Anticlastic behavior of flat plates

The purpose of this paper is to show that the readings from strain gages can be used effectively to compute small transverse deflections in a rectangular plate and, further, show that the theory developed by Lamb for the rectangular-plate problem agrees with experiment. A numerical procedure is developed, based on the trapezoidal rule, which determines the transverse deflections from the readings of strain gages mounted to the top and bottom surface of a rectangular plate subjected to large longitudinal curvatures. It is shown using the strain-gage technique that experiment agrees with Lamb's theory forb ² /Rt ratios up to 50.     

Anticlastic action of flat sheets in bending

A theoretical analysis is made of the stresses induced in thin elastic strips which are bent (by uniform bending moments) into arcs with substantially constant radii of curvatures. Stresses determined experimentally on aluminum and steel strips agree very well with the theory.     

Finite Anticlastic Bending of Hyperelastic Solids and Beams

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.     

Creep and anelasticity in the springback of aluminum

Draw-bend tests, devised to measure springback in previous work, revealed that the specimen shapes for aluminum alloys can continue to change for long periods following forming and unloading. Steels tested under identical conditions showed no such time-dependent springback. In order to quantify the effect and infer its basis, four aluminum alloys, 2008-T4, 5182-O, 6022-T4 and 6111-T4, were draw-bend tested under conditions promoting the time-dependent response (small tool radius and low sheet tension). Detailed measurements were made over 15 months following forming, after which the shape changes were difficult to separate from experimental scatter. Earlier tests were re-measured up to 7 years following forming. The shape changes are generally proportional to log(time) up to a few months, after which the kinetics become slower. In order to understand the basis of the phenomenon, two models were considered: residual stress-driven creep, and anelastic deformation. In the first case, creep properties of 6022-T4 were measured and used to simulate creep-based time-dependent springback. Qualitative agreement was obtained using a crude finite element model. For the second possibility, novel anelasticity tests following reverse-path loading were performed for 6022-T4 and drawing-quality silicon-killed (DQSK) steel. Based on the experiments and simulations, it appears that anelasticity is unlikely to play a large role in long-term time-dependent springback of aluminum alloys.     

Simulation of springback: Through-thickness integration

The number of through-thickness integration points (N_IP) required for accurate springback analysis following sheet forming simulation using shell elements is a subject of confusion and controversy. Li and Wagoner recommended, in 1999, based on a finite element analysis (FEA) of draw-bending springback, the use of 25 integration points (IP), with up to 51 IP required to ensure accuracies of 1%. Several researchers have since reported that N_IP between 5 and 11 are adequate, or even that 7 or 9 IP are optimal, with reduced accuracy for more IP. These apparent contradictions are addressed with an analytical model of elasto-plastic bending under tension, followed by elastic springback. The fractional error in the evaluated bending moment, which is equal to the fractional error in springback, was determined by comparing three numerical integration schemes, with various N_IP, to the closed-form result. The results illustrate the oscillatory nature of numerical integration error with small parametric changes, such that fortuitous agreement can be obtained in isolated simulations where the number of integration points is inadequate. The concept of an assured error limit is introduced as well as a maximum error limit (for a range of generally unknown sheet tensions). The assured error limit varies with the integration scheme, N_IP, bending ratio (R/t), and sheet tension. Guidelines for the number of integration points required for given error tolerances are reported to allow practitioners to choose numerical parameters appropriately.     

Elastic springback in plates and beams formed by bending

A plane-strain solution is presented to predict the springback of plates formed by bending. The solution assumes that the strains are small, that the material is an isotropichardening von Mises material, and either that the stress-strain diagrams for tension and compression are identical or that an average diagram can be employed. The plastic portion of each stress-strain diagram is approximated by a finite number of straight lines. Both analytical solutions and experimental evidence indicate that the much simpler plane-stress solution can be used to predict springback for both beams and plates. Experimental data confirm that the type of die has a pronounced effect on the springback for beams and plates if the material exhibits a yield point.     

The bending moment and springback in pure bending of anisotropic sheets

Finite deformation rigid plastic and elastic–plastic analyses of plane strain pure bending of a plastically anisotropic sheet is presented. An efficient method for finding the exact solution is proposed by extending the previously developed method to the stage of unloading. Using this method the solutions are obtained in closed form or reduced to a numerical treatment of ordinary integrals, or an ordinary differential equation, or transcendental equations. An effect of plastic anisotropy and elastic properties on the bending moment is analyzed. The distribution of residual stresses is illustrated and an effect of material and process parameters on springback is investigated.     

Analytical springback model for lightweight hexagonal close-packed sheet metal

Experiments have shown that magnesium alloy sheet a common hexagonal close-packed metal, exhibits mechanical behavior unlike that of sheets made of cubic metals (X.Y. Lou et al., 2007, Int. J. Plasticity, 24, 44). The unique stress–strain response includes a strong asymmetry in the initial yield and subsequent plastic hardening. In other words, the stress–strain curves in tension and compression are significantly different. A proper representation of the constitutive relationships is crucial for the accurate evaluation of springback, which occurs due to the residual moment distribution through the sheet thickness after bending. In this paper, we propose an analytical model for asymmetric elasto-plastic bending under tension followed by elastic unloading in order to evaluate the bending moment, which is equivalent to the springback amount. To simplify the calculations, the experimentally measured stress–strain curve of the magnesium alloy sheet was approximated with discrete linear hardening in each deformation region, and the material properties were characterized according to several simplifying assumptions. The bending moment was calculated analytically using the approximate asymmetric stress–strain relationship up to the prescribed curvature corresponding to the radius of the tool in sheet metal forming operations. A numerical example showed an unusual springback increase, even with an increase in the applied force; this is an unexpected result for conventional symmetric materials. We also compared the calculated springback amounts with the results of physical measurements. This showed that the proposed model predicts the main trends of the springback in magnesium alloy sheets reasonably well considering the simplicity of the analytical approach.     

On the modeling and simulation of induced anisotropy in polycrystalline metals with application to springback

The presence of initial, and the development of induced, anisotropic elastic and inelastic material behavior in polycrystalline metals, can be traced back to the influence of texture and dislocation substructural development on this behavior. As it turns out, via homogenization or other means, one can formulate effective models for such structure and its effect on the macroscopic material behavior with the help of the concept of evolving structure tensors. From the constitutive point of view, these quantities determine the material symmetry properties. Most importantly, all dependent constitutive fields (e.g., stress) are by definition isotropic functions of the independent constitutive variables, which include these evolving structure tensors. The evolution of these tensors during loading results in an evolution of the anisotropy of the material. From an algorithmic point of view, the current approach leads to constitutive models which are quite amenable to numerical implementation. To demonstrate the applicability of the resulting constitutive formulation, we apply it to the case of metal plasticity with combined hardening involving both deformation- and permanently induced anisotropy. Comparison of simulation results based on this model for the bending tension of aluminum-alloy sheet-metal strips with corresponding experimental ones show good agreement.     

Isolating curvature effects in computing wall-bounded turbulent flows

An adjoint optimization method is utilized to design an inviscid outer wall shape required for a turbulent flow field solution of the So–Mellor convex curved wall experiment using the Navier–Stokes equations. The associated cost function is the desired pressure distribution on the inner wall. Using this optimized wall shape with a Navier–Stokes method, the abilities of various turbulence models to simulate the effects of curvature without the complicating factor of streamwise pressure gradient are evaluated. The one-equation Spalart–Allmaras (SA) turbulence model overpredicts eddy viscosity, and its boundary layer profiles are too full. A curvature-corrected version of this model improves results, which are sensitive to the choice of a particular constant. An explicit algebraic stress model does a reasonable job predicting this flow field. However, results can be slightly improved by modifying the assumption on anisotropy equilibrium in the model's derivation. The resulting curvature-corrected explicit algebraic stress model (EASM) possesses no heuristic functions or additional constants. It slightly lowers the computed skin friction coefficient and the turbulent stress levels for this case, in better agreement with experiment. The effect on computed velocity profiles is minimal.     

Plastic deformation and springback of rectangular bars subject to combined torsion and tension

Springback of rectangular bars under combined torsion and tension is investigated. A theoretical model for springback is developed and evaluated by comparing calculated and experimental results. It is concluded that springback is analytically predictable. Both analysis and test data show that an axial tension always reduces angular springback in a twisted bar. The order of plastic deformation is found to be important : twist followed by pull produces smaller angular springback upon release of torque and force than does a deformation in the reverse order.     

Vorticity and curvature at a stream surface

If S is a stream surface in a flow, we show the relationship between the three components of the vorticity field on S and the curvatures of the streamlines (geodesic torsion, normal curvature and geodesic curvature).     

Temperature invariance and curvature of slowness surfaces

The suggestion, made in [1], that hyperbolic regions of slowness and wave surfaces might afford the means whereby orientations for which the transit times of waves are invariant is further examined.General principles of time invariance, consistency of slowness and of energy flux are developed.At linear approximation in increments of slowness and of temperature, invariance of wave travel-time is shown to depend on the existence of real intersections of the slowness surface and another associated surface of the same degree. In the case of elastic waves, the two surfaces are, in general, sextic.In hexagonal media, the sextic for the slowness of elastic waves is resolvable into quadratic and quartic factors and this decomposition permits the problem to be reduced to the consideration of two quartic plane curves. The possibility of intersection of these curves is discussed in detail for some zinc-like materials.While the mathematical validity of the original suggestion is vindicated, the normal observed magnitudes of the temperature dependences of the stiffnesses, compared with the values of thermal expansion coefficients, are such that their effect outweighs the counter-balancing contribution from negative curvature in the equations. Some anomaly in temperature dependence appears essential for the existence of the intersections required for temperature invariance; however, the strength of such anomaly may be abated if the elastic stiffness tensor occasions hyperbolic regions in the slowness surface.     

PIV through moving shocks with refracting curvature

Particle image velocimetry (PIV) is applied to moving millimeter shock waves whose density jump and small radii of curvature make refraction significant. The motion of the shock front is also much larger than the motion of the corresponding mass at the front. A Lagrangian model of particle displacement in response to a moving shock is developed to investigate the relationship between particle displacements and the actual mass velocity behind the shock. Errors in PIV measurements due to light refraction across a curved, moving shock are investigated in terms of both position and velocity errors using a refraction model developed from geometrical optics. The model is experimentally validated and applied to 1-D slices of data extracted from PIV vector fields, and the resulting measurement errors are quantified.     

Incompatible deformation field and Riemann curvature tensor

Compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kr¨oner and Le-Stumpf are provided.     

Wave propagation in circular cylindrical shells with periodic axial curvature

The propagation of longitudinal and flexural waves in axisymmetric circular cylindrical shells with periodic circular axial curvature is studied using a finite element method previously developed by the authors. Of primary interest is the coupling of these wave modes due to the periodic axial curvature which results in the generation of two types of stop bands not present in straight circular cylinders. The first type is related to the periodic spacing and occurs independently for longitudinal and flexural wave modes without coupling. However, the second type is caused by longitudinal and flexural wave mode coupling due to the axial curvature. A parametric study is conducted where the effects of cylinder radius, degree of axial curvature, and periodic spacing on wave propagation characteristics are investigated. It is shown that even a small degree of periodic axial curvature results in significant stop bands associated with wave mode coupling. These stop bands are broad and conceivably could be tuned to a specific frequency range by judicious choice of the shell parameters. Forced harmonic analyses performed on finite periodic structures show that strong attenuation of longitudinal and flexural motion occurs in the frequency ranges associated with the stop bands of the infinite periodic structure.