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.     

Three-parameter approach for elastic–plastic fracture of the semi-elliptical surface crack under tension

Systematic three-dimensional elastic–plastic finite element analyses are carried out for a semi-elliptical surface crack in plates under tension. Various aspect ratios (a/c) of three-dimensional fields are analyzed near the semi-elliptical surface crack front. It is shown that the developed J–Q annulus can effectively describe the influence of the in-plane stress parameters as the radial distances (r/(J/σ₀)) are relatively small, while the approach can hardly characterize it very well with the increase of r/(J/σ₀) and strain hardening exponent n. In order to characterize the important stress parameters well, such as the equivalent stress σ_e, the hydrostatic stress σ_m and the stress triaxiality R_σ, the three-parameter J–Q_T–T_z approach is proposed based on the numerical analysis as well as a critical discussion on the previous studies. By introducing the out-of-plane stress constraint factor T_z and the Q_T term, which is determined by matching the finite element analysis results, the J–Q_T–T_z solution can predict the corresponding three-dimensional stress state parameters and the equivalent strain effectively in the whole plastic zone. Furthermore, it is exciting to find that the values of J-integral are independent of n under small-scale yielding condition when the stress-free boundary conditions at the side and back surfaces of the plate have negligible effect on the stress state along the crack front, and the normalized J tends to a same value when φ equals about 31.5° for different a/c and n. Finally, the empirical formula of T_z and the stress components are provided to predict the stress state parameters effectively.     

A two-level optimization procedure for material characterization of composites using two symmetric angle-ply beams

A two-level optimization procedure for determining elastic constants E₁, E₂, G₁₂, and ν₁₂ of laminated composite materials using measured axial and lateral strains of two symmetric angle-ply beams with different fiber angles subjected to three-point-bending testing is presented. In the first-level optimization process, the theoretically and experimentally predicted axial and lateral strains of a [(45°/−45°)₆]_s beam are used to construct the strain discrepancy function which is a measure of the sum of the squared differences between the experimental and theoretical predictions of the axial and lateral strains. The identification of the material constants is then formulated as a constrained minimization problem in which the best estimates of shear modulus and Poisson's ratio of the beam are determined to make the strain discrepancy function a global minimum. In the second-level optimization process, shear modulus and Poisson's ratio determined in the first level of optimization are kept constant and Young's moduli of the second angle-ply beam with fiber angles different from 45° are identified by minimizing the strain discrepancy function established at this level of optimization. The suitability of the proposed procedure for material characterization of composite materials has been demonstrated by means of a number of examples.     

Micro-Raman depth analysis of residual stress in machined germanium

Raman spectroscopy has proved to be a useful nondestructive technique for measuring residual stresses in semicondutors. The Raman microprobe is used to investigate the effects of machine parameters on residual stresses in single point diamond turned germanium (Ge). A profiling technique that provides a method of obtaining the residual stress information as a function of depth with depth resolutions of 10.0 nm is discussed. This method is used to analyze the asymmetrically broadened and shifted spectral features in the machined samples. Residual stresses are sampled across ductile feed cuts in (100) Ge wafers, which were single point diamond turned using various feed rates (12.5, 25 nm/rev), rake angles (0°, −10°, −30°), and clearance angles (6°, 16°). In general a region of plastically deformed material that shows slight compressive stresses exist near the surface of the diamond turned sample. The compressive surface stress increases to a maximum at a depth of ≈ 50 nm beneath the surface at which point the stress rapidly changes sign. The rapid sign change is indicative of the transition from plastic to elastic deformation. Deeper probe regions exhibit increasing tensile stresses, which reach a maximum and then relax to zero at greater depths in the sample. A related study of the stress field occurring around Vicker's hardness indents provides a link between theoretical and experimental stress profiles and demonstrates the accuracy of the micro-Raman profiling technique.     

Effect of cyclic loading on the yield surface evolution of 18G2A low-alloy steel

Experimental analysis is presented of the plastic properties of 18G2A steel (notation according to Polish Standards) in the as-received state, and of the same material subjected to cyclic predeformation in different directions of the two-dimensional stress space (σ_xx, τ_xy). The analysis was made by studying the position in stress space and by determination of typical dimensions of the yield surfaces. The initial yield surface has been determined using a number of specimens which were loaded up to the plastic range along different stress directions, and this surface was used as the starting point for comparative studies of yield surfaces of the cyclic prestrained material. Cyclic predeformations were induced by loading at ambient temperature. After predeformation, yield surfaces were determined by the technique of sequential probes of the single specimen. The anisotropic yield condition due to Szczepiński was shown to model the experimental results well. Prior cyclic loading induced the softening effect observed during subsequent monotonic loading of the steel in the plastic strain range considered.     

Numerical simulation of the localization behavior of hydrostatic-stress-sensitive metals

The present paper deals with the numerical simulation of the elastic–plastic deformation and localization behavior of solids which are plastically dilatant and sensitive to hydrostatic stresses. The model is based on a generalized macroscopic theory taking into account macroscopic as well as microscopic experimental data obtained from tests with iron-based metals. It shows that hydrostatic components may have a significant effect on the onset of localization and the associated deformation modes, and that they generally lead to a notable decrease in ductility. The continuum formulation relies on a generalized I₁–J₂–J₃ yield criterion to describe the effect of the hydrostatic stress on the plastic flow properties of metals. In contrast to classical theories of metal plasticity, the evolution of the plastic part of the strain rate tensor is determined by a non-associated flow rule based on a plastic potential function which is expressed in terms of stress invariants and kinematic parameters. Numerical simulations of the elastic–plastic deformation behavior of hydrostatic-stress-sensitive metals show the physical effects of the model parameters and also demonstrate the efficiency of the formulation. Their results are in excellent agreement with available experimental data. A variety of large-strain elastic–plastic problems involving pronounced localizations is presented, and the influence of various model parameters on the deformation and localization behavior of hydrostatic-stress-sensitive metals is discussed.     

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.     

The stability of a vertical cut

Coulomb's problem of the depth to which a trench can be dug in cohesive soil without the sides falling in is investigated. In the present paper the soil is assumed to be perfectly plastic with yield stress c in shear and unit weight γ; internal friction is ignored. Thus the problem will be treated as one of plane strain with the maximum shear stress criterion of failure. Coulomb's own expression for the depth of cut was the upper-bound value 4c/γ (actually 4(c/γ) cot ς, allowing for internal friction, where and ø is the angle of friction). The best upper bound in the literature is 3·83c/γ, given by a slip-circle analysis; the highest lower bound seems to be 2c/γ. Simple distributions of stress which satisfy both equilibrium and the yield condition for the whole field are shown to give a lower bound of 2√2(c/γ). A partial solution is established for 3·2c/γ.     

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.     

Elastic–plastic stress analysis of an orthotropic rotating disc

In this study, the elastic–plastic stress analysis of a curvilinearly orthotropic rotating annular disc is investigated analytically for strain-hardening material behavior. To be able to see the separation of the plastic region, a few angular velocities are taken into consideration for such an analysis. Radial and circumferential stress components are obtained to increase angular velocity. It is seen that the magnitudes of the circumferential stress components are higher than those of the radial stress components. The magnitudes of the residual stress component of the circumferential stress and plastic flow are the highest at the inner surface. The radial displacements in both the elastic and plastic solutions calculated analytically have higher values at the inner surface than those of the outer surface for all the angular velocities.     

Lattice distortions in GaN on sapphire using the CBED–HOLZ technique

The convergent beam electron diffraction (CBED) methodology was developed to investigate the lattice distortions in wurtzite gallium nitride (GaN) from a single zone-axis pattern. The methodology enabled quantitative measurements of lattice distortions (α, β, γ and c) in transmission electron microscope (TEM) specimens of a GaN film grown on (0, 0, 0, 1) sapphire by metal-organic vapour-phase epitaxy. The CBED patterns were obtained at different distances from the GaN/sapphire interface. The results show that GaN is triclinic above the interface with an increased lattice parameter c. At 0.85 μm from the interface, α=90°, β=8905° and γ=11966°. The GaN lattice relaxes steadily back to hexagonal further away from the sapphire substrate. The GaN distortions are mainly confined to the initial stages of growth involving the growth and the coalescence of 3D GaN islands.     

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.     

Strain and Stress Fields During Scratch Tests on Amorphous Polymers: Influence of the Local Friction

In this paper, we present results deduced from three-dimensional finite element simulations of scratching, with spherical indenter geometry at different imposed ratios, a/R in the range of 0.1–0.9. For each simulated ratio a/R, the local friction has been increased from 0 to 1. The paper aims at studying the tangential scratch behaviour of homogeneous polymeric substrates, considered in first approximation as elastic linear-hardening plastic material. For only elastic–plastic contacts, without any strain rate or temperature effects, it focuses on studying some characteristic response due to spherical scratching process as a function of scratching conditions (a/R, μ _loc ) such as the stress and plastic strain fields, including the plastic zone dimension and the definition of an volume average plastic strain.     

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.     

A universal slip-line model with non-unique solutions for machining with curled chip formation and a restricted contact tool

A universal slip-line model and the corresponding hodograph for two-dimensional machining which can account for chip curl and chip back-flow when machining with a restricted contact tool are presented in this paper. Six major slip-line models previously developed for machining are briefly reviewed. It is shown that all the six models are special cases of the universal slip-line model presented in this paper. Dewhurst and Collins's matrix technique for numerically solving slip-line problems is employed in the mathematical modeling of the universal slip-line field. A key equation is given to determine the shape of the initial slip-line. A non-unique solution for machining processes when using restricted contact tools is obtained. The influence of four major input parameters, i.e. (a) hydrostatic pressure (P_A) at a point on the intersection line of the shear plane and the work surface to be machined; (b) ratio of the frictional shear stress on the tool rake face to the material shear yield stress (τ/k); (c) ratio of the undeformed chip thickness to the length of the tool land (t₁/h); and (d) tool primary rake angle (γ₁), upon five major output parameters, i.e. (a) four slip-line field angles (θ, η₁, η₂, ψ); (b) non-dimensionalized cutting forces (F_c/kt₁w and F_t/kt₁w); (c) chip thickness (t₂); (d) chip up-curl radius (R_u); and (e) chip back-flow angle (η_b), is theoretically established. The issue of the “built-up-edge” produced under certain conditions in machining processes is also studied. It is hoped that the research work of this paper will help in the understanding of the nature and the basic characteristics of machining processes.     

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°.     

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.     

Forces developed during the ricochet of projectiles of spherical and other shapes

The oblique impact of spherical, dumb-bell shaped and hemispherical-end cylindrical projectiles against a modelling clay (Plasticine) has been studied and experimental displacement-time curves plotted for both ricocheting and penetrating projectiles. It is shown that the forces of resistance are proportional to the square of the instantaneous velocity. For ricocheting projectiles at each impact angle θ₀ investigated, the coefficient of proportionality, k, was found to be independent of the initial velocity. For penetrating projectiles and for θ₀ < 30° (test range), k is independent of both θ₀ and initial speed v₀.     

A fracture mechanics life prediction methodology applied to dovetail fretting

This work evaluates a fracture mechanics based crack growth life prediction methodology for dovetail fretting fatigue laboratory experiments. The Ti–6Al–4V specimens were configured with angles of 35°, 45° and 55°. Experiments were conducted with constant amplitude loading at R of 0.1 and 0.5 with lives ranging from 100,000 to 10 million cycles. The approach included the contact loads and bulk stress calculated from the finite element method as inputs to the stress and life analysis. Contact stresses were calculated using the contact stress analysis software CAPRI. These stresses were input into a stress intensity factor calculation at the edge of contact. Crack propagation life was calculated from an assumed initial crack size. Analysis showed that propagation consumes a majority of the total life and is insensitive to a large range of initial crack sizes.     

A new weighting function for solving nonlinear oscillation problems

A new approximation technique is presented for solving nonlinear oscillation problems. A solution of form x = x₀ sin ωt is assumed and put into the differential equation, which leads to an algebraic equation which can be solved for ω in terms of x₀. The new technique is to reduce the algebraic equation to an identity at ωt = π/3. This method is compared with other common approximation techniques and with the exact solution for several nonlinear problems. Overall, the new method works better than any other approximation technique. In addition, it has the very practical advantage of being purely algebraic; no integrals need be evaluated, as in several other methods.