Authors:Wolfgang E. Weber; George D. Manolis Pages: 1095 - 1104 Abstract: This work addresses the evaluation of the displacement field in an elastic matrix due to the presence of an embedded homogeneous inclusion under time-harmonic SH-wave loads. We consider the bond between the circular cylindrical inclusion and the surrounding matrix of infinite extent to be partly damaged in the circumferential direction. The material of both inclusion and matrix is modelled as homogeneous, isotropic, and linearly elastic. An analytical approach is introduced here for describing the scattering of SH-waves by inclusions with partly damaged bond. Subsequently, a numerical example serves to illustrate the methodology, which can be extended to the scattering of SV/P-waves in a straightforward manner. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1238-9 Issue No:Vol. 87, No. 7 (2017)

Authors:F. Hache; N. Challamel; I. Elishakoff; C. M. Wang Pages: 1105 - 1138 Abstract: This paper investigates both stability and vibration of nonlocal beams or plates in the presence of compressive forces. Various nonlocal structural models have been proposed to capture the inherent scale effects of lattice-based beams or plates. These nonlocal models are either based on continualization of the difference equations of the original lattice problem (labeled as continualized nonlocal models), or developed from phenomenological nonlocal approaches such as Eringen’s type nonlocality. Considered herein are several continualization schemes that lead to either fourth or sixth order spatial governing differential or partial differential equation. Even if the new continualized nonlocal plate models differ in their mathematical description, they appear to furnish very close macroscopic results as shown from asymptotic expansion arguments. The continualized nonlocal beam and plate models and the phenomenological approaches are also introduced from variational arguments. The key role of boundary conditions is shown especially for Eringen’s nonlocal model that is not necessarily variationally based. For each of them, the buckling load and the natural frequencies are determined for simply supported beams and plates and compared to their counterparts obtained from the lattice model. The small length scale coefficient of the nonlocal beam or plate models is intrinsically constant and problem independent for the continualized approaches, whereas it is calibrated for the phenomenological models based on the equivalence with the reference microstructured model and consequently, depends on the load, the buckling or vibration mode or the aspect ratio. It is found that the nonlocal continualized approaches are more efficient than the nonlocal phenomenological ones. For beam problems, continualized nonlocal and phenomenological approaches such as Eringen’s nonlocal theory can become the same. However, for plate problems, phenomenological approaches may differ significantly from continualized nonlocal ones; thereby offering one the opportunity to have a new class of two-dimensional nonlocal models. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1235-z Issue No:Vol. 87, No. 7 (2017)

Authors:B. Zhang; J. G. Yu; X. M. Zhang Pages: 1139 - 1150 Abstract: By introducing the double orthogonal polynomial method from the Cartesian coordinate system into the cylindrical coordinate system, this paper investigates the guided wave propagation in cylindrical structures with sector cross-sections. Comparison with available reference results, the validity of the presented method is verified. The corresponding phase velocity dispersion curves, displacement distributions, stress curves and the Poynting vectors are illustrated. The influences of the radius-to-thickness ratio and angular measure on the characteristics of the guided wave are discussed, which gives significant guidance on ultrasonic guided wave nondestructive testing for cylindrical structures with sector cross-sections. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1237-x Issue No:Vol. 87, No. 7 (2017)

Authors:Wang Xu; Zhenzhen Tong; Dalun Rong; A. Y. T. Leung; Xinsheng Xu; Zhenhuan Zhou Pages: 1151 - 1163 Abstract: A finite element discretized symplectic method is presented for the determination of modes I and II stress intensity factors (SIFs) for cracked bimaterial plates subjected to bending loads using Kirchhoff’s theory and symplectic approach. The overall plate is meshed by conventional discrete Kirchhoff theory elements and is divided into two regions: a near field which contains the crack tip and is enhanced by the symplectic series expansion and a far field which is far away from the crack tip. Based on the analytical solutions of global displacement, numerous degrees of freedom are transformed to a small set of undetermined coefficients of the symplectic series through a displacement transformation, while those in the far field remain unchanged. The SIFs can be obtained directly from coefficients of eigensolution (Re \(\mu < 1\) ), and no post-processing or special singular element are required to develop for extracting the SIFs. Numerical examples are presented and compared with existing results to demonstrate the efficiency and accuracy of the method. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1239-8 Issue No:Vol. 87, No. 7 (2017)

Authors:Song Qiao; Xinchun Shang Pages: 1165 - 1198 Abstract: Three-dimensional scattering and dynamic stress concentration of Lamb-like waves around a spherical inclusion (inhomogeneity and a cavity) in a thick spherical shell are investigated theoretically and numerically. Two spherical coordinates, located at the spherical shell center and the inclusion center, are established to express the incident and scattered wave potential functions. By a kind of addition formulas, all the potential functions can be transformed into the same coordinate, then the analytical solution of the displacements and stresses are derived, and all the undetermined coefficients are solved by satisfying the boundary condition and the interface condition. In order to describe the 3-D stresses concentration, multiple DSCFs are employed and the 3-D distributions are depicted. The results reveal the influences of inclusion material and the cavity on the distributions of DSCFs, and the influence of incident wave frequency and inclusion position are also calculated. This research is expected to provide theoretical understanding on dynamic analysis and mechanical properties evaluation of the spherical shells. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1240-2 Issue No:Vol. 87, No. 7 (2017)

Authors:Gergana Nikolova; Jordanka Ivanova; Frank Wuttke; Petia Dineva Pages: 1199 - 1211 Abstract: The interface fracture behaviour of a bi-material poroelastic plate with normal to the interface surface-breaking pre-crack with crack-tip approaching the interface subjected to time-harmonic uniaxial uniform load is considered. A viscoelastic isomorphism to Biot’s dynamic poroelasticity is applied to describe the soil material properties, thus replacing the original two-phase poroelastic material by a single-phase viscoelastic one of Kelvin–Voigt type. A viscoelastic shear-lag model for one-dimensional stress–strain state with analytically derived solution for the length of the delamination zone along the interface is proposed. The parametric analysis demonstrates that the debonding length is sensitive to the following key factors: (a) frequency and magnitude of the applied load; (b) material and geometric characteristics; (c) soil porosity as respected soil type; and (d) soil saturation—dry or saturated soils. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1241-1 Issue No:Vol. 87, No. 7 (2017)

Authors:Wenzhen Qu; Yaoming Zhang; Chein-Shan Liu Pages: 1213 - 1226 Abstract: This paper presents new regularized boundary integral equations (BIEs) for elastic displacement gradients in three dimensions and then combines them by the generalized Hooke’s law to calculate the boundary stress. In the new regularized BIEs, two special tangential vectors are designed with the normal vector to construct a transformation system. Based on this system, the displacement gradient in any direction can be transformed into a linear combination of the normal gradient and tangential gradients along the two special vectors. Moreover, a theorem related to some integral properties of the fundamental solution is introduced. Finally, the regularized indirect BIEs are developed by using the above-mentioned technique of linear combination and theorem. The proposed method has some advantages over the direct boundary element method, such as the relaxed continuity requirement of density function, no hypersingular integral, and being available to calculate the displacement gradient in any direction. The numerical implementation of the developed integral equation is provided, and the accuracy and convergence of the approach are also illustrated through four numerical examples. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1242-0 Issue No:Vol. 87, No. 7 (2017)

Authors:Aleksandar Nikolić Pages: 1227 - 1241 Abstract: This paper considers the free vibrations of the non-uniform axially functionally graded cantilever beam with a tip body. It is assumed that the mass center of the tip body is eccentrically displaced in both axial and transverse direction relative to the beam tip. All considerations are carried out within the Euler–Bernoulli beam theory. The in-plane transverse and axial deformations of the beam are considered. It is shown that there is a coupling between the axial and transverse deformations of the cantilever beam due to the tip body mass center eccentricity in the transverse direction. An universal, numerically efficient rigid element method which is able to analyze the cantilever beam with any law of changes of the geometric parameters of the cross section or the characteristics of the material along the beam was formed. Theoretical considerations are accompanied by numerical examples. There is a good agreement of the results obtained with the results available in the literature. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1243-z Issue No:Vol. 87, No. 7 (2017)

Authors:Liquan Wang; Songyu Li; Shaoming Yao; Dong Lv; Peng Jia Pages: 1243 - 1253 Abstract: The initial vertical stiffness of the bonded spherical elastic layer with Poisson’s ratio from 0 to 0.5 is formulated on the basis of two kinematic assumptions. The governing equations of the stiffness are solved by numerical method. The vertical stiffness is not only related to the load area and the thickness of the elastic layer, but also related to the shape factor and inner edge angle of the spherical elastic layer. The analytical method is verified by FE analysis with a 2D axisymmetric model with Poisson’s ratio within (0.49, 0.499999). Poisson’s ratio, radius–thickness ratio and edge angles will influence the stiffness error. The stiffness error increases as the radius–thickness ratio increases and tends to saturate when the radius–thickness ratio exceeds 50. When the inner edge angle is constant, the relative errors of the vertical stiffness will decrease as the outer edge angle increases. When the outer edge angle is constant, the errors will increase and reach a maximum value and then decrease as the inner edge angle increases. For a single elastic layer with the radius–thickness ratio of 20, the relative error of vertical stiffness is within (−10, 12%), which is acceptable in bearing design. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1246-9 Issue No:Vol. 87, No. 7 (2017)

Authors:Feng Yang; Mojia Huang; Jun Liu Pages: 1255 - 1267 Abstract: Most rolled sheet metals are orthotropic aggregates of cubic crystallites. The texture coefficients, characterized by the preferred orientation of the crystallites, are important to set up the yield function. Although the Hosford yield function is more suitable than the Hill yield function for describing both the yielding and plastic deformation of orthotropic material, it suffers from the restriction that the three principal stresses must be coaxial with the orthotropy of materials. This paper proposes the new Hosford yield function for weakly textured sheets of cubic crystal orthotropic metals in any stress state by expanding the introduced orientation-dependent functions to its sixth-order Taylor series expansion. Also, the new yield function, which covers three material parameters and seven texture coefficients, is more general than the existing Hosford yield function. Finally, both the plastic anisotropy of the q-value and the yield stress obtained from the new yield function agree well with experimental results. This yield function can lay a theoretical foundation for analyzing the mechanical properties of metal materials. PubDate: 2017-07-01 DOI: 10.1007/s00419-017-1244-y Issue No:Vol. 87, No. 7 (2017)

Authors:I. Romero Abstract: We prove two results that generalize the two classical theorems of Castigliano to structures with known (stationary) eigenstrains, including those originating from temperature changes, lack-of-fit, phase changes, etc. We show that in these situations the classical theorems of Castigliano still hold, provided that the appropriate modifications are made to the elastic energy and its conjugate. The derivation of these results follows directly from the principle of minimum potential energy and some of the properties of convex functions. Just like the classical theorems, the final results can be conveniently employed to solve simple structures by hand. PubDate: 2017-07-22 DOI: 10.1007/s00419-017-1282-5

Authors:C. G. Provatidis Abstract: Older and contemporary CAD-based interpolations, either for surfaces or for volume blocks, are capable of creating sets of basis functions on which finite-element (Galerkin–Ritz) and global collocation procedures can be supported. For some of these interpolations, this paper investigates the quality of the relevant numerical solution in several 2D and 3D engineering problems. It is shown that the global character of all these CAD interpolations ensures excellent numerical solution, although somewhere the boundary may be slightly violated. The study deals with several benchmark tests that span a large part in the spectrum of engineering analysis, from potential problems (Poisson equation-electrostatics and acoustics) to elasticity ones (beam in torsion, plate bending: statics and dynamics). PubDate: 2017-07-22 DOI: 10.1007/s00419-017-1283-4

Authors:E. G. Kakouris; S. P. Triantafyllou Abstract: A novel phase field formulation implemented within a material point method setting is developed to address brittle fracture simulation in anisotropic media. The case of strong anisotropy in the crack surface energy is treated by considering an appropriate variational, i.e. phase field approach. Material point method is utilized to efficiently treat the resulting coupled governing equations. The brittle fracture governing equations are defined at a set of Lagrangian material points and subsequently interpolated at the nodes of a fixed Eulerian mesh where solution is performed. As a result, the quality of the solution does not depend on the quality of the underlying finite element mesh and is relieved from mesh distortion errors. The efficiency and validity of the proposed method are assessed through a set of benchmark problems. PubDate: 2017-07-14 DOI: 10.1007/s00419-017-1272-7

Authors:Moshe Eisenberger; Isaac Elishakoff Abstract: In this paper, we present a general methodology for solving buckling problems for inhomogeneous columns. Columns that are treated are functionally graded in axial direction. The buckling mode is postulated as the general order polynomial function that satisfies all boundary conditions. For specificity, we concentrate on the boundary conditions of simple support, and employ the second-order ordinary differential equation that governs the buckling behavior. A quadratic polynomial is adopted for the description of the column’s flexural rigidity. Satisfaction of the governing differential equation leads to a set of nonlinear algebraic equations that are solved exactly. In addition to the recovery of the solutions previously found by Duncan and Elishakoff, several new solutions are arrived at. PubDate: 2017-07-13 DOI: 10.1007/s00419-017-1278-1

Authors:Nikolaos I. Ioakimidis Abstract: Several criteria for mixed-mode fracture in crack problems are based on the maximum of a quantity quite frequently related to stress components. This quantity should not reach a critical value. Computationally, this approach requires the use of the first and the second derivatives of the above quantity although frequently the use of the second derivative is omitted because of the necessary complicated computations. Therefore, mathematically, the determination of the maximum of the quantity of interest is not assured when the classical approach is used without the second derivative. Here a completely different and more rigorous approach is proposed. The present approach is based on symbolic computations and makes use of modern quantifier elimination algorithms implemented in the computer algebra system Mathematica. The maximum tangential stress criterion, the generalized maximum tangential stress criterion (with a T-stress term), the T-criterion and the modified maximum energy release rate criterion are used for the illustration of the present new approach in the mode I/II case. Beyond the conditions of fracture initiation, the determination of the fracture angle is also studied. The mode I/III case is also considered in brief. The present approach completely avoids differentiations, similarly the necessity of a distinction between maxima and minima, always leads to a global (absolute) and not to a local (relative) maximum and frequently to closed-form formulae and automatically makes a distinction of cases in the final formula whenever this is necessary. Moreover, its use is easy and direct and the maximum of the quantity of interest is always assured. PubDate: 2017-07-13 DOI: 10.1007/s00419-017-1274-5

Authors:Mesut Huseyinoglu; Orhan Çakar Abstract: Structural modifications in the form of mass, stiffness and damping to a structure change the dynamic properties of that structure. However, in some cases, after modifications are made to the structures, certain specified natural frequencies of the structure are desired to remain unchanged. This study is interested in the determination of necessary stiffness modifications in order to keep a certain number of natural frequencies of the system unchanged despite mass attachments. In particular, two methods based on the Sherman–Morrison formula are developed in order to determine the spring coefficients needed to keep one and more than one natural frequency of the structures unchanged. The developed methods directly use the Frequency Response Functions of the unmodified system relating the modification coordinates only and they need neither a physical model nor a modal model. The numerical simulations show that they are very effective. However, due to the nature of the inverse problem, any solution or practical realistic solution may be not found. The existence of the solution depends on also the modification coordinates chosen. A simple sensitivity approach demonstrated by a 3D graph is proposed to be able to choose a suitable modification. PubDate: 2017-07-12 DOI: 10.1007/s00419-017-1276-3

Authors:Thanasis Zisis Abstract: The problem of calculating the displacements and stresses in a layered system often arises in engineering analysis and design, ranging from the field of mechanical engineering, to the field of materials science and soil mechanics. The present work focuses on the anti-plane response of half-planes and layers of finite thickness bonded on rigid substrates, under a point load, in the context of couple stress elasticity. The theory of couple stress elasticity is used to model the material microstructure and incorporate the size effects into the macroscopic response. This problem in plane strain configuration is referred to as Burmister’s problem. The purpose is to derive the pertinent Green’s functions that can be effectively used for the formulation of anti-plane contact problems in the context of couple stress elasticity. Full-field solutions regarding the out-of-plane displacements, the strains and the equivalent stress are presented, and of special importance is the behavior of the new solutions near to the point of application of the force where pathological singularities and discontinuities exist in the classical solutions. PubDate: 2017-07-12 DOI: 10.1007/s00419-017-1277-2

Authors:Carmine M. Pappalardo; Domenico Guida Abstract: This paper introduces a new coordinate formulation for the kinematic and dynamic analysis of planar multibody systems composed of rigid bodies. The methodology presented in this work is called planar reference point coordinate formulation (RPCF) with Euler parameters. In the planar RPCF with Euler parameters, the rotational coordinates used for describing the body orientation are the redundant components of a two-dimensional unit quaternion that identify a planar set of Euler parameters. It is shown in the paper that the planar RPCF with Euler parameters allows for obtaining consistent kinematic and dynamic descriptions of two-dimensional rigid bodies. In the numerical solution of the equations of motion, the well-known generalized coordinate partitioning method can be effectively utilized to stabilize the violation of the algebraic constraints at the position and velocity levels leading to physically correct and numerically stable dynamic simulations. Furthermore, a standard numerical integration procedure can be employed for calculating an approximate solution of the equations of motion resulting from the planar RPCF with Euler parameters. In the paper, the computer implementation of the proposed formulation approach is demonstrated considering four rigid multibody systems which serve as simple benchmark problems. PubDate: 2017-07-10 DOI: 10.1007/s00419-017-1279-0

Authors:V. V. Zozulya Abstract: Since Green’s matrices are widely used for solution of the theoretical and applied problems in science and engineering, it is important to get efficient methods for their calculation. Therefore, a new efficient algorithm for the calculation of Green’s matrices for the boundary value problem (BVP) for the system of ordinary differential equations (ODE) of the first order has been developed here. For any well-defined BVP, a fundamental matrix has to be constructed first; then, using a simple algorithm the corresponding Green’s matrix is calculated. For the fundamental matrix calculation an approach based on the matrix exponential is used. To demonstrate the effectiveness and robustness of the algorithm, Green’s matrices for elastic bar, Euler–Bernoulli, Timoshenko’s and Vekua’s beams have been calculated. All of the presented calculations have been done using the computed algebra software Mathematica. In the cases of the elastic bar, Euler–Bernoulli, Timoshenko’s beams corresponding Green’s functions have been presented in analytical form as the Mathematica output. In the case of the Vekua’s beams analytical expressions for Green’s functions are relatively long; they have been calculated numerically, by using the proposed algorithm. The Green’s matrices for Timoshenko’s and Vekua’s beams have been verified by comparing the solution of the corresponding BVP obtained using Green’s function method with the numerical solution obtained using Mathematica function NDSolve. Proposed algorithm can be applied for solution of the BVP for any linear and some classes of nonlinear systems of the ODE using the Green’s matrices approach. PubDate: 2017-07-03 DOI: 10.1007/s00419-017-1275-4

Authors:Abdolrasoul Ranjbaran; Mohammad Ranjbaran Abstract: Beam, column, plate, and any other structure, under full or partial compressive loading, are prone to failure by the buckling phenomenon. At the instant of failure, the structure may be in unpredictable elastic, elastic–plastic, full plastic, cracked, or other forms of deterioration state. Therefore, in spite of so much study, there is no definite solution to the problem. In this paper a unified, simple, and exact theory is proposed where buckling is considered as the change of state of structure between intact and collapsed states, and then the buckling capacity is innovatively expressed via states and phenomena functions, which are explicitly defined as functions of state variable. The state variable is determined by calibration of the structure slenderness ratio. The efficacy of the work is verified via concise mathematical logics, and comparison of the results with those of the others via seven examples. PubDate: 2017-06-29 DOI: 10.1007/s00419-017-1273-6