Chapter 7: MECHANICAL PROPERTIES * * c07f05 Linear Elastic

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Chapter 7

• MECHANICAL PROPERTIES

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Chapter Outline

• Terminology for Mechanical Properties
• The Tensile Test Stress-Strain Diagram
• Properties Obtained from a Tensile Test
• True Stress and True Strain
• The Bend Test for Brittle Materials
• Hardness of Materials

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Questions to Think About

• Stress and strain What are they and why are
  they used instead of load and deformation?
• Elastic behavior When loads are small, how much
  deformation occurs? What materials deform least?
• Plastic behavior At what point do dislocations
  cause permanent deformation? What materials
  are most resistant to permanent deformation?
• Toughness and ductility What are they and how
  do we measure them?
• Ceramic Materials What special provisions/tests
  are made for ceramic materials?

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Stress-Strain Test

• specimen

• machine

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Tensile Test
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Important Mechanical Properties from a Tensile
Test

• Young's Modulus This is the slope of the linear
  portion of the stress-strain curve, it is usually
  specific to each material a constant, known
  value.
• Yield Strength This is the value of stress at
  the yield point, calculated by plotting young's
  modulus at a specified percent of offset (usually
  offset 0.2).
• Ultimate Tensile Strength This is the highest
  value of stress on the stress-strain curve.
• Percent Elongation This is the change in gauge
  length divided by the original gauge length.

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Terminology

• Load - The force applied to a material during
  testing.
• Strain gage or Extensometer - A device used for
  measuring change in length (strain).
• Engineering stress - The applied load, or force,
  divided by the original cross-sectional area of
  the material.
• Engineering strain - The amount that a material
  deforms per unit length in a tensile test.

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Elastic Deformation
1. Initial
2. Small load
3. Unload
Elastic means reversible.
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Plastic Deformation (Metals)
1. Initial
2. Small load
3. Unload
Plastic means permanent.
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Typical stress-strain behavior for a metal
showing elastic and plastic deformations, the
proportional limit P and the yield strength sy,
as determined using the 0.002 strain offset
method (where there is noticeable plastic
deformation). P is the gradual elastic to plastic
transition.
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Plastic Deformation (permanent)

• From an atomic perspective, plastic deformation
  corresponds to the breaking of bonds with
  original atom neighbors and then reforming bonds
  with new neighbors.
• After removal of the stress, the large number of
  atoms that have relocated, do not return to
  original position.
• Yield strength is a measure of resistance to
  plastic deformation.

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• Localized deformation of a ductile material
  during a tensile test produces a necked region.
• The image shows necked region in a fractured
  sample

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Permanent Deformation

• Permanent deformation for metals is accomplished
  by means of a process called slip, which involves
  the motion of dislocations.
• Most structures are designed to ensure that only
  elastic deformation results when stress is
  applied.
• A structure that has plastically deformed, or
  experienced a permanent change in shape, may not
  be capable of functioning as intended.

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Yield Strength, sy
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Stress-Strain Diagram
ultimate tensile strength
3
necking
Strain Hardening
SlopeE
Fracture
yield strength
5
2
Elastic region slope Youngs (elastic)
modulus yield strength Plastic region
ultimate tensile strength strain hardening
fracture
Plastic Region
Stress (F/A)
Elastic Region
4
1
Strain ( ) (DL/Lo)
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Stress-Strain Diagram (cont)

• Elastic Region (Point 1 2)
• - The material will return to its original
  shape
• after the material is unloaded( like a
  rubber band).
• - The stress is linearly proportional to the
  strain in
• this region.

or
Stress(psi) E Elastic modulus (Youngs
Modulus) (psi) Strain (in/in)

• Point 2 Yield Strength a point where
  permanent
• deformation occurs. ( If it is passed, the
  material will
• no longer return to its original length.)

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Stress-Strain Diagram (cont)

• Strain Hardening
• - If the material is loaded again from Point
  4, the
• curve will follow back to Point 3 with the
  same
• Elastic Modulus (slope).
• - The material now has a higher yield
  strength of
• Point 4.
• - Raising the yield strength by permanently
  straining
• the material is called Strain Hardening.

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Stress-Strain Diagram (cont)

• Tensile Strength (Point 3)
• - The largest value of stress on the diagram
  is called
• Tensile Strength(TS) or Ultimate Tensile
  Strength
• (UTS)
• - It is the maximum stress which the material
  can
• support without breaking.
• Fracture (Point 5)
• - If the material is stretched beyond Point 3,
  the stress
• decreases as necking and non-uniform
  deformation
• occur.
• - Fracture will finally occur at Point 5.

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The stress-strain curve for an aluminum alloy.
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• Stress-strain behavior found for some steels with
  yield point phenomenon.

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T E N S I L E P R O P E R T I E S
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Yield Strength Comparison
Room T values
a annealed hr hot rolled ag aged cd
cold drawn cw cold worked qt quenched
tempered
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Tensile Strength, TS

• After yielding, the stress necessary to continue
  plastic deformation in metals increases to a
  maximum point (M) and then decreases to the
  eventual fracture point (F).
• All deformation up to the maximum stress is
  uniform throughout the tensile sample.
• However, at max stress, a small constriction or
  neck begins to form.
• Subsequent deformation will be confined to this
  neck area.
• Fracture strength corresponds to the stress at
  fracture.

• Region between M and F
• Metals occurs when noticeable necking starts.
• Ceramics occurs when crack propagation
  starts.
• Polymers occurs when polymer backbones are
  aligned and about to break.

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• In an undeformed thermoplastic polymer tensile
  sample,
• the polymer chains are randomly oriented.
• When a stress is applied, a neck develops as
  chains become aligned locally. The neck
  continues to grow until the chains in the entire
  gage length have aligned.
• The strength of the polymer is increased

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Tensile Strength Comparison
Room T values
Based on data in Table B4, Callister 6e. a
annealed hr hot rolled ag aged cd cold
drawn cw cold worked qt quenched
tempered AFRE, GFRE, CFRE aramid, glass,
carbon fiber-reinforced epoxy composites, with 60
vol fibers.
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Engineering Stress
Tensile stress, s
Shear stress, t
Stress has units N/m2 or lb/in2
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VMSE

• http//www.wiley.com/college/callister/0470125373/
  vmse/index.htm

• http//www.wiley.com/college/callister/0470125373/
  vmse/strstr.htm

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Example 1 Tensile Testing of Aluminum Alloy
Convert the change in length data in the table to
engineering stress and strain and plot a
stress-strain curve.
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Example 1 SOLUTION
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Ductility, EL
Ductility is a measure of the plastic deformation
that has been sustained at fracture
A material that suffers very little plastic
deformation is brittle.
Another ductility measure

• Ductility may be expressed as either percent
  elongation ( plastic strain at fracture) or
  percent reduction in area.
• AR gt EL is possible if internal voids form in
  neck.

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Toughness
Toughness is the ability to absorb energy up to
fracture (energy per unit volume of material).
A tough material has strength and
ductility. Approximated by the area under the
stress-strain curve.
Lower toughness ceramics
Higher toughness metals
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Toughness
Energy to break a unit volume of material
Approximate by the area under the stress-strain
curve.
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Linear Elastic Properties
s E e
Hooke's Law
n ex/ey
Poisson's ratio metals n 0.33
ceramics n 0.25 polymers n 0.40
Modulus of Elasticity, E (Young's modulus)
Units E GPa or psi n dimensionless
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Engineering Strain
Strain is dimensionless.
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Axial (z) elongation (positive strain) and
lateral (x and y) contractions (negative strains)
in response to an imposed tensile stress.
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True Stress and True Strain

• True stress The load divided by the actual
  cross-sectional area of the specimen at that
  load.
• True strain The strain calculated using actual
  and not original dimensions, given by et ln(l/l0).

• The relation between the true stress-true strain
  diagram and engineering stress-engineering strain
  diagram.
• The curves are identical to the yield point.

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Stress-Strain Results for Steel Sample
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Example 2 Youngs
Modulus - Aluminum Alloy
From the data in Example 1, calculate the modulus
of elasticity of the aluminum alloy.
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Example 2 Youngs Modulus - Aluminum Alloy -
continued

• Use the modulus to determine the length after
  deformation of a bar of initial length of 50 in.
• Assume that a level of stress of 30,000 psi is
  applied.

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Youngs Moduli Comparison
Graphite Ceramics Semicond
Metals Alloys
Composites /fibers
Polymers
E(GPa)
Composite data based on reinforced epoxy with 60
vol of aligned carbon (CFRE), aramid (AFRE), or
glass (GFRE) fibers.
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Example 3 True Stress and True Strain Calculation
Compare engineering stress and strain with true
stress and strain for the aluminum alloy in
Example 1 at (a) the maximum load. The diameter
at maximum load is 0.497 in. and at fracture is
0.398 in. Example 3 SOLUTION
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Strain Hardening
An increase in sy due to plastic deformation.
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Strain Hardening (n, K or C values)
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Mechanical Behavior - Ceramics

• The stress-strain behavior of brittle ceramics is
  not usually obtained by a tensile test.
• It is difficult to prepare and test specimens
  with specific geometry.
• It is difficult to grip brittle materials without
  fracturing them.
• Ceramics fail after roughly 0.1 strain specimen
  have to be perfectly aligned.

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The Bend Test for Brittle Materials

• Bend test - Application of a force to the center
  of a bar that is supported on each end to
  determine the resistance of the material to a
  static or slowly applied load.
• Flexural strength or modulus of rupture -The
  stress required to fracture a specimen in a bend
  test.
• Flexural modulus - The modulus of elasticity
  calculated from the results of a bend test,
  giving the slope of the stress-deflection curve.

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The stress-strain behavior of brittle materials
compared with that of more ductile materials
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(a) The bend test often used for measuring the
strength of brittle materials, and (b) the
deflection d obtained by bending
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Flexural Strength

• Schematic for a 3-point bending test.
• Able to measure the stress-strain behavior and
  flexural strength of brittle ceramics.
• Flexural strength (modulus of rupture or bend
  strength) is the stress at fracture.

See Table 7.2 for more values.
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MEASURING ELASTIC MODULUS
Room T behavior is usually elastic, with
brittle failure. 3-Point Bend Testing often
used. --tensile tests are difficult for
brittle materials.
Determine elastic modulus according to
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MEASURING STRENGTH
3-point bend test to measure room T strength.
Typ. values
Flexural strength
Si nitride Si carbide Al oxide glass (soda)
700-1000 550-860 275-550 69
300 430 390 69
Data from Table 12.5, Callister 6e.
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Stress-Strain Behavior Elastomers

• 3 different responses
• A brittle failure
• B plastic failure
• C - highly elastic (elastomer)

--brittle response (aligned chain, cross
linked networked case) --plastic response
(semi-crystalline case)
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Hardness of Materials

• Hardness test - Measures the resistance of a
  material to penetration by a sharp object.
• Macrohardness - Overall bulk hardness of
  materials measured using loads gt2 N.
• Microhardness Hardness of materials typically
  measured using loads less than 2 N using such
  test as Knoop (HK).
• Nano-hardness - Hardness of materials measured at
  110 nm length scale using extremely small (100
  µN) forces.

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Hardness

• Hardness is a measure of a materials resistance
  to localized plastic deformation (a small dent or
  scratch).
• Quantitative hardness techniques have been
  developed where a small indenter is forced into
  the surface of a material.
• The depth or size of the indentation is measured,
  and corresponds to a hardness number.
• The softer the material, the larger and deeper
  the indentation (and lower hardness number).

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Hardness
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.)
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Hardness Testers
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Conversion of Hardness Scales
Also see ASTM E140 - 07 Volume 03.01 Standard
Hardness Conversion Tables for Metals
Relationship Among Brinell Hardness, Vickers
Hardness, Rockwell Hardness, Superficial
Hardness, Knoop Hardness, and Scleroscope Hardness
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Correlation between Hardness and Tensile Strength

• Both hardness and tensile strength are indicators
  of a metals resistance to plastic deformation.
• For cast iron, steel and brass, the two are
  roughly proportional.
• Tensile strength (psi) 500BHR

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Summary
Stress and strain These are
size-independent measures of load and
displacement, respectively.
Elastic behavior This reversible behavior
often shows a linear relation between
stress and strain. To minimize deformation,
select a material with a large elastic
modulus (E or G).
Plastic behavior This permanent deformation
behavior occurs when the tensile (or
compressive) uniaxial stress reaches sy.
Toughness The energy needed to break a unit
volume of material.
Ductility The plastic strain at failure.