Viscoelasticity

1. Viscoelasticity By Vikram chetkuri & Varsha maddala MEEN 5330

2. Abstract Viscoelasticity is time dependent material. In this report we study about the introduction of viscoelasticity, examples of viscoelasticity, various Phenomena, Creep, Relaxation, Recovery, and Viscoelastic models like Maxwell Model, Kelvin Model, Three-parameter viscous models, Burgers Model and Generalized Maxwell and Kelvin Models. MEEN 5330

3. Viscoelasticity • Time dependent material behavior where the stress response of that material depends upon both the strain applied and strain rate at which it was applied.[1] • Examples • Silly Putty • Synthetic polymers,wood human tissue and metals at high temperature They are called as time dependant materials MEEN 5330 3

4. Introduction • The time dependent behavior of materials can be studied by[3] • Creep • Stress relaxation • Constant rate Straining MEEN 5330 4

5. Creep • The slow and continuous deformation of a material under constant stress is called creep One dimension • Creep is generally described in 3 stage • Primary creep- Concave is facing down • Secondary creep-Near constant rate • Tertiary creep-It increases and fractures s (t) = s0H (t) MEEN 5330 5

6. Creep Response Creep response If the load is removed , a reverse elastic strain followed by recovery of a portion of the creep strain will occur at a continuously decreasing rate.[2] MEEN 5330 6

7. Stress relaxation • The gradual decrease of stress when the material is held at constant strain is called the stress relaxation.[3] One dimension e (t) = e0H (t) MEEN 5330 7

8. Viscoelastic models • These models can be used to describe viscoelastic materials and to establish their differential equations • Used to compare stress-strain-time relation of viscoelastic materials.[4] MEEN 5330 8

9. Basic elements • The two basic elements were • Helical spring • Hooks law(σ=Ee) • Dash pot • Piston moving in a cylinder with perforated bottom σ=η (de/dt) MEEN 5330 9

10. Maxwell model • It consists of linear spring and linear dashpot connected in series • σ=E ε1 (spring) & σ= η (dε2 /dt) (Dash pot) • The total strain and strain rate is given by • ε =ε1+ε2 & ε *=ε1*+ε2* MEEN 5330 10

11. Maxwell cont… • Strain- time relaxation can be obtained by ε*(t)=σs0*/E+σd0*/η*t MEEN 5330 11

12. Generalized Maxwell Model A sequence of Maxwell units in series is called a “Generalized Maxwell model” Total strain in generalized Maxwell series model is MEEN 5330 12

13. Maxwell units in parallel Total strain MEEN 5330 13

14. Kelvin/Voigt Model Stress strain relationship ε* + (E/η)ε = σ / η MEEN 5330 14

15. Generalized Kelvin model Kelvin units arranged in series is called a generalized Kelvin model Total strain for this model is MEEN 5330 15

16. Standard Linear Solid • A three-parameter model constructed from two springs and one dashpot is known as the standard linear model. MEEN 5330 16

17. η 2 ε* = σ*{(η2 / E1) + E2} Standard Linear solid cont…… • For the unit one the linear elastic strain is given by, σ=E1 ε 1 (For spring) • There fore the strain rate can be written as, ε1* = σ* /E1 • The differential equation for the standard linear solid from the stress-strain relation would be given by MEEN 5330 17

18. Three parameter Viscous Model • A Three parameter viscous model is constructed by two dashpots and one spring • Strain, ε=ε 1+ε 2 • Strain rate, ε* =ε1* + ε2* MEEN 5330 18

19. ε* = (σ *2 / E2) + σ (η1 + η2)/ η1 η2 Three parameter Viscous Model Cont….. • The linear elastic strain is given by, σ=η1 ε 1*(For spring ) ε 2*= (σ *2 / E2 ) + (σ /η2 ) The Total strain rate in Three-parameter viscous model is given by MEEN 5330 19

20. Example Of Kelvin Model • Problem • The deformation response of a certain polymer can be described by Voigt model. If E=200 Mpa and η= 2 * 1012 Mpa-s. Compute the relaxation time with a steady stress of 10 Mpa. Plot strain rate curve with an increment of one second in time? Solution: Relaxation time can be calculated by = Stress strain relation for the Kelvin model can be estimated by the expression ε= σ/E(1- e –Et/η) MEEN 5330 20

21. Results of Kelvin Model example • The table shows the value of strain for a increment of 1 second in time • The values of strain are tabulated by substituting the values of different times ‘t’ in Stress strain relation for the Kelvin model MEEN 5330 21

22. Graph From the graph we can see Strain increases for certain time and remains constant. Initially stress increases and remains constant at certain point. MEEN 5330 22

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24. Homework problem • Derive the constitutive relation for the viscoelastic model shown below. MEEN 5330 24

25. References References:- 1)http://www.emba.uvm.edu/~iatridis/me301/viscoelasticity_intro.pdf 2) William N.Findley, James S.Lai and Kasif Onaran.., “Creep and Relaxation of Nonlinear Viscoelastic Materials”.Dover Publications, INC., New York. 3)http://silver.neep.wisc.edu/~lakes/VEnotes.html 4) Wilhelm Flugge., “Viscoelasticity”.Springer-Verlag New York.Heidelberg 1975. 5)George E.Mase” Schaum’s Outline Series Continuum Mechanics”.McGraw-Hill. 6) http://ocw.mit.edu/OcwWeb/index.htm 7)Daniel Frederick, Tien Sun Chang “Continuum Mechanics” Allyn and Bacon Series. MEEN 5330 25

26. Thank you MEEN 5330 26