Forgeabilité des aciers inoxydables austéno-ferritiques

tel-00672279, version 1 - 21 Feb 2012
Appendices 183
Appendix B: Strain computations
The calculations presented in this section can also be found in Refs. [98] and [128]. The objective is to
understand how the strains are computed in the software CorrelManuV.
B.1 Definition of the local strain
Let X be the position in a reference orthonormal coordinate system of a material point in the initial
configuration of a three dimensional continuous medium and x its position in the same reference
system in the current configuration. The gradient of the mechanical transformation at X is then:
where
F
�x
� X
� , eq B.1
X
X � Y et x� y .
Z
x
z
The Green-Lagrange strain tensor � GL is defined as:
1
( F F I)
,
2
t
� GL � � � �
eq B.2
where I is the unit tensor and t denotes the transposition. � GL depends non-linearly on F , but for
small strains it may be approximated by its linearized counterpart:
t 1
� lin � � ( F � F)
� I.
eq B.3
2
B.2 Average strain over a domain
Let� be a given regular domain of the initial configuration, which might be multiply connected. Let
�� be its regular edge, with outer normal n , andV� its volume. The average transformation gradient
over� is given by:
F
1
� FdV
� � V �
� �
�
. eq B.4