The quarterly journal of mechanics and applied mathematics最新文献

{"title":"Rate-independent soft crawlers","authors":"P. Gidoni","doi":"10.1093/qjmam/hby010","DOIUrl":"https://doi.org/10.1093/qjmam/hby010","url":null,"abstract":"This paper applies the theory of rate-independent systems to model the locomotion of bio-mimetic soft crawlers. We prove the well-posedness of the approach and illustrate how the various strategies adopted by crawlers to achieve locomotion, such as friction anisotropy, complex shape changes and control on the friction coefficients, can be effectively described in terms of stasis domains. Compared to other rate-independent systems, locomotion models do not present any Dirichlet boundary condition, so that all rigid translations are admissible displacements, resulting in a non-coercivity of the energy term. We prove that existence and uniqueness of solution are guaranteed under suitable assumptions on the dissipation potential. Such results are then extended to the case of time-dependent dissipation.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"545 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77449742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wave Propagation and Homogenization in 2d and 3d Lattices: A Semi-Analytical Approach","authors":"A. A. Kutsenko;A. J. Nagy;X. Su;A. L. Shuvalov;A. N. Norris","doi":"10.1093/qjmam/hbx002","DOIUrl":"https://doi.org/10.1093/qjmam/hbx002","url":null,"abstract":"Wave motion in two- and three-dimensional periodic lattices of beam members supporting longitudinal and flexural waves is considered. An analytic method for solving the Bloch wave spectrum is developed, characterized by a generalized eigenvalue equation obtained by enforcing the Floquet condition. The dynamic stiffness matrix is shown to be explicitly Hermitian and to admit positive eigenvalues. Lattices with hexagonal, rectangular, tetrahedral and cubic unit cells are analyzed. The semi-analytical method can be asymptotically expanded for low frequency yielding explicit forms for the Christoffel matrix describing wave motion in the quasistatic limit.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 2","pages":"131-151"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49953234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Crack Induced by a Thin Rigid Inclusion Partly Debonded from the Matrix","authors":"Y. A. Antipov;S. M. Mkhitaryan","doi":"10.1093/qjmam/hbx003","DOIUrl":"https://doi.org/10.1093/qjmam/hbx003","url":null,"abstract":"The interaction of a thin rigid inclusion with a finite crack is studied. Two plane problems of elasticity are considered. The first one concerns the case when the upper side of the inclusion of length \u0000<tex>$2b$</tex>\u0000 is completely debonded from the matrix, and the crack formed symmetrically penetrates into the medium; its length is \u0000<tex>$2a&gt;2b$</tex>\u0000. In the second model, the upper side of the inclusion is partly separated from the matrix, and \u0000<tex>$a&lt;b$</tex>\u0000. It is shown that both problems are governed by singular integral equations that share the same kernel but have different right-hand sides, and their solution satisfy different additional conditions. Derivation of a closed-form solution to these integral equations is one of the main results of the article. The solution is found by reducing the integral equation to a vector Riemann–Hilbert problem with the Chebotarev–Khrapkov matrix coefficient. A feature of the method proposed is that the vector Riemann–Hilbert problem is set on a finite segment, while the original Khrapkov method of matrix factorization is developed for a closed contour. In the case, when the crack and inclusion lengths are the same, the solution is derived by passing to the limit \u0000<tex>$b/ato 1$</tex>\u0000. It is demonstrated that the limiting case \u0000<tex>$a=b$</tex>\u0000 is unstable, and when \u0000<tex>$a&lt;b$</tex>\u0000 and the crack tips approach the inclusion ends, the crack tends to accelerate in order to penetrate into the matrix. It is shown that the stresses and the tangential derivative of the displacement have the square root singularity at the crack and the inclusion tips. The stresses and the displacement derivative are monotonic at the external singular points, \u0000<tex>$pm a$</tex>\u0000 and \u0000<tex>$pm b$</tex>\u0000 for Models 1 and 2, and oscillate in small neighborhoods of the internal singular points, \u0000<tex>$|x/a-b/a|&lt;varepsilon$</tex>\u0000 and \u0000<tex>$|x/b-a/b|&lt;varepsilon$</tex>\u0000, for the first and second problems, respectively, and \u0000<tex>$0&lt;varepsilonle 10^{-6}$</tex>\u0000. The potential energy release rate and the Griffith crack growth criterion are established for both models.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 2","pages":"153-185"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49953235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sensitivity of Love and Quasi-Rayleigh waves to model parameters","authors":"D. R. Dalton;M. A. Slawinski;P. Stachura;T. Stanoev","doi":"10.1093/qjmam/hbx001","DOIUrl":"https://doi.org/10.1093/qjmam/hbx001","url":null,"abstract":"We examine the sensitivity of the Love and the quasi-Rayleigh waves to model parameters. Both waves are guided waves that propagate in the same model of an elastic layer above an elastic halfspace. We study their dispersion curves without any simplifying assumptions, beyond the standard approach of elasticity theory in isotropic media. We examine the sensitivity of both waves to elasticity parameters, frequency and layer thickness, for varying frequency and different modes. In the case of Love waves, we derive and plot the absolute value of a dimensionless sensitivity coefficient in terms of partial derivatives, and perform an analysis to find the optimum frequency for determining the layer thickness. For a coherency of the background information, we briefly review the Love-wave dispersion relation and provide details of the less-common derivation of the quasi-Rayleigh relation in the Appendix. We compare that derivation to past results in the literature, finding certain discrepancies among them.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 2","pages":"103-130"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbx001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49953233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A straight mixed mode fracture with the steigmann–ogden boundary condition","authors":"Anna Y. Zemlyanova","doi":"10.1093/qjmam/hbw016","DOIUrl":"10.1093/qjmam/hbw016","url":null,"abstract":"A problem of a straight mixed mode non-interface fracture in an infinite plane is treated analytically with the help of complex analysis techniques. The surfaces of the fracture are subjected to surface elasticity in the form proposed by Steigmann and Ogden (Steigmann and Ogden, Proc. R. Soc. A453 (1997); Steigmann and Ogden, Proc. R. Soc. A455 (1999)). The boundary conditions on the banks of the fracture connect the stresses and the derivatives of the displacements. The mechanical problem is reduced to two systems of singular integro-differential equations which are further reduced to the systems of equations with logarithmic singularities. It is shown that modeling of the fracture with the Steigmann–Ogden elasticity produces the stress and strain fields, which are bounded at the crack tips. The existence and uniqueness of the solution for almost all the values of the parameters is proved. Additionally, it is shown that introduction of the surface mechanics into the modeling of fracture leads to the size-dependent equations. A numerical scheme of the solution of the systems of singular integro-differential equations is suggested, and the numerical results are presented for different values of the mechanical and the geometric parameters.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 1","pages":"65-86"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbw016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45303044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Propagation of water waves over an asymmetrical rectangular trench","authors":"Ranita Roy;Rumpa Chakraborty;B. N. Mandal","doi":"10.1093/qjmam/hbw015","DOIUrl":"10.1093/qjmam/hbw015","url":null,"abstract":"Assuming linear theory, the problem of water wave scattering by an asymmetric rectangular trench is investigated by employing Havelock’s expansion of water wave potential. A multi-term Galerkin approximation technique involving ultra-spherical Gegenbauer polynomials has been utilised for solving a first-kind vector integral equation, which is obtained in the analysis of the problem following Havelock’s inversion formulae. Numerical estimates for the reflection and transmission coefficients are depicted graphically for different configurations of the rectangular trench. Numerical results available in the literatures are recovered by using the present method and thereby confirming the correctness of the numerical results presented here.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 1","pages":"49-64"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbw015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45521310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A dynamic contact model for viscoelastic plates","authors":"Mircea Sofonea;Krzysztof Bartosz","doi":"10.1093/qjmam/hbw013","DOIUrl":"https://doi.org/10.1093/qjmam/hbw013","url":null,"abstract":"We introduce a mathematical model that describes the evolution of a viscoelastic plate in frictionless contact with a deformable foundation. The process is dynamic, the contact is with normal compliance and is modelled with a subdifferentiable boundary condition. We derive a variational formulation of the problem which has the form of a second-order evolutionary hemivariational inequality for the displacement field. Then, we establish the existence of a weak solution to the model.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 1","pages":"1-19"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbw013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the summation of divergent, truncated, and underspecified power series via asymptotic approximants","authors":"N. S. Barlow;C. R. Stanton;N. Hill;S. J. Weinstein;A. G. Cio","doi":"10.1093/qjmam/hbw014","DOIUrl":"10.1093/qjmam/hbw014","url":null,"abstract":"A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series’ expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three non-linear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl–Petviashvili (FP) monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and FP monopole.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 1","pages":"21-48"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbw014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48909064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Total transmission of waves through narrow gaps in channels","authors":"D. V. Evans;R. Porter","doi":"10.1093/qjmam/hbw017","DOIUrl":"10.1093/qjmam/hbw017","url":null,"abstract":"The problem of the transmission of plane waves along a uniform parallel-walled waveguide or channel through a periodically spaced array of thin screens with gaps placed across the channel is investigated. Of particular interest is the existence of frequencies at which energy it totally transmitted—shown to occur when there are two or more screens—and how they depend upon the size of the gaps in the screen. Attention focuses on small gaps where the phenomenon of total transmission persists and where an approximate analysis of a system of full coupled integrals equations is reduced to simple linear difference equations that can be solved in closed form. Results show comparisons between exact computations, the small-gap computations and corresponding wide-spacing approximations. A discussion of total reflection is also included when the gaps are no longer assumed to be small.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"70 1","pages":"87-101"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbw017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42972992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The large-time development of the solution to an initial-value problem for the generalised burgers’ equation","authors":"J. A. Leach","doi":"10.1093/qjmam/hbw006","DOIUrl":"https://doi.org/10.1093/qjmam/hbw006","url":null,"abstract":"In this article, we consider an initial-value problem for the generalized Burgers’ equation. The normalized Burgers’ equation considered is given bywhere \u0000<tex>$-frac{1}{2} le delta not= 0$</tex>\u0000, and \u0000<tex>$x$</tex>\u0000 and \u0000<tex>$t$</tex>\u0000 represent dimensionless distance and time respectively. In particular, we consider the case when the initial data has a discontinuous step, where \u0000<tex>$u(x,0)= u_{+}$</tex>\u0000 for \u0000<tex>$x ge 0$</tex>\u0000 and \u0000<tex>$u(x,0)=u_{-}$</tex>\u0000 for \u0000<tex>$x&lt;0$</tex>\u0000, where \u0000<tex>$u_{+}$</tex>\u0000 and \u0000<tex>$u_{-}$</tex>\u0000 are problem parameters with \u0000<tex>$u_{+} not= u_{-}$</tex>\u0000. The method of matched asymptotic coordinate expansions is used to obtain the large-\u0000<tex>$t$</tex>\u0000 asymptotic structure of the solution to this problem, which exhibits a range of large-\u0000<tex>$t$</tex>\u0000 attractors depending on the problem parameters. Specifically, the solution of the initial-value problem exhibits the formation of (i) an expansion wave when \u0000<tex>$delta&gt;-frac{1}{2}$</tex>\u0000 and \u0000<tex>$u_{+}&gt;u_{-}$</tex>\u0000, (ii) a Taylor shock (hyperbolic tangent) profile when \u0000<tex>$delta&gt;-frac{1}{2}$</tex>\u0000 and \u0000<tex>$u_{+}&lt;u_{-}$</tex>\u0000 and (iii) the Rudenko–Soluyan similarity solution when \u0000<tex>$delta=-frac{1}{2}$</tex>\u0000.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"69 3","pages":"231-256"},"PeriodicalIF":0.0,"publicationDate":"2016-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbw006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49954119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}