Page 291 - Biaxial Multiaxial Fatigue and Fracture

An Analysis of Elasto-Plastic Strains and Siresses in Notched Bodies Subjected to Cyclic ... 275

b) Point B:! is connected to the center, 02, of the surface f2.

c) A line is extended through the center of the smaller active surface , 01, parallel to the line
02B2 to find point BI on surface fl.

d) The conjugate points B1 and I32 are connected by the line B&.
e) Surface fl is translated from point 0, to point 01' such that vector 0101' is parallel to line
BlB2. The translation is complete when the end of the vector defined by the stress increment,
Ao, lies on the translated surface fl'.

02a

0 (53a
>

Fig. 6. Geometrical illustration of the translation rule in the Garud incremental plasticity model.
The mathematics reflecting these operations can be found in the original paper of Mroz [9] or
Garud [IO] or in any recent textbook on the theory of plasticity. The Mroz and Garud models are
relatively simple but they are not very efficient numerically, especially in the case of long load
histories with a large number of small increments. If the computation time is of some concern the
model based on infinite number of plasticity surfaces proposed by Chu [ll] can be used in
lengthy fatigue analyses.
The cyclic plasticity models enable the relationship, AE,~P-Ac&<, to be established providing the
actual plastic modulus for given stresfload increment, AG,. In other words the plasticity model
determines which piece of the stress-strain curve (Fig. 5) is to be utilized during given stresfload
increment. Two or more tangent ellipses translate together as rigid bodies and the largest moving
ellipse indicates which linear piece of the constitutive relationship should be used for a given