金属板材冲压成型过程的有限单元法模拟

用有限单元法模拟了金属板材的冲压成型过程。在模拟过程中，应用了同时考虑了薄膜力和弯矩的板壳大变形理论，考虑了板材在塑性阶段各向异性的强化性质，考虑了模具和工件的接触和摩擦条件，分析了金属板在冲压过程中的屈曲现象，建立了增量形式的变分原理。弹塑性薄壳单元被引入，它的位移模式在变分的意义上满足单元边界上一阶导数连续的条件，并有足够的秩来适应单元的有限拉伸、转动和弯曲，计算中采用了罚函数方法，即在模具和工件的接触面上，模具的表面被假设为文克尔地基，采用了修正的库伦摩擦定律，迭代法被用来决定模具与工件的接触条件和金属板的塑性行为。本文最后提供了一个算例。     

Stretch-induced wrinkling of polyethylene thin sheets: Experiments and modeling

This paper presents a study on stretch-induced wrinkling of thin polyethylene sheets when subjected to uniaxial stretch with two clamped ends. Three-dimensional digital image correlation was used to measure the wrinkling deformation. It was observed that the wrinkle amplitude increased as the nominal strain increased up to around 10%, but then decreased at larger strain levels. This behavior is consistent with results of finite element simulations for a hyperelastic thin sheet reported previously (Nayyar et al., 2011). However, wrinkles in the polyethylene sheet were not fully flattened out at large strains (>30%) as predicted for the hyperelastic sheet, but exhibited a residual wrinkle whose amplitude depended on the loading rate. This is attributed to the viscoelastic response of the material. Two different viscoelastic models were adopted in finite element simulations to study the effects of viscoelasticity on wrinkling and to improve the agreement with the experiments, including residual wrinkles and rate dependence. It is found that a parallel network model of nonlinear viscoelasticity is suitable for simulating the constitutive behavior and stretch-induced wrinkling of the polyethylene sheets.     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets

Wrinkles are commonly observed in stretched thin sheets and membranes. This paper presents a numerical study on stretch-induced wrinkling of hyperelastic thin sheets based on nonlinear finite element analyses. The model problem is set up for uniaxial stretching of a rectangular sheet with two clamped ends and two free edges. A two-dimensional stress analysis is performed first under the plane-stress condition to determine stretch-induced stress distribution patterns in the elastic sheets, assuming no wrinkles. As a prerequisite for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio of the sheet and the applied tensile strain in the longitudinal direction. A phase diagram is constructed with four different distribution patterns of the stretch-induced compressive stresses, spanning a wide range of aspect ratio and tensile strain. Next, an eigenvalue analysis is performed to find the potential buckling modes of the elastic sheet under the prescribed boundary conditions. Finally, a nonlinear post-buckling analysis is performed to show evolution of stretch-induced wrinkles. In addition to the aspect ratio and tensile strain, it is found that the critical condition for wrinkling and the post-buckling behavior both depend sensitively on the sheet thickness. In general, wrinkles form only when both the magnitude and the distribution area of the compressive stresses are sufficiently large. The wrinkle wavelength decreases with increasing strain, in good agreement with the prediction by a scaling analysis. However, as the tensile strain increases, the wrinkle amplitude first increases and then decreases, eventually flattened beyond a moderately large critical strain, in contrast to the scaling analysis.     

A comparative analysis of numerical approaches to the mechanics of elastic sheets

Numerically simulating deformations in thin elastic sheets is a challenging problem in computational mechanics due to destabilizing compressive stresses that result in wrinkling. Determining the location, structure, and evolution of wrinkles in these problems has important implications in design and is an area of increasing interest in the fields of physics and engineering. In this work, several numerical approaches previously proposed to model equilibrium deformations in thin elastic sheets are compared. These include standard finite element-based static post-buckling approaches as well as a recently proposed method based on dynamic relaxation, which are applied to the problem of an annular sheet with opposed tractions where wrinkling is a key feature. Numerical solutions are compared to analytic predictions of the ground state, enabling a quantitative evaluation of the predictive power of the various methods. Results indicate that static finite element approaches produce local minima that are highly sensitive to initial imperfections, relying on a priori knowledge of the equilibrium wrinkling pattern to generate optimal results. In contrast, dynamic relaxation is much less sensitive to initial imperfections and can generate low-energy solutions for a wide variety of loading conditions without requiring knowledge of the equilibrium solution beforehand.     

方板对角拉伸分叉、鼓动与起皱

以Ｈｉｌ关于弹塑性材料唯一性的充分性条件为理论基础，采用虚功率增率型原理和Ｍｉｎｄｌｉｎ曲壳单元的弹塑性大变形有限元模型成功地模拟了方板对角拉伸（ＹＢＴ）分叉后继变形的鼓动与起皱过程，并与同类实验结果进行了比较最后还讨论了板厚ｔ和端面拉伸宽度Ｂ对分叉模式和起皱模式的影响为板材成形过程中发生的分叉、鼓动与起皱现象提供一种有效的有限元数学模型     

ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL-Ⅱ: FINITE ELEMENT MODELING AND APPLICATION

Based on the generalized variational principle of magneto-thermo-elasticity of a ferromagnetic thin shell established (see, Analyses on nonlinear coupling of magneto-thermo-elasticity of ferromagnetic thin shell-Ⅰ), the present paper developed a finite element modeling for the mechanical-magneto-thermal multi-field coupling of a ferromagnetic thin shell. The numerical modeling composes of finite element equations for three sub-systems of magnetic, thermal and deformation fields, as well as iterative methods for nonlinearities of the geometrical large-deflection and the multi-field coupling of the ferromagnetic shell. As examples, the numerical simulations on magneto-elastic behaviors of a ferromagnetic cylindrical shell in an applied magnetic field, and magneto-thermo-elastic behaviors of the shell in applied magnetic and thermal fields are carried out. The results are in good agreement with the experimental ones.     

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.     

Constitutive equations for densification of matrix-coated fibre composites during hot isostatic pressing

This work addresses the development of physically based constitutive equations for the consolidation of fibre-matrix-void systems typically arising in the manufacture of matrix-coated fibre metal matrix composite materials. The analyses consider square array packing of the coated fibres under symmetrical in-plane compressive load and take into account the power-law creep of the matrix. Two models have been developed. The first is based on an energy approach in which assumed velocity fields in the deforming matrix are considered and are expressed in terms of an unknown parameter. In this way, the dependence of the deformation rate on volume fraction of voids and fibres is derived through the use of Hill's minimum principle for velocities. The second model makes use of micro-mechanical finite element modelling in which fibre, matrix and void are modelled explicitly. The micro-mechanical finite element model is developed for validation and comparison. Theoretical predictions are examined. The constitutive equation for consolidation derived from Hill's minimum principle shows good agreement with results obtained from micro-mechanical finite element modelling.     

Application of the strain energy density criterion to ductile fracture

The strain energy density criterion due to Sih is used to predict fracture loads of two thin plates subjected to large elastic-plastic deformation. The prediction is achieved with a finite element analysis which is based on Hill's variational principle for incremental deformations capable of solving gross yielding problems involving arbitrary amounts of deformation. The computed results are in excellent agreement with those obtained in Sih's earlier analysis and with an experimental observation.     

夹层梁总体屈曲及皱曲的有限元计算

皱曲是夹层结构的一种短波屈曲模式,通常发生于夹心较厚或夹心刚度较低的情况。由于模型规模的限制,在常规有限元建模时通常将夹层板模拟为二维板单元,这种方法忽略了面板和夹心在厚度方向上的相互作用,无法计算出皱曲模式。针对上述问题,本文首先介绍了一个计算夹层结构总体屈曲和皱曲的统一理论,并将此理论的计算结果作为理论解。为了同时...     

Wrinkled membrane element based on the wrinkling potential

For a membrane structure, wrinkles have an important effect on its mechanical behaviors. Wrinkling level characterizes the development of wrinkles and reflects the performance of a membrane in its service, and it plays a very significant role in the wrinkling analysis. The shell elements combined with the stability theory would be an ideal solution to the wrinkling problem. However, this approach requires a dense mesh and the computation is very time consuming. Also the wrinkling parameters are very sensitive to the size of shell elements. Existing wrinkling models based on membrane elements are derived from the Tension Field Theory which are incapable of describing fully the wrinkling behaviors.A new wrinkling model adopting the wrinkling strain as a measure of the wrinkling level is proposed in this paper to address these issues. According to the analogy between the wrinkling strain and the elasto-plastic strain, a wrinkling potential surface is assumed to exist and its normal direction defines the direction of the wrinkling strain tensor by virtue of the flow rule. Based on the consistent condition of the wrinkling potential surface, a modified constitutive tensor is obtained. To avoid the switching of the wrinkling state in the numerical solution, a new wrinkling criterion is proposed, in which the predominant influence of the previous state is included. Besides, a new approach to determine the wrinkling orientation is given to improve the efficiency of convergence in the slack region. The objectivity of the wrinkling coordinate frame is also demonstrated as an accompanying set of results. Finally, two benchmark problems are analyzed with the proposed wrinkling model, and their results are compared with those in the literatures. Results indicate that the proposed wrinkling model is valid and accurate to characterize the wrinkling level of a membrane and it exhibits efficient convergence even in the slack region.     

Numerical study of the wrinkling of a stretched thin sheet

A thin sheet clamped at opposite ends and stretched develops wrinkles parallel to the direction of the applied tensile strain due to the hindered Poisson lateral contraction at the clamps. To study this phenomenon, a variational model recently proposed by Puntel, Deseri and Fried is adopted. The relevant energy functional includes bending and membranal contributions and is minimized subject to a constraint on the area of the mid-surface of the sheet. A fourth order partial-differential equation is henceforth obtained and numerically implemented using B-splines. Predictions are obtained concerning the number of wrinkles, critical applied stretches, and scaling relationships for wrinkle amplitude and wavelength. Both a linearized version of the boundary-value problem based on the small-slope approximation and a fully nonlinear one are considered: their results are found to be in good agreement for the whole range of applied stretches taken into account. Comparisons with previous analytical results by Puntel, Deseri and Fried, who used different boundary conditions and an Ansatz on the deflection function are also provided. The numerical results substantially confirm the validity of the analytical predictions. The present work provides then an alternative numerical method for the study of wrinkling in thin sheets and supports the use of analytical and semi-analytical solutions as viable options for specific geometries. Though further investigation, particularly experimental, is still needed, extensive comparisons of the results with other studies available in the literature provide confirmation for the scaling laws and signal that predicted values of the critical stretches may only be accurate for higher length-to-width aspect ratios.     

Post-buckling evolution of wavy patterns in trapezoidal film/substrate bilayers

This paper investigates quantitatively the post-buckling response of irregular wrinkles in a trapezoidal film/substrate bilayer. The geometric gradient can change the wrinkling profile to create ribbed and graded structural patterns with variations of wave direction, amplitude and wavelength. The tapered angle and edge dimension are examined numerically using a nonlinear shell/solid coupled finite element model that incorporates a path-following continuation technique, which explores their influences on secondary bifurcations, localization and surface mode transition. For instance, the competition between plate-like and beam-like post-buckling behavior is discussed. An analysis of graded amplitude is also provided based on Fourier envelope equations of beam/foundation models, which gives an insightful understanding of these fading wrinkles that differ from the ones usually observed in rectangular geometric cases. The results of this work can be used to guide the design of geometrically gradient film/substrate systems to achieve desired wavy instability patterns.     

The effect of plastic anisotropy on compressive instability in sheet metal forming

The wrinkling behavior of a thin sheet with perfect geometry is associated with compressive instability. The compressive instability is influenced by many factors such as stress state, mechanical properties of the sheet material, geometry of the body, contact conditions and plastic anisotropy. The analysis of compressive instability in a plastically deforming body is difficult considering all the factors because the effects of the factors are very complex and the instability behavior may show a wide variation for a small deviation of the factors. In this study, the bifurcation theory is introduced for the finite element analysis of puckering initiation and growth of a thin sheet with perfect geometry. All the above mentioned factors are conveniently considered by the finite-element method. The instability limit is found by the incremental analysis and the post-bifurcation behavior is analyzed by introducing the branching scheme proposed by Riks. The finite-element formulation is based on the incremental deformation theory and elastic–plastic material modeling. The finite-element analysis is carried out using the continuum-based resultant shell elements considering the anisotropy of the sheet metal. In order to investigate the effect of plastic anisotropy on the compressive instability, a square plate that is subjected to compression in one direction and tension in the other direction is analyzed by the above-mentioned finite-element analysis. The critical stress ratios above which buckling does not take place are found for various plastic anisotropic modeling methods and discussed. Finally, the effect of plastic anisotropy on the puckering behavior in the spherical cup deep drawing process is investigated. From the results of the finite-element analysis, it is shown that puckering behavior of sheet metal is largely affected by plastic anisotropy.     

Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic–plastic formulation

This paper proposes an extension of the SHB8PS solid–shell finite element to large strain anisotropic elasto-plasticity, with application to several non-linear benchmark tests including sheet metal forming simulations. This hexahedral linear element has an arbitrary number of integration points distributed along a single line, defining the “thickness” direction; and to control the hourglass modes inherent to this reduced integration, a physical stabilization technique is used. In addition, the assumed strain method is adopted for the elimination of locking. The implementation of the element in Abaqus/Standard via the UEL user subroutine has been assessed through a variety of benchmark problems involving geometric non-linearities, anisotropic plasticity, large deformation and contact. Initially designed for the efficient simulation of elastic–plastic thin structures, the SHB8PS exhibits interesting potentialities for sheet metal forming applications—both in terms of efficiency and accuracy. The element shows good performance on the selected tests, including springback and earing predictions for Numisheet benchmark problems.     

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love's thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the generalized differential quadrature (GDQ) method. Hill's method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.     

薄板冲压成型中板料起皱的临界应力分析和预测

针对薄板冲压成型中起皱这一常见的材料失效形式，运用板料压缩失稳理论，提出虚拟刚度变量的概念和板料产生起皱的临界应力计算方法，并借助于计算机仿真技术和有限元计算方法，搜寻并计算得到反映板料各处发生起皱难易程度的临界因子，生成起皱云图来观察材料各部分的稳定状态，预测可能出现起皱的部位，为修改冲压工艺和修模提供依据，通过分析起皱云图得出的结论与实际情况非常接近。     

板材多点成形过程的有限元分析

多点成形过程采用静力隐式格式进行数值模拟是比较合适的。本文建立了用于多点成形过程分析的静力隐式弹塑性大变形有限元方法 ,给出了对稳定迭代收敛过程效果较好的板壳有限单元模型、处理多点不连续接触边界的接触单元方法以及增量变形过程中应力及塑性应变计算的多步回映计算方法。基于这些方法编制了计算软件 ,应用该软件进行了矩形板的液压胀形过程及球形模具拉伸成形过程的有限元分析 ,数值计算结果与典型的实验结果及计算结果吻合很好。最后给出了球形、圆柱形目标形状的实际多点成形过程的数值模拟结果。     

Wrinkling of elastoplastic circular plates during stamping

Circular plates that are stamped into a shallow, biaxially curved die by a matching punch develop radial wrinkles near the periphery when the edge is not clamped. Thin ductile metal plates develop these wrinkles after some plastic deformation occurs at the center of the plate. In comparison with elastic wrinkling, the center deflection to thickness ratio for wrinkling is increased as a consequence of the plastic deformation. In elastoplastic plates, this critical deflection ratio is a decreasing function of the plate thickness parameter ξ_o