UAA School of Engineering

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Plastic Stress-Strain Relationships

• UAA School of Engineering
• CE 334 - Properties of Materials
• Lecture 5

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Outline

• Plastic Stress-Strain Relationships
• Plastic deformation
• - Metal as a ductile material
• What is plastic stress-strain relation?
• Definitions about plastic stress-strain curves

Plastic
?
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Inelastic Material Behavior

• Limitations on the Use of Uniaxial Stress-Strain
  Data
• Tension or compression tests are usually run at
  room temperature in testing machines that have
  head speeds in the range of 0.20 to 10 mm/min.
• Rate of Loading
• If a tension test is run at a high rate of
  loading then the material response is less
  ductile.
• Temperature
• -Temperature Lower than Room Temperature
• A metal tension specimen may fail in a brittle
  manner.
• -Temperature Higher than Room Temperature
  Creep

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Effect of Unloading and Load Reversal

•  
•   The accrual unloading path does not follow the
  ideal linear elastic unloading path
•  The yielding strength in compression is reduced
  below the original value ? Bauschinger Effect.

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Effect of Unloading
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Nonlinear Material Response

• If the unloading path coincides with the loading
  path, the process is reversible and the material
  is said to be Elastic.
• If the unloading path does not follow the loading
  path, the behavior is said to be Inelastic.
• If after the load is released, the permanent
  deformation remains, the behavior is said to be
  Plastic.
• If after load removal, the response continues to
  change with time, its response is Viscoelastic or
  Viscoplastic.
• If a viscous material, after complete unloading,
  the material will return to an unstrained state
  in time, it is Viscoelastic
• In a Viscoplastic material, after complete
  unloading, the response will change with time.
  However, some permanent strain will remain.

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Metal as a Ductile Material

• The Crystals that Make the Metal
• Let's take a look at a paper clip. A typical
  paper clip is made up of 1,000,000,000,000,000,000
  ,000 atoms of iron.
• These atoms are tightly packed and in a
  crystalline structure, a regular arrangement of
  atoms that repeats itself many times. This is
  the crystalline structure for iron atoms. The
  atoms of other metals, such
• as aluminum and zinc, have different
  arrangements.

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Metal as a Ductile Material

• The structure of iron atoms isn't continuous
  throughout the entire paper clip. When a metal
  cools and is transitioning from liquid to solid,
  its atoms come together to form tiny grains, or
  crystals. Even though the crystalline structure
  does not continue from crystal to crystal, the
  crystals are bound to one another. In this
  diagram, each square represents an individual
  atom.

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Metal as a Ductile Material

• The Flaws that Break the Bonds
• When a metal crystal forms, the atoms try to
  assemble themselves into a regular pattern.
• But there are many imperfections within each
  crystal, and these flaws produce weak points in
  the bonds between atoms.
• It is at these points, called slip planes, that
  layers of atoms are prone to move relative to
  adjacent layers if an outside force is applied.

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Edge Dislocation
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Edge Dislocation
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Screw Dislocation
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Screw Dislocation
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Mixed Dislocation
Screw Dislocation
Edge Dislocation
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Mixed Dislocation
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Dislocation Movement -Example Edge Dislocation
Motion
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Obstacles to Dislocation Movement

• Piling up As dislocations move through a
  crystal, they become piled up if they encounter
  some obstacle.
• Locking up When two similar dislocations move
  through a crystal and become joined, creating an
  area of greater potential energy.
• Canceling When two unlike dislocations come
  together and cancel out each other so as to
  eliminate the discontinuity in the crystal.

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Example of Obstacles Carbon in Iron Crystals

• Adding other elements to a metal can counteract
  the effects of the imperfections and make the
  metal harder and stronger.
• Carbon, for example, is added to iron to make
  steel, and tin is added to copper to make bronze.
  

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Carbon in Iron Crystals

• Carbon is an impurity
• The carbon atom creates a stress field that
  blocks the intended movement of the dislocation.
• It takes substantial energy to overcome the
  obstacle.

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Effects of Carbon Content in Steel(Detailed
discussion in the later slides)
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Basic Slip in Crystal Lattice due to Shear
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Mohrs Circle for Uniaxial Tension

• Maximum shear stress is half of maximum tensile
  stress for uniaxial loads.
• Tensile fracture will occur on a plane
  perpendicular to the direction of force.
• Shear fracture will occur on a plane 45 deg. from
  direction of force.

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Discontinuous Yielding

• Particular to mild steel
• Yielding begins suddenly and results in large
  deformations at a more-or-less constant stress.
• A distinct yield point is seen at the beginning
  of the region of discontinuous yielding.

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• Plastic (permanent) Deformation
• Deformation beyond the elastic
  range

Recovered elastic deformation
Permanent plastic deformation or set
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Ductility

• Ductility The ability of a material to undergo
  plastic deformation without fracture.
• Ductility and Percent Elongation (The change in
  gage length in the specimen after fracture)/(gage
  length), assuming the fracture occurs within the
  gage length.
• Ductility and Percent Reduction in Area (The
  change in cross sectional area in the specimen
  after fracture)/(original area).

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Measurement of Ductility

• General Measurement of Ductility
• Absolute Ductility based on
• a. elongation
• b. area reduction
• A0 original area
• An net area at necking section

EIT requested formula!!
Lab report required formula
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More Definitions

• Ultimate Strength The highest stress exhibited
  by the specimen as shown by its stress-strain
  curve.
• Fracture Strength The apparent stress when
  fracture occurs.
• Toughness The amount of energy required to
  rupture a material.
• Modulus of Toughness The amount of work per unit
  volume of a material required to carry the
  material to failure. It can be measured by the
  area under the entire stress-strain curve.(see
  later discussion)

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Ultimate and Fracture Strengths
true stress

• For brittle material, the ultimate and fracture
  strengths coincide.
• For a ductile material, the ultimate strength is
  higher than the fracture strength (as computed
  with conventional stress and strain formulas).

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Modulus of Toughness
Strain energy due to ?? (??)(??) area under ??
in ? ? curve
Modulus of Toughness The amount of work per unit
volume of a material required to carry the
material to failure. It can be measured by the
area under the entire stress-strain curve.
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Low-carbon steel Tensile test
-the important
material properties
FE exam problems

• Yield Strength 250 Mpa
• Strain at yield 0.0013
• The Modulus of Elasticity
• 150 MPa/0.0007520 x 104 Mpa
• Ultimate Strength 500 Mpa

• Fracture Strength 400 MPa
• The ductility
• 0.28/0.0013215
• The percent elongation at
• failure 25

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Bibliography

• Durrant, Olani and Holiday, Brent, An
  Introduction to the Properties of Materials,
  Brigham Young University, 1980.
• Shackelford, James F., Introduction to Material
  Science for Engineers, Macmillan Publishing Co.,
  New York, 1985.