Analysis of sheet metal forming operations by a stress resultant constitutive law

Sheet metal forming is simulated by finite element methods using a stress resultant constitutive law in this paper. A Lagrangian description of axisymmetric and plane-strain shell deformation is first reviewed. Then a stress resultant constitutive law in rate form is presented, where the effect of thickness reduction due to large plastic deformation is considered. A finite element formulation in terms of stress resultants and their work-conjugate generalized strain rates is derived based on the virtual work principle. A hemispherical punch stretching operation and a plane-strain draw operation are simulated by a finite element program based on the finite element formulation. The results of these finite element simulations are in good-agreement with those using the through-the-thickness integration method. The results of the hemispherical punch stretching simulation suggest that the coupling term of moments and membrane forces of the modified Ilyushin yield function should be eliminated to avoid numerical instability under stretching-dominated conditions for this rate-independent plasticity formulation. Further, the results suggest that the hardening rule in a power-law form based on the small-strain approach must be modified to take account for finite deformation effects of combined stretching and bending. Under the plane-strain draw operation, the sheet experiences a large amount of bending before the final stretching. The simulation based on the stress resultant constitutive law can produce this essential aspect of deformation pattern as that of the through-the-thickness integration method, whereas a simulation based on a membrane theory cannot. In conclusion, the results of these simulations indicate that a finite element program based on the stress resultant constitutive law can simulate sheet-forming processes with much shorter computational time than that based on the through-the-thickness integration method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/50100/1/1620370502_ftp.pd

A novel 'boundary layer' finite element for the efficient analysis of thin cylindrical shells

Classical shell finite elements usually employ low-order polynomial shape functions to interpolate between nodal displacement and rotational degrees of freedom. Consequently, carefully-designed fine meshes are often required to accurately capture regions of high local curvature, such as at the ‘boundary layer’ of bending that occurs in cylindrical shells near a boundary or discontinuity. This significantly increases the computational cost of any analysis. This paper is a ‘proof of concept’ illustration of a novel cylindrical axisymmetric shell element that is enriched with rigorously-derived transcendental shape functions to exactly capture the bending boundary layer. When complemented with simple polynomials to express the membrane displacements, a single boundary layer shell element is able to support very complex displacement and stress fields that are exact for distributed element loads of up to second order. A single element is usually sufficient per shell segment in a multi-strake shell. The predictions of the novel element are compared against analytical solutions, a classical axisymmetric shell element with polynomial shape functions and the ABAQUS S4R shell element in three problems of increasing complexity and practical relevance. The element displays excellent numerical results with only a fraction of the total degrees of freedom and involves virtually no mesh design. The shell theory employed at present is kept deliberately simple for illustration purposes, though the formulation will be extended in future work

Mixed Models and Reduction Techniques for Large-Rotation, Nonlinear Analysis of Shells of Revolution with Application to Tires

An effective computational strategy is presented for the large-rotation, nonlinear axisymmetric analysis of shells of revolution. The three key elements of the computational strategy are: (1) use of mixed finite-element models with discontinuous stress resultants at the element interfaces; (2) substantial reduction in the total number of degrees of freedom through the use of a multiple-parameter reduction technique; and (3) reduction in the size of the analysis model through the decomposition of asymmetric loads into symmetric and antisymmetric components coupled with the use of the multiple-parameter reduction technique. The potential of the proposed computational strategy is discussed. Numerical results are presented to demonstrate the high accuracy of the mixed models developed and to show the potential of using the proposed computational strategy for the analysis of tires

Component-wise models for static, dynamic and aeroelastic analyses of metallic and composite aerospace structures

In the framework of structural mechanics, the classical beam theories that are commonly adopted in many applications may be affected by inconsistencies, because they are not able to foresee higher-order phenomena, such as elastic bending/shear couplings, restrained torsional warping and 3D strain effects. Depending on the problem, those limitations can be overcome by using more complex and computationally expensive 2D and 3D models or, alternatively, by adopting refined beam models, to which many scientists have dedicated their research over the last century. % One of the latest contributions to the development of advanced models, including variable kinematic beam theories, is the Carrera Unified Formulation (CUF), which is the main subject of the research discussed in this thesis. According to CUF, the 3D displacement field can be expressed as an arbitrary expansion of the generalized displacements. Depending on the choice of the polynomials employed in the expansion, various classes of beam models can be implemented. In this work, for instance, Taylor-like and Lagrange polynomials are adopted. The former choice leads to the so-called TE (Taylor Expansion) beam models, whereas LE (Lagrange Expansion) beam models with only pure displacement variables are obtained by interpolating the problem unknowns by Lagrange polynomials. The strength of CUF lies in the fact that, independently of the choice of the polynomials, the governing equations are written in terms of fundamental nuclei, which are invariant with the theory class and order. In this thesis, both strong and weak form governing equations for arbitrarily refined CUF models are derived. Subsequently, exact closed-form and approximate solutions are sought. Exact solutions of any beam model with arbitrary boundary conditions are found by formulating a frequency-dependant Dynamic Stiffness (DS) matrix and by using the Wittrick-Williams algorithm to carry out the resulting transcendental eigenvalue problem for free vibration analysis. Conversely, a linear eigenvalue problem is also derived by approximating the strong form governing equations by Radial Basis Functions (RBFs). On the other hand, weak form solutions are discussed by Finite Element Method (FEM), which still deserves important attentions due to its versatility and numerical efficiency. The various problems of the mechanics are addressed, including static, free vibration and dynamic response problems. Based on CUF and the proposed numerical methods, advanced methodologies for the analysis of complex structures, such as aircraft structures and civil engineering constructions, are developed. Those advanced techniques make use of the Component-Wise (CW) and the Multi-Line approaches. The CW method exploits the natural capability of the LE CUF beam models to be assembled at the cross-section level. This characteristic allows the analyst to use only CUF beam elements to model each component (e.g., stringers, panels and ribs) of the structure and purely physical surfaces are employed to construct the mathematical models. In the ML framework, on the other hand, each component of the structure is modelled via TE beam elements of arbitrary order. Compatibility of displacements between two or more components is then enforced through the Lagrange multipliers method. The second part of this thesis deals with aeroelasticity. In particular, the Vortex (VLM) and the Doublet Lattice Methods (DLM) are employed and extended to CUF to develop aeroelastic models. VLM is used to model the steady contribution in the aerodynamic model, whereas DLM provides the unsteady contribution in the frequency domain. The infinite plate spline approach is adopted for the mesh-to-mesh transformation. Finally, the g-method is described as an effective means for the formulation of the flutter stability problem. Particular attention is given to the extension of this methodology to exact DS solutions of CUF beams. Simplified, discrete, dynamic gust response analysis by refined beam models is also discussed. In this work, vertical gusts and one-minus-cosine idealization is addressed. Accordingly, gust loads in terms of time-dependent load factors are formulated. Subsequently, the mode superposition method is briefly introduced in order to solve the linear dynamic response problem in the time domain by using both weak and strong form solutions of CUF models. In the final part of the work, extensions of 1D CUF models for Fluid-Dynamics problems are carried out. CUF approximation of laminar, incompressible, Stokes flows with constant viscosity was introduced in a recent thesis work and it is here extended to the hierarchical p-version of FEM, which makes use of Legendre-like polynomials to interpolate the generalized unknowns along the 1D computational domain. Finally, the structural, aeroelastic and fluid-dynamics formulations are validated by discussing some selected results. In particular, regarding structures, the efficiency of the various numerical approaches when applied to CUF is investigated and simple to complex problems are considered, including metallic and composite wings. The aeroelastic analyses show that classical beam models are not adequate for the flutter detection, and at least a third-order beam model is required. Contrarily, classical beam models can be quite accurate in dynamic gust response analysis if no coupling phenomena occur, i.e. when the response is dominated by only pure bending modes. Regarding fluid-dynamics, it is demonstrated that CUF models can reproduce the results by finite volume codes for both simple Poiseuille and complex non-axisymmetric fluids in cylinders. In general, the capability of the proposed CUF models to provide accurate results with very low computational efforts is firmly highlighted. Similar analyses are possible only by using 3D models, which usually require a number of degrees of freedom that is some two order of magnitude higher

A closed-form solution for asymmetric free vibration analysis of composite cylindrical shells with metamaterial honeycomb core layer based on shear deformation theory

Asymmetric free vibration analysis of composite cylindrical shells with a honeycomb core layer and adjustable Poisson’s ratio is performed analytically in this study. The equations of motion which are a system of coupled partial differential equations are extracted using Hamilton’s principle by employing the first-order shear deformation theory and they are solved analytically. To study the sensitivity of the results to the different parameters of the honeycomb structure, geometrical parameters, and boundary conditions, a parametric study is presented. It is concluded that for the auxetic composite shell with a negative Poisson’s ratio, by decreasing the Poisson ratio, the frequency decreases. Also, it is shown that by employing the composite shells the weight decreases significantly, while the asymmetric frequency will not change remarkably. By adjusting the Poisson ratio, the frequency variations are studied for a composite shell with a honeycomb core layer. The results are compared with the finite element method and some other references

Introduction to the finite element method

2019/202

Nonlinear Formulations of a Four-Node Quasi-Conforming Shell Element

The quasi-conforming technique was introduced in the 1980’s to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented. The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening

The Ritz – Sublaminate Generalized Unified Formulation approach for piezoelectric composite plates

This paper extends to composite plates including piezoelectric plies the variable kinematics plate modeling approach called Sublaminate Generalized Unified Formulation (SGUF). Two-dimensional plate equations are obtained upon defining a priori the through-thickness distribution of the displacement field and electric potential. According to SGUF, independent approximations can be adopted for the four components of these generalized displacements: an Equivalent Single Layer (ESL) or Layer-Wise (LW) description over an arbitrary group of plies constituting the composite plate (the sublaminate) and the polynomial order employed in each sublaminate. The solution of the two-dimensional equations is sought in weak form by means of a Ritz method. In this work, boundary functions are used in conjunction with the domain approximation expressed by an orthogonal basis spanned by Legendre polynomials. The proposed computational tool is capable to represent electroded surfaces with equipotentiality conditions. Free-vibration problems as well as static problems involving actuator and sensor configurations are addressed. Two case studies are presented, which demonstrate the high accuracy of the proposed Ritz-SGUF approach. A model assessment is proposed for showcasing to which extent the SGUF approach allows a reduction of the number of unknowns with a controlled impact on the accuracy of the result

Coupled thermo-mechanical finite element models with node-dependent kinematics for multi-layered shell structures

Node-dependent Kinematic (NDK) shell finite element (FE) formulations are presented for the steady-state thermo-mechanical analysis of laminated structures. The displacements and temperature change are treated as primary variables in the FE models and are directly solved through the coupled thermo-mechanical models. The enforcement of distributed temperature boundary conditions on the top or the bottom surface of hierarchical shell elements is conducted through the Linear Least Squares. The effectiveness of the proposed FE approaches is verified by comparing the results against those from the literature. The application of adaptive refinement approach based on the hierarchical elements and NDK to build FE models with optimal efficiency is demonstrated through numerical examples