IFMASS10

The conditions defining growth of plastic strain through the yielding surface, with the

parameters

and

defining its size and position in stress space are shown in Fig. 3, so

that variation of them allows different kinds of material behaviour to be described.

Accordingly, any change in inelastic strains depends not only on stress state but also

on yield surface, which defines the state of the material and varies as the inelastic strain is

generated. This point out to the fact that the development of inelastic strains is always

accompanied by a change in the microstructure of the material, causing the change in the

material response, including the fracture mechanics parameters.

There are many different criteria which define yield surface change, as the main

conditions for the origin of inelastic strains, but for the purposes of the fracture mechanics

the criteria von Mises and Tresca are generally used.

A Tresca or maximum shear stress criterion is based on the observation that the

materials fracture in shear planes, what is suitable for the application to metals.

{

}

( )

1 2 2 3 3 2

( , ) max

YS p

σ ε

−

− −

(5)

Von Mises criteria, also named maximum distortion energy criteria, takes in conside-

ration that the hydrostatic stresses in material doesn’t affect yielding and doesn’t partici-

pate in fracture, what is an important characteristics of materials, particularly metallic.

(

)

(

)

(

) (

)

( )

1 2

2 3

3 1

YS p

σ ε

⎡

⎤

− + − + −

−

⎢

⎥

⎣

⎦

(6)

In both criteria for

(

)

σ ε

the material deforms elastically, for

(

)

σ ε

plastically.The yield stress

may increase during plastic straining, as a function of a

measure of total plastic strain

. Experiments have shown that if a solid is plastically

deformed, unloaded, and again re-loaded to produce next plastic strains, its resistance to

plastic flow will be increased by strain hardening. So, if the inelastic deformation leads to

the material hardening, than the yield surface increase or move in the stress space.

Obviously, it is possible to describe strain hardening in an appropriate way by

definition how, depending on plastic strain, the yield surface changes its form and

position. Previously described way of yield surface size increase is so called isotropic

hardening, which however, is not adequate for the application in case of cyclic loading,

because it does not account for the Bauschinger effect; that is the hardening by the

loading in one direction has the consequence of softening if loaded in opposite direction.

In this case the application of so-called kinematic hardening is more appropriate, by

which the yield surface moves in load direction without size change.

However, material behaviour in the case of cyclic loading is much more complex and

under elevated temperature further complicated by time dependent inelastic strain (creep).

Let us look back on the flow rule. If the stress state is on the yield surface, and stress

slightly increases, this rule defines direction and size for the development of plastic strain

increment. As already stated this increment only appears for loading increase. Since the

plastic strain increment is dependent on total stress and not on stress increment. this

theory is known as incremental theory of plasticity or flow theory. During loading the

stress follows non-linear line (Fig. 4), and in case of unloading the strains are elastic.

For the purposes of fracture mechanics the theory of non-linear elastic deformation

(deformation theory), by which loading and unloading follows the same curve, is