Recent Advances in Mechanics

An application of the global reciprocity theorem for thermo-anisotropic elastodynamics of inhomogeneous solids is discussed. The theorem relates body forces, surface tractions, displacements, temperatures and strains of two solutions of the field equations, states A and B, by integrals over a volume V and its bounding surface S. The difference between various anisotropies enters via the stress temperature tensor, which is different for different cases of anisotropy. The reciprocity theorem is used to analyze the dynamic response to high-intensity heating of a small surface region of a half-space that is transversely isotropic and whose elastic moduli and mass density depend on the depth coordinate.

By an asymptotic method the solution of boundary value problems of elasticity theory for isotropic, anisotropic, layered beams, plates and shells is built. The first, second and the mixed boundary problems for one-layered and multy-layered beams, plates and shells are solved. The asymptotic method permits us to solve effectively dynamic problems for thin bodies. Free and forced vibrations are considered. General asymptotic solutions are obtained. The conditions of resonance rise are established.

The problem of contact pressure optimization is formulated for the case of rigid punch interacted with elastic medium. Coupling of the punch penetration and action of external loads at the outside regions is taken into account. The shape of the punch is considered as an unknown design variable. The minimized integral functional characterizes the discrepancy between the actual contact pressure and the required pressure distribution. The problem is studied under condition that the total forces and moments applied to the punch and the loads acted at the outside regions are given. It is shown that the considered optimization problem can be splitted and transformed to two successively solved problems. Optimal shapes are found analytically for the punches having rectangular contact domains.

Engineering structures such as aircraft, bridges, dams, nuclear containments and ships, as well as computer circuits, chips and MEMS, should be designed for failure probability < 10^− 6–10^− 7 per lifetime. The safety factors required to ensure it are still determined empirically, even though they represent much larger and much more uncertain corrections to deterministic calculations than do the typical errors of modern computer analysis of structures. The empirical approach is sufficient for perfectly brittle and perfectly ductile structures since the cumulative distribution function (cdf) of random strength is known, making it possible to extrapolate to the tail from the mean and variance. However, the empirical approach does not apply to structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared to structure size. This paper presents a refined theory on the strength distribution of quasibrittle structures, which is based on the fracture mechanics of nanocracks propagating by activation energy controlled small jumps through a nano-structure and an analytical model for the multi-scale transition of strength statistics. Based on the power law for creep crack growth rate and the cdf of material strength, the lifetime distribution of quasibrittle structures under constant load is derived. Both the strength and lifetime cdf’s are shown to be size- and geometry- dependent. The theory predicts intricate size effects on both the mean structural strength and lifetime, the latter being much stronger. The theory is shown to match the experimentally observed systematic deviations of strength and lifetime histograms of industrial ceramics from the Weibull distribution.

This paper presents a complete theory for metal plasticity that includes directional distortional hardening, supplemented by the classical kinematic and isotropic hardenings. Starting from an isotropic yield surface, the distortional hardening will be modeled either by fourth-order tensor-valued internal variable multiplied by a scalar, a scalar-valued internal variable in conjunction with the back stress, or a second-order tensor-valued internal variable. These models are unique because the rate equations for all internal variables, including the fourth order tensor, are derived strictly on the basis of sufficient conditions for the satisfaction of the second law of thermodynamics for positive dissipation, in conjunction with a few simple and plausible assumptions about free energy storage and release in the material. The models are shown to fit experimentally found yield surfaces rather well, in particular the model with the fourth-order tensor. Furthermore, this model is shown to simulate stress controlled biaxial ratchetting better than the same model without distortion of the yield surface.

Analytical solutions of the problems about interrelated vibrations of an elastic (deformable) structure with viscoelastic layer are obtained. The investigations were made to workout the method of the dynamical computations and optimization of technological processes for forming of reinforced concrete articles on the shock-and-vibration machines and shock machines. The analysis of numerical results and experimental data are presented.

The dynamic asymptotic instability of autonomous multi-parameter discrete systems under step compressive loading either of constant direction (conservative load) or of varying direction (follower or nonconservative loading) is thoroughly reconsidered using the efficient - and rather forgotten - Liénard-Chipart stability criterion. Attention is focused on the interaction of nonuniform mass and stiffness distribution with infinitesimal damping. Such parameters alone or combined with others may have a tremendous effect on the Jacobian eigenvalues and thereafter on the local asymptotic dynamic instability which – strangely enough –may occur before static (divergence) instability, even in the case of a positive definite damping matrix. It was also found that such systems when unloaded, although being statically stable, under certain conditions may become dynamically locally unstable to any small disturbance. Hopf bifurcations, double zero eigenvalues, double pure imaginary eigenvalues, loading discontinuity and other phenomena are properly established.

The integrodifferential approach incorporated in variational technique for static and dynamic problems of the linear theory of elasticity is considered. A families of statical and dynamical variational principles, in which displacement, stress, and momentum fields are varied, is proposed. It is shown that the Hamilton principle and its complementary principle for the dynamical problems of linear elasticity follow out the variational formulation proposed. A regular numerical algorithm of constrained minimization for the initial-boundary value problem is worked out. The algorithm allows us to estimate explicitly the local and integral quality of numerical solutions obtained. As an example, a problem of lateral controlled motions of a 3D rectilinear elastic prism with a rectangular cross section is investigated.

In this paper a new efficient algorithm for numerical integration of the equation of motion of a non linear system with restoring forces governed by fractional derivatives in the time domain is devised. This approach is based on the Grunwald-Letnikov representation of a fractional derivative and on the well known Newmark numerical integration scheme for structural dynamic problems. A Taylor expansion is used at every time step to represent the near past terms of the solution; thus, a dual mesh of the time domain is introduced: the coarse mesh is used for the time integration and the fine mesh is used for the fractional derivative approximation. It is shown that with this formulation the problem yields an equivalent non linear system without fractional terms which involves effective values of mass, damping, and stiffness coefficients as a predictive approach and a correction on the excitation. The major advantage of this approach is that a rather small number of past terms are required for the numerical propagation of the solution; and that the calculation of the effective values of mass, damping, and stiffness is performed only once. Several examples of applications are included.

Photoelastic tomography is a non-destructive method of 3D stress analysis. It permits determination of normal stress distribution in an arbitrary section of a 3D test object. In case of axial symmetry also the shear stress distribution can be determined directly from the measurement data. To determine also the other stress components one can use equations of the theory of elasticity. Such a combined application of experimental measurements and numerical handling of the equations of the theory of elasticity is named hybrid mechanics. It is shown that if stresses are due to external loads, the hybrid mechanics algorithm is based on the equations of equilibrium and compatibility. In the case of the measurement of the residual stress in glass the compatibility equation can not be applied. In this case a new relationship of axisymmetric thermoelasticity, the generalized sum rule can be applied.

The method and equipment of the electronic speckle pattern interferometry (ESPI) for studying a stress-deformation state (SDS) of elastic bodies and structures are presented. Examples of use ESPI for measurements of residual stresses in welded structures, microdisplacements related to delamination of thin coatings, diagnostics of shrinkage stresses in coatings, determination of their elastic characteristics, generalization of the ESPI method for measurements of displacements of the nanometer scale fulfilled in IPMech RAS are given.

The compressive response of an axially restrained composite column, which is exposed to a heat flux due to fire is studied by both analytical and experimental means. The column is exposed to fire from one-side and an analytical approach is outlined for the resulting heat damage, the charred layer formation and non-uniform transient temperature distribution. Due to the nonuniform stiffness and the effect of the ensuing thermal moment, the structure behaves like an imperfect column, and responds by bending rather than buckling in the classical Euler (bifurcation) sense. In order to verify the mechanical response, the compressive buckling behavior of the same material subjected to simultaneous high intensity surface heating and axial compressive loading were investigated experimentally in a specially designed cone calorimeter. Experiments on the residual compressive strength following exposure to fire are also conducted for a range of heat fluxes and exposure times.

Sampling moiré method is a newly developed moiré method using an image processor for a grating pattern to measure shape or displacement or strain distributions. A grating pattern on an object is recorded by a digital camera. Though the digitized image shows the grating, a moiré fringe pattern appears by thinning-out the pixels, i.e., sampling the image with a larger pitch than the pixel pitch. If the sampling pitch is changed, the moiré pattern is changed very much. If the phase of the sampling is changed, the phase of the moiré pattern is changed. The phase analysis of moiré pattern provides accurate result of displacement of the grating. If the number of the phase-shifted moiré patterns i.e. the number of pixels for a pitch of grating is larger, the resolution of phase analysis becomes more accurate but the spatial resolution becomes worse. Since the sampling moiré method is useful to analyze the phases of a moiré fringe and a grating from one image of a grating pattern, it is possible to analyze dynamic deformation accurately.

In this paper, the theory of the sampling moiré method is introduced and some applications of the sampling moiré method to displacement measurement of a beam, and shape and strain measurement of a rubber structure are shown.

Recent advances in optoelectronic methodology for microscale measurements are described and their use is illustrated with representative examples of microelectromechanical systems (MEMS) operating at high frequencies and used in demanding environments. Today, the word MEMS is employed to describe a process used as well as the resulting products. Therefore, a MEMS-process is also known as a “microsystem technology” (MST).

Advances in emerging technologies (ETs) of MEMS and nanotechnology, especially relating to the applications, constitute one of the most challenging tasks in today’s micromechanics and nanomechanics. In addition to design, analysis, and fabrication capabilities, these tasks also require advanced test methodologies for determination of functional characteristics of devices produced to enable verification of their operation as well as refinement and optimization of specific designs. In particular, development of miniscule devices requires sophisticated design, analysis, fabrication, testing, and characterization tools. These tools can be categorized as analytical, computational, and experimental. Solutions using the tools from any one category alone do not usually provide necessary information on MEMS and extensive merging, or hybridization, of the tools from different categories is used. One of the approaches employed in this development of structures of contemporary interest, is based on a combined use of the analytical, computational, and experimental solutions (ACES) methodology. Development of this methodology was made possible by recent advances in optoelectronic methodology, which was coupled with the state-of-the-art computational methods, to offer a considerable promise for effective development of various designs. This approach facilitates characterization of dynamic and thermomechanical behavior of the individual components, their packages, and other complex material structures. In this paper, recent advances in optoelectronic methodology for micro- and nano-scale measurements are described and their use is illustrated with representative examples.

A new renaissance in the field of Experimental Mechanics is well underway because of the recent technologies developed for Nano-Engineering. There are many challenges to face and overcome when going from the macro world to the manipulation of nano-objects. In the macro world, with the experience gained in the last century and the development of numerical techniques, Experimental Mechanics has changed its initial role of an analogical tool to solve difficult differential equations to a complementary methodology to support numerical techniques in handling complex boundary condition effects in static or dynamic problems. Experimental Mechanics is also a very important tool in materials science research. With the introduction of Nano-Engineering, Experimental Mechanics has experienced a vast expansion in its applications to understand an almost completely new field where both basic physical properties that have a well established statistical meaning in the macro world and fundamental formulations require a revision and in many cases new theoretical developments.

Since theories must be supported by experimental evidence, Experimental Mechanics represents a necessary basic tool to lay the foundations to understanding properties and behavior of materials at the nano-level. There are well established tools that allow events at the nano-level to be observed: for example, X-rays with new developments in holographic interferometry done with X-rays. Electron microscopy also has been extended to the field of holographic interferometry. Optics with its versatile photons appears also as a promising tool in many cases where X-rays or electron microscopy become difficult or impossible to apply. However, the classical resolution limitations confined for a long while optics to be used in the range of hundreds of nanometers. New recent developments have opened a new window of opportunity for optical techniques to be applied in the nano-range. This chapter will cover these recent developments. The essential theoretical aspects that make it possible to go beyond the classical resolution limits as well as their application in engineering problems such as metrology, surface topography and strain determinations will be presented.

Considerable research and development efforts have been directed towards high strength/high performance concrete with engineered properties, using three main concepts: a low water to binder ratio (w/b), and the partial replacement of cement by fine supplementary cementitious or pozzolanic materials and/or fibers. To better understand how material composition and microstructural modifications determine the concrete structural performance, and to develop new materials with specific properties, researchers at ACBM have taken a materials science approach with an application to nanotechnology to optimize the processing and micro/nanoscale structure of cement based materials. In particular, due to their exceptional mechanical properties, the reinforcing effect of highly dispersed multiwall carbon nanotubes (MWCNTs) and carbon nanofibers (CNFs) in cement paste matrix was investigated. The major challenge however, associated with the incorporation of MWCNTs and CNFs in cement based materials is poor dispersion. In this study, effective dispersion of different length MWCNTs in water was achieved by applying ultrasonic energy and with the use of a surfactant. The excellent reinforcing capabilities of the MWCNTs are demonstrated by the enhanced fracture resistance properties of the cementitious matrix. Additionally, nanoindentation results suggest that the use of MWCNTs can increase the amount of high stiffness C-S-H and decrease the porosity. Besides the benefits of the reinforcing effect, autogenous shrinkage test results indicate that MWCNTs can also have a beneficial effect on the early strain capacity of the cementitious matrix, improving this way the early age and long term durability of the cementitious nanocomposites.

Elastic material properties critically affect the vibration behavior of structures. The value of natural frequencies changes due to change in the plate constants/plate stiffness which is a function of elastic modulus. At each natural frequency, the plate has a unique mode shape of vibration which can be easily differentiated from mode shapes at other natural frequencies. In this paper, a technique for the evaluation of the elastic modulus is proposed which is based on the vibration analysis of the plate using digital speckle pattern interferometry (DSPI) and Rayleigh method. Large numbers of experiments were conducted on square aluminium plate for the boundary condition; one edge is fixed and other edges free. The experimental result reveals that a single observation of frequency at first torsional mode is sufficient to evaluate the elastic modulus for all practical purposes. The evaluated experimental error was found to be less than 1%. Ease in sample preparation, simplicity in evaluation, non destructive nature of the DSPI and speed of DSPI has good prospect to evaluate elastic modulus of a material.

In vibration measurement, traditional time averaged (TA) electronic speckle pattern interferometry (ESPI) method was essentially used for obtaining modal shapes rather for quantitative analysis. In 1996, the authors first reported that the driving force acting on the specimen would be fluctuated due to environmental disturbances and vibration fringe patterns obtained by TA ESPI method can be significantly improved if both the reference and object images were captured under vibration load. This new TA ESPI method was named as amplitude fluctuation (AF) ESPI method. In this paper, the development and successive improvement of the AF ESPI method was first introduced. The effects of environmental noise and vibration characteristics on the ESPI fringe pattern were then investigated. Theoretical derivation on the effect of environmental noise was performed and the time varying brightness of the traditional time averaged (TA) ESPI fringe patterns was successfully explained. In addition, applications of the AF ESPI method were briefly reviewed.

Neutron emission measurements, by means of ³He devices and bubble detectors, were performed during three different kinds of compression tests on brittle rocks: (i) under monotonic displacement control, (ii) under cyclic loading, and (iii) by ultrasonic vibration. The material used for the tests was Green Luserna Granite. Since the analyzed material contains iron, our conjecture is that piezonuclear reactions involving fission of iron into aluminium, or into magnesium and silicon, should have occurred during compression damage and failure. This hypothesis is confirmed by Energy Dispersive X-ray Spectroscopy (EDS) tests conducted on Luserna Granite specimens. It is also interesting to emphasize that the present natural abundances of aluminum (~8%), and silicon (28%) and scarcity of iron (~4%) in the continental Earth’s crust should be possibly due to the piezonuclear fission reactions considered above.

The optical method of caustics constitutes a powerful tool in the hands of the experimentalist for the solution of fracture mechanics problems. According to the method, the area in the vicinity of the crack tip is illuminated by a light beam and the reflected or transmitted light rays form an envelope in space. When this envelope is cut by a screen a highly illuminated curve, the so-called caustic, is formed. By measuring characteristic dimensions of the caustic the stress intensity factor is determined. The method has been based on the assumption that plane stress conditions dominate in the vicinity of the crack tip. However, experimental evidence has shown that the state of stress is not pure plane stress. It changes from plane strain near the crack tip to plane stress at a critical distance away from the tip through a three-dimensional region. In order the caustic to be generated from the light rays reflected or transmitted through the specimen from the plane stress region certain conditions among the dimensions of the optical arrangement, the specimen properties and specimen thickness need to be satisfied. These conditions are investigated in this work so that the method of caustics can safely be used for determination of stress intensity factors in fracture mechanics problems.

We study a series of problems involving multiple connected and cracked bodies. Based on the boundary conditions, we derive the equations describing the stress – strain field of the body. A case of special interest is the one when mixed boundary conditions apply on the crack lips. Specifically, either we can place thin inclusions on the crack lips near its edges or in its central part; these inclusions can be modeled as linearly deformable springs of certain stiffness, based on Winkler’s model. We also examine the case when the central part of the crack is reinforced with a stringer while on the edges of the crack elastic springs are placed in specific distances. On the areas where there are no stringers or thin inclusions, the normal and shear stresses are considered as known. An infinite plate is loaded at infinity with normal and shear strains. We study the interaction of the reinforcements and the weakening of the composite plate by calculating the stress intensity factor (SIF) on the crack lips, as well as the stress – strain field of the composite cracked plate.

Some theoretical and experimental results on fracture mechanics of materials and durability of structural elements are presented. Conceptual bases (statements) of fracture mechanics and strength of cracked materials as well as urgent problems of prospective investigations in this field of science about materials and their strength are formulated. The actual problems of fracture mechanics and strength of materials in service environments are considered.

In the present paper the Preisach model of hysteresis is applied to model cyclic behavior of elasto-plastic material. The problem of axial loading of rectangular cross section will be studied in details. Hysteretic stress-strain loop for prescribed history of stress change is plotted for material modeled by series connection of three unite element. All obtained results clearly show advantages of the Preisach model for describing cyclic behavior of so called stable plastic material. Other effects such are racheting and creep will be studied elsewhere. In this paper extremely low cycle fatigue will also be examined. Extremly low cycle fatigue stands for number of cycles to failure in between 10 and 20. The stress level is larger than the yield stress and the plastic strain is of the same magnitude as the elastic strain. In this paper it is shown that this case is of importance to dampers applied for reconstruction of earthquake damaged structures.

Fracture toughness values are given for an ATH-PMMA composite for filler volume fractions ranging from 0.35 to 0.49 and tested over the temperature range from 0 to 90 °C. The toughness decreased with increasing filler content contrary to expectations. A toughness model based on debonding, and subsequent plastic void growth was extended from a low volume fraction form to accommodate interaction between particles. From the fitted data it was possible to calculate the adhesion energy of the particles and the average particle size. The former was somewhat less than the matrix toughness and the latter agreed quite well with the sizes found from particle size measurements on the filler and surface roughness measurements on the fracture surfaces.