Impact_test and Fracture Test

TAM 224/CEE 210 9–1

9. Impact Testing and Fracture Toughness Overview The objective here is twofold: (a) to learn how to conduct and interpret instrumented impact tests for characterizing the material behavior of both polymeric and metallic materials under impact conditions at various temperatures, and (b) to learn how to conduct and interpret a plane-strain fracture-toughness test on a high-strength metal.

These two important topics in modern engineering design are fairly distinct. Descriptions of the two topics are divided below into subsections 9A—Impact Testing and 9B—Fracture Toughness.

Impact Testing 9A.1. Objective The purpose of impact testing is to determine the nature and extent of material deformation under rapid loading conditions, as well as the maximum forces and energy absorbed during impact. In this lab, the effect of temperature on material response is also studied.

9A.2. Background Impact testing is performed for two reasons. First, impact testing is used to compare the dynamic response of several materials or batches. The results of impact tests are used to compare products manu-factured by different processing routes or as a quality-control parameter for a given process. Second, impact testing is performed to simulate the end-use conditions of a material or product so it can be manufactured to survive impact associated with its intended use.

Until recently, impact testing has been performed without the aid of sophisticated instrumentation. In the standard (uninstrumented) Charpy or Izod test, for example, the only variable measured is the total energy required to break a notched sample in bending. A heavy pendulum of a certain mass m is raised through a height h0 and released, striking and breaking the specimen at the bottom of its swing. The pendulum then attains a final height hf after impact. The absorbed energy Γtot of the specimen, which is assumed to be equal to the loss of potential energy mg h hf( )0 − of the pendulum, is then read directly from a calibrated scale (from zero to 358 J (264 ft·lb)) on the machine. No information about magnitude of

load or time of contact is obtained in an uninstru-mented test.

Instrumented impact testing refers to a newer method now commonly used to determine the load-vs.-deformation response of materials under multi-axial high-speed deformation. A strain-gaged, hard-

ened load cell called a tup is attached to a falling weight and contacts the specimen throughout the loading process (Fig. 1). Detailed information about the specimen’s elastic stiffness, deflection, maximum load, absorbed energy, incipient damage and load at yield or failure is available after the impact event.

An estimate of a transition temperature, if it exists, is sought for each of the materials tested. In polymers, this temperature is called the glass-transition tempera-ture Tg (below which the material is glasslike and above which it is rubbery), and in metals, it is called the brittle-to-ductile transition (BDT) temperature TBDT .

Fig. 1. Instrumented plate-impact test on a polymer.

9–2 Behavior of Engineering Materials TAM 224/CEE 210

9A.3. Apparatus An instrumented drop-tower arrangement (Fig. 2)

is used to test two types of sample—a plate-penetration specimen for polymers (Fig. 3), and a Charpy V-notch specimen for metals (Fig. 4).

A Dynatup model 8210 drop-weight impact machine is used for the plate-penetration tests. The crosshead weight W and height H are variable. The crosshead is equipped with a 16 kN capacity instru-

mented plastic-penetration tup. Impact specimens (Fig. 3) are supported by a pneumatic clamping fixture with a 76 mm (3-in.) diameter cavity for ASTM D-3763 penetration testing. Crosshead velocity V is measured just before impact by a small optical trigger near the sample fixture.

Impact tests are configured so that the falling tup completely penetrates the specimen, relinquishing less

than a third of its energy to the fracture of the test specimen. The tup experiences only a moderate change in velocity during the test, and consequently the strain rate is nearly constant during the loading.

The tup is instrumented with semiconductor strain gages that measure the compressive tup strain continuously throughout the impact with the speci-men; the strain record is then converted by the data-acquisition system into a plot of impact load P as a continuous function of time t. The tup is also fitted with a shear-stress shield so that frictional loads are not included in the data.

For impact testing of metals, a Dynatup model 8250 impact tester, capable of testing either Charpy or Izod specimens, is used to test Charpy samples (Fig. 4). The principle of operation of the tester is the

same as that used for the plate-penetration tests, except that the impact weight and drop height are set to Charpy specifications,1 and the specimen is loaded in three-point bending.

Digital calipers are used to measure all specimens.

Ice baths are used to precool selected specimens to 0°C before testing, and boiling-water baths are used to preheat other specimens to 100°C before testing.

9A.4 Materials Various polymers, such as polymethylmethacry-late, polycarbonate, polyethylene, polyvinyl chloride, acrilonitrile-butadiene-styrene (ABS) copolymer, and epoxy, may be studied in the plate-penetration experiment. These polymer samples are prepared by cutting square specimens from large pre-cast sheets.

Charpy impact specimens are made from construction-grade alloys of steel and aluminum. Standard dimensions for this specimen are: length L = 55 mm, height D = 10 mm, width w = 10 mm. (See

1 In the lab, the actual values used for height H and weight W may be only approximately those of the standardized Charpy test.

x

H

Impactor

Specimen

Velocitygate

Support

V

Flag

W

Instrumentedtup

Fig. 2. Schematic of a Dynatup drop tower.

SpecimenEdge of

clamped area

a

h

Fig. 3. Plate-penetration specimen geometry.

L

D

w

Notch

Fig. 4. Geometry of Charpy impact specimen.

TAM 224/CEE 210 Impact Testing and Fracture Toughness 9–3

Fig. 1.) The notch should have a depth d of 2 mm, an included angle of 45°, and a root radius of 0.25 mm.

9A.5. Experimental Procedure

Plate penetration (polymers) 1. For each specimen, measure the diameter or width

2b and the mean thickness h at the center. Begin filling out Table 1.

2. Adjust the crosshead to the desired height H and weight W.

3. If the sample is to be tested at a temperature other than room temperature, place the sample in a suitable bath and allow time for the sample to reach a uniform temperature.

4. Prepare the LabVIEW® software to acquire the impact data.

5. Place the sample to be tested into the pneumatic clamping fixture, making certain to center it properly. Enable the clamp and put it in the down position. Close the safety door.

6. When the software is ready to run a test, release the crosshead.

7. Obtain graphs of load P vs. time t and absorbed energy Γ vs. time t. Record the impact velocity V, the maximum load Pmax , the time to failure tfail , and the total absorbed energy Γtot .

8. After the test is complete, enable the clamp and put it in the up position. Remove the sample and clear any debris from the fixture.

9. Reassemble the specimen, if necessary, and make a detailed sketch of the fracture.

Charpy impact (metals) 1. Using calipers or a micrometer, measure the

Charpy specimen length L, depth d, width w, and notch depth. Record these and other data in Table 2.

2. If the sample is to be tested at a temperature other than room temperature, place the sample in a suitable bath and allow time for the sample to reach a uniform temperature.

3. Prepare the LabVIEW® software to acquire the impact data.

4. Place the Charpy specimen into the three-point loading fixture, making certain to center it properly. Close the safety door.

5. When the software is ready to run a test, release the crosshead.

6. Obtain graphs of load P vs. time t and absorbed energy Γ vs. time t. Record the maximum load Pmax and the total absorbed energy Γtot .

7. After the test is complete, remove the sample and make a detailed sketch of the fracture.

9A.6. Theoretical Considerations

Analysis of the drop tower experiment In the instrumented drop-tower test, the only quantity measured as a continuous function of time is the force P t( ) that the specimen exerts on the strain-gaged tup. It is important to observe that the specimen itself is not instrumented—therefore, specimens can be prepared easily and tested quickly.

However, the ability to construct load–deflection plots requires that the deflection δ of the specimen be determined somehow as a function of time, and this determination is done indirectly by numerical integration of the equation of motion for the tup.

To see how this is done, consider a free-body diagram of the falling impactor (Fig. 5). Using Newton’s second law, one can show readily that the impactor acceleration a t( ) is given by

a t g P tM

( ) ( )= − , (1)

where M is the mass of the impactor. Let time t = 0 correspond to the initiation of impact. Integrating Eqn. (1) twice with respect to time and employing the initial conditions

Mg

P(t)

x

M

Fig. 5. Free-body diagram

of the impactor.


v t V tx t t( )( )

= == =

atat

00 0

gives the following expressions for the velocity v t( ) and position x t( ) of the impactor:

0

0

( )( ) d

( ) 0 ( ) d

t

t

P tv t V g tM

x t v t t

′ ′= + −

′ ′= +

∫

∫. (2)

As long as the specimen is not punctured or totally fractured, and as long as the specimen remains in contact with the tup, the specimen displacement δ will be equal to x t( ) .

The LabVIEW® software is written to perform the integrations in Eqn. (2) numerically, after the P t( ) data have been collected and stored. The first data file (*.d1) contains the value of load P as a function of time t . The second data file (*.d2) contains, in order,

the following quantities: P t( ), W P t− ( ), a t( ), v t( ) , and x t( ) . While the second data file is being prepared, the absorbed energy Γ( )t is also being calculated as a continuous function of time:

0

( ) d ( ) ( ) dt

t P x P t v t t′ ′ ′Γ = =∫ ∫ . (3)

From this equation, it is seen that once P t( ) → 0 (due to total specimen failure, total penetration, or loss of contact), there is no further increase in absorbed energy, and therefore Γ( )t has reached its maximum or total value Γtot . General material response Typical impact test data for brittle and ductile materials are sketched in Fig. 6. Integrals under these curves or to any point along the curves give the energy Γ absorbed by the specimen up to that point. The total absorbed energy during impact is denoted

by Γtot . From Eqn. (3) it is seen that brittle specimens, even if they sustain large forces, will break at a small displacement δ and will therefore exhibit a low absorbed energy Γtot .

Also, each curve shows a load inflection associated with initial damage. This inflection is the instance of crack formation. An analysis of the data can be used to pick out points of interest, such as maximum load, onset of nonlinearity, or first cracking.

Plate penetration The plate-penetration test, when conducted in a controlled manner, allows valuable comparisons to be made between dissimilar materials, or between differ-ent batches of, or manufacturing methods for, the same material. Such aspects as first crack or damage, relative stiffness, proportional limit, composite failure, ductile or elastomeric failure, and failure mode can be studied for both ordinary and composite materials (ASTM D 3763).

Extrapolation of results to specimens of different thickness, however, is not always straightforward because shear and flexural effects scale differently with thickness (ASTM D 3763). Also, care must be exercised to eliminate or reduce to an acceptable level such spurious dynamic effects as inertial load of the specimen and harmonic oscillations of the specimen and tup (Cheresh and McMichael 1987).

In this laboratory, the effect of temperature on the impact resistance of some common polymers is a subject of primary interest. If impact tests on a given material are conducted over a sufficiently broad temperature range, one can correlate the glass-transition temperature Tg for that material with the temperature dependence of impact resistance. Below the glass-transition temperature, the behavior is brittle, the time to failure is short, and very little energy is absorbed. Near or above the glass-transition temperature, the behavior is rubbery, the time to failure is relatively long, and the energy absorbed is greatly increased.

Every polymer has a glass-transition temperature. However, the value of Tg does not always fall in the 0°C to 100°C range considered in this laboratory. Tough engineering polymers, such as polypropylene, have values of Tg below 0°C, whereas some aerospace polymers, such as the polyimides, have values of Tg well above 100°C. Representative values of Tg are given in a table in the Discussion (Section 9A.8).

Displacement, δ

Impa

ct lo

ad, P

Displacement, δ

Impa

ct lo

ad, PBrittle Ductile

totΓ totΓ

Fig. 6. Two representative impact tests, one showing brittle response and one showing ductile response.


Charpy impact The long-established Charpy impact test is a relatively inexpensive and standardized way to test materials (primarily metals) to rapid failure in bend-ing. Charpy results, like the newer plate-penetration results, are useful for comparative purposes.

It has been found that body-centered-cubic materials, such as steel at ordinary temperatures, exhibit a brittle-to-ductile transition (BDT) tem-perature below which they absorb very little energy

and above which they absorb considerable energy (Fig. 7). Face-centered-cubic materials, such as stain-less steel and aluminum, generally do not exhibit a noticeable BDT. (See, for example, Askeland (1989) or Flinn and Trojan (1990) for additional details.)

9A.7. Analysis of Results

Plate penetration 1. For each polymer tested, plot the values of maxi-

mum load and total absorbed energy as functions of test temperature. Use linear graph paper for this purpose.

2. Sketch the probable dependence of maximum load and absorbed energy on temperature.

Charpy impact 3. Using data points on linear graph paper, plot

energy absorbed (in joules) as a function of temperature (in °C) for the samples tested. If the Charpy specimen dimensions are significantly different from the standard values, make an appropriate notation on the plot.

4. Sketch the probable dependence of energy absorbed as a function of temperature, extrapo-

lating to slightly higher and lower temperatures than those actually used in the lab. Keep in mind that the minimum energy that can be absorbed is zero; the maximum energy that can be absorbed is the initial potential energy WH of the impactor, which in the standard Charpy impact test is 358 J (264 ft·lb).

9A.8. Points for Discussion Note.—Your lab instructor will indicate which of the

following questions are to be addressed in your report.

Plate penetration 1. Compare the energy absorbed, the maximum

load, the time to failure, and the type of fracture for the different samples at the various temperatures. (Refer to your sketches of the failed samples.) What can be said about the ductility of each polymer tested?

2. State whether the glass-transition temperatures for your materials appear to be within the range of temperatures considered in this laboratory. If so, what is your estimate of their values? If not, then is the glass-transition temperature higher than your highest temperature or lower than your lowest one?

3. State how the temperature dependence of absorbed energy correlates with the published values of glass-transition temperature Tg for your polymers. (See accompanying table.)

4. Is there any correlation between Tg and fracture toughness for a polymer? (See table above.)

5. Compare the impact velocity and resulting

FCC

BCC

Ene

rgy

abso

rbed

Temperature, T

TBDT0

Brittle

Ductile

Fig. 7. Brittle-to-ductile transition.

Tg E KIC

Abbr. Polymer name

°C GPa MPa· ÷m

ABS Acrylonitrile-butadiene-styrene copolymer

80 2.1 —

HDPE High-density polyethylene

7 0.75 2–5

LDPE Low-density polyethylene

–3 0.2 1–2

PMMA Polymethyl-methacrylate

105 3.3 1.6

PP Polypropylene –27 1.3 3.5


absorbed energy for the polymer tested at two different heights. How is the ductility of the polymer related to strain rate?

6. If epoxy samples were used, then they consist of a resin mixed with varying amounts of flexibilizer and a curing agent. Suggested ratios of resin and flexibilizer are shown in the following table for each type of sample:

The thin discs (Fig. 2) of each epoxy composition are fabricated by pouring the appropriate resin mixture into a circular mold at room temperature and then curing at an elevated temperature (typically 70°C) for one hour.

Compare the energy absorbed, the maximum load, the time to failure, and the type of fracture for the different flexibilizer concentrations. (Refer to your sketches of the failed samples.) What can be said about the ductility of each epoxy tested?

Charpy impact 7. For each of the materials tested, is there a

noticeable brittle-to-ductile transition? If so, esti-mate the BDT temperature. How is the nature of the fracture surface related to the test temperature, within the range of temperatures used?

8. If 6061-T6 aluminum was used in your test, would you expect 7075-T6 aluminum to behave in a

similar fashion? (Or, if 7075-T6 was used, would you expect 6061-T6 to behave similarly?) Consider such aspects as maximum load, energy absorbed, time to failure, and BDT temperature.

9. Determine whether your instrumented Charpy tests were conducted under standard conditions for the Charpy test.

9A.9. References

Plate penetration American Society for Testing and Materials. 1986.

Standard test method for high-speed puncture properties of plastics using load and displacement sensors. ASTM Standard D 3763–86. Philadel-phia: ASTM.

Ashby, M. F., and D. R. H. Jones. 1986. Engineering Materials 2—An Introduction to Microstructures, Processing and Design. Oxford: Pergamon.

Cheresh, M. C., and S. McMichael. 1987. Instru-mented impact test data interpretation. In Instru-mented Impact Testing of Plastics and Composite Materials, ASTM STP 936. Philadelphia: ASTM, 9-23.

Charpy impact Askeland, D. R. 1989. The Science and Engineering of

Materials, 2nd ed. Boston: PWS-Kent, 159-162.

Callister Jr., W. D. 2003. Materials Science and Engi-neering—An Introduction, 6th ed. New York: Wiley, Section 8.6.

Flinn, R. A., and P. K. Trojan. 1990. Engineering Materials and Their Applications, 4th ed. Boston: Houghton Mifflin, 162-164.

Composition of epoxy samples

Sample Parts by weight

type Resin Flexibilizer

F-0 100 0

F-40 60 40

F-80 20 80


Table 1—Plate-penetration data (Polymers)

Measurement or property Material

Name Symbol Units PMMA

Specimen dimensions and impact parameters

Specimen shape — —

Specimen diameter or edge length

2b mm

Specimen thickness h mm

Crosshead height H m

Crosshead weight W N

Avail. energy (theor.) WH

J

Measurements of time to failure, maximum load, and energy absorbed

Environment

Time

tfail (msec)

Max. load

P (kN)

Total energy

Γ (J)

Time

tfail (msec)

Max. load

P (kN)

Total energy

Γ (J)

Impact data

Freezing water (____°C)

Sketch of fracture surface

Impact data

Room temperature (____°C)


Impact data

Boiling water (____°C)


Test date: Group: Student’s name:


Table 2—Charpy impact data


Name Symbol Units _____ Steel __________ Aluminum

Specimen dimensions and impact parameters

Length L mm

Depth D mm

Width w mm

Notch depth d mm

Notch root radius r mm

Crosshead height H m

Crosshead weight W N

Avail. energy (theor.) WH J

Measurements of maximum load and energy absorbed

Temperature T °C Max. load

(kN)

Energy

(J)

Max. load

(kN)

Energy

(J)

Impact data

Freezing water (____°C)


Impact data

Room temperature (____°C)


Impact data

Boiling water (____°C)




Fracture Toughness

9B.1 Objective The purpose of fracture-toughness testing is to determine the value of the critical stress-intensity value, or plane-strain fracture toughness K CI , of high-strength materials. This material property is used in the design of structural members made of high-strength materials, which are often susceptible to rapid fracture in loading situations that would normally cause lower-strength materials to yield without fracturing.

9B.2. Apparatus An MTS 90-kN (20-kip) capacity servo-hydraulic test frame with swivel-pin grips is used to load a compact tension specimen that has been precracked in fatigue. See Fig. 1. A load cell is used to measure the load P applied to the specimen, and a clip gage is

used to measure the crack-opening displacement (COD) δ of the specimen. LabVIEW® software is used to control the test and to obtain a plot of load P versus crack-opening displacement δ .

9B.3. Material Type 7075-T6 aluminum is generally used to make

the compact tension specimen. This high-strength, lightweight alloy is commonly used in aircraft structures.

9B.4. Experimental Procedure Note.—The laboratory technician will normally have

the specimen installed and COD gage already attached.

1. Observe the fatigue precrack in the vicinity of the machined crack tip of the compact tension speci-men, using a video camera fitted to a microscope (Fig. 1).

2. Prepare the data-acqusition system to plot load P versus crack-opening displacement δ.

3. Load the specimen slowly until failure occurs, observing the load–deflection plot and viewing the deformation of the specimen in the area of the crack tip. After the specimen fails, note the appearances of the machined surfaces, the fatigue-precracked surface, and the final fracture surface. Record comments in Table 3.

4. Using calipers or a micrometer, measure the speci-men and crack dimensions, and record these data in Table 3. See Fig. 2.

Fig. 1. Removing upper pin from a failed

compact tension specimen.

Fatigueprecrack

Thickness=

Notch forCOD gage

d

h

a

B

w

P

P

Fig. 2. Nomenclature for a compact tension specimen. Specimen


9B.5. Theoretical Considerations The stress-intensity factor KI at the tip of the precrack in a compact tension specimen2 is given by

K PB w

fI = ( )λ , (4)

where P is the applied load, w is the specimen width (illustrated in Fig. 2), B is the specimen thickness, and

f ( )λ is a geometrical factor; λ denotes the ratio a w/ , where a is the crack length (including the precrack—see Fig. 2). The function f ( )λ is plotted in Fig. 3.3

For a sufficiently large load PQ, the specimen fails, and a provisional value of fracture toughness KQ is calculated from Eqn. (4), i.e.

KP

B wfQ

Q= ( )λ . (5)

The value KQ may turn out to be the critical stress-intensity factor, or fracture toughness, KIC, but only if some conditions are met.

2The singular nature of the stress field surrounding a mode-I loaded crack, and the visualization of this singularity using photoelasticity, are considered in the Discussion (Section 9B.7).

3The ASTM (1987) formula for f (λ ) is

f (λ ) =(2 + λ )

(1− λ ) 3 2 (0.886 + 4.64λ −13.32λ2 +14.72λ3 − 5.6λ 4 ) .

For proper fracture-toughness testing, λ should be in the range 0.45 <λ <0.55 .

First of all, the correct value of PQ must be established; there are three principal types of load-versus-COD curves that are encountered, as shown in Fig. 4. The dashed line OP5 in the figure is drawn at 0.95 of the slope of the linear portion of the P vs. δ curve. The point of intersection of this secant with the original curve defines point P5. Then, point PQ is determined as follows: If the load at every point on the record that precedes P5 is lower than P5, then the load–deflection curve is of type I, and PQ = P5; but if there is a maximum load preceding P5 that exceeds P5, then the load–deflection curve is of type II or III, and PQ is this maximum load.

Second, the conditions for plane strain at the root of the fatigue precrack must also be checked; if the plastic zone size at the root of the crack is too large, as

illustrated in Fig. 5, then the state of stress is rather more plane-stress-like than plane-strain-like, and the apparent fracture toughness KQ of the material will be larger than the actual plane-strain value KIC that is sought. By analysis and experience, it has been found that plane-strain conditions exist if the thickness B and crack length a are sufficiently large compared with the plastic zone size; the accepted rule (ASTM 1987) is:

2

, 2.5 Q

y

KB a

σ

≥

, (6)

where σy is the yield strength of the material being tested. If B and a meet the minimum criteria in Eqn. (6), as they would for the specimen illustrated in Fig. 6, then

K KC QI = . (7)

0.45 0.50 0.55 0.60

6

8

10

14

12

Ratio of crack length to width, λ = a/w

Fun

ctio

n f(λ

)

a

w

P

P

Fig. 3. Plot of the function f( )λ for

a compact tension specimen.

Loa

d, P

Crack-opening displacement, δ

Type I Type II Type III

P = PP

P

P

P

P

P = PQ 5

Q

5

maxmax

5

Q max

O O O

Fig. 4. Types of load–displacement curves observed in KIC

testing.


Otherwise, KIC < KQ; that is, KQ is an upper bound to the true value of fracture toughness KIC.

9B.6. Analysis of Results 1. Compute the value λ = a w/ , where a is the initial

crack length and w is the effective width of the compact tension specimen. Find the value of the geometry factor f( )λ from Fig. 3.

2. Determine the value of the critical load PQ from the load–deflection plot obtained in the labora-tory. Determine the type (I, II or III) of the loading curve.

3. Compute a provisional value of the critical stress-intensity factor, KQ, given by Eqn. (5).

4. If the conditions in Eqn. (6) are met, then the fracture toughness KIC is equal to the provisional KQ. If the conditions in Eqn. (6) are not met, then a thicker specimen (or one with a longer crack, or both) must be used to determine KIC.

The uniaxial yield strength σy of 7075-T6 alumi-num is about 500 MPa (73 ksi), whereas the yield strength of 6061-T6 aluminum is considerably lower—about 280 MPa (40 ksi), according to Marks’ Handbook (1987).

Provide calculations, and state your final conclusion(s).

9B.7. Points for Discussion Note.—Your lab instructor will indicate which of the

following questions are to be addressed in your report.

1. Discuss the features of the fracture surface of the compact tension specimen, and correlate these features with the condition(s) for plane-strain fracture-toughness testing.

2. A typical load–deflection plot is shown in Fig. 7 for a quenched and tempered 4340 steel. Analyze

the plot to determine, if possible, the fracture toughness of this steel. Dimensions of the sample are w = 50.8 mm (2.00 in.), a = 25.4 mm (1.00 in.), and B = 25.4 mm (1.00 in.). Assume that the yield strength of this 4340 steel is 1500 MPa (220 ksi). Are the conditions for plane-strain testing met?

Fig. 5. Failed compact tension specimen of 6061-T6

aluminum, showing large shear lips and associated lack of plane-strain conditions at the crack tip.

Fig. 6. Failed compact tension specimen of 7075-T6

aluminum, showing a flat fracture surface and associated plane-strain conditions at the crack tip.

0 0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

12

Crack-opening displacement, δ (in.)

Loa

d, P

(ki

ps)

Quenched and tempered4340 steel

Fig. 7. Fracture-toughness load–deflection curve

for a 4340 steel.


Explain your work, and include the calculations. Give answers in both SI and U.S. customary units.

3. Consider how fracture mechanics and photo-elasticity can be used together to investigate the state of stress surrounding a mode-I loaded sharp

crack (Fig 8). Under plane-strain conditions, if the stresses remain elastic, the stress distribution for a sharp-tipped, flat crack in an infinitely large body can be shown to be

I

31 sin sin2 2

31 sin sincos 2 2

22 3sin cos2 22

x

y

xy

z

Kr

θ θσ

θ θσ θτ π θ θσ

ν

− + =

, (8)

where σ z is the out-of-plane normal stress, and KI is the mode-I stress-intensity factor. Recall that the in-plane principal-stress difference σ σ1 2− , which gives rise to fringes in a photoelastic model, can be calculated for any combination of σx , σy , and τxy from the Mohr’s circle equation

2

21 2 2

2x y

xyσ σ

σ σ τ−

− = +

. (9)

Show by combining Eqns. (8) and (9) that the principal-stress difference for a mode-I loaded crack is given by

σ σπ

θ1 2 2− =

Kr

I sin . (10)

(You will need to use the double-angle formula

sin sin cosθθ θ

= 22 2

and the fact that the square root in Eqn. (9) is taken to be positive—that is, σ1 is always greater than or equal to σ2 .)

Finally, recall from Lab 8 that the photoelastic fringe value N is related to the principal-stress difference by the equation

σ σ σ1 2− =

Nfh

, (11)

where fσ is the fringe constant for the photoelastic

material and h is the thickness of the model. Combine Eqns. (10) and (11) to show that the theoretical fringe distribution N r( , )θ is given by

N r hf

Kr

( , ) sinθπ

θσ

= I2

. (12)

This distribution is illustrated in Fig. 9. Note that along any radial line (θ = const ), the value of N increases as 1/ r as the crack tip is approached.

4. Using Eqn. (12), determine the value of the stress-intensity factor KI for the crack in the compact tension specimen that is illustrated in Fig. 10. Note in particular that along a vertical line

Crack

P

P

x

y

σy

σx

τxy

rθ

Fig. 8. Nomenclature for stresses surrounding a

mode-I loaded crack.

Crack

P

P

x

y

N = 1

N = 2

N = 3

(etc.)

N = 0

N = 1

Fig. 9. Theoretical photoelastic fringe pattern for a

mode-I loaded crack in an infinite medium.


emanating from the crack tip, θ π= ± / 2 and consequently r y= ± . Therefore, from Eqn. (12),

Kfh

N yI = σ π2 | | . (13)

For the PSM-1® material that was used, the fringe constant fσ is about 7.0 kN/m (40 psi-in.), and the

model has a thickness of 5.54 mm (0.218 in.). To calculate KI , you will need at least one value of the fringe number and its corresponding location. In Fig. 10 there are several fringes to choose from—note that the whole-order fringes (N = 0 1 2, , ,L) in a light-field photograph are white, whereas the half-order fringes (N = 1

212

121 2, , ,L) are black—but it is recom-

mended that only the crack-tip dominated fringes of order 4 1

2 , 5, and 5 12 be considered, since these

are far enough away from the blunt crack tip to be

relatively insensitive to the actual shape of the tip, yet near enough to the crack tip to be unaffected by specimen boundaries and loading points.

Remember to scale your measurements for fringe location based on the actual size of the specimen. (The image in Fig. 10 was enlarged slightly for printing.)

Express your result for KI in MPa m .

9B.8. References American Society for Testing and Materials. 1987.

Standard E399-87, Test method for plane-strain fracture toughness of metallic materials. Annual Book of ASTM Standards. Philadelphia: ASTM.

American Society for Testing and Materials. 1991. Standard D5045-91a, Plane-strain fracture toughness and strain energy release rate of plastic materials. Annual Book of ASTM Standards. Philadelphia: ASTM.

American Society for Testing and Materials. 1991. Standard B645-91, Plane-strain fracture toughness testing of aluminum alloys. Annual Book of ASTM Standards. Philadelphia: ASTM.

Ashby, M. F., and D. R. H. Jones. 1981. Engineering Materials—An Introduction to their Properties and Applications. Oxford: Pergamon Press, Chapter 14 (esp. pp. 129-132).

Callister Jr., W. D. 2003. Materials Science and Engi-neering—An Introduction, 6th ed. New York: Wiley, Section 8.5.

Dieter, G. E. 1986. Mechanical Metallurgy, 3rd ed. New York: McGraw-Hill, 356-359.

Flinn, R. A., and P. K. Trojan. 1990. Engineering Materials and Their Applications, 4th ed. Boston: Houghton Mifflin, Sections 11.4, 19.1-19.11.

Hertzberg, R. H. 1983. Deformation and Fracture Mechanics of Engineering Materials, 2nd ed. New York: Wiley, 300-301, 675.

Fig. 10. Light-field photoelastic fringe pattern in a

compact tension specimen containing a blunt notch. Outside dimensions of the specimen are 63.5 mm x

63.5 mm (2.5 in. x 2.5 in.).

4 0 1 2 3

0

3

3

3 6


Table 3—Fracture-toughness testing data


Quantity Symbol Units __________ Aluminum

Initial data

Thickness B mm

Width w mm

Fractured length w a− mm

Crack length a mm

Geometry factor f ( )λ —

Fracture toughness calculations

Type of load–deflection curve — —

Maximum load Pmax kN

Failure load PQ kN

Detail of fracture surface or final shape

—

—

Yield strength σ y MPa

Provisional fracture toughness

KQ MPa ⋅ m

Minimum B, a for plane strain

2

2.5 Q

y

Kσ

mm

Derived quantities

Fracture toughness K CI MPa ⋅ m


Printed 7/9/03

Impact_test and Fracture Test

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