THE CHARACTERISTIC OF THE STRAIN RATE SENSITIVITY OF STRESS-STRAIN CURVES IN THE LINEAR VISCOELASTICITY THEORY AND ITS INTERRELATION WITH RELAXATION MODULUS

Abstract

Properties of the stress-strain curves family generated by the Boltzmann-Volterra linear viscoelasticity constitutive equation under uni-axial loadings at constant strain rates are studied analytically. Assuming relaxation modulus is arbitrary, the general expression for strain rate sensitivity index as the function of strain and strain rate is derived and analyzed. It is found out (within the framework of the linear viscoelasticity theory) that the strain rate sensitivity index depends only on the single argument that is the ratio of strain to strain rate. So defined function of one real variable is termed “the strain rate sensitivity function” and it may be regarded as a material function since it is interconvertible with relaxation modulus. It is found out that this function can be increasing or decreasing or non-monotone or can have local maximum or minimum without any complex restrictions imposed on the relaxation modulus. It is proved that the strain rate sensitivity value is confined in the interval from zero to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) whatever strain and strain rate magnitudes are and its values may be close to unity (even for the standard linear solid model). It means that the linear viscoelasticity theory is able to produce high values of strain rate sensitivity index and to provide existence of the strain rate sensitivity index local maximum with respect to strain rate (for any fixed strain). These properties are the most distinctive features of superplastic deformation regime observed in numerous materials tests. The explicit integral expression for relaxation modulus via the strain rate sensitivity function is derived. It enables one to restore relaxation modulus assuming a strain rate sensitivity function is given. The restrictions on the strain rate sensitivity function are obtained to provide decrease and convexity down of the resulting relaxation modulus as a function of time, i.e. to provide necessary properties of a relaxation modulus in the linear viscoelasticity. Thus, the technique is developed to evaluate relaxation modulus using test data for strain rate sensitivity, in particular, using piecewise smooth approximations (by splines, for example) of an experimental strain rate sensitivity function.