PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 86–96 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

89

4

where t is time, s and f are material properties. The material is continuously inhomogeneous along the height, width and length of the beam. The distributions of K , m , s and f in the beam cross-section are expressed as

n

n b K K − b m m − d r d

( ) h K K 2 − h m m 2 − p l ( ) q l

b y

2    +

 +  

(

) p

g

g

K K

h z +

= +

,

(5)

g

3

3

r

b y

2    +

 +  

(

) q

g

g

m m

h z +

= +

,

(6)

g

3

3

n

s b s s − n d

( ) h K K 2 − s p l

b y

2    +

s  +  

(

) s p

g

g

s s

h z +

= +

,

(7)

g

3

3

r

f

f

f

f

−

−

b y

2    +

f  +  

(

) f q

d

g

l

g

f

f = +

h z +

,

(8)

( ) h 2

g

3

3

r

q

b

f

f

where

3 b y b −  

,

(9)

2

2

h z h −   3 .

(10)

In formulae (6), (7) and (8), g K , g m , g s and g f are the values of K , m , s and f in the upper left-hand vertex of the beam cross-section. The values of K , m , s and f in the upper right-hand vertex of the beam cross section are denoted by d K , d m , d s and d f , respectively. l K , l m , l s and l f are the values of K , m , s and f in the lower left-hand vertex of the cross-section. The distribution of K , m , s and f along the width of the cross-section is controlled by the material properties, n , r , s n and f r , respectively. The material properties, p , q , s p and f p , control the distribution of K , m , s and f along the height of the cross-section. The distributions of g K , d K , l K , g m , d m , l m , g s , d s , l s , g f , d f and l f in the longitudinal direction of the beam are described by the following power laws:

g l K K − n gC

n

gF

x

K K

= +

g

,

(11)

g

gF

3

−

n

d l K K K K = + n dC dF d

x

dF

,

(12)

d

3

−

n

l l K K K K = + n lC lF l

x

lF

,

(13)

l

3