An elastic–plastic stress analysis steel-reinforced thermoplastic composite cantilever beam

In the present study an analytical elastic–plastic stress analysis is carried out for a low-density homogeneous polyethylene thermoplastic cantilever beam reinforced by steel fibers. The beam is loaded by a constant single force at its free end. The expansion of the region and the residual stress component of σ_x are determined for 0°, 30°, 45°, 60° and 90° orientation angles. Yielding begins for 0° and 90° orientation angles at the upper and lower surfaces of the beam at the same distances from the free end. However, it starts first at the upper surface for 30° and 45° orientation angles. The elastic–plastic analysis is carried out for both the plastic region which spreads only at the upper surface and the plastic region which spreads at the upper and lower surfaces together. The residual stress components of σ_x and τ_xy are also determined. The intensity of the residual stress component is maximum at the upper and lower surfaces of the beam, but the residual stress component of τ_xy is maximum on or around the x-axis. The beam can be strengthened by using the residual stresses. The distance between the plastically collapsed point and the free end is calculated for the same load in the beam for 0°, 30°, 45°, 60° and 90° orientation angles.     

Interlaminar stresses of a rectangular laminated plate with simply supported edges subject to free vibration

The interlaminar stresses in a laminated rectangular orthotropic plate with four sides simply supported edges during free vibration was determined by using the integration method involving the dynamic inertia terms and displacements. The approximate stresses solutions are obtained under the effect of frequencies of vibration for four-layer symmetric cross-ply laminates with the ply configurations [0°/90°]_s and [90°/0°]_s, angle-ply laminates with the ply configuration [45°/−45°]_s. Numerical results show that the natural frequency has significant effects on the dominant interlaminar stresses in the stacking sequences [0°/90°]_s, [90°/0°]_s and [45°/−45°]_s.     

Linear free vibration analysis of cross-ply laminated cylindrical helical springs

A linear free vibration analysis of symmetric cross-ply laminated cylindrical helical springs is performed based on the first-order shear deformation theory. Considering the rotary inertia, the shear and axial deformation effects, governing equations of symmetric laminated helical springs made of a linear, homogeneous, and orthotropic material are presented in a straightforward manner based on the classical beam theory. The free vibration equations consisting of 12 scalar ordinary differential equations are solved by the transfer matrix method. The overall transfer matrix of the helix is computed up to any desired accuracy. The soundness of the present results are verified with the reported values which were obtained theoretically and experimentally. After presenting the non-dimensional graphical forms of the free vibrational characteristics of (0°/90°/90°/0°) laminated helical spring made of graphite-epoxy material (AS4/3501-6) with fixed–fixed ends, a non-dimensional parametric study is worked out to examine the effects of the number of active turns, the shear modulus in the 1–2 plane (G₁₂), the ratio of the cylinder diameter to the thickness (D/d), and Young's moduli ratio in 1 and 2 directions (E₁/E₂) on the first six natural frequencies of a uniaxial composite helical spring with clamped-free, clamped-simple, and clamped–clamped ends.     

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°.     

Predictive model to estimate the stress–strain curves of bulk metals using nanoindentation

Many studies have shown that finite element modeling (FEM) can be used to fit experimental load–displacement data from nanoindentation tests. Most of the experimental data are obtained with sharp indenters. Compared to the spherical case, sharp tips do not directly allow the behavior of tested materials to be deduced because these produce a nominally-constant plastic strain impression. The aim of this work is to construct with FEM an equivalent stress–strain response of a material from a nanoindentation test, done with a pyramidal indenter. The procedure is based on two equations which link the parameters extracted from the experimental load–displacement curve with material parameters, such as Young's modulus E, yield stress Y₀ and tangent modulus E_T. We have already tested successfully the relations on well-known pure metallic surfaces. However, the load–displacement curve obtained using conical or pyramidal indenters cannot uniquely determine the stress–strain relationship of the indented material. The non-uniqueness of the solution is due to the existence of a characteristic point (ε_c, σ_c); for a given elastic modulus, all bilinear stress–strain curves that exhibit the same true stress σ_c at the specific true strain ε_C lead to the same loading and unloading indentation curve. We show that the true strain ε_c is constant for all tested materials (Fe, Zn, Cu, Ni), with an average value of 4.7% for a conical indenter with a half-included angle θ=70.3°. The ratio σ_c/ε_c is directly related to the elastic modulus of the indented material and the tip geometry.     

Elasticity solution for an axially compressed and encased cylindrical specimen

We present the solution of the linear elasticity equations governing the deformation of an elastic cylinder encased in a tube and subjected to uniform compression on the flat ends. The solutions for the stresses, strains, and displacements in the encased and compressed cylinder are all systematically determined from the basic solution of Lamé's classical elasticity problem of the long tube subjected to internal and external pressures. We first derive the general elastostatic analysis for an encased hollow cylinder, stress-free at the cavity, and later particularize the solution to a solid cylindrical specimen. The effective modulus E_eff of the encased sample is found to be a function of the bulk modulus k and Poisson's ratio ν of the material. E_eff differs from k except for nearly incompressible materials, where E_eff approaches the bulk modulus value. In the incompressible case, we also show how a load applied on the cylinder's flat ends is equivalent to, and can be replaced by, the same load acting on the curved surface. For compressible materials, a more general expression for E_eff is found that also accounts for the case deformation. These results explain the deformation of an axially compressed and encased cylindrical specimen tested in compressibility measuring devices such as those described by Matsuoka and Maxwell [Response of linear high polymers to hydrostatic pressure. Journal of Polymer Science 1958; 32:131–59]. The present analysis thus contributes to a better understanding of how this device works and to the interpretation of measurements taken with it.     

An elastic-plastic stress analysis in silicon carbide fiber reinforced magnesium metal matrix composite beam having rectangular cross section under transverse loading

In this work, an elastic-plastic stress analysis has been conducted for silicon carbide fiber reinforced magnesium metal matrix composite beam. The composite beam has a rectangular cross section. The beam is cantilevered and is loaded by a single force at its free end. In solution, the composite beam is assumed perfectly plastic to simplify the investigation. An analytical solution is presented for the elastic-plastic regions. In order to verify the analytic solution results were compared with the finite element method. An rectangular element with nine nodes has been choosen. Composite plate is meshed into 48 elements and 228 nodes with simply supported and in-plane loading condations. Predictions of the stress distributions of the beam using finite elements were overall in good agreement with analytic values. Stress distributions of the composite beam are calculated with respect to its fiber orientation. Orientation angles of the fiber are chosen as 0°, 30°, 45°, 60° and 90°, The plastic zone expands more at the upper side of the composite beam than at the lower side for 30°, 45° and 60° orientation angles. Residual stress components ofσ _x andτ _xy are also found in the section of the composite beam.     

Mechanical analysis of the scratching of metals and polymers with conical indenters at moderate and large strains

Scratch test provides a convenient mean to study the surface mechanical properties and the tribological performances of materials. The representative strain of the material in this test increases with the attack angle β of the indenter and so for a conical indenter increases as its apical angle 2θ decreases. But the mechanical analysis of this test by analytic models is very intricate. First we perform a preliminary discussion of the various aspects of the problem by considering the plane strain scratching of materials by wedges. After we present the conditions of the numerical simulations of the scratch test with conical indenters with a three-dimensional (3D) finite element code. These simulations provide the scratch geometry (contact surface, elastic recovery), the plastic strain map and the volume average plastic strain, the scratch hardness and the force ratio, the apparent friction coefficient μ₀=F_t/W. So we compare the behaviour of polymeric and metallic materials in scratch test at low and large strain and relate their difference in scratching resistance to their rheological properties. Polymers develop more higher elastic strains than metals a phenomenon which is characterised at low strain by the yield stress to Young's modulus ratio ε_e=σ_y/E. For θ=70.3° where pure ploughing occurs we study the scratching of elastic perfectly plastic solids with various values of ε_e under zero friction. Some comparisons with the behaviour in indentation are performed and we study the influence of friction in the scratching of workhardened steel with the same cone. At high strain the main rheological difference is the workhardening behaviour: it is described by a power law for metals and an exponential law for polymers. For θ decreasing from 70.3 to 20° we compare the behaviour of a cold worked steel to the behavour of polycarbonate, a thermoplastic polymer: a transition from ploughing to ploughing–cutting occurs only for steel.     

Recent developments and new strategies in scanning electron microscopy*

In addition to improvements in lateral resolution in scanning electron microscopy, recent developments of interest here concern extension of the incident beam energy, E₀, over two decades, from ≈ 20 keV to ≈ 0.1–0.5 keV and the possibility of changing the take-off emission, α, of detected secondary electrons. These two degrees of freedom for image acquisition permit a series of images of the same field of view of a specimen to be obtained, each image of the series differing from the others in some aspect. The origins of these differences are explored in detail and they are tentatively interpreted in terms of the change in the secondary electron emission yield δ vs. E₀, δ = f(E₀), and also of the change in δ vs. α, ∂δ/∂α. Various origins for the chemical contrast and topographic contrast have been identified. Illustrated by correlating a secondary electron image and a backscattered electron image, use of the scatter diagram technique facilitates image comparison. The difference between the lateral resolution and the size of the minimum detectable detail is outlined to avoid possible errors in nanometrology. Some aspects related to charging are also considered and possible causes of contrast reversal are suggested. Finally, the suggested strategy consists of the acquisition of various images of a given specimen by changing one parameter: primary beam energy and take-off angle for conductive specimens; working distance or beam intensity for high-resolution experiments; scanning frequency for insulating specimens.     

Influence of Poisson's ratio variation on lateral spring constant of atomic force microscopy cantilevers

Atomic force microscopy (AFM) can be used to measure the surface morphologies and the mechanical properties of nanostructures. The force acting on the AFM cantilever can be obtained by multiplying the spring constant of AFM cantilever and the corresponding deformation. To improve the accuracy of force experiments, the spring constant of AFM cantilever must be calibrated carefully. Many methods, such as theoretical equations, the finite element method, and the use of reference cantilever, were reported to obtain the spring constant of AFM cantilevers. For the cantilever made of single crystal, the Poisson's ratio varies with different cantilever-crystal angles. In this paper, the influences of Poisson's ratio variation on the lateral spring constant and axial spring constant of rectangular and V-shaped AFM cantilevers, with different tilt angles and normal forces, were investigated by the finite element analysis. When the cantilever's tilt angle is 20 degrees and the Poisson's ratio varies from 0.02 to 0.4, the finite element results show that the lateral spring constants decrease 11.75% for the rectangular cantilever with 1muN landing force and decrease 18.60% for the V-shaped cantilever with 50nN landing force, respectively. The influence of Poisson's ratio variation on axial spring constant is less than 3% for both rectangular and V-shaped cantilevers. As the tilt angle increases, the axial spring constants for rectangular and V-shaped cantilevers decrease substantially. The results obtained can be used to improve the accuracy of the lateral force measurement when using atomic force microscopy.