EE s      . i i ij i , ij j     , i (4) , , ik ik c m          i i , , ik ikjl    , jl m i ij j J s E  , , j j E   (2)



, i i   x ( , )

( , ) x

( , )      x c

( , )  x

, ik ik m

Q





.

(8)

From the variation principle one gets also the form of boundary conditions: Essential b.c.: ( , ) ( )     x x on   ,    ( , ) ( ) p p   x x on p  , p  

(9)

( , ) ( )    x x on

  ,

     ,

    

Natural b.c.: