Parameters of Viscoelastic Material
Feihu [email protected]
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Relationship among Parameters of Isotropic Material
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Young’s Relaxation Modulus
• NanoindentationMaterial: Polymethyl Methacrylate (PMMA) and Polycarbonate (PC).
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Young’s Relaxation Modulus
Sneddon(1965) provided load-displacement relation:
(1)Ting (1966) modified Eq.(1) by adding Hereditary Integral Expression:
(2)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Young’s Relaxation Modulus
While under constant-rate load:
(3)Apply time derivative on both sides of Eq.(3):
(4)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Young’s Relaxation Modulus
• Generalized Maxwell Model
(5)
Submit Eq.(5) into Eq.(3):
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Young’s Relaxation Modulus
By fitting the experimental data, the E(t) expression can be gotten!
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Creep Compliance
(6)
(7)
Constant Loading Rate
(8)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Creep Compliance
• Kelvin Model
(9)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linearly Viscoelastic Creep Compliance
Submit Eq.(9) into Eq.(6):
P(t)=v0t
(10)By fitting the experimental data, J(t) can be gotten!
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Linear Shear Relaxation Modulus
• In the Form of Prony SeriesThe shear relaxation function G(t):
(1)gR(t) is a Prony Series:
(2)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Young’s Relaxation Modulus
• Models Created by P.N. EleniTest Material:
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Young’s Relaxation Modulus
Mathematical Models for Compression Test Temperature:25°C Relative Humidity: 50%Constant deformation rate :5mm/min.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Young’s Relaxation Modulus
Parameter Estimation for Compression model
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Young’s Relaxation Modulus
Mathematical Models for Tensile TestTemperature: 25°CRelative Humidity: 50%Loading Speed: 5mm/min.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinearn Young’s Relaxation Modulus
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Young’s Relaxation Modulus
Parameters Estimation for Tensile Model
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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;
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Shear Relaxation Modulus
Combine Eq.(1)~ Eq.(5):
(6)
G(t)= (t)/ (t)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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).
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Nonlinear Shear Relaxation Modulus
For linear viscoelasticity: G’M=G’L, so e3=0.
For nonlinear viscoelasticity:G’M G’L, so e3 0.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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)
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Dynamic Viscosity
Two definitions are introduced:(3)
(4)’=G’’/ ;
M’ is the minimum-rate dynamic viscosity;
L’ is the large-rate dynamic viscosity.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Dynamic Viscosity
For linear viscoelasticity: M’=
L’ , so 3=0.
For nonlinear viscoelasticity:
M’ L’, so 3 0.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Modification of PDMS Mechanical Properties
• By doped with nanoparticles (TiO2 and SiO2):
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Modification of PDMS Mechanical Properties
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
Modification of PDMS Mechanical Properties
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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.
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi
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
Vahvis
ta nap
sautta
malla "
OSTA"PDF-XChange
ww w.m mpro.fiVah
vista
napsa
uttamall
a "OSTA"
PDF-XChange
ww w.m mpro.fi