PSI - Issue 18

Evgeny Lomakin et al. / Procedia Structural Integrity 18 (2019) 549–555 Evgeny Lomakin and Boris Fedulov / Structural Integrity Procedia 00 (2019) 000–000 ( ) = 1 2 � 1 �� � ( ) + 1 �� � ( ) − 1 �� � ( ) � , ( ) = 1 2 � 1 �� � ( ) + 1 �� � ( ) − 1 �� � ( ) � , ( ) = 1 2 � 1 �� � ( ) + 1 �� � ( ) − 1 �� � ( ) � ,

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( ) = 3 2 � 1 �� � ( ) � , ( ) = 3 2 � 1 �� � ( ) � , ( ) = 3 2 � 1 �� � ( ) � .

(3)

Where coefficients �� can be determined as the ratio of yield stress in this direction to the reference one : �� ( ) = �� � / , �� ( ) = �� � / , � �� — yield limit, obtained in experiment where the only one active �� component is applied. In practice, one can use different expressions for the coefficients �� ( ) , for example, piecewise linear functions. Below (Fig. 4), there is an example of the dependence of such coefficients and the corresponding yield surface. �� ( ) = �� � / , �� ( ) = �� � √3/ , �� ( ) = �� � √3/ , �� ( ) = �� � √3/ , (4)

Fig. 4. Example of Hill yield surface with modified coefficients.

It can be seen, that the yield stress depends not only on the direction of load, but also on the type of loading, that is, for example, under tension or compression. Let us consider the general, without any specific variants of dependencies on the parameter , relations in the case of the associated flow rule. If one goes back to a more general form of the criterion of plasticity (2), then the reduced relation can be written as follows: ���� ( ) �� �� = � , (5)