Lecture 5 Limit of LEFM

7/28/2019 Lecture 5 Limit of LEFM

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CEE 770 Meeting 5

Objectives of This Meeting

Learn the limit of applicability of LEFM:

K-dominance

The process zone, and estimating its size

Small-scale yielding, SSY

Dimensional restrictions on process zone size

The concept of Brittleness Number

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LEFM predicts infinite stress at the crack front.

This cannot be true for a real material.

There must be a region near the crack front where nonlinear materialbehavior prevails. This behavior might be governed by elasto-plastic, or

microcracking, or any other dissipative mechanisms.This region is called the fracture process zone.

Process zone formation in Al-Cu-Mg alloy, From Broek,Elementary Engineering Fracture Mechanics, 3d Edit, 1982.

Example of an elasto-plastic process zone

Physical Observation

See Computational Simulation

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Lets Estimate rp

First approximation, call it r*p:

Ignore triaxiality, assume elastic-perfectly plastic behavior,fy = uniaxial yield stress, assume no stress redistribution, then

*2 p

Iyy

r

Kf

==

x

KIy

2 =

2

2

*

2y

f

K

r

I

p

=

yield

x

y

x

KIy

2 =

rp

And an upper bound on r*p

2

2

*

2y

f

K

rIc

p=

(62)Is this an upper or lowerbound to actual value?

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Better Estimates forrp

Include 1D stress redistribution:

**2

2

yf

Kr Icp

=

strainplanein32

2

y

Icpf

Kr

=

stressplanein2

2

y

Icp

f

Kr

=

Include stress redistribution and allow for multiaxiality with the von Misesyield criterion:

Why difference?

Next, we want rp to be small. Compared to what? How small?

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Small Compared to What? How Small?

ForK, a linear-elastic concept, to dominate the solution, the

process zone must be small compared to all significant dimensionsof the structure containing the crack.

By convention and by reasonableness, small is accepted to meanless than about 1/25th.

For example, we would want rp to be less than 1/25th of the crack

length, a

2

2

2

5.23

25

y

Ic

y

Ic

f

Ka

f

Ka or

(63)

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Necessary Geometrical Conditions for Validity of LEFM in Materials

in Which Elasto-Plastic Process Zone Behavior Dominates

W

2a

B

Necessary Conditions:

2

,,

Y

I

fKaWBa c2.5>-

100

where

KIc= Plane Strain Fracture Toughness

fy= Uniaxial yield strength

W-a = ligament

If all these conditions are met, thenSSY conditions are said to prevail

and LEFM applies.

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Brittleness Numbers

An important concept that emerges from this simple calculation is that of theBrittleness Number.

A material in a state with a high brittleness number has a small process zone, eg.

statematerialplastic-elastofor2

y

Icp

f

Kr

so the ratio of yield stress to plane strain fracture toughness, fy/KIc, can beinterpreted as a brittleness number for materials in which the process zone ispredominately dissipating energy in plastic strain energy.

Similarly, brittleness numbers for materials which dissipate fracture energythrough other mechanisms have been defined, eg.

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Lecture 5 Limit of LEFM