Buckling analysis of an elastic bar using the Bubnov-Galerkin method

The factors responsible for the errors often encountered in the stability analysis of elastic systems are found by testing the Bubnov-Galerkin method for the buckling problem of a single-layer flexible elastic bar. Refined formulas are obtained for the maximum deflection of a longitudinally compressed hinged three-layer bar. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 148–156, January–February, 2008.     

Mechanical properties of ultrananocrystalline diamond thin films relevant to MEMS/NEMS devices

The mechanical properties of ultrananocrystalline diamond (UNCD) thin films were measured using microcantilever deflection and membrane deflection techniques. Bending tests on several free-standing UNCD cantilevers, 0.5 μm thick, 20 μm wide and 80 μm long, yielded elastic modulus values of 916–959 GPa. The tests showed good reproducibility by repeated testing on the same cantilever and by testing several cantilevers of different lengths. The largest source of error in the method was accurate measurement of film thickness. Elastic modulus measurements performed with the novel membrane deflection experiment (MDE), developed by Espinosa and co-workers, gave results similar to those from the microcantilever-based tests. Tests were performed on UNCD specimens grown by both micro and nano wafer-seeding techniques. The elastic modulus was measured to be between 930–970 GPa for the microseeding and between 945–963 GPa for the nanoseeding technique. The MDE test also provided the fracture strength, which for UNCD was found to vary from 0.89 to 2.42 GPa for the microseeded samples and from 3.95 to 5.03 for the nanoseeded samples. The narrowing of the elastic modulus variation and major increase in fracture strength is believed to result from a reduction in surface roughness, less stress concentration, when employing the nanoseeding technique. Although both methods yielded reliable values of elastic modulus, the MDE was found to be more versatile since it yielded additional information about the structure and material properties, such as strength and initial stress state.     

Subcritical crack growth in an aging plate with variable elastic modulus

The paper addresses subcritical growth of a crack in a thin isotropic plate made of an aging viscoelastic material with time-dependent elastic modulus. The behavior of the material is described by Arutyunyan’s creep theory. To simulate fracture, a modified Leonov–Panasyuk–Dugdale model and a critical crack opening displacement criterion are used. An equation describing the subcritical growth of the crack is derived assuming that Poisson’s ratio is constant. As an example, the critical loads are determined, and curves of subcritical crack growth are plotted for a specific material. The results are compared with the case of constant elastic modulus     

Boundary film shear elastic modulus effect in hydrodynamic contact. Part I: theoretical analysis and typical solution

Boundary film shear elastic modulus effect is analyzed in a hydrodynamic contact. The contact is one-dimensional composed of two parallel plane surfaces, which are, respectively, rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact is consisted of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. Analysis is thus carried out for a micro Raleigh bearing in this contact. The hydrodynamic contact in this micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity large enough at the film–contact interface. In boundary film area, the viscosity, density and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling, which depend on the film thickness. The analytical approach proposed by Zhang (J Mol Liq 128:60–64, 2006) and Zhang et al. (Int J Fluid Mech Res 30:542–557, 2003) is used for boundary film area. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. Conventional hydrodynamic analysis is used for fluid film area. The present paper presents the theoretical analysis and a typical solution. It is found that for the simulated case the boundary film shear elastic modulus effects on the mass flow through the contact, the overall film thickness of the contact and the carried load of the contact are negligible but the boundary film shear elastic modulus effect on the local film thickness of the contact may be significant when the boundary film thickness is on the 1 nm scale and the contact surfaces are elastic. In Part II will be presented detailed results showing boundary film shear elastic modulus effects in different operating conditions.

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Boundary film shear elastic modulus effect in hydrodynamic contact. Part II: influence of operational parameters

The present paper is the subsequent research of the first part (Theor Comput Fluid Dyn, 2009). It investigates the boundary film shear elastic modulus effect in a hydrodynamic contact in different operating conditions. The hydrodynamic contact is one-dimensional, composed of two parallel plane surfaces, which are respectively rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact consists of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. The hydrodynamic contact in a micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity of the film–contact interface large enough. In boundary film area, the viscosity, density, and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling which depends on the film thickness. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. It is found from the simulation results that the boundary film shear elastic modulus influences are normally negligible on the mass flow through the contact, the carried load of the contact and the overall film thickness of the contact, and the boundary film shear elastic modulus would normally influence the local film thickness in an elastic contact when the local film thickness is on the film molecule diameter scale. It is also found that the boundary film shear elastic modulus effect has the tendency of being increased with the reduction of the width of a micro contact. It is increased with the reduction of the boundary film–contact interfacial shear strength or with the increase of the critical boundary film thickness, while it is strongest at certain values of the contact surface roughness, the width ratio of fluid film area to boundary film area, and the lubricant film shear elastic modulus.

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Elastoplastic bending of a sandwich bar on an elastic foundation

The elastoplastic bending of a sandwich bar with a stiff compressible core on an elastic foundation is studied. The kinematics of the bar, which is asymmetric across the thickness, is described adopting Bernoulli’s hypotheses for the face layers. The displacements of the core are assumed to vary linearly across the thickness. The foundation is described by the Winkler model. A system of equilibrium equations for displacements is derived and solved. Numerical results for a metal-polymer sandwich bar are presented __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 110–120, April 2007.     

Affine Transforms of Three-Dimensional Anisotropic Media and Explicit Formulas for Fundamental Matrices

The concept of classes of algebraically equivalent anisotropic three-dimensional media is introduced: elastic fields in such media are related by simple algebraic expressions. An explicit formula is obtained for a fundamental matrix with ten free constants in the elastic modulus tensor (rather than for five free constants, as in the famous case of transverse isotropy). A hypothesis is formulated and several questions are posed, which are related to the notion of algebraic equivalence under discussion. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 95–102, March–April, 2006.     

Wormlike chain or tense string? A question of resolution

$\bar S$ and a bending elasticity , attached to a frame of length L, can be described – at low resolutions – by the same type of elastic free-energy as a tense string. The corresponding tension is calculated as a function of temperature, L, and . Received August 31, 2001 / Published online May 21, 2002     

Correlation between the high-frequency elastic modulus and the interparticle interaction potential in zirconium oxide colloidal suspensions

In this work, we study the high-frequency elastic modulus of aqueous suspensions made with two kinds of zirconium oxide particles, one commercially available and the other synthesized as monodisperse spheres. The effect of volume fraction of solid, ionic strength (sodium chloride as indifferent electrolyte) and particle geometry is taken into account in the study on this viscoelastic property of the suspensions. Frequency sweeps were performed at a fixed value of the applied shear-stress in order to obtain the frequency-limiting value of the elastic modulus by rheometrical methods. On the other hand, the high-frequency modulus is theoretically calculated independently by means of the models proposed by Buscall and co-workers, Wagner and Bergenholtz and co-workers, which correlate the interaction potential between particles with this rheological parameter. The approach to the interparticle potential is the extended DLVO theory, which considers the electrical repulsion between charged colloidal particles, the van der Waals attraction and the acid–base interaction that can be attractive or repulsive depending on the thermodynamic nature of the solid–liquid interface.     

A Technique for Dynamic Tensile Testing of Human Cervical Spine Ligaments

An experimental study is undertaken to examine the dynamic stress–strain characteristics of ligaments from the human cervical spine (neck). Tests were conducted using a tensile split Hopkinson bar device and the engineering strain rates imposed were of the order of 10²∼10³/s. As ligaments are extremely soft and pliable, specialized test protocols applicable to Hopkinson bar testing were developed to facilitate acquisition of reliable and accurate data. Seven primary ligaments types from the cervical spines of three male cadavers were subjected to mechanical tests. These yielded dynamic stress–strain curves which could be approximated by empirical equations. The dynamic failure stress/load, failure stain/deformation, modulus/stiffness, as well as energy absorption capacity, were obtained for the various ligaments and classified according to their location, the strain rate imposed and the cadaveric source. Compared with static responses, the overall average dynamic stress–strain behavior foreach type of ligament exhibited an elevation in strength but reduced elongation.     

Linear viscoelastic master curves of neat and laponite-filled poly(ethylene oxide)–water solutions

Aqueous solutions composed of dispersed nanoparticles and entangled polymers are shown to exhibit common viscoelasticity over a range of particle and polymer concentrations. Time–temperature superposition and time–concentration superposition are applied to generate rheological master curves for neat and laponite-filled aqueous solutions of poly(ethylene oxide). The shift factors were correlated in terms of temperature and concentration and are found to differ from previous reports for ideal polymer solutions, which can be rationalized with a molecular interpretation of the structure of the laponite–polymer solutions. Laponite addition to the concentrated polymer solution is observed to increase the relaxation time but decrease the elastic modulus, which is a consequence of polymer adsorption and bridging. The addition of small amounts of laponite to stable PEO–water solutions also leads to ageing on the time scale of days.     

Determination of Bone Trabeculae Modulus—An Ultrasonic Scanning and MicroCT (μCT) Imaging Combination Approach

To date, there is no method to measure non-destructively the modulus of trabeculae within cancellous bone, whilst retaining its structural integrity. In this study ultrasonic scanning, coupled with microCT imaging, is employed to determine trabeculae modulus along the three major anatomical axes non-destructively. The proposed method allows cancellous bone specimens to remain intact, for possible use in subsequent studies. Volume rendering of the microCT images allows three-dimensional visualization of cancellous bone specimens to be tested. This facilitates trabeculae selection and accurate measurement of distance traveled by the ultrasonic wave, thus yielding a good degree of confidence in the acoustic velocity measured. For all the three principal anatomical directions, the measured acoustic speeds ranged from 2,115 to 3,077 m/s, giving an average of 2,505 m/s. Average wave velocities in the superior–inferior, medial–lateral and anterior–posterior anatomical directions were found to be 2,295, 2,469 and 2,754 m/s, respectively; the differences corresponding to the three directions do not appear to be significant. Subsequently, the modulus was then determined using elastic wave propagation theory.     

应力平衡和接触状态在Hopkinson杆测试聚合物动态弹性模量中的影响研究

针对用Hopkinson杆试验能否准确测量聚合物动态弹性模量以及其中主要影响因素的问题,本文基于重构试样初始加载阶段的应力波反射透射过程,分别计算了6个特征时间内的前三次反射波和透射波,得到试样的应力平衡度和试样的应力应变曲线。对于所研究的聚合物材料,通过比较重构的应力应变曲线的弹性模量与输入的材料弹性模量,发现在4个特征时间后,误差仅在3%左右。因此试样变形过程中的应力平衡与否不是材料在Hopkinson杆试验中弹性模量测不准的原因。通过环氧树脂试样试验发现,根据Hopkinson杆理论计算的应变结果要大于试样上应变片实测的结果,误差在11%左右。相应的数值模拟研究发现:试样和杆子端面接触状态直接决定着试样弹性模量测量的精度。关于惯性效应和压痕效应的研究也证实它们的影响是可以忽略的。     

Seismic Wave Propagation in Composite Elastic Media

It has been known since the time of Biot–Gassman theory (Biot, J Acoust Soc Am 28:168–178, 1956, Gassmann, Naturf Ges Zurich 96:1–24, 1951) that additional seismic waves are predicted by a multicomponent theory. It is shown in this article that if the second or third phase is also an elastic medium then multiple p and s waves are predicted. Futhermore, since viscous dissipation no longer appears as an attenuation mechanism and the media are perfectly elastic, these waves propagate without attenuation. As well, these additional elastic waves contain information about the coupling of the elastic solids at the pore scale. Attempts to model such a medium as a single elastic solid causes this additional information to be misinterpreted. In the limit as the shear modulus of one of the solids tends to zero, it is shown that the equations of motion become identical to the equations of motion for a fluid filled porous medium when the viscosity of the fluid becomes zero. In this limit, an additional dilatational wave is predicted, which moves the fluid though the porous matrix much similar to a heart pumping blood through a body. This allows for a connection with studies which have been done on fluid-filled porous media (Spanos, 2002).     

Oscillation of an elastic bar rigidly linked to a kinematically excited pendulum

The oscillation of a mechanical system consisting of an elastic bar rigidly linked at the middle to a kinematically excited pendulum is studied. A system of integro-differential equations with appropriate boundary and initial conditions for the deflections of the bar axis and the rotation angle of the pendulum is derived using the Hamilton-Ostrogradsky principle. Given kinematic excitation conditions, the rotation angle is found as a solution to an inhomogeneous Hill equation in the form of a double power series in the amplitude of kinematic excitation. It is shown that the oscillation of the bar is the linear superposition of three oscillations __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 107–115, October 2006.     

Determining the shear modulus of water in experiments with a floating disk

The problem of decaying rotation of a disk floating on the surface of a viscoelastic fluid in a cylindrical container is solved by numerical methods. The motion is found to have the form of decaying oscillations observed previously for water. In addition to the viscosity coefficient, the constructed mathematical model of the viscoelastic fluid has two more independent parameters: shear modulus and time of relaxation of elastic stresses. Elastic parameters of water are determined through comparisons with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 100–103, January–February, 2008.     

Microstructural effects on the dynamic rheology of a discotic mesophase pitch

The dynamic rheology and accompanying microstructure of a synthetic mesophase pitch (AR-HP) is reported. The loss modulus (G″) was found to be higher than the storage modulus (G′) at all frequencies (∼0.1 to ∼100 rad/s) and temperatures (280–305 °C) tested. The slope of the low-frequency terminal zone for G′ was found to be approximately 0.8, much lower than a value of 2 that is observed for flexible chain polymers. Cross-polarized optical microscopy with full-wave retardation plate confirmed the presence of different textures of mesophase pitch processed under different conditions. While loss moduli remained fairly unchanged, finer textures led to significantly lower storage moduli. Consistent with this trend, coarsening of the microstructure during textural relaxation led to an increase in storage moduli. Therefore, for the discotic mesophase pitch, the viscous component was found to remain unaffected by the microstructure, but the elastic modulus was dependent on the orientation of layer-planes and size of the texture.     

Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars

The Saint-Venant torsion problem of linearly elastic cylindrical bars with solid and hollow cross-section is treated. The shear modulus of the non-homogeneous bar is a given function of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsional problem of non-homogeneous bar is expressed in terms of the torsional and Prandtl's stress functions of homogeneous bar having the same cross-section as the non-homogeneous bar.     

Characterization of shock accelerometers using davies bar and strain-gages

This paper proposes a novel method for evaluating the dynamic characteristics of shock accelerometers under high acceleration levels and a wide frequency bandwidth. High accelerations of 10³∼10⁵m/s² can be generated by the reflection of an elastic wave pulse propagating in a metal bar known as the Davies bar. The elastic wave pulse is produced by the collision of a projectile against one end of the bar, and is detected by straingages. The accelerometer to be characterized is attached to the other end of the bar. The one-dimensional theory of elastic waves enables the derivation of an input acceleration to the accelerometer from the measured strain. The dispersion of the elastic waves caused by the lateral inertia of the bar is compensated for by using a two-dimensional analytical solution. This method was validated by an experiment characterizing a piezoelectric-type accelerometer within the frequency band approximately 1 kHz∼70 kHz.     

Characterization of shock accelerometers using Davies bar and laser interferometer

In this paper, we propose a novel method for evaluating the frequency response of shock accelerometers using Davies bar and interferometry. The method adopts elastic wave pulses propagating in a thin circular bar for the generation of high accelerations. The accelerometer to be examined is attached to one end of the bar and experiences high accelerations of the order of 10³∼10⁵ m/s². A laser interferometer system is newly designed for the absolute measurement of the bar end motion. It can measure the motion of a diffuse surface specimen at a speed of 10⁻³ ∼10⁰ m/s. Uncertainty of the velocity measurement is estimated to be±6×10⁻⁴ m/s, proving a high potential for use in the primary calibration of shock accelerometers. Frequency characteristics of the accelerometer are determined by comparing the accelerometer's output with velocity data of the interferometry in the frequency domain. Two piezoelectric-type accelerometers are tested in the experiment, and their frequency characteristics are obtained over a wide frequency range up to several ten kilohertz. It is also shown that the results obtained using strain gages are consistent with those by this new method. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.