Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

Erosion of α-Fe by spherical glass particles

Polycrystalline α-Fe has been eroded at 30° and 90° with glass spheres of average diameters 70 μm and 200 μm in the velocity range 61–122 m s⁻¹. Detailed studies of the influence of the impact variables on the erosion rate as well as scanning electron microscopy studies of the eroded surfaces have been performed. It was observed that “breaking” waves developed on erosion at 30° and hills and valleys at 90°. Several different material loss processes that operate at various positions within waves, hills and valleys have been identified. It was clear that most material loss processes involved extensive localized shear and required the surface to become “conditioned” by a specific number of impacts before material loss began.     

Onset of matrix cracking in angle-ply ceramic matrix composites

The problem of initial damage in angle-ply [−θ_m/0_n/θ_m] and [−θ/θ] ceramic matrix composites subjected to axial tension is considered in this paper. The damage is in the form of matrix cracks that may appear in either inclined (−θ and θ lamination angle) or longitudinal layers. As follows from the analysis, if the lamination angle of the inclined layers is small, the initial failure occurs in the 0-layers of [−θ_m/0_n/θ_m] composites or in [−θ/θ] composites in the form of bridging cracks. However, if the inclined layers form a larger angle with the load direction, they fail due to tunneling cracks. It is shown that the boundary between two different modes of failure in a representative SiC/CAS composite corresponds to a lamination angle equal to 35° in the case of [−θ_m/0_n/θ_m] composites. In the case of [−θ/θ] laminates, the boundary value of the lamination angle is equal to 45°, i.e. bridging cracks form if θ<45° and tunneling cracks appear if θ>45°.     

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

In the conventional finite element method (FEM), the dynamic characteristics of a longitudinally vibrating rod with mass density ρ_r, Young's modulus E_r, cross-sectional area A_r and total length ℓ_r are considered to be the same as those of a helical spring with stiffness constant k_r=A_rE_r/ℓ_r and total mass m_r=ρ_rA_rℓ_r. For a lumped-mass model, the mass matrix of a rod element is a 2×2 diagonal one with each of its non-zero coefficients to be equal to one half of the total rod mass (i.e., 0.5m_r). Furthermore, the dynamic characteristics of a rod on the basis of last “lumped-mass” model have been found to be very close to those on the basis of “consistent-mass” model. Thus, one can easily take into account of the inertial effect of a helical spring using a massless one with “one half of its total mass”, respectively, concentrated at its two ends (in Method 2) instead of modeling it by an elastic rod with uniform mass per unit length (in Method 1). When one more spring-mass system is attached to the beam, the total number of unknown constants increases “one” in Method 2 and “two” in Method 1, thus, Method 2 will reduce more effort than Method 1 for studying the dynamic behaviors of a beam carrying a number of spring-mass systems with mass of each helical spring considered. In this paper, the formulations of Methods 1 and 2 are presented first and then the numerical examples are illustrated to confirm the reliability of the presented theory and the developed computer programs. Finally, the effect concerning mass of each helical spring of the spring-mass systems is studied.     

Twin shear stress yield criterion

A twin shear stress yield criterion is described. This criterion was proposed by the author in 1961[1]. It assumes that yielding begins when the sum of the two larger principal shear stresses reaches a magnitude C. Thus the initial yield function is f = τ₁₃ + τ₁₂ = σ₁ − 1/2(σ₂ + σ₃) = c[(τ₁₃ + τ₁₂) ± c][τ₂₃ + τ₂₁) ± c][τ₃₁ + τ₃₂) ± c] = 0.     

Closed-form solution for wedge cutting force through thin metal sheets

A simple kinematic model is developed which describes the main features of the process of the cutting of a plate by a rigid wedge. It is assumed in this model that the plate material curls up into two inclined cylinders as the wedge advances into the plate. This results in membrane stretching up to fracture of the material near the wedge tip, while the “flaps” in the wake of the cut undergo cylindrical bending. Self-consistent, single-term formulas for the indentation force and the energy absorption are arrived at by relating the “far-field” and “near-tip” deformation events through a single geometric parameter, the instantaneous rolling radius. Further analysis of this solution reveals a weak dependence on the wedge angle and a strong dependence on friction coefficient. The final equation for the approximate cutting force over a range of wedge semiangles 10° ≤ θ ≤ 30° and friction coefficients 0.1 ≤ μ ≤ 0.4 is: F = 3.28σ₀(δ_t)^0.2l^0.4t^1.6μ^0.4, which is identical in form and characteristics to the empirical results recently reported by Lu and Calladine [Int. J. Mech. Sci.32, 295–313 (1990)].This analysis is believed to resolve a controversy recently developed in the literature over the interpretation of plate cutting experiments.     

Rotor dynamic instability analysis on hybrid air journal bearings

Hybrid air journal bearings with multi-array of 1, 2, 3, 4, or 5-row orifice feedings are analyzed for the problem of rotor dynamic instability. The bearing stiffness and damping coefficients are calculated numerically to determine threshold rotor mass under various operating conditions. The hybrid porous air journal bearings are also analyzed for comparison to investigate the similarities in dynamic characteristics between the multi-array of orifice feeding bearings and the porous bearings. The results show that the porous bearing is more stable than the orifice feeding bearing at lower rotation speeds (Λ<0.1) or at higher rotation speeds (Λ>1) with lower feeding parameters (λ_P<10⁻⁸). The 5-row orifice feeding bearing is more stable than the porous bearing at moderate speeds (0.3<Λ<0.6) with lower feeding parameters (λ₀<10⁻⁴).     

Image processing nanoparticle size measurement for determination of density values to correct the ELPI measures

The ELPI is an electrical low-pressure impactor that classifies aerosol particles according to their aerodynamic diameter, and generates the amount of particles impacted on each of the 12 stages. This number depends on the measured current induced by the pre-charged particles and on the density which is given by the user and may not be a priori known. In addition, the density used by the software is considered to be similar for all stages. In this paper, a method to evaluate the density of the particles on each stage is proposed in order to consequently increase the accuracy of the results given by the software. The data needed are the aerodynamic diameter, the equivalent diameter and information on the form. The aerodynamic diameter is a range defined by the cut-off aerodynamic diameter of the stages of the ELPI. To measure the equivalent diameter and evaluate the form, an adapted procedure that used microscopy and image processing tools was set up with the study of two different polydispersed aerosols, silica and fly ash particles from wood combustion. This method was validated with Silica particles (ρ = 2.5 g cm⁻³ with the pycnometer): the density was found to be 2.2 g cm⁻³ and 2.4 g cm⁻³ for stages 2 (d_ae around 76 nm) and 3 (d_ae around 127 nm), respectively. The results match reality for fly ashes from wood combustion as well: ρ = 1.0 g cm⁻³ for the stage 2 and 1.9 for stage 5.     

Determination of the Minority Carrier Surface Thermogeneration Rate in Metal-Oxide-Semiconductor Structures

A metal-oxide-semiconductor (MOS) structure with the common field electrode insulated from the semiconductor by oxide layers h of different thicknesses allows the surface generation rate of minority charge carriers from current I(t) of nonequilibrium depletion state relaxation to be found. At the same time, it is possible in similar structures to observe the kinematics of electron-hole pair generation at the periphery of the field electrode (the edge generation effect). Measurements performed on an n-Si MOS structure with a stepwise change in oxide layer thickness (h ₁ = 100 Å and h ₂ = 3200 Å) allowed the generation rate at both the initial transient (t ∼ 10⁻⁵ s) stage of surface generation (4.34 × 10¹⁰ cm⁻² s⁻¹) and at the basic extremely slow stage (10.4 cm⁻² s⁻¹) to be reliably determined for the first time. The estimated peripheral generation rate of minority charge carriers (holes) was 7.8 × 10¹¹ cm⁻² s⁻¹.__________Translated from Pribory i Tekhnika Eksperimenta, No. 4, 2005, pp. 84–88.Original Russian Text Copyright © 2005 by Chucheva, Zhdan, Akhmedov, Kukharskaya.     

The friction behavior of Ni-, SiO2- and mica sodium silicate-based solid lubrication composites

The friction behavior of Ni⁻-, SiO⁻₂- and mica sodium silicate-based lubricant composites, which included BN, WS₂ and graphite as lubricants, were examined. A ring-on-disk apparatus, in which a solid lubricant composite disk was held against a rotating stainless ring, was used as the test configuration. The tests were run with a load from 62 to 250 N in temperatures from 20 to 800°C in the laboratory environment. The wear surface was characterized by scanning electron microscope and X-ray photo spectroscopy. The major findings were that both mica sodium silicate- and SiO⁻₂-based composites failed at above 500°C due to severe wear and surface damage; in contrast, Ni⁻-based composite showed a stable friction coefficient and low wear from 20 to 800°C.     

Assessments of the kinetic and dynamic loads sustained by stationary tools during high rate plastic forming

A systematic method for evaluating the kinetic and dynamic loads sustained by stationary tools (as opposed to moving tools for which methods already exist) during high rate plastic forming is examined and exemplified by examples. It is essentially based on the momentum theorem for continua for incompressible flow, utilizing kinematically admissible velocity fields. In steady state forming processes (such as rolling, wire drawing, etc.), the difference between the active load (imposed or calculated a priori) and the reactive load, is formulated rigorously, whereas for non-steady processes (forging, impact extrusion, etc.) the formulation gives merely an approximation to the dynamic effects on the tools. The resulting velocity-dependent reactions on the tools are given in terms of two nondimensional numbers, namely, the “kinetic head” (u₀²/σ₀) (called the Euler Number) and the “dynamic head” (ú₀L/σ₀), which includes the machine speed (u₀), machine acceleration ( ), material density , yield strength ₀ and a characteristic dimension of the product, L. The same two non-dimensional heads emerged previously from energy-balance consideration in Ref. [1], while approximating dynamic loads on moving tools, hence a consistency is demonstrated. These heads are unavoidably multiplied by geometrical functions, which typify the specific process under consideration and may amplify (or diminish) the intensity of the dynamic effects. The present work is focussed on quantifying, by the above method, the inherent difference between the reactive load sustained by the non-moving tool (say, a die) and the acting load carried by the moving tool (piston, ram, etc.) In particular cases of very slow processes, these loads are equal by static equilibrium. In some practical processes (like rolling) their difference appears to be relatively small, whereas in others (like impact extrusion) it appears extremely large.     

Measurement of springback

Springback, the elastically-driven change of shape of a part after forming, has been measured under carefully-controlled laboratory conditions corresponding to those found in press-forming operations. Constitutive equations emphasizing low-strain behavior were generated for three automotive body alloys: drawing-quality silicon-killed steel; high-strength low-alloy steel; and 6022-T4 aluminum. Strip draw-bend tests were then conducted using a range of die radii (3<R/t<17), friction coefficients (0<μ<0.20), and controlled tensile forces (0.5<F_b/F_y<1.5). Springback angles and curvatures were measured for bend and bend–unbend areas of the specimen, the latter corresponding to the “sidewall curl” region, which dominates the geometric change and the dependence on process variables. Friction coefficient and R/t (die-radius-to-sheet-thickness) greater than 5 have modest but measurable effects over the ranges tested. As expected, strip tension dominates the springback sensitivity, with higher forces reducing springback. For 6022-T4, springback is dramatically reduced as the tensile stress approaches the yield stress, corresponding to the appearance of a persistent anticlastic curvature. The presence of this curvature, orthogonal to the principal curvature, violates the simple two-dimensional models of springback reported in the literature. The measured springback angles and curvatures are reported both in graphical summary and tabular form for use in assessing analytical models of springback.     

Stress and failure probability analysis for a transversely isotropic, brittle, elastic cylinder subjected to internal pressure and axisymmetric thermal loading

Procedures are developed for the determination of the stresses in and thence the probability of failure of a transversely isotropic cylinder made of a brittle material and loaded by an internal pressure and an axisymmetric radial temperature gradient. As examples of the application of the procedures a cylinder is analysed first with isotropic material properties, then with various degrees of anisotropy including both the “fibrous” and “laminar” types. The treatment is non-dimensional; results are presented graphically in the form of failure probability “contours”. For the dimensions and materials considered it is shown that the probability of failure is affected only slightly by the fibrous form of anisotropy but markedly by the laminar form when the thermal loading predominates.     

Yield loci of anisotropic aluminium sheets

AA8014 aluminium sheets were tested in uniaxial, equibiaxial (bulge test) and plane-strain tension at an almost constant strain rate of 2 × 10⁻³s⁻¹. The results were compared with predictions at different levels of strain, based on macroscopic and crystallographic theories of yielding. While most of the previous investigations compare the results at a strain level of 0.10, the present comparison was made in early stages of the deformation, which might be considered as the real yielding strain of the material. It was concluded that, at the high strain levels Hosford's method, which is a modification of Hill's “old” criterion, gives a proper fit to the experimental data. However, at the lower strain levels the experimental results were very close to the crystallographic loci.     

Application of a continuum constitutive model to metallic foam DEN-specimens in compression

The behavior of double-edge notched specimens of metallic foams in compression is studied numerically. To model the constitutive behavior of the metallic foam, a recently developed phenomenological, pressure-sensitive yield surface [1] is used. Compressive yielding in response to hydrostatic stress is incorporated through a dependence on the plastic Poisson ratio ν^p. Results are presented in terms of limit load F_lim, as a function of notch depth, a/W, and the plastic Poisson ratio ν^p. For incompressible plastic behavior, ν^p=0.5, the results show notch-strengthening due to constrained plastic deformation near the crack/notch-tip. For fully compressible plastic behavior (no lateral expansion on uniaxial compression, ν^p=0), no notch-effect is observed. The validity of using a continuum model for the analysis of metallic foam notched specimens is discussed.     

New cylinder liner surfaces for low oil consumption

Anticipated emission legislation and reduced fuel consumption are the main driving forces when developing new engines. Optimization of the active surfaces in the piston system is one possible way to meet the above demands. In this study the effects of surface topography and texture direction of the ring/liner contact on oil film thickness and friction were simulated and experimentally tested. “Low wear” results from the experimental wear tests with “glide honed” smooth liner surfaces supported the “low friction” simulation results. In addition a new wear volume sensitive surface roughness parameter, R_ktot, based on the Abbot–Firestone bearing area curve was introduced.     

Plate-like wear particle formation in a lubricated ball-on-plate friction pair

The mechanism of formation of plate-like wear particles in a ball-on-plate lubricated friction pair has been examined for wear constants of K < 10⁻¹⁰ (mm³ mm⁻¹ N⁻¹). The plate Vicker's hardness was 2.80–3.00 kN/mm², the sliding speed 1.74 m s⁻¹ and the load 50 N. The following mechanism is suggested: scratching of the surface and formation of ridges at the scratch border, lateral deformation of ridges and formation of thin sheets, and cracking and separation of plate-like particles from these sheets.     

The influence of complex surface geometry on contact damage in curved brittle coatings

The effects of complex geometry on contact damage in bi-layer systems composed of curved brittle coating layers on compliant polymeric substrate is investigated. Previous studies of this problem utilise relatively simple flat or singly curved (domed) model structures. It is not known the extent to which conclusions driven from such observations may extended to more complex (practical) geometry. Glass plates of 1000 μm thick are used as representative of the brittle coating layer, and epoxy filler under layer as representative of under layer support. A series of doubly curved specimens (having curvatures of 4 and 8 mm) are produced to allow investigation of the influence of complex curvature on the evolution of damage. The specimens are tested by indentation with spheres of 4 mm radius loaded along the convex axis of symmetry. For comparison, some specimens loaded parallel to the axis of symmetry but off-centre. The study explores the influence of supporting geometries on the conditions to initiate and propagate subsurface “radial” cracks, which are believed to be responsible for catastrophic failure of brittle-coating-based structures in certain applications, such as dental crowns. It is demonstrated that critical loads for initiation of radial cracks and the subsequent crack propagation are insensitive to complex geometry, so that simple monotonic indentation “axis and/or off-axis loading” with minimum geometrical complication “flat, simply curved” remains an appropriate route to study the evolution of radial cracks in practical brittle coating structures.     

A full factorial investigation of the erosion durability of automotive clearcoats

A full factorial experimental investigation has been carried out into factors affecting the resistance of a commercial acrylic/melamine automotive clearcoat to erosion by silica sand particles. The factor variables and their ranges were: particle size 125–425 μm; temperature 30°C–65°C; impact angle 30°–90°; particle velocity 35 m s⁻¹–55 m s⁻¹; and the baking process applied to the coating. An empirical linear regression model for the erosion response of the coating with R²_adj=97.5% was generated from the data. The regression coefficients of this model quantify the relative strengths of the effects of each of the factors. Several interactions between the factor variables were identified. In particular, the glass transition of the coating, which occurs at 40°C, has a significant effect on its response to erosion. The study has allowed the combinations of conditions that would be of most concern for automotive paint users to be identified.     

Nano-abrasion machining of brittle materials and its application to corrective figuring

A method of ultraprecision abrasion machining named “Nano-abrasion machining” is proposed for optical finishing of brittle materials. The fundamental characteristics and its applicability for corrective figuring to improve form accuracy of optics of brittle materials are investigated. It is experimentally ascertained that the material removal rate and surface roughness are suitable for optical finishing. However, the cross-sectional profile of the machined spot that is dependent on the collision angle is a combination of V- and W-shape, which is unsuitable for the corrective figuring. Therefore, circular motion machining is introduced and a preferable profile with an axis-symmetric V-shape is realized. The machining method is applied to corrective figuring of optical glass of BK7. The NC program is generated with a computer program developed by modifying the scanning motion and the form accuracy is predicted. According to the simulation results, corrective figuring is performed. The flatness is improved from PV = 151 to 29 nm. From the experimental results, it is clarified that the nano-abrasion machining is applicable to corrective figuring of brittle materials.