4 SHORT DESCRIPTION OF NONLINEAR THEORY OF
Note 4.15. Because the model is constructed relative to the set of characteristics and has the goal to study these characteristics, it is always simpler than the real
object. This corresponds to the general philosophical principle that the real object is more reach than its model.
Mechanics works with the mathematical and physical models. The mathematical models study the functions and equations. The physical models are divided into physically realized and speculative models. The first kind of model deals with the physical experiments, whereas the second one is understood as the transitional model for constructing the mathematical models.
The speculative models correspond usually to real physical laws. Sometimes, such models have an essential resemblance with the real object. The big set of mechanical models can be related just to such kind of models. For example, the model of a continuum or the model of viscous fluid can be thought of this kind of model.
One and the same real object can have many nonequivalent models and can be described by different mathematical models. For example, the composite material can be studied with the use of continuous or discrete models, deterministic or stochastic models.
The chosen model can not be considered as something unchanging. The identification of the model assumes the determination-improving of the parameters and the structure of the model. Thus, the model can be rebuilt.
Comment 4.2. While being tailored to some new model in mechanics, it is necessary to follow the gene- ral metaphysical recommendations and principles.
One of the most popular principles is the so-called Occam’s razor. The most known formulation of this principle is as follows
Non sunt multiplicanda entia sine necessitate Entities are not to be multiplied without necessity.
In modern science, Occam’s razor is usually understood as a general principle that states that if there are several logically consistent explanations for a phenomenon that explain it equally well, then, ceteris paribus, one should prefer the simplest of them. The content of
FOUNDATIONS OF MECHANICS OF MATERIALS PART 1
SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 1
it is not necessary to introduce new laws without the need to explain some new phenomenon if this phenomenon can be fully explained by the old laws.
The well-known mechanician and philosopher Ernst Mach formulated the stronger version of Occam’s razor into physics, which he called the principle of economy stating
Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses.
Wikipedia stated that the razor’s statement other things being equal, simpler explanations are generally better than more complex ones is amenable to empirical testing. Another interpretation of the razor’s statement would be that simpler hypotheses are generally better than the complex ones.
The procedure to test the former interpretation would compare the track records of simple and comparatively complex explanations. If one accepts the first interpretation, the validity of Occam’s razor as a tool would then have to be rejected if the more complex explanations were more often correct than the less complex ones (while the converse would lend support to its use). If the latter interpretation is accep- ted, the validity of Occam’s razor as a tool could possibly be accepted if the simpler hypotheses led to correct conclusions more often than not.
Occam’s razor forms the basis of methodological reductionism, also called the principle of frugality, or the law of economy (lat. Lex parsimoniae). However, it is important to remember that Occam’s razor is not an axiom, but a presumption, that is, it does not prohibit more complex explanations in principle, but only recommends the procedure for considering hypotheses.
Note 4.17. In philosophy, the term “razor” refers to a tool that helps to discard (shave) improbable, implausible explanations. And since the shaving tool is a razor, the same name has been transferred to the truth-establishing tool.
Note 4.18. There are the so-called anti-Occam’s razors. Below two of them are shown.
Hanlon’s razor is a presumption according to which,
when searching for the causes of unpleasant events, human errors should be assumed first of all, and only secondarily, someone’s conscious malicious actions. It is usually expressed by the phrase: Never attribute to malice that which can be adequately explained by stupidity.
Hitchens’ razor is an epistemological razor. It says that
the burden of proof regarding the truthfulness of a claim lies with the one who makes the claim; if this burden is not met, then the claim is unfounded, and its opponents need not argue further in order to dismiss it.
Another variant is as follows
what can be asserted without evidence can also be dismissed without evidence.
Further reading
4.1. Blekhman, II, Myshkis, AD & Panovko, YaG 1976, Applied Mathematics. Subject, Logics, Features of Appro aches, Naukova Dumka, Kiev. (In Russian)
4.2. Fu, YB 2001, Nonlinear Elasticity: Theory and Applications. London Mathematical Society Lecture Note Series,Cambridge University Press, Cambridge.
4.3. Fung, YC 1965, Foundations of Solid Mechanics. Prentice Hall, Englewood Cliffs.
4.4. Germain, P 1973, Cours de mécanique des milieux continus. Tome 1. Théorie générale Masson et Cie, Editeurs,Paris. (in French)
4.5. Green, AE & Adkins, JE 1960, Large Elastic Deformations and Nonlinear Continuum Mechanics. Oxford University Press, Clarendon Press, London.
4.6. Gurtin, ME 1981, An Introduction to Continuum Mechanics. Academic Press, New York.
4.7. Guz, AN 1986, Elastic Waves in Bodies with Initial (Residual) Stresses. In 2 vols. V.1.
General Problems. V.2. Regularities of Propagation. – Kyiv: Naukova Dumka. (In Russian).
4.8. Hanyga, A 1983, Mathematical Theory of Nonlinear Elasticity. Elis Horwood, California.
4.9. Holzapfel, GA 2000, Nonlinear Solid Mechanics: A Continuum Approach for Engineering.
Birkhauser, Zurich. 2nd ed. 2006, Wiley, Chichester.
4.10. Lur’e, AI 1990, Nonlinear Theory of Elasticity. North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam.
4.11. Novozhilov, VV 1948, Osnovy Nielinieinoi Uprugosti (Foundations of nonlinear elasticity).
Gostekhizdat, Moscow. (In Russian)
4.12. Novozhilov, VV 1953, Foundations of the Nonlinear Theory of Elasticity. Graylock Press, Rochester-New York.
4.13. Ogden, RW 1997, Non-Linear Elastic Deformations. Dover Civil and Mechanical Engineering, Mineola.
4.14. Ratner, LW 2003, Non-Linear Theory of Elasticity and Optimal Design. Elsevier, London.
4.15. Rushchitsky, JJ 2014, Nonlinear Elastic Waves in Materials. Series “Foundations of Engineering Mechanics”. Springer, Heidelberg.
4,16. Storakers, B & Larsson, P-L 1998, Introduktion till finit elasticitetteori. Hallfasthetslara, KTH. (In Swedish)
4.17. Taber, LA 2004, Nonlinear Theory of Elasticity: Applications in Biomechanics. Birkhauser, Zurich.
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SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 1
Questions
4.1. The displacement (4.2) is a primary notion in the description of deformation. But further in constructing the theory of deformation, the new notion – the gradient of displacement – is used, which is only the linear approximation of the change of displacement. Comment the possible contradiction between using this linear approximation and constructing the nonlinear models.
4.2. Write a list of the strain tensors that are utilized in the mechanics. Comment a difference among them.
4.3. Find the mathematical definition of the tensor of arbitrary rank. Describe a difference between covariant and contravariant indexes of tensors. Estimate in which volume the tensor analysis is used in mechanics.
4.4. Find the definition of the conjugate pairs (ordered pairs) that consist mainly of one stress and one strain tensors. Write the most used in mechanics ordered pairs.
4.5. Write three examples of the real materials which are described in some simple problems of mechanics by generally elastic, hypoelastic, and hyperelastic models.
4.6. Thinking the Hooke law as one of the most important in mechanics of elastic materials, read the history of this Hooke brilliant scientific invention.