By George L. Ph.D., D.Sc. Clark
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In addition, it should be noted that in any approximation to the response functional be it order fluids of the integral type, order fluids of the differential type or differential fluids of grade n the number of material constants rises rapidly with increasing order n of the approximation posing seemingly insurmountable experimental difficulties in determining these constants. Siginer  notes that although it is next to impossible to determine a large number of parameters from a single experiment of rheometry, it may be possible to determine the parameters sequentially from a series of rheometrical experiments at the lower orders, and outlines a series of experiments involving free surface rheometry and pulsating flow experiments in tubes to determine the material constants of the integral fluid of order three with fading memory.
13) except certain symmetries of T given the symmetry of Ct. Green and Rivlin [97, 98] and Green et al.  in a series of papers developed an integral expansion of the functional I for finite strains, which led to the integral type of fluids of order n. They assume that a suitable function space endowed with a suitable topology can be assigned to the domain of the response functional I, and that I is continuous with respect to that topology. Given these premises, Stone–Weierstraβ theorem can be used to show that the response functional I can be uniformly approximated by integral polynomials, that is the response functional can be expanded into a uniformly approximating series of multiple integral terms with tensor polynomial integrands as: 1 1 X s¼0 1 I ½GðX; sÞ ¼ 1 ð ð1 1 ð Sn ¼ K1 ðsÞ GðsÞ ds þ 0 K2 ðs1 ; s2 Þ Gðs1 ÞGðs2 Þ ds1 ds2 0 0 1 ð1 ð ð1 þ K3 ðs1 ; s2 ; s3 Þ Gðs1 ÞGðs2 ÞGðs3 Þds1 ds2 ds3 þ Á Á Á 0 0 0 ð2:14Þ where the even order kernel tensors Ki ultimately define the material functions and therefore the behavior of the fluid.
Both pioneers considered not only problems in non-linearly viscous fluids but in finitely strained non-linear elastic materials as well, which is the hallmark of continuum mechanics from the point of view of unified field theories. ” Revolutionary ideas were introduced of which the occurrence of the second-order normal stress effects in both non-linear fluid and solid mechanics is a good example. 1 Constant Stretch History Flows The Reiner–Rivlin CE is the most general frame-invariant constitutive equation for steady extensional flows.
Applied X-rays by George L. Ph.D., D.Sc. Clark