Annals of Warsaw University of Life Sciences - SGGW

�uu� 1
�ij � i, j � j, i , � ij � C ijks�
ks , � ij, j � 0 , (1)
2
Where, i u , � ij and � ij are respectively, the displacement, stress and strain. The
repeated indices imply summation; a comma stands for differentiation and C ijks are the
elastic constants which are assumed to be fully symmetric. A general solution satisfying Eq.
(1) has been presented [5, 6] as
where
� Af �z� Af �z� , � Bf �z��Bf �z� u �
A � �aaa�, �bbb� 1
2
� � � � � � � � ��T
z � f z , f z , f z
1
1
2
3
2
3
� (2a)
B � ,
3
1
2
3
f , z � x � p x , � � 1,
2,
3.
(2b)
�
u and � , are 1
u 1 , u2,
u3
and the stress
functions �� 1 , �2,
�3
�.
The stress function � i is related to the stresses by
38
1
� 2
3� column vectors denoting the displacements � �
� i1
� ��i,
2 , i2
i, 1 � �
� . (2c)
The superscript T denotes the transpose and the overline represents the conjugate of a
complex number. The material eigenvalues � p , and eigenvectors � a , b� are determined by
the following eigenrelations
N � � p�
, (3a)
where
and
� �
�N1
N 2 �
N � �
T � ,
�N3
N1
�
N �T
R
-1 T
1 � ,
�a�
� � � � ,
�b�
-1
N 2 T �
ik i1k1
C Q � , ik i1k
2 C
� N ,
T
2
-1 T
N3 RT R �Q
�
� N
(3b)
R � , ik i2k
2 C T � . (3c)
� � z f , � =1, 2, 3, are three holomorphic functions of complex variables z � , which will be
determined by the boundary conditions set for each particular problem. The surface traction
vector t can be calculated by using Cauchy’s formula [4], i.e., ti � � ijm
j where m j is the unit
normal to the surface boundary. Usually, the stress components along any coordinate axes are
calculated using the transformation law of second order tensors. An alternative approach to
determining stress components of the rotated coordinate axes has been introduced [6]. Let (n,
m) be the unit vector tangent and normal to a surface boundary then we have
where
�
�
mm
nn
T
T
� m ����, n , � mn � n ����, n , � m3
� �� , n � , 3
T
� �n
����, m , � nm � �m
����, m � � mn
T
, n3
� ���
, m �3 T
T
n �����cos� , sin�
, 0�,
������sin� , cos�,
0�
T
3
� , (4a)
m , (4b)