Construction of Nonlinear Normal Modes by Shaw-Pierre via Schur Decomposition

In the paper the simplification of construction of nonlinear normal vibration modes by Shaw-Pierre in power series form is considered. The simplification can be obtained via change of variables in the equations o f motion of dynamical system under consideration. This change of variables is constructed by means of so-called ordered Schur matrix decomposition. As the result of the transformation there is no need in solving nonlinear algebraic equations in order to evaluate coefficients of nonlinear normal mode

Non-Iterative Rauscher Method for 1-DOF System: a New Approach to Studying Non-Autonomous System via Equivalent Autonomous One

In the paper a new non-iterative variant of Rauscher method is considered. In its current statement the method can be used in analysis of forced harmonic oscillations in 1-DOF nonlinear system. It is shown that three different types o f equivalent authonomous dynamical systems can be built for a given 1-DOF non-autonomous one. Two of them (1st and 2nd type) have wider set of solutions than that of the initial system. These solutions correspond to various values of amplitude and phase of external excitation. Solutions of the equivalent system of 3rd type are exclusively periodic ones. Based on the equivalent system of 3rd type such a function W(x,x') can be constructed that its level curves correspond to periodic orbits of the initial non-autonomous system. This function can be built a priori via computation of the invariant manifold of the equivalent system of 1st type. Using the same approach the Rauscher expansions cos(Qt)=C(x,x'), sin(Qt)=S(x,x') can also be constructed. It is also shown that equivalent systems can be investigated by means of harmonic balance method which allows construction o f W(x,x'), C(x,x') andS(x,x') in semi-analytical manner

Evaluation of elastic and adhesive properties of solids by depth-sensing indentation

To describe properly interactions between contacting solids at micro/nanometre scales, one needs to know both adhesive and mechanical properties of the solids. Borodich and Galanov have introduced an effective method (the BG method) for identifying both characteristics from a single experiment on depth-sensing indentation by a spherical indenter using optimal fitting of the experimental data. Unlike traditional indentation techniques involving sharp indenters, the Borodich-Galanov methodology intrinsically takes adhesion into account. It is essentially a non-destructive approach. These features extend the scope of the method to important applications beyond the capabilities of conventional indentation. The scope of the original BG method was limited to the classic JKR and DMT theories. Recently, this restriction has been overcome by introducing the extended BG (eBG) method, where a new objective functional based on the concept of orthogonal distance curve fitting has been introduced. In the present work, questions related to theoretical development of the eBG method are discussed. Using the data for elastic bulk samples, it is shown that the eBG method is at least as good as the original BG method. It is shown that the eBG can be applied to adhesive indentation of coated, multilayered, functionally graded media

Adhesive contact problems for a thin elastic layer : Asymptotic analysis and the JKR theory

Contact problems for a thin compressible elastic layer attached to a rigid support are studied. Assuming that the thickness of the layer is much less than the characteristic dimension of the contact area, a direct derivation of asymptotic relations for displacements and stress is presented. The proposed approach is compared with other published approaches. The cases are established when the leading-order approximation to the non-adhesive contact problems is equivalent to contact problem for a Winkler–Fuss elastic foundation. For this elastic foundation, the axisymmetric adhesive contact is studied in the framework of the Johnson–Kendall–Roberts (JKR) theory. The JKR approach has been generalized to the case of the punch shape being described by an arbitrary blunt axisymmetric indenter. Connections of the results obtained to problems of nanoindentation in the case that the indenter shape near the tip has some deviation from its nominal shape are discussed. For indenters whose shape is described by power-law functions, the explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius

Estimation of the elastic modulus and the work of adhesion of soft materials using the extended Borodich–Galanov (BG) method and depth sensing indentation

© 2018 Elsevier Ltd The depth-sensing indentation (DSI) is currently one of the main experimental techniques for studying elastic properties of materials of small volumes. Usually DSI tests are performed using sharp pyramidal indenters and the load-displacement curves obtained are used for estimations of elastic moduli of materials, while the curve analysis for these estimations is based on the assumptions of the Hertz contact theory of non-adhesive contact. The Borodich–Galanov (BG) method provides an alternative methodology for estimations of the elastic moduli along with estimations of the work of adhesion of the contacting pair in a single experiment using the experimental DSI data for spherical indenters. The method assumes fitting the experimental points of the load-displacement curves using a dimensionless expression of an appropriate theory of adhesive contact. Earlier numerical simulations showed that the BG method was robust. Here first the original BG method is modified and then its accuracy in the estimation of the reduced elastic modulus is directly tested by comparison with the results of conventional tensile tests. The method modification is twofold: (i) a two-stage fitting of the theoretical DSI dependency to the experimental data is used and (ii) a new objective functional is introduced which minimizes the squared norm of difference between the theoretical curve and the one used in preliminary data fitting. The direct experimental validation of accuracy and robustness of the BG method has two independent steps. First the material properties of polyvinyl siloxane (PVS) are determined from a DSI data by means of the modified BG method; and then the obtained results for the reduced elastic modulus are compared with the results of tensile tests on dumbbell specimens made of the same charge of PVS. Comparison of the results of the two experiments showed that the absolute minimum in relative difference between individual identified values of the reduced elastic modulus in the two experiments was 3.80%; the absolute maximum of the same quantity was 27.38%; the relative difference in averaged values of the reduced elastic modulus varied in the range 16.20.. 17.09% depending on particular settings used during preliminary fitting. Hence, the comparison of the results shows that the experimental values of the elastic modulus obtained by the tensile tests are in good agreement with the results of the extended BG method. Our analysis shows that unaccounted factors and phenomena tend to decrease the difference in the results of the two experiments. Thus, the robustness and accuracy of the proposed extension of the BG method has been directly validated