Large Longitudinal Retarded Elastic Deformation of Rubberlike Network Polymers. II. Application of a General Formulation of Nonlinear Response

Leaderman, Herbert; McCrackin, Frank L.; Nakada, Osamu

doi:10.1122/1.548948

Cited by 42 publications

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“…The above mentioned concepts of orthogonality are quite natural in the spaces considered and they are important for certain applications [81,123,159], however, there are several other concepts of orthogonality defined and developed by Birkhoff, Roberts, James and Day (see Sect. 1).…”

Section: Approximate Orthogonalitiesmentioning

confidence: 99%

Orthogonalities and functional equations

Sikorska

2014

Aequat. Math.

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Abstract. In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations.Mathematics Subject Classification. 39B52, 39B82, 46B20, 47B49, 15A86.

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Section: Approximate Orthogonalitiesmentioning

confidence: 99%

Orthogonalities and functional equations

Sikorska

2014

Aequat. Math.

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“…Using th e approximation of Na kada 's general formulation desc ribed by Leaderman , McCrackin, and Nakada [7] , th e ordinates of th e inv erted recover y c urves as d esc rib ed above s hould be greate r in the early portion of th e ti me scale than the ordinates of th e loa ded portion of th e creep c urves minus th e vis· co us s train. Th e diffe re nce between th e c urv es s hould ri se to a maximum within th e firs t few minutes and gradually dec rease to zero as time in creases.…”

Section: )mentioning

confidence: 99%

“…If on e appli es a n approximati o n of Na k ada's ge neral formulatio n described by Lead erm an , McC rac kin , a nd N ak ada [7] , th e equation re prese nting th e re o tarde d e lasti c r es po nse takes th e foll owin g form :…”

Section: Da-(e )mentioning

confidence: 99%

“…It is also noted in figure 8 th at as a fu ncti on of stress th e co mbination strain divid ed by th e stress is a s trai ght lin e for all values of t. This indi cated th at ove r all ranges of t , the co mbination elas ti c s train obeys th e same functional relati on to th e s tress. Since th e reco ve ry c urves we re found to be s upe rimposabl e upon th e loading c urves minu s th e vi sco us porti on, it appeared th at th e nonlin ear material did not obey th e ap proximation of nonlinear ge neraliza ti on pre viously noted [7]. It was the n qu es ti oned wh e th er th e r e tarded behavior of amalgam was trul y nonlin ear or whe ther the nonlin ear behavior co uld be a ttribute d to th e geo me try of th e specim e n be in g observed.…”

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confidence: 99%

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Viscoelastic behavior of dental amalgam

Oglesby¹,

Dickson²,

Rodriguez³

et al. 1968

J. RES. NATL. BUR. STAN. SECT. C. ENG. INSTR.

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Meas u re men ts ma de on de nt a l a malga m in tens io n indi ca te th at a ma lga m ex h ib its three t ypes of vi scoelasti c ph e nome na: (1) in st anta neo us e las ti c s tra in , (2) re ta rd ed e las ti c st ra in (tra nsie nt c ree p), a nd (3) visco us s train (s teady state c reep). T he co mbin a ti on of e lasti c plu s re ta rd e d s train ca n be rep re· se nt ed by a n eq uation of th e form f = A(T + 8 '(J' wh er e A a nd 8 a re fun c ti ons of time but not of the stress, (J . T he visco us s train ra te ca n be re p rese nted b y a n equ a ti on of th e for m €v = K(J'" wh ere K a nd m a re cons ta nts of th e ma te ri a l. By a pplyin g a nonlin ea r gene ra li zati o n of th e Bo lt zma nn s upe rposi ti on prin c ipl e to a ge ne ral eq ua ti on descri bin g the c ree p be hav ior of a ma lgam, the res ults of c reep tests ca n be d irectl y re la te d to th e resu lts of st ress-s train tes ts.Key Words: A ma lgam; c ree p; d e nt al; steady-state c reep; s tress-stra in ; tra ns ie nt c ree p; viscoe lasti c.

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“…Although the above orthogonality is important for certain applications [5], [9], [13], in this note we consider orthogonalities which are standard in inner product and normed spaces. To some extent our results are less general than those in the above papers since the standard orthogonality is weaker than lattice theoretic orthogonality.…”

Section: ±_ L Y) If {T:x(t)y(t)j£0} Is 0 In Case (I) or Of Measure mentioning

confidence: 99%

Orthogonality and nonlinear functionals

Gudder¹,

Strawther²

1974

Bull. Amer. Math. Soc.

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Let X be a vector space of functions. In practice, Zis taken to be (i) the set of continuous functions on some type of topological space or (ii) a set of measurable functions on a measure space. We say that x, y e X are orthogonal in the lattice theoretic sense (x ±_ L y) if {t: [13], in this note we consider orthogonalities which are standard in inner product and normed spaces. To some extent our results are less general than those in the above papers since the standard orthogonality is weaker than lattice theoretic orthogonality. On the other hand, some of our theorems apply to more general vector spaces than the above and furthermore we have obtained results for a more general class of functionals which we call orthogonally monotone functionals. Finally, we use our results to solve a nonlinear functional equation and give an application for the solution.An orthogonality vector space is a real vector space X with dim X^.2 on which there is defined a relation x_]_y such that (01) 0_L*, xj_0 for all xeX y (02) if x_[_y and x, y^O then x, y are linearly independent, (03) if x±y then ouc±py for all a 9 peR 9 (04) if B is a two-dimensional subspace of X, then for every Ojéx e B, there exists Ojéy e B such that x±_y and x+y J_x-y.It is easily seen that any real vector space of dimension ^2 is an orthogonality vector space if we define 0J_x, *_i_0 for all x and if x, y5^0 then xA_y iff x, y are linearly independent. It is also clear that an inner product space is an orthogonality vector space under the standard orthogonality AMS (MOS) subject classifications (1970). Primary 46B99, 46C05, 47H15; Secondary 46F05, 81A12.

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Large Longitudinal Retarded Elastic Deformation of Rubberlike Network Polymers. II. Application of a General Formulation of Nonlinear Response

Cited by 42 publications

References 0 publications

Orthogonalities and functional equations

Orthogonalities and functional equations

Viscoelastic behavior of dental amalgam

Orthogonality and nonlinear functionals

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