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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( , ) P   x x on P  , (10) Additional boundary conditions are prescribed in the gradient theory than in the classical counterpart. Besides the normal derivative of temperature, the high order heat flux P is introduced and the heat flux  is modified / p    n , k i ik P n nm  ,   , ( ) ( ) c c i i ik k c n m             x x x π , (11) with k i ik n m    , and i n and i  are the Cartesian components of the unit normal and tangent vector on  , respectively. The term ( ) : ( 0) ( 0) c c c        x x x denotes a jump at possible corners on the oriented boundary contour  . One can see above that electric fields are independent of elastic and thermal fields, while the thermal fields are affected by the electric fields via the Joule heating source, and the elastic fields are influenced by the temperature field. Next, the gradient theory in the uncoupled thermoelasticity is developed. The constitutive equations for Cauchy stresses ij  , and higher-order stresses jkl  , in this theory are given as (Sladek et al. 2017) e ij ijkl kl ijkl kl ij c c         , e jkl jklmni nmi g    , (12) where c and g denote the elastic and the higher-order elastic coefficients, respectively. The elastic part of total strains and their gradients are given as e ij ij ij       , , e e ijk ij k    (13) where kl  are the linear thermal expansion coefficients and ij ijkl kl c    are the stress-temperature modulus . Only one additional material parameter called as the internal length material parameter l is used in the simplified gradient elasticity to express the higher order elastic coefficients (Lazar and Kirchner 2007) 2 jklmni jkmn li g l c   . (14) Applying the variation principle to elastic energy it is possible to derive governing equations (Sladek et al. 2017) for gradient uncoupled thermoelasticity. The complete system of governing equations for considered electro-thermo elastic problem is given by Eqs. (1), (8) and (15) ( ) P P p    , P p    .

, ij j ijk jk       x x , , ( , ) ( , ) 0

(15)

and the admittible boundary conditions (9), (10) are to be supplemented by: Essential b.c.: ( , ) ( ) i i u u   x x on u  , u   ( , ) ( ) i i s s   x x on s  , s  

(16)

0  

( ) ( )    x x on

  ,

   , for

Natural b.c.: ( , ) ( ) i i t t   x x on t  ,

t u    , t

u   

( , ) R   x x on R  , ( ) ( ) H H  x x on H  , ( ) i i R

R s    , R s   

(17)

0   ,

H     , H

    , for

i i H J n  and

: s u

, j i j n u    n , / i

: i k j ijk R n n   ,

where

i