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Stress Analysis in Viscoelastic Materials

MEEN 5330, Fall 2006

Presented by:SURESH KUMAR ANBALAGANSAI PRASEN SOMSETTYVEERABHADRA REDDY KAJULURIJAMAL MOHAMMEDSUBRAHMANYA SRI VITTAL CHEDERE

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Introduction

Hooke’s LawViscoelastic MaterialRelation with Hooke’s LawExamples

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Some phenomena in viscoelastic materials

A) Instantaneous elasticityB) Creep under constant stressC) Stress relaxation under constant

strainD) Instantaneous recoveryE) Delayed recoveryF) Permanent set

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CREEP

Slow progressive deformation of a material under constant stress

Anelastic Material

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THREE STAGES OF CREEP

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STRESS RELAXATION

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Basic Elements

These models allow the mathematical representation of viscous and elastic properties of viscoelastic materials.Basic elements: (Spring and Dashpot)F = -KXF is restoring force by spring, K is spring constant and X is spring elongation.F/A = -(K/A)*Xσ = E ε E is modulus of elasticity.

Spring.

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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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Maxwell model

For series connection the total strain is given byε = ε1 + ε2 strain rate is given by ε *=ε1*+ε2*ε *= σ*/E + σ /ηε * is (dε/dt)

Spring and dashpot in series

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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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Generalized Maxwell model

The generalized Maxwell model consists of a series of Maxwell model as shown below,

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Generalized Maxwell model cont’d

The total strain is given by

N

i i

N

i

G11

.. 11

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Generalized Maxwell model cont’d

Consider the following setup of Maxwell model units,

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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,

N

i

ii

tG

1

.

1

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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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Example for a Generalized Maxwell model cont’d

SolutionFor a generalized Maxwell model, we have the stress – strain relation as follows

N

i

ii

tG

1

.

1

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Example for a Generalized Maxwell model cont’d

For N=2, the above equation becomes,

We know the relaxation time relation as,

222111 /1///1//

GG

111 /1/ G

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Example for a Generalized Maxwell model cont’d

Using the above relation the equation for the strain changes to,

Which when expanded becomes

122112 /1/1)/(/1 ttt GG

)//()(//)( 221121212121 GGGG

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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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Generalized Kelvin Model cont’d

The total strain, for an N number of Kelvin model units is given as follows

N

i tiiG1

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Standard Linear Solid

For a spring,σ = E1 ε1ε1* = σ*/E1For the Kelvin model,σ = E2 ε2 + η2 ε2*η2 ε2* = σ - E2 ε2ε2* = 1/ η2 (σ – E2ε2)

three parameter linear solid model

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Standard Linear Solid cont’d

for a series combination of linear solid,ε = ε1 + ε2strain rate ε* = ε1* + ε2*ε* = σ*/E1 + 1/ η2 (σ – E2ε2)

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Three parameter Viscous model

For a dashpotσ = η ε*ε1* = σ/ η1For the Kelvin model,σ = E2 ε2 + η2 ε2*η2 ε2* = σ - E2 ε2ε2* = 1/ η2 (σ – E2ε2)

three parameter viscous model

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Three parameter Viscous model cont’d

for a series combination of the viscous model,ε = ε1 + ε2strain 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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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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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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REFERENCES