Issue 39

M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17 170 l l l r l rz r V e S S r u S S dV F l ns r r z ¬ ¨ ¬¬¬¬¬¬¬¬¬¬ 1, 2, 3, , T V U V W § · w w } ¨ ¸ w w © ¹ ³ (3-8) l l l rz z z V e S S r w S dV F l ns r z ¬ ¨ ¬¬¬¬¬¬¬¬¬¬¬ 1, 2, 3, , U W V § · w w } ¨ ¸ w w © ¹ ³ (3-9) where ns is the number of shape functions of the element e and l S is the component of the vector S . ^ ` m m m m S N N N N 1 2 3 4 1 2 3 4 1 2 3 4 , , , , , , , ¬, , , , ‡ ‡ ‡ ‡ < < < < (3-10) For axisymmetric problems in cylindrical coordinates relations between the stresses and displacements can be expressed in the form below [21]. r z rz u u w u r r z r u u w u r r z r u u w w r r z z u w z r 2 ¬ ¬2 ¬ 2 T V O P V O P V O P W P § · w w w ¨ ¸ w w w © ¹ § · w w ¨ ¸ w w © ¹ § · w w w ¨ ¸ w w w © ¹ § · w w ¨ ¸ w w © ¹ (3-11) By substituting Eqs. (3.11) into Eqs. (3.8) and (3.9) we have l l l r V e V e l S S u u w u u u w u r r r z r r r r z r r u S dV r dV F S u w z z r l ns ¬ ¬ ¨ 2 2 1, 2, 3,..., O P O P U P § · § · § · § · § · w w w w w w ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ w w w w w w ¨ ¸ © ¹ © ¹ © ¹ © ¹ ¨ ¸ § · § · w w w ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ w w w © ¹ © ¹ © ¹ ³ ³ (3-12) l l l z V e V e S S u w u u w w r w S dV r dV F l ns r z r z r r z z ¬ ¬ ¨ ¬ ¬ ¬ 2 ¬ ¬¬¬¬¬¬¬ 1, 2, 3, , U P O P § § · § · § · § · · w w w w w w w } ¨ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¸ ¨ ¸ ¨ ¸ ¨ w w w w w w w © ¹ © ¹ ¹ © ¹ © ¹ © ³ ³ (3-13) By substituting displacements (Eqs. (3-6) and (3-7)) into Eqs. (3-12) and (3-13), and some manipulations, equations are obtained which we can assemble them to a matrix form as below. > @ ^ ` > @ ^ ` ^ ` M K F ' ' (3-14) In this equation > @ M and > @ K are the mass and stiffness matrices, respectively. Also ^ ` ' and ^ ` F are the nodal displacements and force vectors, respectively. Generally, for the fictional element e which is enriched with both Heaviside and crack tip enrichment functions, these matrices and vectors can be written as follows: