Fracture Mechanics - UTEPme.utep.edu/cmstewart/documents/ME5390/Lecture 8 - LEFM... · 2020. 11. 4. · plane strain fracture toughness and critical stress intensity factors, K Ic

$Page 1: Fracture Mechanics - UTEPme.utep.edu/cmstewart/documents/ME5390/Lecture 8 - LEFM... · 2020. 11. 4. · plane strain fracture toughness and critical stress intensity factors, K Ic$
LEFM Testing

Fracture Mechanics

LEFM TestingPresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Fall 2020

Outline

• Introduction

• Plane Strain Fracture Toughness (KIc) Testing

• Specimen Size Requirement

• KIc Testing Procedure

• Analysis of Load-Displacement Records

• Plane Stress Fracture toughness (Kc) Testing

• Practical Uses of Data

Introduction

Introduction

• In the previous lectures, we reviewed the analytical relationships for the determination of stress intensity factors in elastic specimens.

• The stress intensity factors, (KI,KII,KIII) are functions of load, crack size, and specimen geometry.

• In this lecture, we focus on the experimental determination of the plane strain fracture toughness and critical stress intensity factors, KIc and Kc respectively.

Introduction

• Ideally a critical stress intensity factor, Kc, can be used to predict the fracture behaviour in an actual structure.

• Beyond a certain thickness, when the material is predominantly in plane strain and under maximum constraint, the value of Kc tends to a constant lower limit, KIc, the plane strain fracture toughness.

• KIc may be considered a material property, but does depend on the test temperature and loading rate.

Introduction

• After considerable study and experimental verification the American Society for Testing and Materials (ASTM) published a standard method for KIc testing (last revision is ASTM E 399-90).

• Of more general interest is the establishment of a test methodology for Kc determination. • Many engineering applications are subjected to transitional or fully plane

stress conditions.

• Only one method, the Feddersen method, has the versatility required for structural design.

Plane Strain Fracture Toughness (KIc) Testing

Plane Strain Fracture Toughness (KIc) Testing• During the period in which fracture toughness testing developed (late

1950s and the 1960s) the most suitable analyses for characterizing the resistance to unstable crack growth were those of LEFM.

• Under the supervision of the ASTM E-24 Fracture Committee numerous specimen designs and test methods for KIc determination were considered.

• ASTM E399 - Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness of Metallic Materials


• A variety of thick samples are accepted as test specimens for Kic-Testing: (a) compact specimen, (b) disk-shaped compact specimen, (c) single-edge-notched bend SE(B) specimen, (d) Center Crack Tension (CCT) specimen, and (e) arc-shaped specimen.


• Two key standard specimens, are the

• single edge notched bend (SENB or SE(B) in the last revision)

• compact tension (CT or C(T) in the last revision)


• ASTM Standard Notched Bending Specimen (SENB)


• ASTM Standard Notched Bending Specimen (SENB)


• ASTM standard compact tension specimen (CT)


• ASTM standard compact tension specimen (CT)


• Experiments have shown that it is impractical to obtain a reproducible sharp, narrow machined notch that will simulate a natural crack well enough.

• Therefore the specimens must be fatigue pre-cracked.

Fatigue Pre-cracking


• To ensure that cracking occurs correctly the specimens contain starter notches.

• Several possibilities are listed in the ASTM standard, but the most frequently used is a chevron notch, owing to its good reproducibility of symmetrical in-plane fatigue crack fronts.

Chevron notch crack starter.

Specimen Size Requirement


• The specimen size requirements for KIc ensure• the specimen dimensions are large enough compared with the plastic zone

size, ry.

• An overall elastic behavior of the specimen, so the stress intensity approach is appropriate.

• a predominantly plane strain state will be present at the crack tip.


• The relevant dimensions are:

1. The crack length, a.

2. The specimen thickness, B.

3. The remaining uncracked ligament length, W - a.


• After considerable experimental work the following minimum specimen size requirements were established in ASTM E399:

2

, 2.5 Ic

ys

Ka B

0.45 0.55a

W

a, B, (W-a) are nearly equal

2

1

2

Icy

ys

Kr

=

Plane Stress

Such that,

2.5 2 yB r


• It is important to note that the specification of a, B, and W (and all the other specimen dimensions) requires that the KIc value must already be known or at least estimated.

There are three general ways of sizing test specimens before Kic

is actually obtained:1. Overestimate KIc based on experience with similar

materials and empirical correlation with other types of notch toughness test, for example the Charpy impact test.

2. Use specimens that have as large a thickness as possible.3. For high strength materials the ratio of (σys/E) can be used

according to the following table, which was drawn up by the ASTM.

KIc Testing Procedure


1. Determine the critical dimensions of the specimen (a, B, W).2. Select a specimen type (notch bend or compact tension) and prepare

shop drawings, e.g. specifying a chevron notch crack starter.3. Specimen manufacture.4. Fatigue precracking.5. Obtain test fixtures and clip gauge for crack opening displacement

measurement.6. Testing.7. Analysis of load-displacement records.8. Calculation of conditional KIc (KQ).9. Final check for KIc validity.


1. Determine the critical dimensions of the specimen (a, B, W)

2. Select a specimen type and prepare shop drawings, e.g. specifying a chevron notch crack starter

The chevron notch forces fatigue cracking to initiate at the center of the specimen thickness and thereby increases the probability of a symmetric crack front.


3. Fabricate the SpecimenMETCUT, Laboratory Devices Company, Laboratory Testing Inc., Tensile Testing Metallurgical Laboratory, Metal Samples Company, UTEP Machine Shop

rolled plate and forgings disks and hollow cylinders


4. Fatigue Pre-crack the specimen

➢ The purpose of chevron notching and fatigue

pre-cracking is to generate an ideal plane

crack with essentially zero tip radius to agree

with the assumptions made in stress intensity

analyses.

➢ There are several requirements pertaining to

fatigue pre-cracking. The most important is

that the maximum stress intensity Kmax during

the final stage of fatigue cycling shall not

exceed 60% of KIc result.

maxK

minK

Ic QK K=

max 0.6 IcK K


5. Obtain test fixtures and displacement measurement devices

• Recommended test fixtures for notch bend and compact tension specimen testing are described in the ASTM standard.

• These fixtures were developed to minimize friction and have been used successfully by many laboratories.

• Other fixtures may be used provided good alignment is maintained and frictional errors are minimized.



Clip gauge / Extensometer

• An essential part of a KIc test is accurate measurement of the crack mouth opening displacement (CMOD) as a function of applied load.

• The displacement is measured with a so-called clip gauge /extensometer which is seated on integral or attachable knife edges on the specimen.



Linear Variable Differential Transformer (LVDT)

Potential Drop Method for monitoring crack growth DC or AC

Other methods to measure displacement and/or crack length include

• Change in Compliance (measured during periodic unloading, discussed extensively in the text book)

• Potential Drop Method• Optical Methods (digital image

correlation)• etc.



Simultaneous measurement of crack-mouth-opening displacement (CMOD) and load-line displacement on an SE(B) specimen. The CMOD is inferred from a clip gage attached to knife edges, while the load-line displacement can be determined from a comparison bar arrangement; the bar and outer coil of the LVDT remain fixed, while the inner rod moves with the specimen.



Digital Image Correlation



Electromagnetic Linear MotorElectro-Mechanical Screw FrameServo-Hydraulic

Moderate Frequency

High Force

High Power

Low Frequency

Moderate Force

Moderate Power

High Frequency

Low Force

Low Power


6. Conduct the Test.

7. Analysis the Load-Displacement Curves.

➢ Non-linearity is due to Plastic Deformation (Type I & II)

➢ Sudden crack extension and arrest called “Pop-in”

➢ P5 is measured via 5% secant offset, also denoted as PS

➢ PQ is the maximum load occurring before or simultaneously with P5

➢ The Final Requirement is that max 1.10 QP P

Type I Non-linear

Type II“Pop-In”

Type IIIPerfectly Linear-Elastic Brittle

Three Types of Behavior


8. Calculate Conditional Value of KIc (KQ)

9. Check of Validity of KIc = KQ

For the CT specimen,

Q

Q

P aK f

WB W

=

max 1.10 QP P 0.45 0.55a

W

2

, 2.5 Ic

ys

KB a

Note: Each specimen type has a different KQ equation

Then,

If,

Ic QK K=


• Some structural materials are not well-suited to KIc testing where the test requirements are too stringent based on the material properties.

• Other test standards have more relaxed requirements• ASTM 1820

• BS 7448

• E1290 –CTOD

• E561 – R Curve Testing

• With progress in fracture mechanics analysis techniques, Standards are revised, reapproved, replaced, or discontinued.

ASTM E399 -12E3

Organization Standard No. Publication Year Edition

Example

• Consider a structural steel with σYS = 350 MPa (51 ksi). Estimate the specimen dimensions required for a valid KIc test. Assume that this material is on the upper shelf of toughness, where typical KIc values for the initiation of microvoid coalescence in these materials are around 200 MPa (m)^1/2

2

, 2.5 Ic

ys

KB a

0.45 0.55a

W

2 2

200, 2.5 2.5 0.816

350

Ic

ys

K MPa mB a m

MPa

= =

0.5 1.63 !!!!!!!a

W mW

=

Assuming,

Specimen will be very large and expensive to fabricate. It will be hard to find a universal test machine capable of fracturing this specimen

Analysis of Load-Displacement Records

Analysis of Load-Displacement Records

• Plots of load versus displacement may have different shapes.

• Initially the displacement increases linearly with load, P.

• In many cases there is either a gradually increasing nonlinearity, type I, or sudden crack extension and arrest (called ‘pop-in’) followed by nonlinearity, type II. Nonlinearity is caused by plastic deformation and stable crackgrowth before fast fracture.

• If a material behaves almost perfectly elastically (as is rarely the case) a diagram like that type III is obtained.

Principal types of load-displacement plots obtained during KIc testing.

Plane Stress Fracture Toughness (Kc) Testing


• There is no standard for Plane Stress Fracture Toughness

• Only Rules of Thumb!

• The Federson Approached,

0

0

stable

unstable

apparrent stress intensity

i i

c c c

e c

aK a f

W

aK a f

W

aK a f

W

=

=

=


• where Ki is the critical stress intensity for the onset of (stable) crack growth and Kc is the critical stress intensity for fracture.

• Ke is an apparent stress intensity, since it relates the initial crack size to the fracture stress: it has an engineering significance because irrespective of whether slow stable crack growth subsequently occurs the value of Ke defines the residual strength of a plate containing a crack of a given initial size.


• For Small Cracks

• For Large Cracks

• For a Valid Test:

• Assuming Yielding will not occur and minimal stable crack growth

c →

0c →

2, 2 ;

3 3

ys Wa

Practical Uses of Data


• KIc data can be useful in two general ways.

• First, they may be used directly for choosing between materials for a particular application, especially high strength aerospace materials.

•

• More generally, since it is desirable (if possible) to avoid plane strain/low energy fracture, KIc values may be used as a basis for a screening criterion to ensure plane stress/high energy fracture.


• Several criteria have been proposed. One of the simplest is the through-thickness yielding criterion

• which gives the desired increase in toughness with increasing yield strength and sheet thickness in order to obtain plane stress fracture.

• This criterion is useful when σys and KIc of a material are known. In this case the thickness B has to be less than or equal to

• in order to avoid low energy fracture.


• Material Selection Chart

• The lines give minimum values of Kic/σysnecessary for through-thickness yielding to occur in a sheet of given thickness, e.g. for a sheet 3 mm thick the minimum value of KIc/σys is 0.055.

Summary➢ At this point, we come to the end of our discussion on LEFM.

➢ It is important to note, LEFM is unable to account properly for plasticity.

➢ Many engineering materials combine high toughness with low yield strength, so that the required thickness for a valid KIc test may reach the order of magnitude of a metre!

➢ Obviously, KIc tests on such materials are neither practical nor useful, if only because the materials would never be used in such thicknesses.

➢ Also, excessive plasticity in these materials will rule out Kc testing of plates with thicknesses representative for actual structures.

➢ Another method called Elastic-Plastic Fracture Mechanics (EPFM) must be introduced

Homework

1. Nothing is Week.

2. Focus on Project 3a.

References

• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd Edition, Spon Press

• Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press.

• Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall

• Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture Mechanics of Engineering Materials, 5th Edition, Wiley.

• https://www.fracturemechanics.org/

https://www.fracturemechanics.org/

CONTACT INFORMATION

Calvin M. Stewart

Associate Professor

Department of Mechanical Engineering

The University of Texas at El Paso

500 W. University Ave, Suite A126, El Paso, TX 79968-0521

Ph: 915-747-6179

[email protected]

me.utep.edu/cmstewart/

mailto:[email protected]

http://me.utep.edu/cmstewart/

Documents

Fracture Mechanics - UTEPme.utep.edu/cmstewart/documents/ME5390/Lecture 8 - LEFM... · 2020. 11. 4. · plane strain fracture toughness and critical stress intensity factors, K Ic