Chapter 13
Initial Concepts of Plasticity Theory
13.1 Second-Rank Tensor in Euclidean Space
Let us select two arbitrary coordinate systems (x 1 , x 2 , x 3 ) and (x
1 , x
2 , x
3 ) in the
Euclidean space. Let us designate unit vectors as ( #» e 1 , #» e 2 , #» e 3 ) and ( #» e
1 , #» e
2 , #» e
3 ).
The vector #» a in the first coordinate system can be written as
#» a =
3
i=1
a i
#» e i = a i
#» e i , (i = 1, 2, 3),
where a i are the components of the vector #» a in the first coordinate system; the
summing symbol in the second form of writing is omitted since it is suggested
on default in vector and tensor designations that summing is done over repeated
indexes.
Similarly to the previous formula, the vector #» a in the second coordinate system
is represented as follows:
#» a = a
i
#» e
i ,
the components (a
i ) of the vector #» a in the second system of coordinates are
expressed through the components (a i ) of this vector in the first system under the
formulas
a
i = β ik a k , β ik = #» e
i
#» e k = cos(
#» e
i , #» e k ).
(13.1)
In the same manner, the vector components in the first coordinate system are
expressed through the components in the second system using the formulas
a i = β ki a
k , β ki = #» e
k
#» e i = cos(
#» e
k , #» e i ).
(13.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_13
165