1. Stress Analysis in Viscoelastic Materials
MEEN 5330, Fall 2006
Presented by:
SURESH KUMAR ANBALAGAN
SAI PRASEN SOMSETTY
VEERABHADRA REDDY KAJULURI
JAMAL MOHAMMED
SUBRAHMANYA SRI VITTAL CHEDERE
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2. Introduction Hooke’s Law Viscoelastic Material
Relation with Hooke’s Law
Examples
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3. Some phenomena in viscoelastic materials
A) Instantaneous elasticity
B) Creep under constant stress
C) Stress relaxation under constant strain
D) Instantaneous recovery
E) Delayed recovery
F) Permanent set
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4. CREEP Slow progressive deformation of a material under constant stress
Anelastic Material
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5. THREE STAGES OF CREEP
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6. STRESS RELAXATION
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7. Basic Elements
These models allow the mathematical representation of viscous and elastic properties of viscoelastic materials.
Basic elements: (Spring and Dashpot)
F = -KX
F is restoring force by spring,
K is spring constant and
X is spring elongation.
F/A = -(K/A)*X
σ = E ε
E is modulus of elasticity.
Spring.
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8. Basic Elements Piston cylinder arrangement with a perforated bottom
A viscous lubricant between the cylinder and piston walls
force ‘F’ is proportional to the velocity
F = K (dX/dt)
σ = η (dε / dt)
η is Viscosity coefficient
Dashpot
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9. Maxwell model For series connection the total strain is given by
ε = ε1 + ε2
strain rate is given by
ε *=ε1*+ε2*
ε *= σ*/E + σ /η
ε * is (dε/dt)
Spring and dashpot in series
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10. Kelvin/Voight Model σ = σ1+ σ2 (total stress) σ1 = E ε (for spring)
σ2 = η ε* (for dashpot)
σ = E ε + η ε*
Spring and Dashpot in parallel
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11. Generalized Maxwell model
The generalized Maxwell model consists of a series of Maxwell model as shown below,
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12. Generalized Maxwell model cont’d
The total strain is given by
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13. Generalized Maxwell model cont’d
Consider the following setup of Maxwell model units,
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14. Generalized Maxwell model cont’d
The above diagram shows a generalized model of Maxwell model units connected in parallel.
The total strain is given by,
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15. Example for a Generalized Maxwell model
consider the following arrangement of Maxwell model units. Determine the stress-strain relations for the given arrangement.
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16. Example for a Generalized Maxwell model cont’d
Solution
For a generalized Maxwell model, we have the stress – strain relation as follows
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17. Example for a Generalized Maxwell model cont’d
For N=2, the above equation becomes,
We know the relaxation time relation as,
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18. Example for a Generalized Maxwell model cont’d
Using the above relation the equation for the strain changes to,
Which when expanded becomes
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19. Generalized Kelvin Model
The generalized Kelvin model is represented using a number of Kelvin model units connected in series as shown
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20. Generalized Kelvin Model cont’d
The total strain, for an N number of Kelvin model units is given as follows
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21. Standard Linear Solid For a spring, σ = E1 ε1 ε1* = σ*/E1
For the Kelvin model,
σ = E2 ε2 + η2 ε2*
η2 ε2* = σ - E2 ε2
ε2* = 1/ η2 (σ – E2ε2)
three parameter linear solid model
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22. Standard Linear Solid cont’d
for a series combination of linear solid,
ε = ε1 + ε2
strain rate
ε* = ε1* + ε2*
ε* = σ*/E1 + 1/ η2 (σ – E2ε2)
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23. Three parameter Viscous model
For a dashpot
σ = η ε*
ε1* = σ/ η1
For the Kelvin model,
σ = E2 ε2 + η2 ε2*
η2 ε2* = σ - E2 ε2
ε2* = 1/ η2 (σ – E2ε2)
three parameter viscous model
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24. Three parameter Viscous model cont’d
for a series combination of the viscous model,
ε = ε1 + ε2
strain rate
ε* = ε1* + ε2*
ε* = σ/ η 1 + 1/ η2 (σ – E2ε2)
ε* = σ (1/ η 1 + 1/ η2) - E2ε2/ η2
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25. CONCLUSION
The study of the stress-strain analysis of viscoelastic materials helps us understand the procedure and the steps involved in determining the stress-strain relations for a whole bunch of different models.
This in time will prove to be useful in solving any kind of stress-strain analysis problems that exhibit the kind of behavior as discussed in this report.
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26. Homework problem
Determine the stress - strain equation for a four parameter model (four parameter model is a Maxwell and a Kelvin/Voight Model connected in series).
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27. REFERENCES
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