Issue 39

M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17

where in these equations r , U , u , f , t , V and W are radius, density, displacement, body force, time, normal and shear stress, respectively. Subscripts r and z refer to radial and longitudinal directions of cylinders, respectively. Also, in axisymmetric problems we have [17]

r W

z W

u 0¬¬¬¬¬,¬¬¬¬

0¬¬¬¬¬¬,¬¬¬¬¬¬

0

(3-3)

T

T

T

To solve equations of motion, discontinuous-Galerkin-based extended finite element method is employed. The extended finite element model of the problem is obtained by discretizing the solution domain into a number of arbitrary elements. The formulation of the XFEM for displacement components can be written as [18]

¬

¬

¦

n N cr ¦ ‹

n n n n N r z H r z H r z b t ¬ , ¬ , ¬ , ª º

u r z t , ,

N r z a t ¬ ,

(3-4)

¬

¼

n

n

all nodes ¬

¦ ¦ ‹

N r z F r ¬ , ¬ , ˆ ª

ˆ ,

M

M

º ¼

F r ¬

c

t

¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

¬

n

m

m n n nm

m n N tip

N is the set of nodes inside a

where cr N is the set of nodes that the discontinuity has in its influence domain, while tip predefined area around the crack tip (Fig. 1). H r z ¬ , is Heaviside eichment function and m enrichment functions [19]. r ˆ and M are the usual crack-tip polar coordinates. Also, n a ¬ , n b ¬ and nm

F represents crack tip c ¬ are vectors of the

nodal unknowns. ^ n a t ¬

` T

^

` T

^

` T

u

w

u

w

u

w

¬

¬

a t a t ,

n b t

b t b t ,

c

t

c

nm nm t c ,

t

,

,

(3-5)

n

n

n

n

nm

Figure 1 : Selection of enriched nodes for crack. Circled nodes are enriched by the discontinuity function whereas the squared nodes are enriched by the crack tip enrichment functions.

Displacement components for the element (e) with n nodal points are approximated by compact forms as follows

u

u

u

(3-6)

‡

<

u N r z a t ,

r z b t ,

r z c ,

t

h

h

h

h

hm

hm

w

w

w

(3-7)

‡

<

w N r z a t ,

r z b t ,

r z c ,

t

h

h

h

h

hm

hm

h = 1, 2, …, ne and m = 1, 2, 3, 4, where ne is the number of nodes in an element. Also ‡ and < exhibit the enriched parts of displacements related to crack path and crack tip enrichment, respectively [20]. Applying the weighted residual integral to the equations of motion with respect to the weighting functions l S r z , , the formal Galerkin approximations reduce to

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