Resonant-triad instability of a pre-stressed incompressible elastic plate

The dynamic stability properties of a pre-stressed incompressible elastic plate are studied in this paper with respect to perturbations in the form of one near-neutral mode and two non-neutral modes interacting resonantly. The pre-stresses are assumed to be an all-round pressure. With the aid of a novel derivation procedure, the evolution equations governing the scaled amplitudes of the three modes are found to be given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaCa% aaleqabaGaaGOmaaaakiaadgeadaWgaaWcbaGaaGymaaqabaGccaGG% VaGaamizamaaBaaaleaacqaHepaDdaahaaadbeqaaiaaikdaaaaale% qaaOGaeyypa0JaeyOeI0Iaam4yamaaBaaaleaacaaIWaaabeaakiaa% dgeadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGJbWaaSbaaSqaai% aaigdaaeqaaOGaaiiFaiaadgeadaWgaaWcbaGaaGymaaqabaGccaGG% 8bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyAaiabeo7aNnaaBa% aaleaacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIYaaabeaa% kiqadgeagaqeamaaBaaaleaacaaIZaaabeaaaaa!5308!\[d^2 A_1 /d_{\tau ^2 } = - c_0 A_1 - c_1 |A_1 |^2 - i\gamma _1 \bar A_2 \bar A_3 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadg% eadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaamizaiabes8a0jabg2da% 9iabeo7aNnaaBaaaleaacaaIYaaabeaakiqadgeagaqeamaaBaaale% aacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIZaaabeaaaaa!4324!\[dA_2 /d\tau = \gamma _2 \bar A_1 \bar A_3 \] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadg% eadaWgaaWcbaGaaG4maaqabaGccaGGVaGaamizaiabes8a0jabg2da% 9iabeo7aNnaaBaaaleaacaaIZaaabeaakiqadgeagaqeamaaBaaale% aacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIYaaabeaaaaa!4325!\[dA_3 /d\tau = \gamma _3 \bar A_1 \bar A_2 \], where a bar denotes complex conjugation, is a slow time variable and c ₀, c ₁, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaigdaaeqaaaaa!387B!\[\gamma _1 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaikdaaeqaaaaa!387C!\[\gamma _2 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaiodaaeqaaaaa!387D!\[\gamma _3 \] are real constants. These equations are solved exactly for the special case when A ₂ and A ₃ have constant amplitudes but time-dependent phases. A series of new post-buckling states, which does not exist when the perturbation is monochromatic, are found. We show that two nonneutral modes can interact resonantly to produce a much larger near-neutral mode, and in particular, two O() non-neutral modes may induce a much larger % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaacI% cacqaH1oqzdaahaaWcbeqaamaalyaabaGaaGOmaaqaaiaaiodaaaaa% aOGaaiykaaaa!3B87!\[O(\varepsilon ^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} )\] oscillation or static post-buckling state. In this sense, resonant-triad interaction is a powerful mechanism in producing high levels of strain and stress in a pre-stressed elastic plate.     

The proof of Fermat's Last Theorem

ＴＨＥＰＲＯＯＦＯＦＦＥＲＭＡＴ＇ＳＬＡＳＴＴＨＥＯＲＥＭＷｏｎｇＣｈｉａｈｏ（汪家訸）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ１０,１９９５）Ａｂｓｔｒａｃｔ：（ｉ）Ｉｎｓｔｅａｄｏｆｘ￣ｎ＋ｙ￣ｎ＝ｚ￣ｎ,ｗｅｕｓｅａｓｔｈｅｇｅｎｅｒａｌｅｑｕａｔｉｏｎｏ...     

On the traction problem in linear elastostatics

A sharp uniqueness class is determined for the traction problem of linear elastostatics in exterior domains and in the half space. In particular, it is shown that this problem has a most one solution in the class of all vector fields u such that either u=o(r) or % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbaK% aaaaa!3782!\[\hat \nabla \]u=o(1), as r+, with w and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbaK% aaaaa!3782!\[\hat \nabla \]u respectively rigid displacement and symmetric part of u.     

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＦＯＲＡＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＦＩＲＳＴＯＲＤＥＲＳＹＳＴＥＭＣｈｅｎＳｏｎｇｌｉｎ（陈松林）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ８,１９８４;ＲｅｖｉｓｅｄＡｐｒｉｌ１５,１９...     

The limiting case of zero shear modulus in linear elasticity

In this paper we study the behavior of an isotropic homogeneous material with strain-stress relation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaai% ikaiaadweacaGGPaGaeyypa0JaaGOmaiabeY7aTjaadweacqGHRaWk% cqaH7oaBcaGGOaGaaeiDaiaabkhacaqGGaGaamyraiaacMcacaWGjb% aaaa!462B!\[\sigma (E) = 2\mu E + \lambda ({\text{tr }}E)I\] when tends to zero by using a decomposition of a Hilbert space and a minimization problem of an energy functional in some subspace. The existence and uniqueness of the solution of the differential system in this limiting case subject to certain boundary condition is also obtained in this work.     

A Reissner-Sagoci problem for a non-homogeneous elastic solid

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular punch which is bonded to the surface of a non-homogeneous isotropic elastic halfspace. The non-homogeneity corresponds to an axial variation of the linear elastic shear modulus according to the exponential form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaacI% cacaWG6bGaaiykaiabg2da9iaadEeadaWgaaWcbaGaaGymaaqabaGc% cqGHRaWkcaWGhbWaaSbaaSqaaiaaikdaaeqaaOGaaeiiaiaabwgada% ahaaWcbeqaaiaab2cacqaH+oaEcaqG6baaaaaa!439C!\[G(z) = G_1 + G_2 {\text{ e}}^{{\text{ - }}\xi {\text{z}}} \]. A Hankel transform development of the governing equations yields a set of dual integral equations which in turn can be reduced to a Fredholm integral equation of the second kind. A numerical evaluation of this integral equation yields results which can be used to estimate the torque-twist relationship for the circular punch.     

A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

Ａ－ＨＩＧＨ－ＯＲＤＥＲＡＣＣＵＲＡＣＹＥＸＰＬＩＣＩＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＳＯＬＶＩＮＧＴＨＥＥＱＵＡＴＩＯＮＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＰＡＲＡＢＯＬＩＣＴＹＰＥＭａＭｉｎｇｓｈｕ（马明书）（ＲｅｃｅｉｖｅｄＪｕｎｅ２,１...     

The traction boundary value problem for the elastostatic semi-infinite strip; existence of solution,and completeness of the Papkovich-Fadle eigenfunctions

The semi-infinite strip x0, –1y1 is in equilibrium under no body forces, with the sides y=±1, x>0 free of tractions, and on the end x=0% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS% baaSqaaiaadIhacaWG4baabeaakiaacIcacaaIWaGaaiilaiaadMha% caGGPaGaeyypa0JaamOzaiaacIcacaWG5bGaaiykaiaacYcaaaa!4298!\[\sigma _{xx} (0,y) = f(y),\]% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS% baaSqaaiaadIhacaWG4baabeaakiaacIcacaaIWaGaaiilaiaadMha% caGGPaGaeyypa0Jaam4zaiaacIcacaWG5bGaaiykaiaacYcaaaa!4299!\[\sigma _{xx} (0,y) = g(y),\]where f(y), g(y) are independent, self-equilibrating tractions prescribed for y[–1, 1]. A rigorous proof is given that if f, g are of bounded variation on [–1, 1], then this traction boundary value problem posesses a solution, and the stress field of this solution may be expanded, even on x=0, as a convergent series of the Papkovich-Fadle eigenfunctions. Thus these eigenfunctions are complete for the expansion of such data {f, g}.On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grant No. A9117.     

Nonlinear parametric identification using periodic equilibrium states

The subject of this paper is the development and assessment of a new nonlinear parametric identification method for dynamic systems using periodic equilibrium states or outputs. The method consists of a modified Bayesian point estimation technique which can be used in combination with a time discretization method or a shooting method to obtain the periodic equilibrium states. It is assumed that the specific excited and measured periodic solution can be computed directly from a static initial guess. An important feature of this estimator is the possibility to estimate the best parameters based on all experiments of the complete experimental set-up. The choice of using periodic states appears to be computationally efficient compared to using transient states. The new method is applied to multiple sets of nonlinear shaker-test measurements of a uniaxially loaded F-16 nose landing gear damper, for which a standard 1 dof mechanical model and a more comprehensive 2 dof thermo-mechanical model are postulated. Finally, the predictive power of the method is assessed by comparison of predictions for a transient drop-test load case of the best 2 dof model with corresponding parameters and real drop-test measurements.Notation a,b partial derivative of a with respect to vector b - % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaca% WGHbaaleqabaGaaiOlaaaaaaa!37CD!\[\mathop a\limits^. \] total derivative of a with respect to time t - a total derivative of a with respect to the dimensionless time - â predicted value of a - mean value of a - ã approximate equations a - e residuals - f output equations - f _e external excitation frequency - F force - g ordinary differential equations - l number of parameters - n number of samples in each experiment - N number of experiments - % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuGacqWFneVtdaWgaaWcbaGa% amyyaaqabaGccaGGOaGaamOyaiaacYcacqWFce-qcaGGPaaaaa!4710!\[{\cal N}_a (b,{\cal C})\] a-dimensional normal distribution (mean b and covariance C) - p pressure - q degrees of freedom - s number of outputs - t time - T absolute temperature     

Stochastic averaging using elliptic functions to study nonlinear stochastic systems

In this paper, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Itô differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The second order nonlinear differential equation that is examined in this work can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGaf8hEaGNbamaacqGHRaWkcaWGJbadcaaIXaGc% cqWF4baEcqGHRaWkcaWGJbadcaaIZaGccqWF4baEdaahaaWcbeqaai% aaiodaaaGccqGHRaWkcqaH1oqzcaWGMbGaaiikaiab-Hha4jaacYca% cqWFGaaicuWF4baEgaGaaiaacMcacqGHRaWkcqaH1oqzdaahaaWcbe% qaaiaaigdacaGGVaGaaGOmaaaaruWrL9MCNLwyaGGbcOGaa43zaiaa% cIcacqWF4baEcaGGSaGae8hiaaIaf8hEaGNbaiaacaGGSaGae8hiaa% IaeqOVdGNaaeikaiaadshacaqGPaGaaiykaiabg2da9iaaicdaaaa!645D!\[\ddot x + c1x + c3x^3 + \varepsilon f(x, \dot x) + \varepsilon ^{1/2} g(x, \dot x, \xi {\text{(}}t{\text{)}}) = 0\] where c ₁ and c ₃ are given constants, (t) is stationary stochastic process with zero mean and 1 is a small parameter. This method involves the laborious manipulation of Jacobian elliptic functions such as cn, dn and sn rather than the usual trigonometric functions. The use of a symbolic language such as Mathematica reduces the computational effort and allows us to express the results in a convenient form. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. Finally, this method was applied to study some standard second order systems.     

On a counterexample to a conjecture of Saint Venant

The elliptic boundary value problem % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi% slcqGHuoarcaWG1bGaeyypa0dccaGae8hiaaIaaGymaiab-bcaGiab% -bcaGiab-bcaGiaabMgacaqGUbGaaeiiaiabfM6axjaabYcaaeaaae% aacaWG1bGaeyypa0JaaGimaiab-bcaGiab-bcaGiab-bcaGiaab+ga% caqGUbGaaeiiaiabgkGi2kabfM6axjaabYcaaaaa!4E11!\[\begin{gathered}- \Delta u = 1 {\text{in }}\Omega {\text{,}} \hfill \\\hfill \\u = 0 {\text{on }}\partial \Omega {\text{,}} \hfill \\\end{gathered}\]is considered. The Saint Venant's conjecture for convex plane domains , having symmetry about two orthogonal axes, is that the maximum of |u| occurs only at the points on which are nearest to the origin. G. Sweers constructed one such domain and claimed that either the conjecture fails for or for ={(x, y);u(x, y) >}, which again is convex. We give a totally different proof of this claim. Our proof brings out clearly the reason for the failure of the conjecture and also allows us to construct many more such domains.     

Bifurcation scenarios of the noisy duffing-van der pol oscillator

This paper presents a numerical study of the bifurcation behavior of the noisy Duffing-van der Pol oscillator% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttLeary% qr1ngBPrgaiuGacuWF4baEgaWaaiaaiccacqWF9aqpcaaIGaGaaiik% aerbtLhBMfwzUbacgiGaa4xSdiaaiccacqGHRaWkcaaIGaGaeq4Wdm% 3ccaaIXaGcceqGxbGbaiaaliaaigdakiGacMcacqWF4baEcaaIGaGa% ci4kaiaaiccacqaHYoGycuWF4baEgaGaaiaaiccacqGHsislcaaIGa% Gae8hEaG3aaWbaaSqabeaacaaIZaaaaOGaaGiiaiabgkHiTiaaicca% cqWF4baEdaahaaWcbeqaaiaaikdaaaGccuWF4baEgaGaaiaaiccaci% GGRaGaaGiiaiabeo8aZTGaaGOmaOGabe4vayaacaGaaeOmaiaabYca% aaa!5F62!\[\ddot x = (\alpha + \sigma 1{\rm{\dot W}}1)x + \beta \dot x - x^3 - x^2 \dot x + \sigma 2{\rm{\dot W2,}}\]where , are bifurcation parameters, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceqGxbGbaiaali% aaigdakiqabEfagaGaaSGaciOmaaaa!35B4!\[{\rm{\dot W}}1{\rm{\dot W}}2\] are independent white noise processes, and ₁, ₂ are intensity parameters. A stochastic bifurcation here means (a) the qualitative change of stationary measures or (b) the change of stability of invariant measures and the occurrence of new invariant measures for the random dynamical system generated by (1). The first type of bifurcation can be observed when studying the solution of the Fokker-Planck equation, this stationary measure is a quantity corresponding to the one-point motion. More generally, if one is interested in the simultaneous motion of n points (n1) forward and backward in time, then the second type of bifurcation arises naturally, capturing all the stochastic dynamics of (1). Based on the numerical results, we propose definitions of the stochastic pitchfork and Hopf bifurcations.     

Application of melt flow index for rheological characterization of branched and linear polymers

A procedure for evaluating rheological characteristics, such as the master curves log/ ₀ vs. log % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xd9Gqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGaceWFZo% Gbaiaaaaa!3B59!\[\dot \gamma \] ₀ and flow curves, using the melt flow index is described for branched and linear polymers. Experimental data on the melt flow index and branching degree are needed for this purpose, as well as some polymer constants, i.e. coefficients of the ₀ vs. MFI relation and coefficients of fluidity dependence on molecular characteristics. An example is given for bisphenol A polycarbonate.     

The semi-infinite strip x≥0, −1≤y≤1; completeness of the Papkovich-Fadle eigenfunctions when Φxx(0,y), Φyy(0,y) are prescribed

For the problem of bending of a semi-infinite strip x0, –1y1, with the sides y=±1 clamped, we give a proof that the end-data% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaarmWu51MyVXgaiuGacqWFgpGzdaWgaaWcbaGaaeiEaiaabIha% aeqaaGqbaOGae4hiaaIaaiikaiaaicdacaGGSaGae4hiaaIaamyEai% aacMcacqGFGaaicqGH9aqpcqGFGaaicaWGMbGaaiikaiaadMhacaGG% PaGaaiilaaqaaiab-z8aMnaaBaaaleaacaqG5bGaaeyEaaqabaGccq% GFGaaicaGGOaGaaGimaiaacYcacqGFGaaicaWG5bGaaiykaiab+bca% Giabg2da9iab+bcaGiaadAgacaGGOaGaamyEaiaacMcacaGGSaaaaa% a!5D6D!\[\begin{array}{l} \phi _{{\rm{xx}}} (0, y) = f(y), \\ \phi _{{\rm{yy}}} (0, y) = f(y), \\ \end{array}\] where f(y), g(y) are arbitrary independent functions prescribed on (–1,1), may be expanded as a series of the bi-orthogonal Papkovich-Fadle eigenfunctions for the strip. This represents an advance on the standard work of R. T. C. Smith [6], who proved such an expansion, but under conditions which are often not satisfied in practice. In particular we are able to solve this bi-harmonic boundary value problem when f, g do not satisfy the side conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaacaWGMbGaaiikaiabgglaXkaaigdacaGGPaqedmvETj2BSbac% faGae8hiaaIaeyypa0Jae8hiaaIaamOzamaaCaaaleqabaGaai4jaa% aakiab-bcaGiaacIcacqGHXcqScaaIXaGaaiykaiab-bcaGiabg2da% 9iab-bcaGiaaicdacaGGSaaabaGaam4zaiaacIcacqGHXcqScaaIXa% Gaaiykaiab-bcaGiabg2da9iab-bcaGiaadEgadaahaaWcbeqaaiaa% cEcaaaGccqWFGaaicaGGOaGaeyySaeRaaGymaiaacMcacqWFGaaicq% GH9aqpcqWFGaaicaaIWaGaaiilaaaaaa!6222!\[\begin{array}{l} f( \pm 1) = f^' ( \pm 1) = 0, \\ g( \pm 1) = g^' ( \pm 1) = 0, \\ \end{array}\]and when the conditions of consistency% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qmaeaacaWGNbGaaiikaiaadMhacaGGPaqedmvETj2BSbacfaGa% e8hiaaIaamizaiaadMhacqWFGaaicqWF9aqpcqWFGaaidaWdXaqaai% aadMhacaWGNbGaaiikaiaadMhacaGGPaGae8hiaaIaamizaiaadMha% cqWFGaaicqGH9aqpcqWFGaaicaaIWaaaleaacqWFsislcqWFXaqmae% aacqWFXaqma0Gaey4kIipaaSqaaiabgkHiTiaaigdaaeaacaaIXaaa% niabgUIiYdaaaa!5A1B!\[\int_{ - 1}^1 {g(y) dy = \int_{ - 1}^1 {yg(y) dy = 0} } \]are not satisfied.The present completeness proof thus answers questions raised recently (in the mathematically equivalent context of Stokes flow) by Joseph [3], and Joseph and Sturges [5], who showed that if the side conditions (A), (B) are relaxed then the corresponding eigenfunction series may still converge; but they left open the more difficult question of whether these series still converge to the data.The method of proof used here also succeeds in proving a corresponding completeness theorem for the Williams eigenfunctions for the wedge with the data.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaadaabciqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaa% daqadiqaamaalaaabaGaaGymaaqaaiaadkhaaaqedmvETj2BSbacfi% Gae8NXdygacaGLOaGaayzkaaaacaGLiWoadaWgaaWcbaGaamOCaiab% g2da9iaaigdaaeqaaGqbaOGae4hiaaIaeyypa0Jae4hiaaIaamOzai% aacIcacqaH4oqCcaGGPaGaaiilaaqaamaaeiGabaWaaSaaaeaacqGH% ciITdaahaaWcbeqaaiaaikdaaaGccqaHgpGzaeaacqGHciITcqaH4o% qCdaahaaWcbeqaaiaaikdaaaaaaOWaaeWaceaadaWcaaqaaiaaigda% aeaacaWGYbaaaiab-z8aMbGaayjkaiaawMcaaaGaayjcSdWaaSbaaS% qaaiaadkhacqGH9aqpcaaIXaaabeaakiab+bcaGiabg2da9iab+bca% GiaadEgacaGGOaGaeqiUdeNaaiykaiaacYcaaaaa!6B9C!\[\begin{array}{l} \left. {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = f(\theta ), \\ \left. {\frac{{\partial ^2 \phi }}{{\partial \theta ^2 }}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = g(\theta ), \\ \end{array}\]prescribed on –<<, (where 2 is the wedge angle).Department of Mathematics, University of ManchesterOn leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A 9259 and A9117.     

Polyconvex energies and symmetry requirements

For polyconvex stored energy mappings % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabiqaaiGacaGaamqadaabaeaafiaakeaadaqiaaqaaiabeo8aZbGa% ayPadaaaaa!4654!\[\widehat\sigma \], customary spatial and material symmetry requirements are shown to impose restrictions which are effective in the case of solids. Next, a generalized notion of material symmetry group is introduced, and it is shown that polyconvexity and infinite growth of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabiqaaiGacaGaamqadaabaeaafiaakeaadaqiaaqaaiabeo8aZbGa% ayPadaaaaa!4654!\[\widehat\sigma \](F) as det F 0+ imply that all symmetry transformations must have unit determinant.     

Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts

In high shear rate capillary rheometry the combined effect of pressure dependent viscosity and dissipative heating becomes significant. Analytical expressions are derived to treat curved Bagley plots and throttle experiments. End effects are taken into account by using an effective length over radius ratio. The non-adiabatic case is described using a lump heat transfer coefficient ? following Hay et al. (1999). The latter enters into the dissipative heating coefficient % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba % WaaqaafaaakeaacqaH1oqzdaWgaaWcbaGaemiCaahabeaakiabg2da % 9iabeg8aYnaaCaaaleqabaGaeyOeI0IaeGymaedaaOWaaeWaaeaacq % WGJbWydaWgaaWcbaGaemiCaahabeaakiabgUcaRmaalyaabaGaeu4M % dWeabaGafmyBa0MbaiaaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacq % GHsislcqaIXaqmaaaaaa!4D6C! e_p = r ^{- 1} ( c_p + L \mathord

Radiation conditions in asymmetric elasticity

In the present paper the radiation conditions of the Sommerfield type for a linear homogeneous and isotropic micropolar elasticity are discussed. A regular solution (u, ) of the fundamental system of field equations in an infinite domain has been defined using the radiation conditions for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\], and formulae of the Betti type for an infinite domain with a cavity have been derived.

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Zusammenfassung Betrachtet werden die Ausstrahlungsbedingungen der Sommerfeldschen Art für lineare homogene mikropolare Elastizitätstheorie. Die reguläre Lösung (u, ) der grundlegenden Gleichungen für einen unendlichen Raum wird mit Hilfe der Bedingungen für Paare % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] und % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\] definiert, wobei % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\] ist. Es wird gezeigt, dass derartige reguläre Lösung mit Hilfe der Formel Bettischer Art für einem unendlichen Raum mit einem Hohlraum dargestellt werden kann.

     

Effects of viscous dissipation on heat transfer in an oscillating flow past a flat plate

Theoretical investigation has been carried out of laminar thermal boundary layer response to harmonic oscillations in velocity associated with a progressive wave imposed on a steady free stream velocity and convected in the free stream direction. Series solutions are derived both to velocity and temperature field and the resulting equations are solved numerically. The functions affecting the temperature field are shown graphically for different values of Prandtl number. It is observed that there is more reduction in the rate of heat transfer for P _r<1 and a rise in the rate of heat transfer for P _r>1 due to the presence of oscillatory free-stream.Nomenclature u, v velocity components in the x and y direction - x, y Cartesian coordinate axes - t time - U, U ₀ instantaneous value of and mean free stream velocity - density of fluid - kinematic viscosity - T, T _w, T temperature of the fluid, wall and free stream fluid - c _p specific heat at constant pressure - thermal diffusivity - amplitude of free stream velocity - frequency - _p non-dimensional temperature (T–T /T _w–T ) - P _r Prandtl number (c _p/K) - E _c Eckert number (U ₀ ² /c _p(T _w–T )) - a parameter ( ) - ₀ boundary layer thickness of the oscillation of a harmonic oscillation of frequency ( ) - ordinary boundary layer thickness ( ) - time-averaged, time-independent external velocity - A, B, C, D, E, K, L, M, N, P functions used in expansion for u and - Nu Nusselt number (hx/k) - T _w–% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8V4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcadaGcaa% qaaiaadAhacaWG4bGaai4laiqadwfagaqeaaWcbeaakiaacMcaaaa!3CA6!\[(\sqrt {vx/\bar U} )\] - k thermal conductivity     

Boundary conditions at the edge of a thin or thick plate bonded to an elastic support

At the clamped edge of a thin plate, the interior transverse deflection ω(x ₁, x₂) of the mid-plane x ₃=0 is required to satisfy the boundary conditions ω=?ω/?n=0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for ω must always have the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakqaabeqaaiaabEhacaqGTaWa% aSaaaeaacaGG0aGaae4vamaaCaaaleqabaGaamOqaaaaaOqaaiaaco% dadaqadaqaaiaacgdacqGHsislcaqG2baacaGLOaGaayzkaaaaaiaa% bIgadaahaaWcbeqaaiaackdaaaGcdaWcaaqaaiaabsgadaahaaWcbe% qaaiaackdaaaGccaqG3baabaGaaeizaiaabIhafaqabeGabaaajaaq% baqcLbkacaGGYaaajaaybaqcLbkacaGGXaaaaaaakiabgUcaRmaala% aabaGaaiinaiaabEfadaahaaWcbeqaaiaadAeaaaaakeaacaGGZaWa% aeWaaeaacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGOb% WaaWbaaSqabeaacaGGZaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaa% caGGZaaaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaa% saaiaacodaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0Jaaiimaiaa% cYcaaeaadaWcaaqaaiaabsgacaqG3baabaGaaeizaiaabIhaliaacg% daaaGccqGHsisldaWcaaqaaiaacsdacqqHyoqudaahaaWcbeqaaiaa% bkeaaaaakeaacaGGZaWaaeWaaeaacaGGXaGaeyOeI0IaaeODaaGaay% jkaiaawMcaaaaacaqGObWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGG% YaaaaOGaae4DaaqaaiaabsgacaqG4bqbaeqabiqaaaqcaauaaKqzGc% GaaiOmaaqcaawaaKqzGcGaaiymaaaaaaGccqGHRaWkdaWcaaqaaiaa% csdacqqHyoqudaahaaWcbeqaaiaabAeaaaaakeaacaGGZaWaaeWaae% aacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaWba% aSqabeaacaGGYaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGGZa% aaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaasaaiaa% codaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0JaaiimaiaacYcaaa% aa!993A!\[\begin{gathered}{\text{w - }}\frac{{4{\text{W}}^B }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4{\text{W}}^F }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^3 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4\Theta ^{\text{F}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\end{gathered}\]with exponentially small error as L/h→∞, where 2h is the plate thickness and L is the length scale of ω in the x ₁-direction. The four coefficients W ^B, W^F, Θ ^B , Θ ^F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h→∞ (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for Θ ^B , W ^F, we prove that these constants take large values when the support is ‘soft’ and so may still have a strong influence even when h/L is small. The coefficient W ^F is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are w=0,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaabsgacaqG% 3baabaGaaeizaiaabIhaliaacgdaaaGccqGHsisldaWcaaqaaiaacs% dacqqHyoqudaahaaWcbeqaaiaabkeaaaaakeaacaGGZaWaaeWaaeaa% caGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaSaaae% aacaqGKbWaaWbaaSqabeaacaGGYaaaaOGaae4DaaqaaiaabsgacaqG% 4bqbaeqabiqaaaqcaauaaKqzGcGaaiOmaaqcaawaaKqzGcGaaiymaa% aaaaGccqGH9aqpcaGGWaGaaiilaaaa!5DD4!\[\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} = 0,\]which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.     

Global solution to the incompressible viscous-multipolar material problem

In the paper we give a proof of the global existence of the weak solution to the initial-boundary-value problem describing an incompressible elasto-viscous-multipolar material in finite geometry. A brief introduction to the physical background of viscous-multipolar materials is given. We suggest the hypothesis% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaxadabaGaeu4OdmfaleaacaWGPbGaaiilaGqaciaa-bcacaWG% QbGaa8hiaiabg2da9iaa-bcacaaIXaaabaGaaG4maaaakiaa-bcada% abdiqcaasaaOWaaSaaaKaaGeaacqGHciITcqqHOoqwcaGGOaGaamOr% aiaacYcacqaH4oqCcaGGPaaabaGaeyOaIyRaamOraOWaaSbaaSqaai% aadMgacaWGbbaabeaaaaqcaaIaa8hiaiaadAeakmaaBaaaleaacaWG% QbGaamyqaaqabaaajaaqcaGLhWUaayjcSdGaa8hiaiabgsMiJkaado% gakmaaBaaaleaacaWFVbaabeaakiaadwgacaGGOaGaamOraiaacYca% ieaacaGFGaGaeqiUdeNaaiykaiaa+bcacqGHRaWkcaGFGaGaam4yam% aaBaaaleaacaaIXaGaa4hiaiaacYcaaeqaaaaa!686E!\[\mathop \Sigma \limits_{i, j = 1}^3 \left| {\frac{{\partial \Psi (F,\theta )}}{{\partial F_{iA} }} F_{jA} } \right| \leqslant c_o e(F, \theta ) + c_{1 ,} \] which enables one to obtain a priori estimates.