Elastic Behavior

1
Elastic Behavior

• Dr. Richard Chung
• Department of Chemical and Materials Engineering
• San José State University

2
Learning Objectives

• Describe the relationship between mechanical
  properties and structure
• Evaluate material loading conditions in terms of
  elastic and permanent deformation
• Assess whether a material will undergo a fracture
  process based on the elastic properties and
  elastic energies released from the material
• Demonstrate the knowledge of isotropic and
  anisotropic behavior developed in polycrystalline
  material and polymeric material
• Explain and apply the concepts of damping in
  organic and inorganic materials

3
Elastic Properties

• Elastic behavior of single crystalline solids
  anisotropic
• Elastic behavior of noncrystalline (amorphous)
  solids isotropic
• Structure orientation is the key!
• How about a polycrystalline material?
• Elastically isotropic! Why?
• How about a heat-treated or cold worked
  polycrystalline material?
• Elastically anisotropic! Why?
• How about a polymeric material? Isotropy or
  anisotropy?
• Temperature involvement, chain linking and/or
  interlocking, side groups interference(molecular
  architecture)

4
Elastic moduli

• Modulus relates to chemical bonding
• Covalent bond gt polar covalent gt ionic bond gt
  hydrogen bond gt van der Waals bond
• Ceramics gt metals (and refractory metals) gt
  polymers
• Elastic modulus value is decided by the type of
  chemical bonding!

5
(No Transcript)
6
Selected from Michael F. Ashby and David R.H.
Jones, Engineering Materials I An Introduction
to Their Properties and Applications, Pergamon
Press, Oxford, 1980, page 31.
7
Volume Change w.r.t. Elastic and Permanent
Deformation

• During elastic tensile deformation the volume
  increases, whereas during the compressive
  deformation the volume decreases. Why?
• Poissons ratio (?) is a material elastic
  property.
• For metals, the Poissons value is generally in
  the order of 1/3
• If ?1 0.005 and ? 0.333, the volume change ?
  0.00167
• The volume change decreases with the increase of
  the Possions value.

8
(No Transcript)
9
Multiaxial Loading on a Material

10

• Assume ?1 ?2 ?3 ?
• K is the bulk modulus of the material
• For a material with Poissons value (?) 1/3,
  the bulk and Youngs Moduli will be very close.

11
Isotropic Material

• Only two of the four (E, K, G, ?) independent
  elastic constants of linear elasticity are needed
  for an isotropic material

12
Interatomic Spacing vs. Modulus

• S is the spring constant, a ratio of applied
  force to the displacement (N/m) r0 is the
  interatomic spacing (m)
• Covalent solids S 20-200 N/m
• Ionic and metals S 15-100 N/m
• Van der Waals S 0.5 2 N/m
• When an external force is applied to a material,
  for example, along a 100 direction, the
  intreatomic spacing will increase in this
  direction. However, the atomic spacing along the
  other two orthogonal directions 010 and 001
  will decrease.

13
(No Transcript)
14
(No Transcript)
15
Deformation in Three Dimensions
16

• Assume ?1 ?2 ?3 ?
• K is the bulk modulus of the material
• For a material with Poissons value (?) 1/3,
  the bulk and Youngs Moduli will be very close.
• Only two of the four (E, K, G, ?) independent
  elastic constants of linear elasticity are needed
  for an isotropic material.

17
Anisotropic Linear Elasticity

• If a crystal does not show any symmetry, the
  anisotropic linear elasticity can be described as
  follows
• As crystal symmetry increases, the number of
  independent elastic constants decreases.
• Cubic crystals need only three independent
  elastic constants C11C22C33 and C44C55C66

18

• The formulas can be reduced to a much simpler
  form
• For another expression between strain and stress
• Similarly,

.
19

• In a hkl direction, the modulus of a crystal
  can be expressed as

20
(No Transcript)
21
(No Transcript)
22
Rubber Elasticity

• What is rubber elasticity?
• Why can elastomers be stretched to 200?
• Why do some polymers and elastomers demonstrate
  rubber elasticity?
• What is the difference between linear elasticity
  and rubber elasticity?

23
Elastomeric Behavior

• Shown in noncrystalline, long-chain polymers with
  many kinks and bends.
• Definitions of isomers and gutta-percha
• Cis-polyisoprene and trans-polyisoprene

24
Natural Rubber (Selected from Callister Text, pp.
461)

• (a) Cis-isoprene
• (b)Trans-isoprene

25
Natural Rubber Constructed in A Helical Chain
ConfigurationReproduced from the textbook Fig.
2.9

• (a)Trans-isoprene (b) Cis-isoprene

26
Linear Elasticity vs. Rubber Elasticity

• Linear elasticityDue to stretching and
  distortion of primary bonding
• Rubber elasticity Due to entropy change,
  structure uncoiling, and cross-linking.

27
Temperature Effects on Modulus

• The stiffness (modulus) of a material in linear
  elasticity decreases when temperature increases.
• C is around 0.5
• The stiffness of a rubber increases with
  temperature

28
Polymer Elasticity and Viscoelasticity

• Polymer moduli are sensitive to time and
  temperature
• Polystyrene is time and temperature dependent
  material. The modulus of Polystyrene (amorphous)
  decreases with temperature van der Waals
  intermolecuar bonding
• How about time effect?

29
Creep Characteristics

• Stress relaxation the longer the time, the
  higher the strain, but the lower the modulus.
• Higher temperatures help the separation between
  polymer chains thus reduce the linear elastic
  modulus

30
Determination of Modulus w.r.t Time Change
31
Review of Theoretical Models
32
Inelastic Deformation

• Slip of crystal planes
• Sliding of chain molecules
• Time dependent creep deformation
• Time independent plastic deformation

33
Four Rheological Models for Deformation Behavior

• Elastic Pkx
• Plastic if pltPo x0
• Steady State Creep
• Transient Creep

34
(No Transcript)
35
Coefficient of Tensile Viscosity

• Youngs modulus
• Strain rate
• Coefficient of tensile viscosity

36
Plastic Deformation

• Spring and slider models are used to explain
  dislocation motion (sliding) between planes of
  atoms in the crystal grains of metals and
  ceramics
• Monotonic straining in a single direction
• The stress remain constant (?o) beyond yielding

37
(No Transcript)
38
Loading and Unloading

• Elastic, perfect plastic deformation
• Elastic, linear hardening
• Nonlinear hardening

39
Creep Deformation

• Steady-state creep
• Transient creep

40
(No Transcript)
41
Relaxation Behavior

• Stress decreases with time at constant strain

42
(No Transcript)
43
Models Used in Viscoelastic Material

• Voigt Model/P Voigt Model/S

44
Strain-time Relationship
45
(No Transcript)
46
(No Transcript)
47
(No Transcript)
48
Mechanical Damping

• Time dependent elasticity mechanical hysteresis
• Materials attenuation is a phenomenon in which
  the amplitude of a vibration signal will be
  lessened with little or no distortion. In other
  words, the material will absorb the force or
  energy in during the vibration. A rigid
  body/structure will have a good resistance to a
  vibration. 

49
A Simple Frequency-time Curve

• Frequency

Time
50
Two Moduli

• Unrelaxed modulus, Eu
• Defined as linear elastic strain
• Relaxed modulus, Er
• Defined as time-dependent visoelastic strain

51
Strain vs. Cyclic Stress
52
Summary

• Linear elastic moduli are associated with the
  interatomic force (the separation among atoms).
• Most materials demonstrate their linear elastic
  elasticity up to Tm.
• Their moduli generally decrease with temperature.
• Polymers (including elastomers) show extensive
  viscoelstic behavior near Tg.
• Viscoelastic deformation is typically associated
  with time-dependent inter molecular chain sliding
  and time-dependent interatomic separation.

53
Summary (contd)

• Mechanical damping is a time-dependent elastic
  behavior. The dashpot (viscoelastic) component is
  not applicable.
• Under a cyclic loading, the applied frequency
  (1/t) is an important factor. The stress-strain
  behavior depends on in-phase and out-of-phase
  conditions.
• Energy loss per cycle is determined by the ratio
  Er/Eu