Summary

 This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions.

A new modeling framework for analyzing steady-state elastic–viscoplastic single crystal problems at micron scale is presented. The model takes as offset the higher-order gradient plasticity theory by Kuroda and Tvergaard (J Mech Phys Solids 54:1789–1810 2006), and it is strongly inspired by an existing steady-state framework for conventional elastic–viscoplastic materials. Details on the modeling framework are laid out, and the numerical challenges and stability issues are discussed. Subsequently, the modeling framework is demonstrated on Mode I steady-state crack growth in HCP single crystals, where the impact of size effects on the active plastic zone that surrounds the crack tip is investigated. The focus is on the plastic zone shape and size, as well as its influence on the macroscopic fracture toughness. In this way, the chosen benchmark problem for the new modeling framework serves as an extension to the conventional study of the corresponding problem without size effects presented in Juul et al. (J Mech Phys Solids 101:209–222 2017). The tendency is that the plastic zone is smeared out (seen as less distinct features) when increasing the material rate sensitivity. This is in-line with already published work in the literature. It is shown that the size effect has a limited effect on the qualitative features of the plastic zone, whereas it clearly suppresses the magnitude of the features due to the added gradient hardening effect. Moreover, the hardening effect tends to lower the shielding ratio, and hence a key result is that single crystal materials appear less crack resistant than conventional studies suggest at the micron scale.

Summary

This paper presents some results on the parameter estimation of a fixed-distance joint. Such joints are of interest in various fields of applied mechanics such as mechanisms, vehicle and human body dynamics. The corresponding geometrical parameters are positions of the points of connected bodies, which are kept at constant distance during the movement. The problem is solved using kinematical information on a certain number of points belonging to the connected bodies. The conditioning problem, and, consequently, the accuracy of the results, depends on the relative movement of the bodies and also on possible a priori information. The case of planar motion is investigated in more detail; a numerical example is provided and discussed.

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.

In order to obtain the equations of motion of vibratory systems, we will need a mathematical description of the forces and moments involved, as function of displacement or velocity, solution of vibration models to predict system behavior requires solution of differential equations, the differential equations based on linear model of the forces and moments are much easier to solve than the ones based on nonlinear models, but sometimes a nonlinear model is unavoidable, this is the case when a system is designed with nonlinear spring and nonlinear damping. Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained; homotopy perturbation method (HPM) is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic; the enhanced HPM is used to solve the nonlinear shock absorber and spring equations.

Summary

Failure of earth structures or laboratory specimens of soils is often characterized by the existence of bands or surfaces at which strain localizes. Numerical simulation of shear band development has attracted the attention of many research groups during past years. Here it is proposed that adaptive remeshing techniques can be applied to better simulate strain localization problems in geotechnique. The algorithm has been previously applied to compressible fluid dynamics problems to capture discontinuities such as shocks. A new refinement functional has been introduced to improve quality of produced meshes. Finally, the algorithm is applied to solve inception and development of shear bands on both homogeneous stress fields and non-homogeneous stress fields.

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.

Summary

A numerical method based on the assumption of a generalized plane strain (GPS) state is presented for calculating the stress and strength ratio distributions of the rotating composite flywheel rotor of varying material properties in the radial direction. The rotor is divided into many rings and each ring has constant material properties. All the rings are assumed to expand and have the same axial strain. A three-dimensional finite element method is then used to verify the accuracy of the present method. This method gives a better solution for most of the rotors than other methods of a plane stress or plane strain state. After verification, the effects of material properties on the total stored energy (TSE) of the composite flywheel rotor are investigated. For this purpose, the material properties of the rotor, i.e. circumferential and radial Young's moduli, ply angles and mass densities are expressed by power functions of the radius, and the rotor is analyzed. The analysis shows that TSE can be most effectively increased by changing the circumferential Young's moduli along the radius, which amounts to over 300% of TSE of the constant material properties. The variation of ply angles along the radius can increase TSE by about 30% at most. The method of changing the mass densities along the radius could be also effective but its effects are not so noticeable in the rotor where the circumferential stiffness is properly arranged.

Summary

 We consider the air contained in a pneumatic tyre with the purpose of investigating its inertial oscillations. We model the tyre as a torus limited by a membrane in contact with the ground. According to this model, we prove that the flow within this torus may be considered as one at low Mach number and that it is ruled by oscillations of incompressible rotating fluid. Investigating such inertial oscillations, we show that the geostrophic oscillation is resonant, and we study the resonance phenomenon.