An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage

A general closed-form solution for the so-called rebound indentation test’ is obtained for a cylindrical flat-ended punch indenting a linear viscoelastic layer lying on a rigid substrate. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in a displacement controlled regime) and for the indentation displacement (in a load-controlled regime) and we consider in detail the case of standard viscoelastic solid. Our results indicate that the rebound displacement (in other words the indentation displacement in the load-controlled stage) is independent of the relaxed elastic modulus and Poisson's ratio, and also of the layer's thickness. Our analytical solution can be used for layered samples of arbitrary materials exhibiting viscoelastic properties; however, since the rebound indentation test has been recently suggested for assessing the viability of biomedical materials, we have applied our theoretical framework to the identification of materials parameters from experiments on articular cartilage. In this context, we have found a pretty good agreement for the rebound deformation, even until the strain becomes relatively large.     

An approximate analytical solution of the cylindrical detonation wave generated by the linear explosion

The solution of the cylindrical detonation wave generated by the linear explosion was obtained by numerical method in ref.[1].In this paper,when the ratio of specific heatγ》1 by using the enlargement coordinate method,the first-order analytical solutions are obtained.The perturbation parameter isε=1/γ~2 .The correction of these solutions is checked at the end of this paper.     

具有幂率速度移动表面边界层问题解析近似解

利用微分方程相似变换,摄动渐进展开和Padé逼近方法对幂率速度移动表面边界层问题进行了研究,得到了问题的解析近似解,对相应的流动特性进行了探讨.     

An analytical poroelastic model for laboratorial mechanical testing of the articular cartilage (AC)

The articular cartilage (AC) can be seen as a biphasic poroelastic material. The cartilage deformation under compression mainly leads to an interstitial fluid flow in the porous solid phase. In this paper, an analytical poroelastic model for the AC under laboratorial mechanical testing is developed. The solutions of interstitial fluid pressure and velocity are obtained. The results show the following facts. (i) Both the pressure and fluid velocity amplitudes are proportional to the strain loading amplitude. (ii) Both the amplitudes of pore fluid pressure and velocity in the AC depend more on the loading amplitude than on the frequency. Thus, in order to obtain the considerable fluid stimulus for the AC cell responses, the most effective way is to increase the loading amplitude rather than the frequency. (iii) Both the interstitial fluid pressure and velocity are strongly affected by permeability variations. This model can be used in experimental tests of the parameters of AC or other poroelastic materials, and in research of mechanotransduction and injury mechanism involved interstitial fluid flow.     

Influence of nonlinearities on the accuracy of the analytical solution for the shaft loaded blister test

Finite element analysis (FEA) is employed to study the effects of nonlinearities on the accuracy of the analytical solution for the shaft loaded blister test. The FEA model was validated using constrained blister test measurements showing a good correlation between the experimental and the FEA data. The analytical solution is then compared with the energy release rate obtained from J-integral evaluation in the FEA. For small and large shaft displacements deviations larger than 20% are encountered which is explained with the violation of the membrane limit condition and the onset of plasticity for larger displacements, respectively. Simplifications of the analytical solution are discussed using a random sampling method and it is shown that the thickness ratio between film and substrate can be neglected for thin films on rigid substrates. Further, values for the angular quantity, ω, which is required to calculate the mode mix phase angle are tabulated for the case of thin, elastic films on stiff substrates using a crack surface opening displacement extrapolation method.     

An exact analytical solution of the unsteady magnetohydrodynamics nonlinear dynamics of laminar boundary layer due to an impulsively linear stretching sheet

In this paper, we investigate the theoretical analysis for the unsteady magnetohydrodynamic laminar boundary layer flow due to impulsively stretching sheet. The third-order highly nonlinear partial differential equation modeling the unsteady boundary layer flow brought on by an impulsively stretching flat sheet was solved by applying Adomian decomposition method and Pade approximants. The exact analytical solution so obtained is in terms of rapidly converging power series and each of the variants are easily computable. Variations in parameters such as mass transfer (suction/injection) and Chandrasekhar number on the velocity are observed by plotting the graphs. This particular problem is technically sound and has got applications in expulsion process and related process in fluid dynamics problems.     

An analytical solution for the laminar flow over a flat plate with similarity preserving suction

An exact analytical solution is presented for the laminar boundary-layer flow over a semi-infinite flat plate subjected to a type of similarity preserving suction. The solution is developed for the case of a plate immersed in either a uniform compressible stream with viscosity proportional to temperature or a uniform incompressible stream with constant viscosity. The problem is formulated in Crocco's variables. It is described by a second-order, non-linear, ordinary differential (and singular) boundary-value problem for the shear stress as a function of the velocity in the boundary layer. A unique solution is shown to exist and to possess a power series representation for all magnitudes of suction. The series is constructed explicitly and provides a transcendental equation for the shear stress at the plate (the important skin friction) which can be solved to any desired accuracy. Examples of upper and lower bounds for the wall shear are presented for several magnitudes of suction and confirm the reasonable accuracy of results obtained heretofore only by numerical solutions of the problem. In addition to the intrinsic value of the technique developed, it can be the basis of accurate checks for the numerical solution of more complex problems.     

Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties

The contact problem for the impression of spherical indenter into a non-homogeneous (both layered and functionally graded) elastic half-space is considered. Analytical methods for solving this problem have been developed. It is assumed that the Lame coefficients vary arbitrarily with the half-space depth. The problem is reduced to dual integral equations. The developed methods make it possible to find the analytical asymptotically exact problem solution, suitable for a PC. The influence of the Lame coefficients variation upon the contact stresses and size of the contact zone with different radius of indenter as well as values of the impressing forces are studied. The effect of the non-homogeneity is examined. The developed method allows to construct analytical solutions with presupposed accuracy and gives the opportunity to do multiparametric and qualitative investigations of the problem with minimal computation time expenditure.     

Development of a mathematical model and analytical solution of a coupled two-beam array with nonlinear tip forces for application to AFM

Utilising arrays of cantilevers has been identified as a method of increasing AFM sensitivity and throughput beyond the limitations of current non-contact AFM. In order to develop the technology, we investigate the change in eigenstates of arrays due to interactions that occur between cantilever tip and sample using a model of a base-coupled array of cantilevers with individual mass and stiffness properties influenced by a quadratic nonlinear force to represent tip–sample interactions. An analytical expression is developed for a coupled two-beam array utilising Multiple Scale Lindstedt–Poincare perturbation theory to assess how a selected parameter space alters the array dynamics. Experiments are carried out using a macroscale set-up to validate the developed model. We show that the model captures the eigenstates of the system with good qualitative accuracy and that the perturbation expansion performs well in both the weakly (far from surface) and strongly (near the pull-in point) nonlinear regimes. The results demonstrate that utilising changes in eigenstates due to force interactions could have significant benefits to non-contact AFM with regard to measurement sensitivity and speed.     

An efficient solution methodology to study the response of a beam on viscoelastic and nonlinear unilateral foundation: Static response

Many engineering materials and foundations such as soils demonstrate nonlinear and viscoelastic behaviour. Yet, it is challenging to develop static and dynamic models of systems that include these materials and are able to predict the behaviour over a wide range of loading conditions. This research is focused on a specific example: a pinned–pinned beam interacting with polyurethane foam foundation. Two cases, when the foundation can react in tension and compression as well as only in compression, are considered. The model developed here is capable of predicting the response to static as well as dynamic forces, whether concentrated or distributed. Galerkin’s method is used to derive modal amplitude equations. In the tensionless foundation case, the contact region changes with beam motion and the estimation of the co-ordinates of the lift-off points is embedded into the solution procedure. An efficient solution technique is proposed that is capable of handling cases where there are multiple contact and non-contact regions. Depending on the loading profiles a high number of modes may need to be included in the solution and to speed up computation time, a convolution method is used to evaluate the integral terms in the model. The adaptability of the solution scheme to complicated loading patterns is demonstrated via examples. The solution approach proposed is applicable to dynamic loadings as well and in these cases the automated treatment of complicated response patterns makes the convolution approach particularly attractive. The influence of various parameters on the static response is discussed.     

The approximate analytical solution for the buckling loads of a thin-walled box column with variable cross-section

1.IntroductionThin-walledboxcolumnswithvariablecross-sectionareextensivelyusedascompressionmembersforhighbridgepiers,waterandtelevisiontowersandothersimilarstructures.Atpresent,however,fewpapersdealingwiththecriticalloadsoftorsional-fie-curalbucklingforthiskindofstructurescanbeseen.Inthispaper,bymeansoftheenergyprincipleandtheGalerkin'smethod,theapproximateexpressionsforcalculatingthecriticalloadsoffie-curalandtorsionalbucklingaredevelopedrespectively.Anapplicablecomputerprogramisworkedout,an…     

An approximate solution to the problem of a circular cylindrical shell with a circular hole subjected to an arbitrary self-equilibrated edge loading

Summary In this investigation, we shall use Donnell's equation of shallow circular cylindrical shell to obtain an approximate solution for stresses in an infinitely long circular cylindrical shell with a circular hole about which there acts a self-equilibrated load. To satisfy the edge conditions, we first shall expand the appropriate solutions of the differential equation into series containing powers, products of powers and logarithm, of the characteristic parameter ₀ defined in (1). We then shall represent the unknown coefficients of the solutions in similar series including new constants which depend only on the material property and the type of loading. We determine the new unknown constants through simple algebraic equations. In conclusion, we shall present two examples for symmetric and anti-symmetric loadings, including expressions which illustrate their accuracy, and expressions for stress concentration.

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Übersicht Es wird eine Näherungslösung für die Spannungen in einer unendlich ausgedehnten kreiszylindrischen Schale mit einem kreisförmigen Loch mit Hilfe der Donnel-Gleichung für flache Schalen angegeben. Der Lochrand wird durch eine im Gleichgewicht befindliche Last beansprucht. Zur Erfüllung der Randbedingungen werden die Eigenlösungen der Differentialgleichung in Reihen entwickelt, die neben Potenzen auch Produkte von Potenzen und logarithmischen Gliedern des charakteristischen Parameters ₀ enthalten, der in Gl. (1) definiert ist. Die unbekannten Koeffizienten der Lösungen werden in entsprechender Weise durch Reihen dargestellt. Sie enthalten neue Konstante, die nur noch von den Materialeigenschaften und der Belastungsart abhängen. Diese Konstanten werden durch einfache algebraische Gleichungen bestimmt. Zwei Beispiele für symmetrische und antimetrische Belastung werden durchgerechnet, wobei die Spannungskonzentration und die Genauigkeit des Verfahrens erkennbar werden.

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The author is indebted to Dr. E. F. Casebeer for invaluable editorial suggestions.     

An analytical model of a matrix heat exchanger with longitudinal heat conduction in the matrix in application for the exchanger dynamics research

The paper presents an analytical method for the solution of the problem of a fixed matrix heat exchanger with axial heat conduction within the matrix. The small parameter method and Laplace transform have been applied. A general solution has been obtained for the unsteady state in the form of function series, using single and double convolutions of functions, as well as a particular solution for both the uniform and non-uniform initial temperature of the matrix and for an arbitrary function of the fluid temperature at the inlet. Particular solutions have been used in the study of the matrix dynamics in determining dynamic characteristics for the standard input signals in the form of: Dirac pulse, Heaviside function and the function of sinusoidal variable temperature of the fluid at the inlet. The results obtained both illustrate and enable the assessment of the effect of axial heat conduction in the matrix on the dynamic properties of the heat exchanger.

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Asymptotic solution of the equations for a three-dimensional multicomponent laminar boundary layer with large injection

The asymptotic method of outer and inner expansions is used to analyze the flow of a multicomponent gas in a three-dimensional boundary layer on a smooth blunt body with large injection. Asymptotic expressions are derived for the friction coefficients, the heat and diffusion fluxes of the components on the surface of the body, and the velocity, temperature, and concentration profiles of the components across the layer of injected gases. It is shown that with large injection the limiting (bottom) streamlines on the surface of the body coincide in the first approximation with the vectorial lines of the pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 47–56, March–April, 1975.The author is indebted to G. A. Tirskii for a discussion of the work.     

Influence of the material hardening and compressibility on the solution of elastoplastic deformation problems for a space with a cylindrical cavity

The present paper deals with plane deformation problems (ɛ _z = 0) concerned with elastoplastic deformation of a space with a cylindrical cavity in the case where the load is given either at infinity or on the cavity surface. It is assumed that the material obeys the relations of the theory of flow with isotropic hardening and the von Mises plasticity condition. The effects of the elastic compressibility (Poisson’s ratio) and the coefficient of linear hardening on the stress-strain state are studied. The influence of the linear hardening is shown to be small, while that of the elastic compressibility is shown to be quite significant.     

An analytical solution for the solidus and liquidus position in the solidification of materials with a phase change range applied to the continuous casting of steel

In continuous casting processes it is important to know how the growth of the solidification layer thickness depends on the properties of the material to be casted. With help of a known approximate solution for the one dimensional solidifcation of a body with fixed wall temperature the solidified layer thickness can be calculated. In this paper it has been done for steel. Also a temperature transition range for the solidification process has been taken into account. The results are given in a few dimensionless numbers representing the relation between the solidification front position and time. The data obtained can be used as an estimation for the effects that occur in practice with steel casting processes.     

An accurate solution for the surface of elastic layer under normal concentrated load acting on a rigid horizontal base

On the basis of ref.^[1], this paper deduces an accurate solution for the surface of elastic layer under normal concentrated load acting on a rigid horizontal base, and gives numerical results, which suit civil engineers for reference.