Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using . To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem. The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. (More simply, the Knudsen number is how many times its own diameter a particle will travel on average before hitting another particle). Problems with Knudsen numbers at or above unity are best evaluated using statistical mechanics for reliable solutions.
He also produced celebrated work on the Continuum hypothesis, showing that it cannot be disproven from the accepted set theory axioms, assuming that those axioms are consistent.
The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial , and therefore results in approximate . Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. That said, under the right circumstances, the continuum hypothesis produces extremely accurate results.
The smallest infinite cardinal number is (aleph-naught). The second smallest is (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between and implies that .
This is why, for example, we can be more confident of research results that are consistent with a causal-directional hypothesis, than is the case of findings that are consistent with a non-directional hypothesis.
By comparing humans to prisoners in a cave, Plato argues that what we see are shadows projected on the wall, only that we mistake them for real knowledge which are named “forms”....
Lakes, R. S., "Strongly Cosserat elastic lattice and foam materials forenhanced toughness", , 12, 17-30 (1993).
Some foams exhibit size effects and other phenomena not describable by classical elasticity. These foams are describable by Cosserat elasticity, which is a continuum theory with more freedom than classical elasticity. Cosserat solids have a characteristic length which is greater than zero. Strongly Cosserat elastic materials are considered to be those materials for which the Cosserat characteristic length is substantially greater than the structure size and for which the coupling number is large. Such materials are predicted to exhibit superior toughness. A mechanically isotropic lattice model is presented for the study of foams. Ordinary open cell foams are shown to be weakly Cosserat elastic. If cell rib properties are modified, strongly Cosserat elastic effects can occur in the foam. Anisotropic laminate and fibrous materials can also be made to exhibit strongly Cosserat elastic effects.
Lakes, R. S., "Experimental micro mechanics methods for conventional and negative Poisson's ratio cellular solids as Cosserat continua", , 113, 148-155 (1991).
Continuum representations of micromechanical phenomena in structured materials are described, with emphasis on cellular solids. These phenomena are interpreted in light of Cosserat elasticity, a generalized continuum theory which admits degrees of freedom not present in classical elasticity. These are the rotation of points in the material, and a couple per unit area or couple stress. Experimental work in this area is reviewed, and other interpretation schemes are discussed. The applicability of Cosserat or micropolar elasticity to cellular solids and fibrous composite materials is considered as is the application of related generalized continuum theories. New experimental results are presented for foam materials with negative Poisson's ratios.
A heavily twisted bar of negative Poisson's ratio foam exhibits minimal warp, in contrast to expectation of classical elasticity as shown in the image.
Lakes, R. S., Nakamura, S., Behiri, J. C. and Bonfield, W., "Fracture mechanics of bone with short cracks", , 23, 967-975 (1990).
Tensile fracture experiments were performed upon specimens of wet mature bovine Haversian bone, with short, controlled notches. Stress concentration factors were found to be significantly less than values predicted using a maximum stress criterion in the theory of elasticity. Results were also modeled with the aid of linear elastic fracture mechanics. Agreement of experiment with theory was better in this case, however deviations were seen for short notches. Two mechanisms were evaluated for the behavior: plasticity near the crack tip, and effects of the Haversian microstructure, modelled by Cosserat elasticity, a generalized continuum theory. Plastic zone effects were found to be insignificant. Cosserat elasticity, by contrast, predicted stress concentration factors which better approximated observed values. To explore strain redistribution processes, further experiments were conducted upon notched specimens in torsion at small strain. They disclosed a strain redistribution effect consistent with Cosserat elasticity. These microelastic effects interpreted within a micropolar continuum model, are attributed to the Haversian architecture of bone.
Chen, C. P. and Lakes, R. S., "Dynamic wave dispersion and loss properties of conventional and negative Poisson's ratio polymeric cellular materials", , 8(5), 343-359 (1989).
This article describes experimental investigations of the dynamical behaviour of conventional and negative Poisson's ratio foamed materials in torsional vibration. Dispersion of standing waves and cut-off frequencies were observed. Consequently, foamed materials do not obey the classical theory of elasticity or viscoelasticity. The dynamical effects were attributed to micro-vibrations of the cell ribs in a structural view and were associated with Mindlin microstructure elasticity or micromorphic elasticity in a continuum view. Cut-off frequencies were lower in re-entrant foams with negative Poisson's ratios than in the conventional foams from which they were derived. An analytical structural model was developed in which the ribs of the conventional foams were modeled as free-free vibrating beams. The predicted cut-off frequencies were comparable to those observed experimentally. Such stop bands can block waves.
Gödel showed that both the axiom of choice and the generalized continuum hypothesis are true in the constructible universe, and therefore must be consistent.