A constitutive equation for the dynamic deformation behavior of polymers

Abstract

A constitutive equation based on the generalized concept of thermally activated flow units is developed to describe the stress–strain behavior of polymers as a function of temperature, strain-rate, and superposed hydrostatic pressure under conditions in which creep and long-term relaxation effects are negligible. The equation is shown to describe the principal features of the dynamic stress–strain behavior of polytetrafluoroethylene and, also, the yield stress of polymethylmethacrylate as a function of temperature and strain rate. A key feature of the model, not utilized in previous constitutive equation descriptions, is an inverse shear stress dependence of the shear activation volume. In contrast to metal deformation behavior, an enhanced strain hardening with increasing strain at higher strain rates and pressures is accounted for by an additional rate for immobilization of flow units. The influence of hydrostatic pressure enters through a pressure activation volume and also through the flow unit immobilization term. The thermal activation model is combined with a temperature dependent Maxwell–Weichert linear viscoelastic model that describes the initial small strain part of the stress strain curve.

Appendix A: obtaining parameters for the viscoplastic element

Each stress–strain curve not already reported as true stress-true strain is transformed to true stress-true strain with the formulae

$$ \begin{array}{c} \sigma = S (1+ e) \\ \varepsilon = \ln (1+ e) \\ \end{array} $$

(A1)

where S and e are load and elongation, respectively, and σ and ɛ are true stress and true strain, respectively. While this transformation applies only if the flow is volume conserving, the error in applying it to the entire stress–strain curve is generally small. Then, for each curve, a guess is made for the initial linear viscoelastic modulus, and the elastic strain is subtracted from the total strain to obtain the viscoplastic strain. Each true stress–viscoplastic strain curve is then fitted to an equation of the form

$$ \sigma = \sigma_0 + K\sqrt {(1 - e^{- \omega \varepsilon })/\omega } $$

(A2)

and σ₀ is obtained from the intercept (or extrapolated intercept) with the σ-axis at zero viscoplastic strain. K and ω could be obtained with a non-linear least squares fit to the data, but a quicker and better result is obtained simply by solving for K and ω by taking two points on the curve, one near the end, at large strain, and one at half that strain.

To obtain B, β ₀, and β ₁, write

$$ \ln \sigma_0 = \ln B - \beta_1 T \ln ({\dot {\varepsilon}}_{0\beta }/\dot {\varepsilon}) $$

(A3)

where ${\beta_0 = \beta_1 \ln {\dot {\varepsilon}}_{0\beta}}.$ Pick an ${{\dot {\varepsilon}}_{0\beta }},$ and plot ${\ln \sigma_0}$ against ${T \ln ({\dot {\varepsilon }}_{0\beta}/\dot{\varepsilon})}.$ With the best value of ${{\dot {\varepsilon }}_{0\beta}},$ all the data points will lie on or close to a single straight line whose slope is ${-\beta_1}$ and intercept is ln B. Similarly, to obtain B ₀, α ₀, and α ₁, write

$$ \ln K = \ln B_0 - \alpha_1 T \ln ({\dot {\varepsilon}}_{0\alpha }/\dot {\varepsilon}). $$

(A4)

Finally, the dependence of ω on strain rate and pressure may be determined by a linear regression and the pressure dependence of B and B ₀ may be determined by a series of linear regressions with differing trial exponents.