Monitoring the degradation in shear and bulk moduli of rubber for inclusion in visco-elastic FE models.

This article is organized into four sections, the first two appearing this month and the last two in January 2001 issue. The first section discusses aspects related to the theory of viscoelasticity. Time and frequency domains are presented, along with calibration of models for finite element analysis (FEA). The second edition is an overview of compressive stress relaxation (CSR) tests developed in the rubber industry. Section three presents a newer method to monitor shear and bulk moduli of rubber under recycling temperature and the circulation of fluids. Section three also presents results of modeling samples used in testing. Data collected under uniaxial tension and compression, planar tension and volumetric compression, along with decay in shear and bulk moduli with time and temperature, were used in analyzing a seal in section four.

Theory of viscoelasticity

Many engineering materials are rate-dependent while behaving elastically. Following deformation, they return to their original undeformed state but at non-uniform speeds. For prescribed stresses, these materials exhibit creep, a recoverable phenomenon that also occurs in metals, but only significantly at high temperatures. Polymers are viscoelastic even at temperatures below -200 [degrees] C. For prescribed strains, polymers exhibit stress relaxation. Stresses drop to zero for viscoelastic fluids (such as glass and plastics at high temperatures or non-vulcanized rubber). Long term stresses remain in viscoelastic solid (examples being plastics at low temperatures and vulcanized rubber).

Linear visco-elasticity

Linear time dependency means a proportionality of the relaxation rate with the instantaneous stress. In the linear viscoelastic theory, creep and stress-relaxation curves are proportional. Classical viscoelasticity uses the small strain theory, i.e., the instantaneous stress is proportional to the strain. "Finite strain" linear viscoelasticity uses the hyper-elastic theory of elasticity with a relaxation rate proportional to the stress. This is the simplest method for viscoelasticity at large strains, and a small amount of experimental data is sufficient to calibrate the mathematical modeling.

Temperature dependency

Viscoelastic properties are strongly dependent on temperature. Polymers are viscoelastic at all temperatures as (1) their instantaneous response is temperature-dependent (moduli depending on temperature) and (2) their rate of relaxation is dependent on temperature, in which case these materials are referred to as thermo-rheologically simple (TRS).

Frequency domain response

For a sinusoidal strain [Gamma](t) = [[Gamma].sub.o] [e.sup.iwt] being the angular frequency, the stress response in a viscoelastic material is out of phase to take the form: [Sigma](t) = [[Sigma].sub.o] [e.sup.i(wt+[Sigma])] where [Delta] is the loss angle. In other words, the strain lags behind the stress by an angle [Delta].

For a purely elastic material [Delta] = 0 whereas [Delta] [much greater than] 0 for polymers (such as rubbers). It is convenient to separate the viscoelastic response of materials into "in-" and "out-of-phase" components. It is also useful to introduce a "complex" modulus:

G*(w) = [Sigma] (t)/[Gamma](t) = [[Sigma].sub.o][e.sup.i(wt+[Sigma])]/[[Gamma].sub.o][e.sup.iwt] or [[Sigma].sub.o]/[[Gamma].sub.o] cos [Delta] + i [[Sigma].sub.o]/[[Gamma].sub.o] sin [Delta]

Storage and loss moduli, [G.sub.s] or [[Sigma].sub.o]/[g.sub.o] cos [Delta] and [G.sub.1] or [[Sigma].sub.o]/[[Gamma].sub.o] sin [Delta] characterize the in- and out-of-phase response of materials. Complex moduli for uniaxial, shear and hydrostatic deformation are:

E*(w) = [E.sub.s] + i[E.sub.1] = E' + iE",

G*(w) = [G.sub.s] + i[G.sub.1] = G' + iG" and

K*(w) = [K.sub.s] + i[K.sub.1] = K' + iK".

Real and imaginary parts in the forms listed are storage and loss moduli. These depend on frequency for unfilled rubbers, but their ratio is independent of the strain amplitude; [G.sub.1]/[G.sub.s] or tan [Delta] is the loss tangent. For filled rubbers, storage and loss moduli depend on the strain amplitude and test data should be very near the amplitude to model.

Hysteresis 

For cyclic loading, viscoelastic materials dissipate energy: for a strain input [Gamma](t) = [[Gamma].sub.o] sin(wt), the stress output is [Sigma](t) = [[Sigma].sub.o] sin (wt+[Delta]). Strain energy U takes the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At very high cycling frequencies, instantaneous properties characterize the viscoelastic response of materials. At very low cycling frequencies, long-term properties characterize such a response.

Material damage

There is a difference between the response of rubber during a first stretch or after repeated straining; rubber gets damaged with stretching besides hysteresis. Before measuring rubber properties, one should consider the end-use application of the product that will be made out of the rubber being tested. Pre-conditioning rubber prior to data collection, if any, should derive from such a consideration.

Dampening characteristics of rubber

In practical applications, damping characteristics of rubber are very important. These are dependent on chemical constituents of the rubber and are often tailored to specific requirements. Fillers (such as carbon black and silica) strongly influence both elastic and viscoelastic behavior of rubber.

The relaxation rate in unfilled rubber is proportional to the stress level. It is thus possible to characterize the behavior of unfilled rubber with finite strain viscoelasticity. Unfortunately, the relaxation rate is not proportional to the stress in filled rubbers.

Linear viscoelastic theory

Classical linear viscoelasticity derives a relaxation modulus from a one-step strain test (a relaxation test). Figure 1 shows the tensile mode of deformation.

[ILLUSTRATION OMITTED]

In classical linear viscoelasticity, the strain [Epsilon](t) equals [[Epsilon].sub.o]H(t) and stress at time t:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Epsilon](-[infinity]) = 0 and E(t) is the "relaxation modulus."

The relaxation modulus can be obtained from a one-step strain test. In such a case, the stress-strain relation can be inverted and strain at time t is characterized by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Sigma](-[infinity]) = 0 and J(t) is the "creep function."

The creep function can also be obtained from a one-step stress test (a creep test) in which case:

[Sigma](t) = [[Sigma].sub.o]H(t) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or J(t) = [Epsilon](t)/[[Sigma].sub.o]

E(t) and J(t) are related through:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The energy dissipated under controlled stress or strain is the same.

In finite strain viscoelasticity, the stress relaxation equation has to be written entirely in terms of stress. Using integration by parts:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that creep and stress relaxation are conjugate.

Time domain viscoelasticity

Advanced finite element codes such as ABAQUS and MARC use isotropic linear viscoelasticity and treat relaxation moduli for deviatoric and volumetric behavior separately:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where S = [Sigma] + pI and p = -1/3 tr([Sigma])

The user must supply [G.sub.o], [K.sub.o], G(t) and K(t).

Prony series representation

G(t) and K(t) are often expressed in terms of Prony series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where instantaneous moduli [G.sub.o] and [K.sub.o] can be determined from [E.sub.o] and [v.sub.o], the instantaneous Young's (or elasticity) modulus and Poisson's ratio.

Relaxation test data

Figure 2 shows a simple shear test in which [[Gamma].sub.o] is the instantaneous shear strain. The normalized shear relaxation modulus is defined through [g.sub.R](t) = G(t)/[G.sub.o] also equal to [Tau](t)/[[Tau].sub.o] or [Tau](t)/[G.sub.o][[Gamma].sub.o]. Due to linearity between shear stress and strain, only one curve is needed.

[ILLUSTRATION OMITTED]

Likewise, the normalized bulk relaxation modulus can be obtained as [k.sub.R](t) = K(t)/[K.sub.o] or p(t)/[p.sub.o]. Constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are obtained from separate nonlinear least square fits to experimental data; results are combined into a single set of Prony parameters. If test data are collected at similar time intervals, one fit results in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is important that a proper number of terms be used in the Prony series - too few terms yield a poor fit and too many cause ill conditioning. Also, experimental data must cover the time domain of interest. Prony series can only represent the behavior of rubber over the fitted time domain, i.e., extrapolation does not work.

For nearly incompressible materials, Poisson's ratio approaches the half. Uniaxial relaxation or creep data can define a shear modulus as G(t) [approximately equals] 1/3 E(t); volumetric test data or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not needed in such a case.

Isotropic finite strain viscoelasticity

In the stress relaxation equation:

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For hyper-elastic materials E(t) depends on both magnitude and direction of straining. Finite strain viscoelasticity considers [Epsilon] (t) as E(t)/[E.sub.o], a dimensionless relaxation modulus that is independent of the magnitude and direction of straining. Only one dimensionless relaxation curve is required to characterize a viscoelastic behavior.

Similarly to the isotropic linear viscoelasticity, independent moduli for deviatoric and volumetric behavior are required in finite strain viscoelasticity. Such moduli depend on the choice of strain energy function. Polynomial energy function has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where coefficients [C.sub.jk]([Tau]) and 1/[D.sub.k]([Tau]) are defined by Prony series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are dimensionless shear and bulk relaxation moduli, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] derive from instantaneous shear and volumetric behavior. Constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] define the visco and hyper-elastic sections in finite element models. Note that if a material is incompressible, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and volumetric test data is not needed.

Ogden's energy function has the form:

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where coefficients [[micro].sub.k] and 1/[D.sub.k]([Tau]) are also defined by Prony series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are dimensionless shear and bulk relaxation moduli and, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] derive from instantaneous shear and volumetric behavior. Again, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] define the viscoelastic response of a material in FEA; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] define hyper-elasticity. Also, no volumetric data is needed for an incompressible material as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Harmonic viscoelasticity

The stress-relaxation relation:

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is inadequate for a Fourier transform since (for solids) E(t) [is not equal to] 0 as t [right arrow] [infinity]. A dimensionless relaxation function e(t) = E(t)/[E.sub.[infinity]]-1 where [E.sub.[infinity]] is a long-term modulus making e(t) [right arrow] 0 as t [right arrow] [infinity] is rather used.

Substituting for e(t) in the stress relaxation equation and letting t-[Tau]' = [Tau] yields:

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Harmonic excitation

Consider the application of a sinusoidal strain [Epsilon](t) = [Epsilon][e.sup.iwt] where w is the angular frequency. Substituting [Epsilon](t) in the long-term stress relaxation equation yields:

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Therefore, [Sigma](t) = E*[Epsilon](t) where E*(w) = [E.sub.[infinity]](1-iwe*(w)) is the complex modulus and e*(w) the Fourier transform of e(t).

Similarly, e*(w) = Re(e*) + i Im(e*), E*(w) = [E.sub.[infinity]](1-w Im(e*)) + i [E.sub.[infinity]](1-w Re(e*)) where [E.sub.[infinity]](1-w Im(e*)) is the storage modulus [E.sub.s](w) and [E.sub.[infinity]](1-w Re(e*)) the loss modulus [E.sub.1](w).

Previous formulas relating storage and loss moduli to Fourier transform of the dimensionless relaxation function define frequency domain viscoelasticity. In such a case, Re(e*) = 1-[E.sub.s](w)/[E.sub.[infinity]] and Im(e*) = 1-[E.sub.1](w)/[E.sub.[infinity]] are required for modeling.

Frequency domain vicoelasticity

Isotropic viscoelasticity is implemented in finite element software such as ABAQUS and MARC. This assumes relaxation moduli for deviatoric and volumetric behavior to be independent. One must supply [G.sub.[infinity]], [K.sub.[infinity]], [G*.sub.[infinity]] and [K.sub.[infinity]] for modeling.

Storage and loss moduli are related to Fourier transforms of non-dimensional shear and bulk relaxation functions g(t) and k(t) though:

[G.sub.s](w) = [G.sub.[infinity]](1-wIm(g*)),

[G.sub.1](w) = [G.sub.[infinity]]wRe(g*),

[K.sub.s](w) = [K.sub.[infinity]](1-wIm(g*)) and

[K.sub.1](w) = [K.sub.[infinity]]wRe(g*).

where g*(w) and k*(w) are Fourier transforms of g(t) and k(t), g(t) = G(t)/[G.sub.[infinity]]-1 and k(t) = K(t)/[K.sub.[infinity]]-1.

Long-term moduli [G.sub.[infinity]] and [K.sub.[infinity]] can be obtained from [E.sub.[infinity]] and [v.sub.[infinity]] defining the elastic stage of materials. Storage and loss moduli are needed for analysis: Re([w.sub.i][g.sib.i]*), Im([w.sub.i][g.sub.i]*), Re([w.sub.i][k.sub.i]*), Im(w.sub.i][k.sub.i]*) and [f.sub.i] (or w/2[Pi]) which define a formula parameter for real and imaginary parts of non-dimensional relaxation functions.

Isotropic finite strain viscoelasticity

Like with isotropic linear viscoelasticity, independent moduli for deviatoric and volumetric behavior are required. These are effective tangent moduli, which depend on the choice of strain energy function.

Frequency dependent behavior is specified in the way described earlier. Long-term moduli specification depends on the hyper-elastic form; i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the polynomial strain energy function, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [[Alpha].sub.i] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for Ogden's.

Time-temperature correspondence

For many viscoelastic materials, a TRS model describes accurately the dependence of relaxation time on temperature. In such a model, relaxation at a given temperature T is described by a temperature-independent relaxation function:

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with A(T) a time reduction factor that depends on temperature.

The reduction factor A is often expressed as a logarithmic time shift: h(T) = -log(A(T)).

Such a time shift can be used to extrapolate relaxation data to long or short times. Relaxation tests allow the measurement of shear and bulk moduli at different temperatures, for instance at T = [T.sub.o] and T = [T.sub.o] [+ or -] [Delta]T where [T.sub.o] is a reference temperature for which h([T.sub.o]) = 0.

The measured curves make it possible to determine the shift functions h([T.sub.o]-[Delta]T) and h([T.sub.o]+[Delta]T). These can, in turn be used to extrapolate the relaxation curves well beyond the measured time domain.

By carrying out relaxation experiments over a wide enough range of temperatures, a complete relaxation curve spanning many decades in time can be obtained.

The default shift function in finite element codes is the Williams-Landell-Ferry (WLF):

H(T) = [C.sub.1](T-[T.sub.o])/([C.sub.2]+(T-[T.sub.o]))

where [C.sub.1] and [C.sub.2] are constants to define from test data and [T.sub.o] the glassy transition temperature. Finite element users can code other forms of A(T) through subroutines.

Industry developed CSR tests

Several tests have been developed to assess the degradation in physical properties of rubber with time, temperature and the presence of fluids. These are known as the Shawbury Wallace, Weckham-Ferrance, Joh and Jamak techniques.

Akron Rubber Development Laboratory (ARDL) published shear tests under quasi-static loading. The lab also published a procedure to measure the decay in physical properties of rubber in hot oil: A rubber button is compressed to 25% of its height then immersed in oil at 150 [degrees] C. The sample is then removed from oil, allowed to cool to the ambient before the force to further compress it by 0.2 mm is recorded. Such operator-dependent procedure is often repeated at intervals of 200 hours or larger.

In the Weckham-Ferrance and current industrial CSR tests, the effect of stressing is not understood, nor is thermal-cycling, the presence of fluids, the shape and size of samples, and straining beside compression.

Monitoring the degradation in forces to compress ring gaskets was developed as early as 1991 at Joh Rubber in Germany Strain gauges were used to measure deflections of a beam under reaction of rubber. The test was rather to validate the design of ring gaskets of given cross-sections and not to characterize materials.

The system by Joh progressed to use a jig from Jamak. A load cell replaced Joh's strain gauges for repeatable usage. Figure 3 presents a dismantled Jamak jig by a load cell and Joh's holding frame.

[ILLUSTRATION OMITTED]

Both a washer and a solid disk are used in the Jamak CSR fixture. The washer is 19.050 and 9.525 mm in outer and inner diameters and 3 mm thick. This is often compressed to 80% of its height as a load cell monitors the decay in the force to keep the sample compressed.

Some of today's issues with CSR testing are drift in calibration of cells in oven environments and their permeation by fluids. A newer system by the Swedish Elastocon keeps the cells at ambient.

Note that viscoelastic tests in the rubber industry are compressive, while finite element codes such as ABAQUS and MARC require decay in shear and bulk moduli to define Prony series.

ARDL developed techniques to measure the shear stress versus shear strain under quasi-static conditions. These are unfortunately tensile and would be difficult to set in time. Figures 4 and 5 present a dual lap shear test at ARDL.

[ILLUSTRATIONS OMITTED]

Figures 6 and 7 present the quadruple lap shear test engineered by the Akron-based lab. Rubber can bound the mating hardware, however, sizes remain to be proved in tested samples. (Parts 3 and 4 will appear in the January, 2001 issue.)

[ILLUSTRATIONS OMITTED]

Dr. Ben Chouchaoui is a graduate of Polytechnic School of Montreal and the University of Waterloo in Canada. He specializes in materials and FEA and currently runs the Windsor Industrial Development Laboratory concentrating on material and process testing and stimulation to aid in product design and manufacturing.