Parameters of Viscoelastic Material

Feihu Zhaofeihu.zhao@tut.fi

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Relationship among Parameters of Isotropic Material

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Linear Viscoelasticity Vs. Nonlinear Viscoelasticity

• Linearwhen the function is separable in both creep response and load. It occurs at small deformation.All linear viscoelastic models can be represented by a Volterra equation connecting stress and strain.

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Linear Viscoelasticity Vs. Nonlinear Viscoelasticity

• Nonlinearwhen the function is not separable. It is usually happens when the deformations are large or if the material changes its properties under deformations.Nonlinear viscoelasticity has not been well explored. There is still no universal principle to explain it.

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Linearly Viscoelastic Young’s Relaxation Modulus

• NanoindentationMaterial: Polymethyl Methacrylate (PMMA) and Polycarbonate (PC).

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Linearly Viscoelastic Young’s Relaxation Modulus

Sneddon(1965) provided load-displacement relation:

(1)Ting (1966) modified Eq.(1) by adding Hereditary Integral Expression:

(2)

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Linearly Viscoelastic Young’s Relaxation Modulus

While under constant-rate load:

(3)Apply time derivative on both sides of Eq.(3):

(4)

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Linearly Viscoelastic Young’s Relaxation Modulus

• Generalized Maxwell Model

(5)

Submit Eq.(5) into Eq.(3):

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Linearly Viscoelastic Young’s Relaxation Modulus

By fitting the experimental data, the E(t) expression can be gotten!

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Linearly Viscoelastic Creep Compliance

• NanoindentationSneddon (1965) derived the indentation load-displacement relation:

(1)Where,f(x) is the shape function for an axisymmetric indenter with x=r/a.

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Linearly Viscoelastic Creep Compliance

The indentation load on an axisymmetric indenter:

(2)For the indentation by a conial indenter, f(x)=axtan , Eq.(1) becomes:

h=(1/2)a tan (3)Submit Eq.(3) into Eq.(2):

(4)

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Linearly Viscoelastic Creep Compliance

Add hereditary integral expression part:

(5)If indent at a constant rate, the load will beP(t)=v0tH(t), with v0 being the loading rate andH(t) the Heaviside unit step function.

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Linearly Viscoelastic Creep Compliance

(6)

(7)

Constant Loading Rate

(8)

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Linearly Viscoelastic Creep Compliance

• Kelvin Model

(9)

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Linearly Viscoelastic Creep Compliance

Submit Eq.(9) into Eq.(6):

P(t)=v0t

(10)By fitting the experimental data, J(t) can be gotten!

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Linear Shear Relaxation Modulus

• In the Form of Prony SeriesThe shear relaxation function G(t):

(1)gR(t) is a Prony Series:

(2)

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Linear Shear Relaxation ModulusSubmit Eq.(2) into Eq.(1):

(3)Where, gi and i are the parameters of materials;G0 is the shear modulus at t=0.G(t) can be gotten by curve fitting! In Reference[7] (PDMS):

(4)

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Nonlinear Young’s Relaxation Modulus

• Models Created by P.N. EleniTest Material:

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Nonlinear Young’s Relaxation Modulus

Mathematical Models for Compression Test Temperature:25°C Relative Humidity: 50%Constant deformation rate :5mm/min.

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Nonlinear Young’s Relaxation Modulus

Parameter Estimation for Compression model

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Nonlinear Young’s Relaxation Modulus

Mathematical Models for Tensile TestTemperature: 25°CRelative Humidity: 50%Loading Speed: 5mm/min.

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Nonlinearn Young’s Relaxation Modulus

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Nonlinear Young’s Relaxation Modulus

Parameters Estimation for Tensile Model

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Nonlinear Shear Relaxation Modulus

• Michelle L. Oyen’s ModelIt is based on the linear shear relaxation modulus. For a linear viscoelastic BoltzmanHereditary Integration formula:

(1)Difference between the linear modulus:

(2)where, e( ) is the form of stress-strain response;

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Nonlinear Shear Relaxation Modulus

(3)Submit Eq.(2) into Eq.(1):

(4)

Assume e( ) =m 2. In addition, the condition of ramp-and –hold relaxation can be described by:

(5)

Where, tR is the total ramp time.

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Nonlinear Shear Relaxation Modulus

Combine Eq.(1)~ Eq.(5):

(6)

G(t)= (t)/ (t)

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Nonlinear Shear Relaxation Modulus

• Randy H. Ewoldt MethodMaterials: a biopolymer hydrogel and a wormlike micelle solutionLoading Condition: Large Amplitude Oscillatory Shear Stress (LAOS) Extract the elastic part from the model:

(1)

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Nonlinear Shear Relaxation Modulus

Employ the Chebyshev Plolynomial:(2)

Where, x= 0;

Tn(x) is the n-th order Chebyshev Polynomial of the first kind: Tn(cos )=cos(n ), Tn(sin )=sin(n )(-1) (n-1)/2;en 0) is the elastic Chebyshev coefficient:en=Gn’ (-1) (n-1)/2, n: odd.

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Nonlinear Shear Relaxation Modulus

Two definitions are introduced:(3)

(4)

GM’ is the minimum-strain modulus (or tagent modulus at =0);GL’ is the large-strain modulus (or secant modulus evaluated at the maximum imposed strain).

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Nonlinear Shear Relaxation Modulus

For linear viscoelasticity: G’M=G’L, so e3=0.

For nonlinear viscoelasticity:G’M G’L, so e3 0.

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Dynamic Viscosity

• Randy H. Ewoldt MethodThe experiment conditions, such as: material and loading conditions etc. are the same with measurement of nonlinear shear relaxation modulus.The viscous stress:

(1)

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Dynamic Viscosity

Employ the Chebyshev Plolynomial:(2)

Where, y= ’/ ’0;

Tn(y) is the n-th order Chebyshev Polynomial of the first kind: Tn(cos )=cos(n ), Tn(sin )=sin(n )(-1) (n-1)/2;

n 0) is the elastic Chebyshev coefficient:

n=Gn’’ (-1) (n-1)/2, n: odd.

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Dynamic Viscosity

Two definitions are introduced:(3)

(4)’=G’’/ ;

M’ is the minimum-rate dynamic viscosity;

L’ is the large-rate dynamic viscosity.

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Dynamic Viscosity

For linear viscoelasticity: M’=

L’ , so 3=0.

For nonlinear viscoelasticity:

M’ L’, so 3 0.

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Modification of PDMS Mechanical Properties

• Through changing the ratio of prepolymer(A) and crosslink agent(B):In Wu Yuanzi’s paper, he stated that through changing the ratio of A and B, the mechanical properties can be changed.

But no metered analysis in the paper

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Modification of PDMS Mechanical Properties

• By doped with nanoparticles (TiO2 and SiO2):

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Modification of PDMS Mechanical Properties

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Modification of PDMS Mechanical Properties

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Reference[1] P.Du, X.Zheng, et al. :Extended Timoshenko Beam Formula for

Cellular Contraction Force Calculation, 14th International Conference on Miniaturized System for Chemistry and Life Sciences 3-7 Oct., 2010, Groningen, The Netherlands.

[2] Gang Huang, Hongbing Lu,: Measurement of Young’s Relaxation Modulus using Nanoindentation, Mech Time-Depend Mater (2006) 10:229-243.

[3] G.Huang, H.Lu,: Measurement of Two Independent ViscoelasticFunctions by Nanoindentation, Experimental Mechanics (2007) 47:87-98.

[4] Li Xiyuan, Zhang Shaoyang, et al.,: A General Hereditary Integral Expression Under Two Time Dimensions, Journal of Xi’an Jiao Tong University (1997) 04-005.

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Reference[5] O.C.Zienkiewicz and R.L.Taylor, The Finite Element Method (Volume

2: Solid Mechanics), 5th Edition, 2000.[6] George E. Mase,: Schaum’s Outline of Theory and Problems of

Continuum Mechanics (Chinese Version), 1986. [7] I-Kuan, Kuang-Shun Ou, et al.,: Viscoelastic Characterization and

Modeling of Polymer Transducers for Biology Application, Journal of Microelectromechanical Systems, Vol. 18, No. 5, October 2009.

[8] : ,

,2009 25 .[9] P.N.Eleni, I. Katsvou, et al.,: Mechanical Behavior of Facial

Prosthetic Elastromers after Outdoor Weathering, Dental Materials, 25 (2009) I493-I502.

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Reference[10] Michelle. Oyen, Robert F. Cook, et al,: Uniaxial and Biaxial

Mechanical Behavior of Human Amnion, J. Mater. Res., Vol. 20, Nov 2005.

[11] Randy H. Ewoldt, A. E. Hosoi, and Gareth H. McKinley,: New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear (LAOS).

[12] Gale A. Holmes, Richard C. Peterson, et al.,: The effect of Nonlinear Viscoelasticity on Interfacial Shear Strength Measurement, Reproduction of Reprint from Standard Technical Publication 1357 American Society for Testing and Materials, 100Barr Harbor Drive, West Conshohocken, PA 19428-2959.

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Reference[13]Wu Yuanzi, Ma Hongwei,: Poly(dimethylsiloxane) Elastromer with

Tethered Peptide Ligands for Cell Adhesion Studies, available at: www.paper.edu.cn.

[14] Wang Peng, Chen Xiang, et al.,: Mechanical modification of PDMS by blending with nano particles as hot embossing mold, Journal of Functional Materials and Devices, Vol.15, No.5, Oct., 2009

[15] http:// imechanica.org

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