06.Mechanical.properties

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Textbook: William D. Callister Jr., Materials Science and Engineering: An Introduction, 6th edition, TA403.C23 2003 Chapter 6 Mechanical Properties Chapter 8 Failure Chapter 7 Dislocations & Strengthening Mechanisms (if time is allowed) Instructor’s coordinates: Prof. Shi San-Qiang 石三强 (Room FG603) Department of Mechanical Engineering Office hour: 16:30~18:00, every Monday Email: mmsqshi@polyu.edu.hk Phone: 2766 - 7821 Fax: 2365 - 4703

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Lectures: Monday 10:30 - 11:30, AG710 Tuesday 11:30 - 12:30, AG710 Wednesday 17:30 - 18:30, AG710 Lab Arrangement: Time: Oct. 30 and Nov. 6, 2012 See Lab Arrangement Sheet Lab work: Tensile tests, room DE006 Tutorials: Oct. 29 and Nov. 9: group 1 Nov. 2 and Nov.12: group 2 Lecture notes and tutorial questions are sent to your email.

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A design problem: How to determine the diameter (or dimensions) of a chair leg? To answer the above question, what do you need to know?

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Chapter 6: Mechanical Properties Why study mechanical properties ? Mechanical properties -> design -> qualification of mechanical/design engineers

§6.1 Introduction This chapter covers: - concept of stress-strain - stress-strain behavior of materials - mechanical properties - scatter of materials data and safety factor

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§6.2 Concepts of Stress and Strain Schematic of mechanical testing: tension, compression, shear, and torsion.

Tension Compression Shear Torsion

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Tension tests: - a specimen is deformed gradually with increasing load, to fracture - cross section of the specimen is usually circular - standard diameter is ~12.8 mm - reduced section length is at least 4 times of this diameter - the specimen is elongated at a constant rate - a standard tensile specimen is shown below:

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Engineering stress: is defined as instantaneous load divided by original area of the cross section - one common unit is MPa (1 MPa = 106 N/m2) (Pa=N/m2) - another common unit is psi (1 psi = 1 lb/in2) Engineering strain: is defined as elongation over original length - it is dimensionless - sometimes it is given in percentage

σ =F

A0

ε =−l ll

i 0

0

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Compression tests - similar to tension tests, except that (1) elongation becomes contraction, and (2) load direction is reversed - conducted usually when in-service load is compressive Shear and torsional tests: Shear stress is defined as shear force over an area: Shear strain γ is defined as the tangent of shear angle

τ =F

A0

shear torsion*

Compression 1. iftan <<≈= θθθγ

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Geometric consideration of the stress state in tension test - stress state is a function of orientation of applied planes - on horizontal plane, it is tensile only - on plane pp’, it is not a pure tensile anymore - force balance requires:

σ σ θ' cos= 2

τ σ θ θ' cos sin=

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§6.3 Stress-Strain Behavior Hooke’s law Tension: - the stress is proportional to strain, as shown in the figure below. - the proportionality constant is the modulus of elasticity (Young’s modulus). It is ~ 40 GPa to 400 GPa (G = 109, Pa=N/m2) - this is true for linear elastic deformation, i.e., when stress is small. * linear elastic deformation (Hooke’s law) is nonpermanent. Shear: - similar as in normal stress-strain - G is called shear modulus - γ = tan θ - when θ << 1, γ ≈ θ.

σ ε= E

τ γ= G

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Non-linear elastic behavior: - tangent modulus is used sometimes, and it is defined as the local slop. - secant modulus is also used, and it is defined as the slop of a straight line connecting origin with the local point.

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Origin of elastic deformation: - stretching of atomic bonds corresponds to deformation (fig below) - tangent of force curve corresponds to the modulus

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Origin of elastic deformation - stretching of atomic bonds corresponds to deformation (fig below) - tangent of interatomic force curve corresponds to the modulus

0rdrdFE

∝

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Temperature dependence of the modulus

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Table 6.1 Room-Temperature Elastic and Shear Moduli, and Poisson’s Ratio for Various Metal Alloys

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§6.5 Elastic Properties of Materials • When a material elongates along z under a uniaxial tension, it will contract along x and y directions. The Poisson’s ratio is defined as: • Theoretical value of the Poisson’s ratio is 0.25, and the maximum is 0.50. • Relationship between Young’s and shear moduli in isotropic solid:

νεε

εε

= − = −x

z

y

z

E G= +2 1( )ν

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Remember that E =

zεσ

Poisson’s ratio ν

ν = εZ = εX =

Z

Y

Z

X

εε

εε −=−

0LL∆

= 0

0

LLLf −

0dd∆

0

0

ddd f −

=

18

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Positive Poisson’s ratio Negative Poisson’s ratio

http://silver.neep.wisc.edu/~lakes/Poisson.html

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§6.6 Tensile Properties Elastic limit: is usually 0.005 in strain. Beyond this, the deformation is plastic, and typical plastic behaviors are shown on the left below.

Yield strength

Typical metals Some steels

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• Yielding point and yield strength: - a convention is that the offset strain is 0.002 the stress at X (fig) is the yield strength σy. - when elastic is not linear, yielding is defined to occur at a fixed strain (e.g., 0.005). - when upper and lower yielding points exist, the yield strength is taken to correspond to the lower yielding point. - yield strength: 35 MPa for Al to 1400 MPa for high-strength steels.

X

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Concept Check: Cite the primary difference(s) between elastic and plastic deformation.

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• Tensile strength (TS) - stress-strain behavior after yielding is shown in the figure below. - tensile strength is the stress at point M. - necking starts at this point - fracture occurs if this stress is maintained. - tensile strength: 50 MPa for Al to 3000 MPa for the high-strength steels

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Concept Check: On the tensile engineering stress-strain curve in the earlier page, plot a compressive engineering stress-strain curve for the same alloy. Explain any differences between tensile and compression.

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Ductility: a measure of the degree of plastic deformation that has been sustained at fracture. - a material is brittle if it fractures with little plastic deformation - ductile vs brittle (see fig) Quantitative characterization of ductility - percent elongation gauge length is ~ 50mm. - percent reduction of area

%EL l ll

f=−

×0

0

100

~5% 100

AAARA%

o

fo ×−

=

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Ductility as a function of temperature

Higher temperature --> more ductile.

Iron

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Resilience: capacity of absorbing energy during elastic deformation, and then recovering it during unloading. - quantitative measure: modulus of resilience Ur, and it is defined as the strain energy per unit volume required to stress an unloaded state up to the point of yielding. - graphically, it is the area in fig (left) - mathematically, it is: * high yield strength and low moduli of elasticity --> resilient materials, and they are good to be used as springs.

y

r 0U d

ε= σ ε∫ Ur y y=

12

σ εHooke

UEry=

σ2

2

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Toughness: a measure of ability to absorb energy before fracture. - under dynamic loading with a notch, notch toughness is used - when a crack is present, fracture toughness is used - under static loading, it is like the ductility, except the final stress is the fracture stress. Toughness = area under the stress-strain curve up to fracture - ductile materials are usually tougher.

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Table 6.2 Typical Mechanical Properties of Several Metals and Alloys in an Annealed State

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Mechanical properties for plastic polymers: • Modulus of elasticity and ductility are defined in the same way as for metals. • Yield point (or yield strength) for plastic polymers is defined as the maximum stress in the curve. • Tensile strength (TS) corresponds to the stress at which fracture occurs, as shown below. Strength of polymers usually refers to tensile strength.

TS might be smaller than σy

polymer

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§6.7 True Stress and Strain True stress: load divided by the instantaneous cross-sectional area. True strain: integration of instantaneous strains. Relationship with engineering stress (strain) if volume is conserved: * note: these relationships are good up to necking point only.

σT iF A= /

σ σ εT = +( )1 ε εT = +ln( )1

“Corrected” refers to correction of tensile stress due to necking (3D stress)

)/Lln(LL

dLε 0i

L

LT

i

0

== ∫

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Strain hardening - ideally, plastic deformation continues without increase of stress - in reality, the stress and strain during plastic deformation up to necking obey: the exponent n is called strain-hardening component (table below)

σ εT TnK=

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§6.8 Elastic Recovery During Plastic Deformation Upon unloading, after plastic deformation, a fraction of the deformation recovers elastically, as shown in the figure below. - initial yield strength σy0 - yield strength σyi after the elastic recovery

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§6.9 Compressive, Shear, and Torsional Deformation - it is in general similar to tensile deformation - compression does not induce necking - compression leads to different fracture mode

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Apart from tensile test, there are many other types of mechanical tests, such as impact test, fatigue test, creep test. we will look at these tests in “Chapter 8 Failure”.

We now briefly discuss Hardness Test.

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• Resistance to permanently indenting the surface. • Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties.

Adapted from Fig. 6.18, Callister 6e. (Fig. 6.18 is adapted from G.F. Kinney, Engineering Properties and Applications of Plastics, p. 202, John Wiley and Sons, 1957.)

e.g., 10mm sphere

apply known force (1 to 1000g)

measure size of indent after removing load

d D Smaller indents mean larger hardness.

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§6.10 Hardness

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Different Types of Hardness Test

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Comparison of hardness scales

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TS MPa HB( ) .≈ ×345

Correlation between hardness and tensile strength - using HB, the tensile strength is roughly proportional to hardness for steel.

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Equipments for Hardness Measurement

A portable and fast hardness gauge, for testing aluminum, mild steel, brass and copper with thickness range of 0.025 to 1/4 inch.

For hardness determination of plastics and elastomers

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Equipments for Hardness Measurement

Brinell Hardness Tester

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Equipments for Hardness Measurement

Rockwell Hardness Testers

Digital type

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Equipments for Hardness Measurement

Micro-Hardness Testers

Room: GH702

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Equipments for Hardness Measurement

Nano-Hardness Testers Room: GH702

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§6.11 Variability of Material Properties - uncertainties exist in experimental measurement - inhomogeneities may exist in samples - typical value of a property is usually taken as the average of many measurements - degree of scatter is measured by the standard deviation

xx

n

ii

n

= =∑

1

sx x

n

ii

n

=−

−=∑ ( )2

1

1

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§6.12 Safety Factor - A safe stress or working stress is taken to be 1/N times of the yield strength, and N is usually between 1.2 and 4 - The factor N is the safety factor.

σσ

wy

N=

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§6.00 Summary • Stress-strain relationship (two types) • Mechanical tests: tension, compression, shear, and torsion • Materials properties: elastic modulus, Poisson’s ratio, yield strength, tensile strength, ductility, modulus of resilience, toughness, and hardness • Hardness measurement: Rockwell, Brinell, Knoop, and Vickers • Relationship between hardness and tensile strength • Scatter of materials data --> safety factor

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Homework Assignments: Questions 6.20 and 6.30. Due on next week.