# Elastic -Plastic Fracture Mechanics/CH_6- Elastic-Plastic...The J-integral is shown to be a...

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Chapter 6 Elastic -Plastic Fracture Mechanics

The high stresses at the crack tip cannot be sustained by, practically, any material. Thus, if the material does not fracture, a plastic zone (or damage zone or process zone) is formed around the crack tip. The damage is specific to the materials but it can be said that in general terms, for a ductile material, the damage is in the form of plastic deformation and for brittle materials in the form of microcracking.

Until now is has been assumed that the size of the plastic zone near the crack tip is relatively small as compared to the specimen dimensions. Thus, the effect of this zone has been neglected and the strain field surrounding the crack tip is dominated by linear elastic fracture mechanics asymptotic field derived in chapter 4. In this chapter the effects of a non-negligible plastic zone is considered in the case of Mode I only. First, in order to determine at what point it is necessary to include the influence of the plastic zone in the stress analysis, two approximations of the size of the plastic zone are presented. Afterwards, two elastic-plastic fracture criteria are discussed, the crack tip opening displacement and the J-integral. The J-integral is shown to be a generalization of the linear elastic release rate to elastic-plastic fracture. The properties of the J and the stress field near the crack tip for elastic-plastic fracture are then derived. Lastly, calculations of J for certain geometries are given as examples.

6.1 One dimensional estimation of Size of Plastic Zone

In order to obtain a first estimate the size of the plastic zone, we apply the von Mises yield criterion,

( ) ( ) ( )2 2 2 2Y

( ) ( ) ( )2 2 2 2Y

1 2 2 3 3 1 2 + + = (6.1)

where (i =1,2,3) are the principle stresses. Thus, we assume that the region in which, i

1 2 2 3 3 1 2 + +

has plastically deformed. Note that the von Mises yield criterion is dependent only upon the principle stresses. This yield criterion is applicable since plastic flow is usually a shear deformation with constant volume, i.e. independent of the principle stress components. Thus, we can define a at the onset of plastic deformation and then assume that all further plastic deformation occurs at this constant .

Y

Y

For this first approximation we consider only the stress field on the line = 0, as shown in Figure 6.1, and define L as the characteristic length of the plastic zone. By comparing the stress field with the von Mises yield criterion, we will determine the length L for which the material in the region x L has exceeded the yield limit. The asymptotic stress field near the crack tip are given by (4.36). On the line = 0 these reduce to,

Irr r 02 r

= = =

K (6.2)

March 2006 6-1

Note that and are the principle stresses rr 1 and 2 for the line = 0. Applying the plane

strain condition (i.e., ( )3 1 2 ; being the material's Poisson ratio) to eq. (6.1), one has, = +

( ) ( ) ( )( ) ( ) (

( )( ) ( )

Y 1 2 2 1 1 2

2 2 2 2 21 2 1 2 1 1

2 2 2 21 2 1 2

2 1 1

4 1 2 1 2 1 1 2 1

2 1 2 2 2 1

= + +

= + + + + + + + + +

= + + +

)

2 2 22

2

2

(6.3)

For the special case of 1 =

( )2 2 2 2 = + +

(such as along the line = 0), eq. (6.3) is simplified to,

( )( )

Y 1

2 2 2Y 1

Y 1

2 4 4 2 2 2 2 2

2 8 8 2

1 2

= +

=

(6.4)

Substituting the principle stress of eq. (6.2) into eq. (6.4), the yield criterion is given by,

( ) IY 1 2 2 L =

K

( )

(6.5)

Thus, the length over which plastic deformation occurs is given by,

22I

Y

1 2 K2

= L (6.6)

Figure 6.1 Schematic of plastic zone ahead of crack tip.

The calculation of L, is only an approximation since the presence of damage will modify the stress field and thus the size of the zone. However, it can serve as a parameter to compare with the overall dimensions of the specimen, and determine the limits of the linear elastic solution. The four principle cases are summarized in Figure 6.2.

March 2006 6-2

Case I, elastic fracture, occurs when L is much smaller than any other of the specimen dimensions. For this case, the plastic zone is negligible and the stress field is dominated by linear elastic fracture mechanics (LEFM). This is the case studied in chapter 4 and 5.

Case II, contained yielding, the effects of the plastic zone are no longer negligible. For this case, the strain field is dominated by elastic-plastic fracture mechanics, the subject of this chapter. We will see later in this chapter that the fracture can be well described by the Jc (non-linear critical energy release rate). The final two cases will not be considered in these notes, however they are included here to show other possibilities.

Case III, full yielding, involves large deformations due to the large plastic zone. This case is less useful since typically the structure is no longer capable of supporting the applied loads.

Case IV, diffuse dissipation, is different from the other three, as there are no distinct elastic and plastic zones. Instead, the fracture is dominated by non-linear elastic, time dependent phenomena, such as creep or viscoelasticity.

Figure 6.2 Modes of fracture in specimens with different extend of plastic deformation.

March 2006 6-3

6.1.2 Two dimensional approximation (2D)

While the length L gives us an indication of the size of the plastic zone relative to the specimen dimensions, it is also useful to know the actual shape of the plastic zone around the crack tip. Using the same approach as in the previous section, it is not difficult to obtain its shape.

In cylindrical coordinates the principle stresses 1 and 2 for an arbitrary stress field are given by,

22rr rr

1 2 r2 2,

+ + = +

(6.7)

Near the crack tip, where the stress field is dominated by the terms of eq. (4.36) we obtain after some algebraic manipulation,

I1,2 cos 1 sin2 22 r

= K

(

(6.8)

and )3 1 2 + = in the case of plane strain.

Note that for the case of = 0, the last equation reduces to (6.2). Substituting eq. (6.8) into (6.3) and simplifying leads to the following yield condition,

( )2K1 2 2 2 2I

Y cos 4 1- -3cos2 r 2 2 = +

Thus, solving for the radius of the plastic zone at which this condition is satisfied rp(), gives,

( ) ( )2

2

2 2 2I

pY

K1r cos 4 1 3cos2 2

= + (6.9)

The shape of the plastic zone is plotted in Figure 6.3(a) for = 1/3 and = 1/2. Also plotted is the limiting case of plane stress (i.e.

= 0 ). As the state of stress changes through the thickness

of a thick specimen (i.e. plane stress plane strain), the shape of the plastic zone also changes. Figure 6.3(b) shows a typical shape of the plastic zone across a specimen.

Note that the shapes of the plastic zone shown above do not take into account the effects of the specimen boundaries. Thus, they are derived for a crack in an infinite plate. For an actual specimen, one must often consider the finite width in order to determine the region of plastic deformation. Figure 6.4 shows examples of the shape of the plastic zone for three standard specimens.

March 2006 6-4

Figure 6.3 Plastic zone shapes; (a) as a function of ,. (b) through the specimen thickness.

Figure 6.4 Effect of finite width of specimen on shape of plastic zone; (a) double edge notch in tension, (b) center cracked specimen in tension, (c) edge crack in bending.

6.1.1 Irwins approximation of the plastic zone

It was mentioned earlier that due to the materials yielding at the crack tip the stress distribution is not given by the asymptotic filed (4.36) and shown in Figure 6.3 by the curve (1). To account for the effects of plastic deformation on the redistribution of the stress field and obtain a simple approximation, perfect plasticity is assumed thus, the stress ahead of the crack tip equals Y up to a distance . After that distance, the distribution is obtained by a translation of the asymptotic field shown by the curve (2).

2r

March 2006 6-5

The two distributions, before and after yielding, should result in equal forces since equilibrium should be assured. With references to Figure 6.5, force equilibrium results in,

1r

Y 2K

r

IY

0

dx r2 x

= (6.10)

where the distance , is the intersection of the filed (1) and the horizontal line at given by, 1 Y

2

IK

2r

IY 1

Y1

K 1 r 22 r

= =

(6.11)

Y

Figure 6.5 A crack in an infinite plate with plastic zones ahead of the crack tip.

Integrating (6.10) and using (6.11) one obtains, r1 = . Thus, the plastic zone extends over a distance equal to given by, 1 2 1r r 2r+ =

2

0

I1

Y

K1 2r

= (6.12a)

The simple analysis shown above is for the case of plane stress i.e., 3 =0

which is realistic for thin plates. In plane strain, yielding is confined due to the effects of 3 > and a smaller plastic zone ahead of the crack tip is developed. In this case, Irwin proposed the following expr

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