Anisotropic Sheet Research Articles

One approach to determination of the stresses in an anisotropic sheet material with a crack

One approach to determination of the stresses in an anisotropic sheet material with a crack

Reflecting Rubber Characteristics of Anisotropic Sheets and Measurement of Complex Permittivity Tensor

Complex values of the permittivity tensor of robber sheets which are manufactured by the rolling process were estimated by the least-squares method using the reflection coefficient measured for normal incidence. First, the frequency characteristics of the reflection coefficients of a rubber sheet backed by thin aluminum were measured from 8.5 to 10.5 GHz by changing the rolling angle relative to the incident electric field. Next, the elements of the permittivity tensor were determined by the least-squares method from many data of measured reflection coefficients of the samples. Five kinds of rubber sheets containing carbon particles or fibers were selected, and circular pieces 30 cm in diameter were measured by this method. The complex permittivity tensors including off-diagonal elements were thus obtained, and the principal directions of the tensor were calculated from the measured permittivity tensor. The following facts were found through this analysis: The permittivity element in the rolling direction is about five times larger and the off-diagonal elements are small compared with the diagonal elements. The principal direction of the real part is different from that of the imaginary part for a certain kind of rubber sheet mixed with carbon particles.

Babinet's Principle for an Anisotropic Resistive Surface Using Different Approaches

Extension of Babinet's principle in electromagnetic theory to an inhomogeneous anisotropic (with non-zero off-diagonal terms) resistive sheet is made. Different methods are employed for this purpose so that a better understanding can be gained. Some basic properties of electromagnetic waves, such as their odd and even symmetry, are also reviewed for clarity of presentation. Some errors in previously published results are also corrected.

Assessment of Work-Hardening Characteristics and Limit Strains of Anisotropic Aluminum Sheets in Biaxial Stretching

In this work, commercially pure aluminum sheets in both the as-received and annealed conditions are tested in uniaxial and biaxial tension. Biaxial stretching is performed in dies giving different degrees of stress biaxiality. Resulting effective stresses and plastic strains are estimated according to the original Hill’s theory in the two situations where planar anisotropy is either neglected or taken into consideration. In both situations discrepancies between biaxial and uniaxial flow curves are observed. By analyzing the above uniaxial and biaxial test results according to the flow rule associated with a yield function recently proposed by Hill, a new material index describing the anisotropic behavior has been evaluated. This new material behavior description realized a satisfactory agreement between work-hardening characteristics of the tested aluminum sheets under various biaxial stress systems. The same tested aluminum sheet materials have been then tested in order to determine their forming limit curves. Correlation between these curves and the theoretical predictions of limit strains according to various instability analyses, is sought through the use of the above description of material work-hardening.

An analytical study of the Kelvin–Helmholtz instabilities of compressible, magnetized, anisotropic, and isotropic tangential velocity discontinuities

An analytical study is made of the linear stability of tangential velocity discontinuities in anisotropic and isotropic compressible, magnetized plasmas. A new analytical technique leads to entirely new results for both cases, and recovers previously obtained numerical results for the isotropic case. For both the anisotropic and isotropic vortex sheets, the structure of unstable, standing wave modes is mapped out in the (inverse plasma beta, Mach number) plane for modes at arbitrary angles to the flow and the magnetic field. Solutions are obtained for unstable standing and traveling modes of the anisotropic vortex sheet propagating parallel or antiparallel to the magnetic field, and at small angles to these directions. Numerically computed unstable standing modes obtained previously for the isotropic case are recovered analytically and extended to more general configurations. Also, it is shown that modes propagating along or opposite to the magnetic field direction, and at a general angle to the flow, do not exhibit standing wave instability for (plasma beta)≤1, for the isotropic vortex sheet.

A rigid-plastic finite-element formulation for the analysis of general deformation of planar anisotropic sheet metals and its applications

General three-dimensional deformation of a planar anisotropic rigid-plastic sheet metal which obeys Hill's quadratic yield function is considered in the present analysis. The explicit expressions of the yield criterion and the related equations are given in the surface curvilinear co-ordinate system. A convective co-ordinate system is used in order to take into account the effect of geometric change during one step in the incremental analysis. The expression of the effective strain increment during one step is obtained in closed form by introducing assumptions on the deformation path and by proper consideration of the rotation of the axes of anisotropy during deformation. Considering the equilibrium at the deformed state, a variational formulation is derived to determine the deformation during a step. The corresponding finite-element equations are found in order to analyze the general deformation of planar anisotropic sheet metals. Two computational examples are chosen and computed by using the developed FEM program in order to verify the present formulation. The flange deformation in deep drawing of a circular diaphragm of planar anisotropy is analyzed and compared with the existing solution. The effects of planar anisotropy in the circular diaphragm bulge test are also investigated.

Simulation of the collisionless tearing instability in an anisotropic neutral sheet

A two‐dimensional magnetoinductive particle‐in‐cell code is used to compare the stability of isotropic and anisotropic current layers in a collisionless plasma. We find, in agreement with recent theories, that temperature anisotropy with T⊥ > T∥ significantly enhances the instability. The simulation growth rates agree well with analytical predictions of growth rates in the linear phase. Nonlinear growth rates are significantly larger than linear growth rates and are apparently the result of coalescence instability. For a fixed system length the influence of coalescence on the growth rates increases with anisotropy. This is evidently because of a shift in the most unstable mode to shorter wavelength with increasing anisotropy. Thus anisotropic current layers tend to produce large amplitude small wavelength islands which rapidly merge.

The combined effect of stress state and grain size on hydrogen embrittlement of titanium

Abstract : The influence of grain size on the hydrogen embrittlement of titanium sheets is shown to depend on stress state. Results are presented which indicate that the deleterious effect of a large grain size on the susceptibility to hydrogen embrittlement is more severe in equibiaxial tension than in uniaxial tension. This effect may be understood in terms of the sensitivity of both void nucleation (due to hydride fracture) and void link-up due to the combined effects of large grain/hydride fracture) and void link-up due to the combined effects of large gain/hydride sizes and the large multidirectional, normal stresses that develop during deformation of plastically anisotropic sheets in equibiaxial tension. (Author)

Kelvin–Helmholtz instabilities of high-velocity magnetized, anisotropic shear layers

A study is made of the linear stability of finite-thickness, anisotropic compressible shear layers v=ẑvz(x), with a parallel uniform magnetic field, B=ẑB0. The stability of such sheared flows, described by the double adiabatic equations of Chew, Goldberger, and Low [Proc. R. Soc. London Ser. A 236, 112 (1956)], and involving the nonlocal coupling of the firehose and mirror modes caused by the velocity shear, is relevant in a number of astrophysical, geophysical, and space plasma configurations. The scalar perturbation quantities have the form f (x) exp[i(kzz −ωt)]. The dimensionless variables characterizing a shear layer with a given velocity profile, assumed to be a linear profile in the present work, are the sonic Mach number, M ≡ (2vzm/S⊥), the ratio of the magnetic field energy density to the perpendicular thermal energy density, q2 ≡ (vA/S⊥)2, and the anisotropy parameter, r2 ≡ (S∥/S⊥)2. Here, vz(x=±∞)=±vzm, S∥, and S⊥ are the sound speeds parallel and perpendicular to the magnetic field, and vA is the Alfvén speed. The dimensionless variable characterizing the perturbation is the wavenumber B ≡ kzL, where L is the shear layer thickness. It is shown that the resonance of the sound and firehose modes drives unstable standing and traveling waves for a shear layer having a vortex sheet profile [where vz(x) is a step function]. For the vortex sheet, the unstable standing wave modes first appear at M=2(3)1/2r for r>(1+q2)1/2/2, and the unstable traveling wave modes first appear at M=0 for r<((1+q2)1/2/2). Numerical methods are used to generate values of ωr and ωi>0 (corresponding to unstable wave motion) for the ‘‘linear’’ shear layer in the (B, M) plane for various values of q and r. The coverage of the (B, M) plane is for B≤5, M≤10, and for discrete values of q≤0.5 and r≤2. Two regimes of instability are found to occur in the (B, M) plane with the structure of the unstable modes of the ‘‘linear’’ layer being very different from that for the anisotropic vortex sheet. The unstable modes are standing waves with ωr=0, and traveling waves with ωr≠0.

Prediction of the forming limit diagrams of anisotropic sheets in linear and non-linear loading

The formability of sheet metals is often evaluated from strain analysis using the concept of forming limit diagrams (FLDs). Numerous experimental results show that the formability of the material strongly depends on the loading history and on the plastic anisotropy of the rolled sheet. The limiting strains at the onset of localized necking can be predicted from a plastic instability analysis introduced by Marciniak and coworkers. However, the hypothesis of proportional loading is no longer valid for complex industrial stampings. Multistage forming operations often involve non-linear strain paths, and abrupt changes in the strain ratio can be observed. In the present work the occurrence of plastic instabilities in the general case of a rate-sensitive anisotropic material with orthotropic symmetry under non-proportional loading is analysed. Theoretical FLDs are determined for proportional loading, for a sequence of two proportional loadings, for a sequence of two proportional loadings and for non-proportional loading (curved strain paths). The simulations are carried out for a wide range of loading conditions, from uniaxial tension in the rolling direction to uniaxial tension in the transverse direction through biaxial stretching. The influence of the anisotropic behaviour of the sheet is studied using Hill's theory of plastic anisotropy for both linear and non-linear strain paths in terms of the anisotropic coefficient measured during uniaxial tensile stretching in three different directions referred to the rolling direction (r o, r 45 and r 90). A good correlation is obtained between the theoretically and experimentally determined FLDs, and it is shown that important increases in formability may be achieved through careful specification of strain paths, blank-holder pressure and anisotropic blank orientation.

Examination of Hill's latest yield criterion using experimental data for various anisotropic sheet metals

Using experimental data published earlier [Vial et al., Int. J. Mech. Sci. 25, 899 (1983)], values of the exponent m in the Hill (1979) yield criterion [Hill, Math. Proc. Cambridge Phil. Soc. 85, 179 (1979)] are calculated for each of the four special cases suggested. With these findings, stress-strain relations for plane-strain compression are derived and predictions using these derived equations are compared with experimental results. Comparisons between prediction and experiment are reasonable in all cases and it is suggested that the discrepancies could arise because of the assumption of planar isotropy (via the use of an average r- value ) and because the exponent m apparently varies with induced strain.

A generalized quadratic flow law for sheet metals

A planar quadratic flow law is proposed for anisotropic sheet materials. This law is similar to the anisotropic strength criterion of Tsai and Wu. It has six experimentally determinable coefficients as compared to four in Hill’s flow law and, thus, allows more experimental information to be accommodated. However, the resulting strain increment vector, while unique, is not necessarily normal to the flow surface.

Integrated structural analysis system with on-line coordinate-measuring machines for car body panels

This paper deals with computer graphics applications for developing new types of steel sheets for car body panels, which would contribute to car milage improvement by weight reduction. This computer system in which structural analysis system is implemented, has been primary developed for CAD/CAM application to ROLL PASS design of grooved rolls with which rails, seamless pipes, beams and other steel sections are produced. On line 3-D coordinate measuring machines, graphic displays, a digitizer, a high speed/accuracy drafting machine and an on line ultrasonic thickness meter are the major devices of the computer graphics system of which mini-computers are linked to main frame computers where general purpose Finite Element programs, e.g. NASTRAN, MARC, are installed for structural analysis of pressed steel sheets for car body panels. The functions of this system could be categorized into the following three: 1. (1) Profile measurement and analysis of pressed steel sheets to visualize and quantify even minute unfavourable shape due to inadequate material properties and pressing conditions. 2. (2) An on line ultrasonic thickness meter built in the graphic system plays an important role for analysing plastic deformation behaviour by graphical representation of panel thickness data which is also valuable for geometric description of the structure. 3. (3) An integration of data preparation facility through extensive use of on line 3-D coordinate measuring machines and every variety for Finite Element programs in bigger computing resources are the key feature of this system. These computer analyses, of press formability, vibration behaviour and stiffness performance which would vary with material properties and pressing conditions could add invaluable information to research and development of both forming technique and new steel quality. This paper also contains a discussion on performance properties of newly developed steel sheets, i.e. anisotropic steel which shows high stiffness because of a virtual Young's modulus and non-uniform thickness steel sheets with slight convexo-convex sectional profile for reducing material volume without decreasing the stiffness. Computer Graphics and computer utilization in automobile industry have often been reported so far, from clay model measurement to car body design and die manufacturing. Compared with this situation, profile measurement and analyses for material development of steel sheets are still being done with dial gauges or offline position sensors where tedious data processing work is still required. The Computer Graphics system described in this paper features on-line 3-D coordinate measuring machines and an on-line ultrasonic thickness meter which enable one to do effective research work for improvement of press formability and performance of steel sheets. Panel stiffness and dynamic performance analyses by FEM are to be the major objectives of this paper. As the fruitful results derived from an extensive use of this integrated system anisotropic high Young's modulus steel sheets and intentionally slight-convexo-convex profiled (non flat) steel sheets have become into practical use for energy saving car body building.

Crack growth under static and fatigue loading in glassy polymers oriented by cold-rolling

Static and fatigue crack growth in PC and fatigue crack growth in PVC have been studied using anisotropic sheets oriented by cold-rolling. Tests were carried out at room temperature for samples with various degrees of rolling reduction. In static loading for PC, a slight rolling reduction considerably improves the resistance to crack propagation in the case where the crack grows perpendicularily to the rolling direction. The measured values of crack opening displacement were compared with the Dugdale model, taking into account the effect of anisotropy. In fatigue loading for both polymers used, a power law relationship between crack growth rate dc dN and stress intensity factor range ΔK, i.e., dc dN=A(ΔK) m where A and m are constants, covers most of the data in spite of the differences in degrees of anisotropy. However, the constants A and m are dependent on the degree of rolling reduction. In PVC, the rolling reduction changes the fatigue fracture mode from a discontinuous growth type to continuous one. All the results show that the rolling reduction has an important effect on crack behaviour.

On the elasto-plastic stress concentration at a circular hole in an anisotropic sheet

The classical problem of determining the stress concentration factor at a circular hole embedded in an infinite sheet subjected to remote uniform tension is investigated. A finite strain elasto-plastic deformation theory based on Hill's new anisotropic flow theory [7] is used. It is shown that the governing field equations can be reduced to a single first order differential equation from which the stress concentration factor is obtained by a standard numerical method. The solution covers the entire elasto-plastic range and is valid for any strain hardening function. Comparison with experimental results, for a few materials, shows good agreement. With a pure power hardening law and within the framework of small strain plasticity, our results agree with those obtained from a more general solution discovered by Budiansky [8].

Limit strain predictions for strain-rate sensitive anisotropic sheets

The combined effects of material strain-rate sensitivity and anisotropy on necking or “limit” strain predictions are examined for thin sheets with transversely isotropic properties. Various rate dependent constitutive laws based on flow theory and deformation theory of plasticity are considered. The strong effect of material strain-rate sensitivity in increasing the amount of straining prior to localized necking is first emphasized. We then discuss the joint influence of rate dependence and anisotropy on the theoretical limit strains and forming limit curves. Both strain-rate sensitivity and the local shape of the anisotropic yield surface are shown to significantly affect the predicted limit strains. A necking-band bifurcation analysis is also carried out to reveal in an explicit manner the remarkable sensitivity of overall forming limit diagram shapes to the parameters in the anisotropic yield function.

On the stress state dependence of the strain-hardening of anisotropic sheet steel

On the stress state dependence of the strain-hardening of anisotropic sheet steel

Stress concentration around holes in anisotropic sheets

The formulation of stress concentration problems of plane anisotropic elasticity in terms of integral equations is discussed. First the singular solutions of a concentrated force and a dislocation are formulated so that they remain valid in the case of double roots. The distribution of singularities on smooth curves is then treated, and general formulae for the stress discontinuities are given. Finally, the integral equations arising from stress boundary conditions are discussed, and the characteristics and numerical solution of one of the types are treated in detail.

Temperatures in an Anisotropic Sheet Containing an Insulated Elliptical Hole

Temperatures in an Anisotropic Sheet Containing an Insulated Elliptical Hole

A reappraisal of the wedge-drawing test

The wedge-drawing test is re-evaluated in the light of improved methods of lubrication. The test is used to study the radial drawing of anisotropic sheet materials and the experimental results correlate well with theoretical predictions based on the orthotropic theory of plasticity. In the theoretical analysis planar anisotropy is considered as well as normal anisotropy. The materials tested were soft and half-hard aluminium, soft 70/30 brass, killed steel and titanium. It is clearly demonstrated that the wedge-drawing test does not simulate the instability conditions existing in deep-drawing. While in deep-drawing increasing R- values can lead to large increases in limiting drawing ratio, this is not the case in wedge-drawing.