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# Theory of the Yield Stress Dependence on Strain-Rate

## Summary

This article discusses how the constitutive elements in a material model influence the strain-rate dependence of the predicted stress-strain response. There are many different theoretical approaches that have been developed for this, including the Eyring energy activation model, the Argon energy activation model, and the Power-Law flow model.

## Eyring Energy Activation Equation

Yielding (and plastic flow) of glassy polymers is controlled by one or more energy activation mechanisms. The classical Eyring’s equation (from 1936!) is one way to predict the strain-rate dependence of the yield:

$$\dot{\varepsilon} = \dot{\varepsilon}_0 \, e^{-\Delta H/kT} \, \text{sinh}\left[ \displaystyle\frac{\sigma V}{2kT} \right].$$

Here $$\dot{\varepsilon}$$ is the strain rate, $$\varepsilon_0$$ is a pre-exponential factor (needed for dimensional reasons), $$\Delta H$$ is the activation energy, k is Boltzmann’s contant, T is the absolute temperature, $$\sigma$$ the driving stress for viscoplastic shear transformations, and V is the activation volume. One way to interpret this equation is that the first factor $$\dot{\varepsilon}_0 \, e^{-\Delta H/kT}$$, which is the Arrhenius equation, specifies the attempt frequency for forward plasticity events, and the sinh-part specifies the probability that an attempt will lead to an actual microstructure change causing plasticity.

It is common to “linearize” this equation by considering cases where $$\sigma V / 2kT > 1$$, for which we can replace the sinh function with an exponential function:

$$\dot{\varepsilon} = \dot{\varepsilon}_0 \exp\left( -\displaystyle\frac{\Delta H – \sigma V} {kT} \right)$$.

This equation can also be written:

$$\sigma = \displaystyle \frac{\Delta H} {V} + \frac{kT}{V} \cdot \text{ln} \left[ \frac{\dot{\varepsilon}}{\dot{\varepsilon}_0} \right],$$

showing that there is a linear relationship between yield stress and the logarithm of the strain rate. Also note that this equation for the yield stress becomes functionally identical to the Johnson-Cook equation for the strain-rate dependence if we replace the strain rate with the plastic strain rate. Finally, I have always been a bit troubled by the linearized form of this equation since it gives a non-zero flow rate even when the stress is zero. That is one reason some researchers prefer the sinh function over the exp function.

## Argon Energy Activation Equation

Ali Argon (who was on my thesis committee at M.I.T.) developed (in 1973) a micro-mechanism inspired model for yielding of glassy polymers at low temperatures. In classic Argon style, he proposed that viscoplastic deformations are introduced by rotational motions of molecular segments, and he used that approach to derive an expression for the activation free enthalpy. His final expression for the strain-rate dependence of viscoplastic deformation rate was:

$$\dot{\varepsilon} = \dot{\varepsilon}_0 \exp\left[ -\displaystyle \frac{\sigma_0 V}{kT} \left(1 – \left(\frac{\sigma}{\sigma_0}\right)^{5/6} \right) \right]$$.

Note that if the power exponent was 1 instead of 5/6, then this equation becomes the same as the Eyring equation. This equations gives an almost linear relationship between yield stress and the logarithm of the strain rate.

## Power-Law Flow Equation

Many modern constitutive models of glassy polymers (for example, the PolyUMod TN and TNV, and Abaqus PRF) are based on a micro-structure representation of the material that contains multiple parallel networks, each having a power-law flow equation of the following (slightly simplified) type:

$$\dot{\gamma} = \dot{\gamma}_0 \cdot \left( \displaystyle \frac{\tau}{\hat{\tau}} \right)^m,$$

where $$\dot{\gamma}$$ is the shear deformation rate, $$\dot{\gamma}_0$$ is a constant introduced for dimensional consistency, $$\tau$$ is the driving shear stress, and m  is a power exponent. For thermoplastics, the m parameter is often between 8 and 20. This equation gives a viscoplastic flow rate for all levels of the driving stress, in other words, the model does not have a yield stress below which no plasticity will occur. This can be compared to the classical metal plasticity equation (also called the Cowper-Symonds overstress power law) where:

$$\dot{\varepsilon}^{p} = D \cdot \displaystyle \left( \frac{\sigma – \sigma_0}{\sigma_0} \right)^n$$,

and the equation only applies to cases where $$\sigma > \sigma_0$$, which gives true plasticity.

Let’s go back to a one-network version of the power-law model. This model has a neo-Hookean spring in series with a power-law flow element and the predicted yield stress as a function of the applied strain rate is plotted in the figure to the right. The data was generated using MCalibration with the following values: mu=100 MPa, kappa=500 MPa, and tauHat=10 MPa, and shows that for small m values the yield stress increases non-linearly with the logarithm of the strain rate. For large m-values the dependence between yield stress and logarithm of strain rate becomes almost linear.

To continue the parametric investigation, let’s consider the influence of the parameter tauHat. The figure shows that the slope of the yield stress vs. strain rate curves are strongly dependent on the variable tauHat. In this study I used: mu=100 MPa, kappa=500 MPa, and m=15.

Now consider a parametric study of different shear modulus mu values. The figure shows that the modulus does not influence how the yield stress depend on the strain rate. Here I used: kappa=500 MPa, tauHat=10 MPa, and m=15.

As a final study, I examined the influence of having 2 parallel networks. In this case the stiffness of the first network (muA) directly scales the yield stress, since the two networks are in parallel. Here I used: kappa=500 MPa, muB=50 MPa, tauHat=10 MPa, m=15.

By using multiple networks it is therefore possible to get a bi-linear (or higher order) dependence between strain rate and the predicted yield stress.

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