Stress Concentration Factor and Fracture Toughness in Materials Science, Study notes of Mechanics of Materials

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PRINCIPLES OF LINEARPRINCIPLES OF LINEARELASTIC FRACTUREELASTIC FRACTURE MECHANICS (LEFM)MECHANICS (LEFM)MECHANICS (LEFM)MECHANICS (LEFM)

Chapter 2Chapter 2

1

OO

VERVIEWVERVIEW

Type of Loading Modes

Stress Intensity Factor Approach

Experimental Determination of Fracture ToughnessFracture Toughness

Energy Balance Approach – StrainEnergy Release Rate

Limitation of LEFM and K-dominance

Case Studies

Analysis of cracked bodyAnalysis of cracked body

In deriving the solutions, there are twoalternative approaches:

the stress intensity/concentration approach, andand

the energy balance approach.

Both approaches are equivalent in certaincircumstances as demonstrated later inthe course

StressStress-

-intensity Approach

intensity Approach

The stressThe stress-

-intensity (concentration) factor (K) is used to

intensity (concentration) factor (K) is used to

determine the fracture toughness of most materials.determine the fracture toughness of most materials.

A stress concentration is a site on the surface or in aA stress concentration is a site on the surface or in amaterial where;material where;

the stress is locally greater than the nominal grossthe stress is locally greater than the nominal grossstress.stress.

the lines of force bunch together is an area of high stress.the lines of force bunch together is an area of high stress.

the lines of force bunch together is an area of high stress.the lines of force bunch together is an area of high stress.

The severity of a particular stress concentration depends onThe severity of a particular stress concentration depends onthe; geometry and the type of loading.the; geometry and the type of loading.

Stress ConcentrationStress Concentration

Stress distribution near fillets in flat bar

When a structural member contain a discontinuity, such as hole or a sudden changein cross section, high localised stresses may occur near the discontinuity.

However the information are only useful for the specific design only

Hence the ratio of stress is used;

; Dimensionless, no unit

This ratio, K is referred stress concentration factor for the given discontinuity.

K may be computed once and for all in terms of ratios of the geometric parameters involved.

The results obtained may be expressed in the form of tables of graphs.

Stress distribution near circularhole in flat bar under axial loading.

Stress distribution near fillets in flat bar under axial load

loading.

Stress Concentration Factor (K)Stress Concentration Factor (K)

Example 2.1:Determine the largest axial load P that can be safely supported bya flat steel bar consisting of two portions both 10 mm thick andrespectively, 40 mm and 60 mm wide, connected by fillets of radiusr = 8 mm. Assume an allowable normal stress of 165 MPa.

Stress Concentration Factor (K)Stress Concentration Factor (K)

Cont’dCont’d

The presence of sharp corners, notches, or cracks serves to concentrate the appliedstress at these points.

Inglis showed, using elasticity theory, that the degree of stress magnification at theedge of the hole in a stressed plate depended on the radius of curvature of the hole.

The smaller the radius of curvature, the greater the stress concentration.

Inglis found that the “stress concentration factor”, K, for an elliptical hole is equalto:

........ Eq. 2.2a

c ρ

K

2

1

=

where c is the hole radius and

ρ

is the radius of

curvature of the tip of the hole.

For a very narrow elliptical hole, the stressconcentration factor may be very much greater thanone.

For a circular hole, Eq. 2.2a gives K = 3 (as shown inFig. 2.2.1). It should be noted that the stressconcentration factor does not depend on the absolutesize or length of the hole but only on the ratio of thesize to the radius of curvature.

Most cracks are long and have sharp tip.

Equations that describe the elastic stress field in the vicinity of a crack tipfor tensile stresses normal to the plane of the crack (Mode I deformation)formulated by Irwin and Williams (1957).

Equation 1;

StressStress-

-intensity Factor (K)

intensity Factor (K)

sin

sin

cos

θ

θ

θ

π π

σ

σ

a r

x

= applied stress

z

= 0 ...plane strain (thin sheet)

z

x

y

) ...plane stress (thick sheet)

cos

cos

sin

sin

sin

cos

θ

θ

θ

π π

σ

τ

θ

θ

θ

π π

σ

σ

π

r a r a^ r

xy

y

Examples 2.2 If a through crack of dimension 2.5 cm is placed inthe material and fracture takes place at a stress of700 MPa. What is the critical stress intensity factorof this material? SolutionSolution

a

K

m

MPa

a

K

I

×

×

Solution for stress intensity factor in Equation 2is valid only for an

infinite plate

Stress intensity factor depends on the

applied

stress, the crack shape, size, andorientation, and the structuralconfiguration

of structural components.

A Roman numeral subscript indicates the mode of fracture and the three modes of fracture are illustrated in the image to the right.

StressStress-

-intensity Factor (K)

intensity Factor (K)

Cont’dCont’d

fracture and the three modes of fracture are illustrated in the image to the right.

The stress intensity factor for

finite plate

may

be represented by the following equation:

K

I

is the fracture toughness in

is the applied stress in MPa or psi

a

is the crack length in meters or inches

C

is a crack length and component geometry factor that is different for each specimen,

dimensionless.

a

C

K

I

π

σ

=

StressStress-

-intensity Factor (K)

intensity Factor (K)

Cont’dCont’d

Example 2.2:Example 2.2:

Solution:Solution:

Critical StressCritical Stress-

-intensity Factor, K

intensity Factor, K

CC

All brittle materials contain a

population of small

cracks and flaws

that have a variety of sizes, geometries

and orientations.

When the magnitude of a tensile stress at the tip of oneof these flaws exceeds the value of this critical stress, acrack forms and then propagates, leading to failure.

Condition for crack propagation:

Condition for crack propagation:

If K becomes K

C

, then crack propagate.

K

IC

; plane strain fracture toughness; normally reported

as material property

Fracture toughness - good diagrams

http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Mechanical/FractureToughness.htm

K

≥

K

c

Stress Intensity Factor

--Depends on load & geometry.

Fracture Toughness

Depends on the material,

temperature,

environment & rate of loading.

KK -

• Parameters

Parameters

Stress Concentration factor (Dimensionless)

Stress Intensity factor ( Pa.m

1/

/ Pa.√m)

ave

t

K

σ^ σ

max

=

Critical Stress Intensity factor or FractureToughness ( Pa.m

1/

/ Pa.√m)

a

K

I

π

σ

=

a

K

f

IC

π

σ

=