FAILURE OF MATERIALS AND FAILURE OF MATERIALS · - Ductile and brittle fracture - Modes of fracture...

$: FAILURE OF MATERIALS AND FAILURE OF MATERIALS · - Ductile and brittle fracture - Modes of fracture - Ductile to brittle transition Fatigue - Types of fatigue loading - Fatigue mechanism$
Pruthvi Loy, Chiranth B. P. 1 SJEC, Mangaluru

FAILURE OF MATERIALS AND FAILURE OF MATERIALS

Fracture

- Ductile and brittle fracture

- Modes of fracture

- Ductile to brittle transition

Fatigue

- Types of fatigue loading

- Fatigue mechanism

- Fatigue testing and S-N diagram

- Fatigue properties

- Factors affecting fatigue life

Creep

- Creep phenomenon

- Stages and properties of creep

- Stress relaxation

3.1 INTRODUCTION

The failure of a material is its loss of ability to function normally; the structural or a machine

element may fail to perform their intended functions in three general ways - excessive elastic

deformation, excessive plastic deformation, and fracture. The usual causes of material failures

are - improper material selection, improper processing, inadequate design, excessive loading and

deformation, corrosion, etc.

The commonly observed failure modes are: Fracture, Fatigue and Creep.

Fracture - at room temperature, continued loading over the yielding point may lead to

strain hardening followed by fracture (ductile/brittle fracture). It occurs over short period

of time, and is distinguishable.

Fatigue – occurs when a material has been subjected to alternating or fluctuating stress

(progressive fracture).

Creep - occurs when a material has been statically loaded at an elevated temperature for

a long time (delayed fracture).

Module 1 3. Failure of Materials


3.2 FRACTURE

Fracture is the separation or fragmentation of a solid body into two or more parts under the

action of stress. The process of fracture can be considered to be made up of two components:

crack initiation and crack propagation.

3.2.1 Types of fracture

Fractures can be classified into two general categories;

Ductile fracture

Brittle fracture

The fracture is termed ductile or brittle depending on the ability of a material to undergo plastic

deformation during the fracture.

3.2.1(a) Ductile fracture

In ductile fracture considerable amount of plastic deformation takes place before fracture which

is characterized by the reduction in cross-sectional area called as necking. The most ductile

material when loaded in tension may experience reduction in cross-section close to 100% as it

fractures; this kind of failure is called as rupture. However most of the ductile metals fractures

with the formation of a cup and cone; which is usually preceded by a moderate amount of

necking. A cup and cone fracture is also known as fibrous fracture, as the fracture surface has an

irregular and fibrous appearance.

Figure 3.1: Different stages in ductile fracture

The different stages in a ductile fracture are;

a) Necking

b) Formation of micro voids

c) Coalescence of micro voids to form a crack

d) Propagation of crack by shear deformation

e) Fracture (cup and cone fracture)



3.2.1(b) Brittle fracture

Brittle fracture in metals is characterized by a rapid rate of crack propagation with little or no

preceding plastic deformation. The direction of crack propagation is very nearly perpendicular to

the direction of applied tensile stress. This fracture is also said to be transgranular because crack

propagates through grains; the fracture surface has a grainy appearance.

Brittle fracture is studied much extensively than ductile fracture, as it offers no indication prior to

failure which is a major concern in structural or machine elements.

Griffith theory of brittle fracture:

The observed fracture strength of most of the brittle materials is found to be lower than that is

theoretical calculated, this discrepancy is due to the presence of several micro cracks as

explained by Griffith.

Consider a thin plate with a pre-existing crack (the cracks are

assumed to have elliptical cross section).

Let,

= tensile stress

c = half length of the crack

r = radius of curvature of crack at its tail

When the tensile stress is applied, the stress is distributed such that the maximum stress occurs at

the tip of the flaws.

Maximum stress,

When the magnitude of the tensile stress at the tip of one of these flaws exceeds the value of this

critical stress, a crack forms and then propagates which results in fracture.

max = 2 𝐶

𝑟



A certain amount of energy is always stored in the material which is its elastic strain energy; this

energy is released as the crack propagates.

Also, new surfaces are created as the crack propagates which increases the surface energy of the

system.

According to Griffith theory, “A crack will propagate when the released strain energy is just

sufficient to provide the surface energy to create the new surface”.

The total energy, U = UE + US

According to Griffith, the crack will propagate under a constant applied stress if an incremental

increase in crack length produces no change in total energy of the system. Therefore, fracture is

associated with the consumption of energy.

i.e.,

= 0 (the increased surface energy is compensated by a decrease in strain energy)

Therefore,

+ 4c ] = 0

+ 4 = 0

=

Stress required to propagate a crack as a function of its size.

(The stress necessary to cause fracture varies inversely with length of

existing cracks, thus largest crack determines the strength of

material.)

Where,

E - modulus of elasticity

- specific surface energy

c - one half the length of an internal crack

Elastic strain energy, UE = 𝜋𝑐

𝐸

Surface energy per unit area, US = 4c𝛾

U = 𝜋𝑐

𝐸 + 4c𝛾

= 𝛾𝐸

𝜋𝑐



Note:

1. A modified Griffith equation is suggested for metals considering the plastic work

required for crack propagation.

Where, - plastic work required to propagate the crack

2. For a thick plate,

Where, - Poisson‟s ratio

Fracture toughness:

The resistance offered by the material against the propagation of cracks when the material

already contains cracks is known as fracture toughness.

A crack opening mode (mode-I) of deformation and fracture is observed for tensile loaded

materials. Further, two extreme cases are observed in this mode; for a thin plate the stress state is

plane-stress, whereas for thick plate plane-strain condition exists. The parameter, Stress intensity

factor (K) is often used to describe fracture toughness for plane-stress condition.

Stress intensity factor

Where, - geometric constant

The plane-strain condition represents the more severe stress state and the values of K are lower

than for plane-stress specimens. The plane strain fracture toughness is designated as KIC.

Critical stress intensity factor

Where, - critical crack length

= 2γ𝐸

1 𝜋𝑐

= 𝛾+𝑝 𝐸

𝜋𝑐

𝐾 = 𝜋𝑐

𝐾𝐼𝐶 = 𝜋𝑐𝑐



3.2.2 Fracture modes

There are three basic modes of fracture which are also known as crack deformation modes as

fracture is believed to occur through the presence of a pre-existing crack and its propagation.

Mode I – Opening mode

Mode II – Sliding mode

Mode III – Tearing mode

Mode I corresponds to fracture where the crack surfaces are displaced normal to themselves.

This is a typical tensile type of fracture. In mode II, crack surfaces are sheared relative to each

other in a direction normal to the edge of the crack. In mode III, shearing action is parallel to the

edge of the crack.

Mode I Mode II Mode III

Figure 3.2: Crack deformation modes

3.2.3 Ductile to Brittle Transition

The deformation of a material depends on many conditions such as stress state, rate of loading,

crystal structure, ambient temperature, etc. These further affect their ductile/brittle behavior. A

ductile material may deviate from its normal behavior and undergo fracture in a brittle manner;

such a change from the ductile to brittle type of fracture is promoted by,

The presence of complex state of stress (for example, due to a notch)

An increase in the rate of loading

Crystal structure - materials with BCC or HCP crystal structure can be expected to

experience brittle fracture under normal conditions, whereas materials with FCC crystal

structure are expected to experience ductile fracture.

Decrease in temperature – mild steel acts as a ductile material at room temperature but

becomes brittle at sub-zero temperature as shown in Figure 3.3. As the temperature is

lowered, the impact energy drops suddenly over a relatively narrow temperature range,

below which the energy has a constant but small value; that is, the mode of fracture is

brittle.



Figure 3.3: Ductile to Brittle Transition

Not all metal alloys display a ductile-to-brittle transition. Those having FCC crystal structures

(Al and Cu based alloys) remain ductile even at extremely low temperatures. However, BCC and

HCP alloys experience this transition. For these materials the transition temperature is sensitive

to both alloy composition and microstructure.

3.3 FATIGUE

In most of the service applications the engineering materials are subjected to dynamic loading

(automobile crank-shaft, turbine blades, aircraft wings, etc.). Under these circumstances the

material experiencing alternating or fluctuating stresses may fail progressively at a stress level

considerably lower than the tensile or yield strength for a static load. Failures occurring under

these conditions are called fatigue failures. Fatigue accounts for at least 90% of all service

failures due to mechanical causes.

Three basic factors are necessary to cause fatigue failure

1. A minimum tensile stress of sufficiently high valve.

2. A large enough variation of fluctuation in the applied stress.

3. Sufficiently large number of cycles of the applied stress.

3.3.1 Types of fatigue loading

The general types of fluctuating stresses that can cause fatigue are

Reversed stress cycle (Example: a rotating shaft with a bending load applied)

Repeated stress cycle (Example: connecting rod, turbine blade)

Random stress cycle (Example: automobile shocks, aircraft wings)



(a) Reversed stress cycle

(b) Repeated stress cycle

(c) Random stress cycle

Figure 3.4: Stress Cycles that account for fatigue failure



Fluctuating stress parameters:

Mean Stress – the stress amplitude alternates about a mean stress, defined as the average

of the maximum and minimum stresses in the cycle.

Range of stress – the difference between the maximum and minimum stress.

Stress amplitude – one half of the range of stress.

Stress ratio – the ratio of minimum and maximum stress amplitudes.

3.3.2 Fatigue mechanism

A fatigue failure can be recognized from the appearance of the fracture surface which shows two

distinct regions; the first region being smooth or burnished as a result of the rubbing of the

bottom and top of the crack corresponds to the slow growth of crack (crack propagation). The

second is granular, due to the rapid failure of the material.

Figure 3.5: Fatigue fracture surface

𝝈𝒎 =𝝈𝒎𝒂𝒙 + 𝝈𝒎𝒊𝒏

𝟐

𝝈𝒓 = 𝝈𝒎𝒂𝒙 𝝈𝒎𝒊𝒏

𝝈𝒂 =𝝈𝒓𝟐

𝑹 =𝝈𝒎𝒊𝒏

𝝈𝒎𝒂𝒙



The mechanism of fatigue failure may be viewed on a microscopic level in three steps:

1. Crack initiation

2. Crack propagation

3. Failure

The fatigue crack nucleates at the stress concentration. The potential sites for crack initiation are

the surface scratches caused by handling, or tooling of the material; threads (as in a screw or

bolt); slip bands or dislocations intersecting the surface as a result of previous cyclic loading or

work hardening, etc.

As a crack begins to propagate, the size of the crack also begins to grow. The rate at which the

crack continues to grow depends on the stress level applied. The region of a fracture surface that

formed during the crack propagation step may result in characteristic pattern of concentric rings

known as beach marks or striations radiating outward from the point of crack initiation, where

each beach mark band represents a period of time over which crack growth occurred.

Finally, failure occurs when cross section is no longer able to carry the applied load. This stage

happens very quickly.

3.3.3 Fatigue testing

A specimen of rotating beam type is often used to measure fatigue; the test arrangement is as

shown in figure 3.6. A load is applied on the rotating beam to facilitate rotating-bending test; the

beam is subjected to stress reversals wherein alternate tensile and compressive stresses act upon

the material. The test is conducted for different loads and the number of cycles to failure is

noted.

Figure 3.6: Fatigue testing apparatus (rotating - bending test)



S-N curve

The S-N curve is a plot of stress against the number of cycles to failure. It gives information on

the fatigue behavior of a material as well as its properties.

To construct S-N Curve, a test specimen is tested on a rotating beam machine for its number of

cycles to failure under certain load. After the first specimen breaks away, the second one is tested

with decreased load, similarly third, fourth ... etc., are tested with loads further decreased. The

result of each test (number of cycles to failure) is plotted against the applied stress. It can be

observed that the number of stress cycles that a material can endure before failure increases with

the decreasing load (stress).

Figure 3.7: S-N Curve

Some materials display a peculiar characteristic wherein a limiting stress is reached below which

the material can endure infinite number of stress cycles without failure. This is known as fatigue

limit or endurance limit and the S-N curve is represented by a horizontal line beyond this point.

Figure 3.8: Typical S-N curves for Mild steel and Aluminium



A typical S-N curve for mild steel and aluminium is as shown in figure 3.8; mild steel has an

endurance limit but for aluminium the S-N curve slopes gradually downwards with increasing

stress, representing that it doesn‟t have a true fatigue limit.

3.3.4 Fatigue properties

a) Fatigue Life (N):

It is the total number of cycles required to bring about final fracture in a specimen at a given

stress. Fatigue life for a given condition is a property of the individual specimen and is

arrived at after testing a number of specimens at the same stress.

b) Fatigue Strength (n):

It is the stress at which a material can withstand repeatedly „N‟ number of cycles before

failure. In other words it is the strength of a material for a particular fatigue life.

c) Fatigue limit or Endurance limit (e):

The limiting value of stress below which fatigue failure does not occur is called fatigue

limit. It represents the largest value of cyclic stress that will not lead to fatigue failure

irrespective of the number of cycles. In practice only ferrous materials show fatigue limit

whereas non-ferrous metals (Al, Cu, etc) and alloys do not indicate any fatigue limit.

3.3.5 Factors affecting fatigue life

a) Metallurgical factors

Fatigue strength of a material is proportional to its tensile strength, therefore methods to

improve tensile strength also improves fatigue strength. It includes strengthening

mechanism such as alloying, heat treatment, age hardening, grain refinement etc.

b) Stress Concentration

Fatigue strength is reduced by the introduction of stress raisers such as fillets, keyways,

screw threads, holes etc. The fatigue cracks in structural parts usually start at such

geometrical irregularities; hence it is better to avoid such unnecessary stress raisers.

c) Size

The fatigue strength of large specimens is lower than that of smaller ones. The large samples

have a large surface area and hence may offer more stress raisers owing to surface

irregularities.

d) Surface effects

Practically all fatigue failures start at the surface, i.e., the fatigue properties are very

sensitive to surface conditions. The surface factors that affect the fatigue properties of the

specimen are,



(i) Surface roughness

(ii) Variation of fatigue strength along the surface

(iii) Changes in residual stress condition of the surface

e) Corrosion

The environment in which the component undergoes stress reversals has a marked effect in

fatigue life. i.e., the environment has a corrosive effect.

f) Temperature

An increase in temperature around the room temperature causes an increase in the crack

growth rate. Thus, fatigue strength decreases.

3.4 CREEP

A material subjected to a constant load within its elastic limit at constant temperature for long

periods of time may undergo time dependent deformation. This slow and progressive

deformation of a material under constant loading is called Creep.

Creep deformation is extremely sensitive to temperature. It is usually considered important at

elevated temperatures (temperatures greater than 0.4 Tm, where Tm is the melting temperature).

Study of creep is important for machine components operating at high temperature such as gas

turbines, ic engines, furnaces, etc.

3.4.1 Creep curve

Figure 3.9: Creep curve



Creep behavior of a material is studied using creep test. In creep test, a constant load is applied to

a tensile specimen maintained at a constant temperature and the strain of the specimen is

determined as a function of time. Creep test data is presented as a plot between time and strain

known as creep curve as shown in figure 3.9.

Upon loading, the material shows a sudden strain which is its elastic strain ( ). On further

loading three distinct regions can be observed in a creep curve:

Primary creep

Secondary creep

Tertiary creep

The first stage of creep known as primary creep represents a region of decreasing creep rate; that

is, the slope of the curve diminishes with time. The decreasing creep rate is due to strain

hardening. It is also known as transient creep. The second stage of creep known as secondary

creep is a period of nearly constant creep rate or minimum creep rate observed due to balance

between the competing processes of strain hardening and recovery. It is usually referred to as

steady state creep and is the stage of longest duration. The third stage or tertiary creep occurs at a

faster rate. Tertiary creep occurs when there is an effective reduction in cross sectional area

either because of necking or internal void formation.

The important parameter from a creep test is the minimum or steady-state creep rate (d/dt); it is

the important design parameter that is considered for long-life applications, for many relatively

short-life creep situations time to rupture is considered. Thus, the instantaneous elastic strain,

steady-state creep rate and time to fracture can be ascertained as the creep properties observed

during a creep test.

Note:

The creep behavior of materials observed due to the effects of strain hardening and recovery are;

Transient creep

Viscous creep

Transient creep is characterized by a decreasing rate of deformation. As the material is held at

the constant load it undergoes deformation which enhances its strength due to strain hardening,

thus further deformation becomes difficult leading to decrease in rate of deformation. It is

observed at all temperatures, even at absolute zero and hence referred to as cold creep.

Viscous creep is characterized by a constant or steady creep rate. At elevated temperatures the

recovery process comes into effect along with the strain hardening. During recovery the strength

of the material decreases so that more plastic deformation takes place. Thus both strain

hardening and recovery effects balance each other to maintain a constant creep rate. It is also

known as hot creep.



Figure 3.10: Transient and Viscous creep

3.4.2 Creep mechanism

The prominent deformation mechanism involved in creep

Dislocation glide – involves dislocation moving along slip planes by overcoming barrier

with help of thermal activation. This occurs at high stresses.

Dislocation climb – involves movement of dislocations which overcome barriers by

thermally assisted mechanisms like diffusion of vacancies or interstitials. Occurs at

moderate stresses.

Diffusion creep – involves flow of vacancies and interstitials under the influence of

applied stress. It occurs at relatively low stresses.

Grain boundary sliding – involves sliding of grains against each other.

3.4.3 Factors affecting creep properties

Creep properties of a material are usually characterized at a constant stress and a constant

temperature. A change in either of these parameters causes change in creep behavior of the

material. With increase in either stress or temperature

(a) the instantaneous elastic strain increases

(b) steady-state creep rate increases and

(c) time to rupture decreases.

The other factors that affect the creep characteristics of a material are its melting temperature,

elastic modulus, and grain size. In general, the higher the melting temperature, the greater the

elastic modulus, and the larger the grain size, the better is a material‟s resistance to creep.

Relative to grain size, smaller grains permit more grain-boundary sliding, which results in higher

creep rates.



3.4.4 Stress relaxation

Stress relaxation refers to the behavior of stress reaching a peak and then decreasing over time

under a fixed level of strain. In other words it is the time dependent decrease in the value of

stress in those components which are not allowed to elongate.

Figure 3.11: (a) Stress relaxation (b) Effect of stress relaxation

For example, the mechanical fasteners such as bolts, rivets etc. that are required to hold rigid

bodies in tight contact show stress relaxation after long period of time wherein the stress acting

in such members decreases. The effect of stress relaxation is that it converts all the elastic

deformation to plastic deformation with the passage of time. Thus, the elastic strain in a material

remains as a plastic strain (permanent deformation) upon unloading.

Note:

Creep vs Stress relaxation

There are two similar, yet distinct, mechanisms by which the amount of plastic strain increases

over time:

Creep - increase in plastic strain under constant stress.

Stress relaxation - decrease in stress under constant strain.

Expression for stress relaxation

Consider a specimen subjected to tensile stress i at elevated temperature. The total strain of the

specimen is given by:

= e + p

where, e – elastic strain, and p – plastic strain



Under conditions of a steady creep the specimen will tend to elongate (plastic strain increases

over time). However, if the deformation is restricted, the specimen will still experience increase

in plastic strain but the total strain remains constant, i.e., the initial elastic strain experienced by

the specimen converts into plastic strain.

Therefore,

=

[ + ] = 0

= –

But, =

and

= B (minimum creep rate)

Where, B and n are constants and E is Young‟s modulus.

[

] = – B

1

= – B

1

= – B

Integrating the above equation

1

∫

= – B ∫

On rearranging

∫

= – B E ∫

1

1 = – B E t + C ------------------------------------------------------------------- (1)

Where, C is a constant of integration the value of which can be found by applying boundary

conditions.

At t=0, = i

1

1 = – B E (0) + C



Therefore, C = 1

1

Substituting the value of C in equation (1), we get

1

1 = – B E t – 1

1

This is the expression for stress relaxation which can be calculated for any given time under an

initial stress of i. It can be observed from the above equation that the stress on the specimen

goes on decreasing with time ( 1/t).

Problems:

1. A Steel bolt, clamping two rigid plates together is held at 600 C with an initial

tightening stress of 75 MPa. The minimum creep rate is 2.8 x 10-8

m/m/hr at a stress of 30

MPa. Determine the stress remaining after one year in the steel bolt. Take E=2x105 MPa

and n=3.

Solution :

i = 75 MPa

= 30 MPa

= 2.8 x 10

-8 m/m/hr

E = 2 x 105 MPa

n = 3

Relaxed stress ( ), after 1 year (365 x 24 = 8760 hrs) is,

1

=

1

+ ( – 1) B E t

1

=

1

+ (3 – 1) 1.04 x 10-12

x 2 x 105 x 8760

= 30 MPa

1

𝜎𝑛 = 1

𝑖𝑛 + (𝑛 – 1) B E t

We know that,

𝑑 𝑝

𝑑𝑡 = B σ𝑛

B = .8 x 1 8

3

B = 1.04 x 10-12



2. A component is subjected to initial tension of 14 MPa and relaxation of stress occurs at

elevated temperature. Find the time required for the stress to become 12.5 MPa given n=8

and E=210 GPa. The minimum creep rate is 7.2 x 10-9

mm/mm/day at a stress of 7 MPa.

Solution :

i = 14 MPa

= 7 MPa

= 7.2 x 10

-9 mm/mm/day

E = 210 GPa = 2.1 x 105 MPa

n = 8

The time required for a stress relaxation of 12.5 MPa is,

1

=

1

+ ( – 1) B E t

1

1 . 8 = 1

1 8 + (8 – 1) 1.25 x 10-15

x 2.1 x 105 x t

= 6.2 days

3. The critical stress intensity factor KIC of a material used for a component is 26 MPa m1/2

.

What is the applied stress that will cause fracture if the component contains an internal

crack of length 1.72 mm. Assume the geometric constant to be unity.

Solution :

KIC = 26 MPa m1/2

2c = 1.72 mm

c = 0.86 x 10-3

m

= 1

WKT, KIC = f

Fracture strength (f),

f =

We know that,

𝑑 𝑝

𝑑𝑡 = B σ𝑛

B = . x 1 9

8

B = 1.25 x 10-15



f =

1 .8 x 1

f = 500 MPa

4. If the specific surface energy for aluminium oxide is 0.9 J/m2, compute the critical stress

required for the propagation of an internal crack of length 0.4 mm. Take E = 393 GPa for

aluminium oxide.

Solution :

= 0.9 J/m2

2c = 0.4 mm

c = 0.2 x 10-3

m

E = 393 GPa = 393 x 109 N/m

2

Fracture stress (f),

f =

f = x 3 3 x 1 9 x .

x . x 1

f = 33.6 x 106 N/m

2

f = 33.6 MPa

References:

1. Mechanical Metallurgy – G. E. Dieter

2. Fundamentals of Materials Science & Engineering – William D. Callister

3. Material Science and Metallurgy – K.R.Phaneesh

4. Material Science and Metallurgy – Kesthoor Praveen

FAILURE OF MATERIALS AND FAILURE OF MATERIALS · - Ductile and brittle fracture - Modes of fracture...

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