Through-thickness Shear Stresses Research Articles

Investigation of the effect of anisotropy on Generalized Forming Limit Diagram

In contemporary industrial practices, various methods are employed to subject raw metal sheets to deformation in order to fabricate requisite components. These sheets exhibit a defined capacity for deformation during the forming process. Over recent decades, a plethora of experimental and numerical methodologies have emerged to ascertain these forming limits. Initially, a forming limit diagram (FLD) was devised, predicated on the phenomenon of necking, under the assumption that forming takes place under plane-stress conditions. However, in certain complex processes like hydroforming and incremental forming, necking can manifest at sites where normal and through-thickness shear stresses act upon the sheet in addition to the in-plane stresses, rendering the plane-stress assumption inadequate for predicting forming limits in such scenarios. Thus, it becomes imperative to derive a diagram that can accurately forecast forming limits in these processes. This study aims to establish a Generalized Forming Limit Diagram (GFLD) through numerical means. GFLDs were constructed utilizing two distinct yield functions, namely Von-Mises and Hill48, for isotropic and anisotropic states, respectively. The findings reveal that normal compressive stress and through-thickness shear strain augment the formability of sheet metals. Furthermore, the outcomes illustrate that accounting for anisotropy introduces variances between diagrams in some regions of the FLD curve while the discrepancies are minor within the central regions.

Prediction of FLD for sheet metal by considering through-thickness shear stresses

In this study, the effect of through-thickness shear (TTS) stress has been examined on the prediction of forming limit diagrams (FLDs). Determination of the FLD is based on the Marciniak–Kuczynski (M–K) model with some modifications on the stress states for consideration of the TTS stress effects. For solving the equations, the Newton–Raphson method has been used. Furthermore, the Nakazima test has been simulated to investigate the stress state which occurs in the sheet during the test. Results showed that the formability of sheet metal could be better as the through-thickness stress increased. Also, implementation of TTS stress in the present model, the corresponding FLD has better agreement than the conventional FLD with the experimental test result.

Mechanism on increased sheet formability induced by tangential adhesive stress in sheet flexible forming process employing viscoplastic pressure-carrying medium

Abstract In flexible die forming processes employing semi-solid or elastic materials as pressure-carrying medium, there exist tangential adhesive stress or friction at medium/blank interface, which may improve sheet formability and has been experimentally verified. However, the mechanism behind this phenomenon has not been clarified yet. In order to fill this gap, analytical investigations were implemented in this paper based on previous studies involving through-thickness shear stress. Concretely, the analytical model of axisymmetric shell bulging process considering tangential adhesive stress and the extended M-K model under non-planar stress state were established at first. Accordingly, the relationship between tangential adhesive stress and through-thickness shear strain was deduced together with the relationship between through-thickness shear stress (strain) and sheet forming limit. On the basis of that, the viscous pressure forming (VPF) process was taken as an example to exhibit the characteristic that forming limit changes with tangential viscous adhesive stress and through-thickness shear stress (strain). It is shown that the tangential viscous adhesive stress has positive correlation with through-thickness shear strain which results in increased sheet formability. Tube bulging tests employing viscous medium with different molecular weight as pressure-carrying medium were conducted to give an experimental evidence of the proposed analytical results.

Effects of normal and through-thickness shear stresses on the forming limit curves of AA3104-H19 using advanced yield criteria

In the present study, the effects of normal and through-thickness shear (TTS) stresses are simultaneously considered to predict the generalized forming limit diagram (GFLD) of AA3104-H19 using Yld96 [1], Yld2004 [2] and Yld2011 [3] anisotropic yield criteria. For this purpose, considering the general stress state, including the normal and through-thickness shear stresses, the modified Marciniak–Kuczynski (M–K) model is employed. The Newton–Raphson numerical method is applied to solve the nonlinear equations set, and then the forming limit diagram (FLD) and forming limit stress diagrams (FLSD) are computed. First, the forming limit curves are presented by considering the through-thickness shear stress, and then the generalized forming limit curves are presented by applying the normal and through-thickness shear stresses simultaneously. The results show that the through-thickness shear stress has positive sensitivity on the limit strains. Also, the normal stress is more efficient than the through-thickness shear stress on the formability of the plate. Comparison between the results indicates that the predicted limit strains by the Yld2011 anisotropic yield function are lower than those predicted by Yld96.

Property-adjustable forming medium induced extension of sheet metal formability under variable magnetic field

Deformation states of sheet metals are determined by the in-plane stress distribution, which is closely related to forming load transmitted by forming medium. Mechanical properties of conventional forming medium, such as mineral oil, gas, and polyurethane, can hardly be controlled during the whole forming process. However, if the mechanical properties of forming medium can be freely altered, the stress state of sheet metal may become controllable. As a result, the magneto-rheological (MR) fluid was employed as a property-adjustable forming medium in this paper, which attempted to extend sheet formability in a controllable way. Effects of magnetic flux density (B) and iron particle content (φ) on yield stress of MR fluid in compression mode were analyzed at first. Magneto-rheological pressure bulging (MRPB) tests for Al1060-O sheet were conducted under different sets of parameters. Effects of B and φ on forming load were presented to show the adjustable mechanical properties of forming medium. Extended forming limit diagram (e-FLD) was proposed to describe the variation of sheet forming limit. The results indicate that sheet forming limit can be extended by monotonously increasing B in a certain range. The appropriate range of B is correlated to φ. The through-thickness shear stress caused by the transverse friction between MR fluid and sheet metal is considered to be responsible for this increase. The proposed process is demonstrated to be an efficient method to control the sheet formability, and more practical application may be expectable.

Higher-order beam model for stress predictions in curved beams made from anisotropic materials

A higher-order beam model for analyzing the flexural response of curved multilayered beams with constant curvature and arbitrary constant thickness is developed. The new model is derived from the Hellinger–Reissner mixed variational statement and predicts inherently equilibrated 3D stresses from an equivalent single-layer model. As a starting assumption, the hoop stress is formulated as a series of higher-order stress resultants multiplied by Legendre polynomials. The governing equations are derived in a generalized manner such that the modeling order can be adjusted and is not defined a priori. Hence, the highest order Legendre polynomial determines the modeling order. The through-thickness shear and normal stresses are derived by integrating the generalized hoop stress in Cauchy’s polar equilibrium equations. As a result, all stress fields are based on the same set of variables, thereby considerably reducing the computational effort. The three stress fields, and two displacements in the radial and hoop directions are used in the Hellinger–Reissner functional to derive a new set of stress-displacement relations. The enforcement of the classical membrane and bending equilibrium equations of curved beams in the Hellinger–Reissner functional guarantees that all interlaminar and surface traction equilibrium conditions are satisfied exactly. A validation study of a composite laminate using a high-fidelity 3D finite element model shows that the stresses are captured very accurately by the present model, but with much less computational effort than the finite element model. As a result, the developed model can provide rapid and accurate insights into the expected damage onset behavior of curved laminates.

Dynamic Relaxation Nonlinear Viscoelastic Bending Analysis of Higher-Order Annular Sector Plates

Using higher-order shear deformation theory (HSDT), nonlinear analysis of polymeric plate under uniform transverse pressure is studied considering material nonlinearity for viscoelastic material. The multiaxial stress state of the Schapery single integral equation describes the material behavior by recursive-iterative formulation. The finite difference Dynamic Relaxation (DR) method is utilized to study the annular sector plate, including geometric nonlinearity with in-plane fixed and free clamped and hinged edge constraints. The plate is considered as a laminate of isotropic layers, in which the viscoelastic properties of each layer have their own parameters during the loading. The plate deflection is predicted for different thicknesses and also various pressures at the beginning and end of loading and compared with elastic finite element results. The normal and through-thickness shear stresses are also determined in different time steps.

Stresses in Half-Elliptic Curved Beams Subjected to Transverse Tip Forces

In-plane bending of curved beams produces substantial through-thickness normal and shear stresses that can result in structural failures. A half-elliptic curved beam, having a known prescribed variable radius of curvature, is studied as an extension of the previously published circular arc beam. The equations for the normal, tangential, and shear stresses are developed for a curved beam outlined by two confocal half ellipses loaded by a pair of concentrated perpendicular forces on its ends. Closed-form analytical solutions for the stresses are found using an elasticity approach, where the solution is found by using selected terms of the biharmonic equation in elliptic coordinates. For the case of an elliptic beam with an aspect ratio of very close to unity, the solution closely agrees with published circular beam solutions. For other elliptic beam aspect ratios, the calculated stresses display good correlation to detailed finite element model solutions for thickness to semi-axis ratios < 0.1. A parametric study revealed that the maximum normal stress is located at the midplane for high-aspect ratio (a/b ≥ 1) half-elliptic beams, but shifts toward the load tip for low aspect ratio (a/b < 1) beams due to local curvature effects. Moreover, the peak shear stress location moves toward the midplane and the magnitude greatly increases as the aspect ratio is increased. Thus, there are large normal and shear stress interactions occurring near the midplane for high-aspect ratio half-elliptic beams, which is not observed for circular beams. These stress interactions can produce unique failures in materials having low shear strength and through-thickness strength. The current closed-form solution is an improvement on previously published approximate solutions.

Failure Mode Transition in Air-Backed Plates from Near Contact Underwater Explosions

Recent experiments involving near-contact underwater explosions on air-backed plates suggest the following failure mode categories: (1) holing and petaling, (2) complete or partial edge tearing due to shock only, (3) complete or partial edge tearing due to shock and bubble collapse, and (4) large deformation without rupture. Finite-element analysis was used to further investigate the detailed response and failure of the plates, and determine the limit between center plate holing and edge failure. When compared with experimental results, finite-elements showed good agreement with the failure modes of the plates and reasonable agreement with the experimental deformations. It was found that the linear interaction criteria (LIC) failure between plastic strain and through-thickness shear stress produced results closer to experiments than the quadratic interaction criteria (QIC). For the 18~gauge specimens it was found that the through-thickness shear dominated the failure initiation for very close proximity charges, with the direct strain becoming dominant as the standoff was increased. For the thinner 20 and 22 gauge plates the direct strain was always found to be the dominant factor in the failure criteria.

Failure Mode Transition in Air-Backed Plates from Near Contact Underwater Explosions

Recent experiments involving near-contact underwater explosions on air-backed plates suggest the following failure mode categories: (1) holing and petaling, (2) complete or partial edge tearing due to shock only, (3) complete or partial edge tearing due to shock and bubble collapse, and (4) large deformation without rupture. Finite-element analysis was used to further investigate the detailed response and failure of the plates, and determine the limit between center plate holing and edge failure. When compared with experimental results, finite-elements showed good agreement with the failure modes of the plates and reasonable agreement with the experimental deformations. It was found that the linear interaction criteria (LIC) failure between plastic strain and through-thickness shear stress produced results closer to experiments than the quadratic interaction criteria (QIC). For the 18~gauge specimens it was found that the through-thickness shear dominated the failure initiation for very close proximity charges, with the direct strain becoming dominant as the standoff was increased. For the thinner 20 and 22 gauge plates the direct strain was always found to be the dominant factor in the failure criteria.

Bistable composite laminates: Effects of laminate composition on cured shape and response to thermal load

This paper develops a finite element (FE) approach using commercial ANSYS V11.0 software to accurately predict the cured shape of bistable composites by including the influence of manufacturing imperfections, such as resin rich areas and ply-thickness variations. Laminate composition was characterised by optical microscopy and their cured shapes measured using a Peak Motus motion analysis system. The FE model accurately predicts observed differences between laminate curvature in the two stable states. Localised reversal of curvature resulting from through-thickness shear stress is also predicted. Structural response to thermal loading was experimentally characterised showing a temperature dependent deflection rate and a residual curvature caused by non-reversible residual stresses. FE-predictions show good agreement with experiment over the range 20–110 °C. The presented data highlights the importance of manufacturing processes and materials selection in the design of thermally stressed multi-stable composite structures.

Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory

In the present article, axisymmetric bending and stretching of functionally graded (FG) circular plates subjected to uniform transverse loading based on fourth-order shear deformation plate theory (FOST) have been studied. Using a fourth-order shear deformation theory, the solutions for deflection and rotation functions of FG plates are presented in terms of the corresponding quantities for a homogeneous plate using the classical plate theory (CPT), from which solutions one can easily obtain the FOST solutions for axisymmetric bending of FG circular plates. It is assumed that the effective mechanical properties of the functionally graded plates through the thickness are continuous functions of the volume fractions of the constituent parts which are themselves defined by a power-law function. Numerical results for maximum deflection and shear stress are presented for various percentages of ceramic–metal volume fractions. These results are also compared with those obtained from the first-order shear deformation plate theory of Mindlin (FST), the third-order shear deformation plate theory of Reddy (TST) as well as the exact three-dimensional elasticity solution. It is found that although the maximum deflections obtained using FOST and TST are close to each other, the through-thickness shear stress is predicted more accurately by the FOST formulation than by the TST.

The Effect of Eccentric Loads on the Macroscopic Strain and Stress Distributions in Woven Fabric Composite Iosipescu Specimens

Thispaper documentsthe resultsof work undertaken to evaluate the effects of eccentric loading conditions on the surface shear strains and stress distributions in layered quasi-laminar composite on opposite faces of the specimen as well as more pronounced shear stress gradients through-thickness. It has also been shown in this research that the effect of eccentric loads on the through-thickness shear stress and strain gradients in the Iosipescu specimens depends very strongly on the loading block geometries.