Showing papers on "Tangent stiffness matrix published in 2004"

Numerical finite element formulation of the Schapery non-linear viscoelastic material model

Abstract: This study presents a numerical integration method for the non-linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three-dimensional (3D) non-linear viscoelastic material model is integrated within a displacement-based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non-linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.

The Finite Element Method for Three-Dimensional Thermomechanical Applications

Abstract: Preface.Nomenclature.1 Displacements, Strain, Stress and Energy.1.1 The Reference State.1.2 The Spatial State.1.3 Strain Measures.1.4 Principal Strains.1.5 Velocity.1.6 Objective Tensors.1.7 Balance Laws.1.7.1 Conservation of mass.1.7.2 Conservation of momentum.1.7.3 Conservation of angular momentum.1.7.4 Conservation of energy.1.7.5 Entropy inequality.1.7.6 Closure.1.8 Localization of the Balance Laws.1.8.1 Conservation of mass.1.8.2 Conservation of momentum.1.8.3 Conservation of angular momentum.1.8.4 Conservation of energy.1.8.5 Entropy inequality.1.9 The Stress Tensor.1.10 The Balance Laws in Material Coordinates.1.10.1 Conservation of mass.1.10.2 Conservation of momentum.1.10.3 Conservation of angular momentum.1.10.4 Conservation of energy.1.10.5 Entropy inequality.1.11 The Weak Form of the Balance of Momentum.1.11.1 Formulation of the boundary conditions (material coordinates).1.11.2 Deriving the weak form from the strong form (material coordinates).1.11.3 Deriving the strong form from the weak form (material coordinates).1.11.4 The weak form in spatial coordinates.1.12 The Weak Form of the Energy Balance.1.13 Constitutive Equations.1.13.1 Summary of the balance equations.1.13.2 Development of the constitutive theory.1.14 Elastic Materials.1.14.1 General form.1.14.2 Linear elastic materials.1.14.3 Isotropic linear elastic materials.1.14.4 Linearizing the strains.1.14.5 Isotropic elastic materials.1.15 Fluids.2 Linear Mechanical Applications.2.1 General Equations.2.2 The Shape Functions.2.2.1 The 8-node brick element.2.2.2 The 20-node brick element.2.2.3 The 4-node tetrahedral element.2.2.4 The 10-node tetrahedral element.2.2.5 The 6-node wedge element.2.2.6 The 15-node wedge element.2.3 Numerical Integration.2.3.1 Hexahedral elements.2.3.2 Tetrahedral elements.2.3.3 Wedge elements.2.3.4 Integration over a surface in three-dimensional space.2.4 Extrapolation of Integration Point Values to the Nodes.2.4.1 The 8-node hexahedral element.2.4.2 The 20-node hexahedral element.2.4.3 The tetrahedral elements.2.4.4 The wedge elements.2.5 Problematic Element Behavior.2.5.1 Shear locking.2.5.2 Volumetric locking.2.5.3 Hourglassing.2.6 Linear Constraints.2.6.1 Inclusion in the global system of equations.2.6.2 Forces induced by linear constraints.2.7 Transformations.2.8 Loading.2.8.1 Centrifugal loading.2.8.2 Temperature loading.2.9 Modal Analysis.2.9.1 Frequency calculation.2.9.2 Linear dynamic analysis.2.9.3 Buckling.2.10 Cyclic Symmetry.2.11 Dynamics: The alpha-Method.2.11.1 Implicit formulation.2.11.2 Extension to nonlinear applications.2.11.3 Consistency and accuracy of the implicit formulation.2.11.4 Stability of the implicit scheme.2.11.5 Explicit formulation.2.11.6 The consistent mass matrix.2.11.7 Lumped mass matrix.2.11.8 Spherical shell subject to a suddenly applied uniform pressure.3 Geometric Nonlinear Effects.3.1 General Equations.3.2 Application to a Snapping-through Plate.3.3 Solution-dependent Loading.3.3.1 Centrifugal forces.3.3.2 Traction forces.3.3.3 Example: a beam subject to hydrostatic pressure.3.4 Nonlinear Multiple Point Constraints.3.5 Rigid Body Motion.3.5.1 Large rotations.3.5.2 Rigid body formulation.3.5.3 Beam and shell elements.3.6 Mean Rotation.3.7 Kinematic Constraints.3.7.1 Points on a straight line.3.7.2 Points in a plane.3.8 Incompressibility Constraint.4 Hyperelastic Materials.4.1 Polyconvexity of the Stored-energy Function.4.1.1 Physical requirements.4.1.2 Convexity.4.1.3 Polyconvexity.4.1.4 Suitable stored-energy functions.4.2 Isotropic Hyperelastic Materials.4.2.1 Polynomial form.4.2.2 Arruda-Boyce form.4.2.3 The Ogden form.4.2.4 Elastomeric foam behavior.4.3 Nonhomogeneous Shear Experiment.4.4 Derivatives of Invariants and Principal Stretches.4.4.1 Derivatives of the invariants.4.4.2 Derivatives of the principal stretches.4.4.3 Expressions for the stress and stiffness for three equal eigenvalues.4.5 Tangent Stiffness Matrix at Zero Deformation.4.5.1 Polynomial form.4.5.2 Arruda-Boyce form.4.5.3 Ogden form.4.5.4 Elastomeric foam behavior.4.5.5 Closure.4.6 Inflation of a Balloon.4.7 Anisotropic Hyperelasticity.4.7.1 Transversely isotropic materials.4.7.2 Fiber-reinforced material.5 Infinitesimal Strain Plasticity.5.1 Introduction.5.2 The General Framework of Plasticity.5.2.1 Theoretical derivation.5.2.2 Numerical implementation.5.3 Three-dimensional Single Surface Viscoplasticity.5.3.1 Theoretical derivation.5.3.2 Numerical procedure.5.3.3 Determination of the consistent elastoplastic tangent matrix.5.4 Three-dimensional Multisurface Viscoplasticity: the Cailletaud Single Crystal Model.5.4.1 Theoretical considerations.5.4.2 Numerical aspects.5.4.3 Stress update algorithm.5.4.4 Determination of the consistent elastoplastic tangent matrix.5.4.5 Tensile test on an anisotropic material.5.5 Anisotropic Elasticity with a von Mises-type Yield Surface.5.5.1 Basic equations.5.5.2 Numerical procedure.5.5.3 Special case: isotropic elasticity.6 Finite Strain Elastoplasticity.6.1 Multiplicative Decomposition of the Deformation Gradient.6.2 Deriving the Flow Rule.6.2.1 Arguments of the free-energy function and yield condition.6.2.2 Principle of maximum plastic dissipation.6.2.3 Uncoupled volumetric/deviatoric response.6.3 Isotropic Hyperelasticity with a von Mises-type Yield Surface.6.3.1 Uncoupled isotropic hyperelastic model.6.3.2 Yield surface and derivation of the flow rule.6.4 Extensions.6.4.1 Kinematic hardening.6.4.2 Viscoplastic behavior.6.5 Summary of the Equations.6.6 Stress Update Algorithm.6.6.1 Derivation.6.6.2 Summary.6.6.3 Expansion of a thick-walled cylinder.6.7 Derivation of Consistent Elastoplastic Moduli.6.7.1 The volumetric stress.6.7.2 Trial stress.6.7.3 Plastic correction.6.8 Isochoric Plastic Deformation.6.9 Burst Calculation of a Compressor.7 Heat Transfer.7.1 Introduction.7.2 The Governing Equations.7.3 Weak Form of the Energy Equation.7.4 Finite Element Procedure.7.5 Time Discretization and Linearization of the Governing Equation.7.6 Forced Fluid Convection.7.7 Cavity Radiation.7.7.1 Governing equations.7.7.2 Numerical aspects.References.Index.

Extreme stiffness systems due to negative stiffness elements

Abstract: When an elastic object is pressed, we expect it to resist by exerting a restoring force. A reversal of this force corresponds to negative stiffness. If we combine elements with positive and negative stiffness in a composite, it is possible to achieve stiffness greater than (or less than) that of any of the constituents. This behavior violates established bounds that tacitly assume that each phase has positive stiffness. Extreme composite behavior has been experimentally demonstrated in a lumped system using a buckled tube to achieve negative stiffness and in a composite material in the vicinity of a phase transformation of one of the constituents. In the context of a composite system, extreme refers to a physical property greater than either constituent. We consider a simple spring model with pre-load to achieve negative stiffness. When suitably tuned to balance positive and negative stiffness, the system shows a critical equilibrium point giving rise to extreme overall stiffness. A stability analysis of a viscous damped system containing negative stiffness springs reveals that the system is stable when tuned for high compliance, but metastable when tuned for high stiffness. The metastability of the extreme system is analogous to that of diamond. The frequency response of the viscous damped system shows that the overall stiffness increases with frequency and goes to infinity when one constituent has a suitable negative stiffness.

Enrichment of Enhanced Assumed Strain Approximations for Representing Strong Discontinuities : Addressing Volumetric Incompressibility and the Discontinuous Patch Test

Abstract: SUMMARY We present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary, intra-finite-element discontinuities in the deformation map. Special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit. Given an underlying mesh and an arbitrary failure surface, we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method (X-FEM). With regard to the richer space of functions spanned by the gradient of the enriched approximation, we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain. In particular, in those elements intersected by the failure surface, we construct enhanced strain approximations which are orthogonal to piecewise-constant stress fields. Contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields. Importantly, the present formulation gives rise to a symmetric tangent stiffness matrix, even in localized elements. The present modification also allows for the satisfaction of a discontinuous patch test, wherein two different constant stress fields (on each side of the failure surface) lie in the solution space. We demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface, particularly for nearly incompressible material response. Additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics. Copyright c � 2000 John Wiley & Sons, Ltd.

Modeling of wave propagation in layered piezoelectric media by a recursive asymptotic method

Abstract: In this paper, a simple asymptotic method to compute wave propagation in a multilayered general anisotropic piezoelectric medium is discussed. The method is based on explicit second and higher order asymptotic representations of the transfer and stiffness matrices for a thin piezoelectric layer. Different orders of the asymptotic expansion are obtained using Pade approximation of the transfer matrix exponent. The total transfer and stiffness matrices for thick layers or multilayers are calculated with high precision by subdividing them into thin sublayers and combining recursively the thin layer transfer and stiffness matrices. The rate of convergence to the exact solution is the same for both transfer and stiffness matrices; however, it is shown that the growth rate of the round-off error with the number of recursive operations for the stiffness matrix is twice that for the transfer matrix; and the stiffness matrix method has better performance for a thick layer. To combine the advantages of both methods, a hybrid method which uses the transfer matrix for the thin layer and the stiffness matrix for the thick layer is proposed. It is shown that the hybrid method has the same stability as the stiffness matrix method and the same round-off error as the transfer matrix method. The method converges to the exact transfer/stiffness matrices essentially with the precision of the computer round-off error. To apply the method to a semispace substrate, the substrate was replaced by an artificial perfect matching layer. The computational results for such an equivalent system are identical with those for the actual system. In our computational experiments, we have found that the advantage of the asymptotic method is its simplicity and efficiency.

A return map algorithm for general isotropic elasto/visco‐plastic materials in principal space

Abstract: We describe a methodology for solving the constitutive problem and evaluating the consistent tangent operator for isotropic elasto/visco-plastic models whose yield function incorporates the third stress invariant . The developments presented are based upon original results, proved in the paper, concerning the derivatives of eigenvalues and eigenprojectors of symmetric second-order tensors with respect to the tensor itself and upon an original algebra of fourth-order tensors obtained as second derivatives of isotropic scalar functions of a symmetric tensor argument . The analysis, initially referred to the small-strain case, is then extended to a formulation for the large deformation regime; for both cases we provide a derivation of the consistent tangent tensor which shows the analogy between the two formulations and the close relationship with the tangent tensors of the Lagrangian description of large-strain elastoplasticity. Copyright © 2004 John Wiley & Sons, Ltd.

Contact formulation via a velocity description allowing efficiency improvements in frictionless contact analysis

Abstract: A velocity description, based on the consideration of contact from the surface geometry point of view, is used for a consistent formulation of contact conditions and for the derivation of the corresponding tangent matrix. Within this approach differential operations are treated as covariant derivatives in the local surface coordinate system. The main advantage is a more algorithmic and geometrical structure of the tangent matrix, which consists of a “main”, a “rotational” and a pure “curvature” term. Each part of the tangent matrix contains the information either about the internal geometry of the contact surface or about the change of the geometry during incremental loading and can be estimated in a norm during the analysis. Representative examples with contact and bending of shells modelled with linear and quadratic elements over some classical second order geometrical figures serve to show situations where keeping all parts of the tangent matrix is not necessary.

Finite Element Computation of Woven Ply Laminated Composite Structures up to Rupture

Abstract: For unidirectional ply laminates, the great diversity of the damage mechanisms and their patterns of evolution make it extremely difficult to estimate the strength margins. In the case of woven ply laminates, the number of damage mechanisms is fairly small (no transverse rupture occurs and the material has a greater resistance to delamination) and the behaviour of the material is fairly simple to model up to rupture. In this study, a numerical model for woven ply laminated composite structures up to rupture is developed. The implementation is performed in a Euler Backward scheme and the consistent tangent stiffness matrix is calculated. Comparison with some experiments on structures are made and the model predicts these experiments well.

Interphasic formulation for the prediction of delamination

Abstract: A stress-strain delamination constitutive law based on a continuum damage mechanics formulation is postulated for the thin resin layer that exists between adjacent composite laminae to predict initiation and growth of delamination. Delamination is predicted to initiate based on a interlaminar maximum stress obtained from a multiaxial stress criterion and the delamination front is predicted to advance based on a critical fracture energy obtained from a mixed mode fracture criterion. Damage is assumed to be irreversible via the inclusion of a thermodynamically consistent damage parameter. The damage parameter is directly related to the dissipated fracture energy. Although the implementation of the constitutive law in a commercial finite element code is straightforward, the nonlinear solver often fails to converge to a solution using the conventional tangent stiffness matrix. Convergence is achieved by using a stiffness matrix with a modification that improves its condition number. With this novel constitutive law and stiffness modification, the delamination front and the direction of delamination propagation are predicted for various fracture test configurations such as the double cantilever beam, end load split, and end crack torsion.

On analysis of the elasto‐viscoplastic response of single crystals with anisotropic damage: constitutive modelling and computational aspects

Abstract: Based on the concept of continuum damage mechanics, an anisotropic damage model for single crystals under the theory of crystal plasticity is presented. Damage and inelastic deformations are incorporated in the proposed model which is developed within the framework of thermodynamics with internal state variables. The dependence of the plastic anisotropy on the damage evolution has been considered. The anisotropic damage is characterized kinematically here through a second-order damage tensor which is physically based. The proposed model can successfully describe the interaction between the evolution of micro-structure of single crystals such as lattice orientation and the hardness development of each slip system and the process of material degradation. The Newton-Raphson iterative scheme is used to integrate the constitutive equations that work directly with the evolution equations for the elastic deformation gradient. The consistent algorithmic tangent stiffness for the present algorithm is formulated. The prescribed algorithm together with the consistent algorithmic tangent stiffness has been implemented into the ABAQUS finite element code by using user subroutine. Using the loading processes with homogeneous deformations and simulation of the classical tensile test of a notched bar illustrate the basic aspects of the model described. Numerical simulations show the validation and performance of the present model and algorithm.

Derivation of the higher-order stiffness matrix of a space frame element for geometric nonlinear analysis of structrues

Abstract: Summary A physical concept, the rigid body rule, is applied for the derivation of the higher-order stiffness matrix of a space frame element. The derivation has a physical meaning that is the higher-order stiffness matrix can be derived by regarding there is a set of incremental nodal forces existing on the element, then the element undergoes a small rigid body rotation. The incremental forces should keep their magnitude and follow the rigid body motions. Then taking advantage of the existing geometric stiffness matrix derived by researchers, the higher- order stiffness matrix can be analogy derived without any difficulty. The derived higher-order stiffness matrix has explicit expressions. It can be used at the forces recovery stage in the geometric nonlinear analysis of frame structures. Meanwhile an effective numerical method, the Generalized Displacement Control (GDC) method, was adopted to trace the load-defection curves of the structures. Some numerical examples were tested by taking the proposed higher-order stiffness matrix into consideration in the nonlinear analysis of the structures.

Spatial beam analysis with large displacement for deployable truss structures

Abstract: A formula of multibody finite element analysis based on the corotational formulated finite element method and the direct coordinate partitioning for deployable truss structures is proposed. A beam element used in linear finite element analyses is a basic element in the formula. A corotational frame is defined by node coordinate systems attached to a beam element. A tangent stiffness matrix contains a virtual rotation displacement of the corotational frame. By the appropriate transformation of this virtual displacement, a simple geometric stiffness for the formula is obtained. The formula makes it possible to analyze the deploying motions of large deformed deployable truss structures, to be applicable to general finite elements, and to increase convergence speed of the Newton method in a solution algorithm. This improvement is achieved by neglecting virtual work done by rigid body motion in the process of employing the geometric stiffness to the conventional formula. A validity of this formula is confirmed through results of a numerical deployment analysis of a deployable beam structure.

Nonlinear Analysis of Single-Layer Reticulated Spherical Shells Under Static and Dynamic Loads

Abstract: A nonlinear finite element technique is developed for analyzing the nonlinear static and dynamic responses as well as the nonlinear stability of single-layer reticulated shells under external loads, in which the nonlinear three-dimensional beam elements are employed. Using the updated Lagrangian formulation, we derive a tangent stiffness matrix of three-dimensional beam element, considering the geometric nonlinearity of the element. Moreover, the modified Newton-Raphson method is employed for the solution of the nonlinear equilibrium equations, and the Newmark-β method is adopted for determining the seismic response of single-layer reticulated shells. An improved arc-length method, in which the current stiffness parameter is used to reflect the nonlinear degree of such space structures, is presented for determining the load increment for the structural stability analysis. In addition, an accurate incremental method is developed for computing the large rotations of the space structures. The developed appro...

Branch Switching for Pitchfork Bifurcation in Nonlinear Finite Element Method

Abstract: A general branch-switching approach is proposed, based on pseudo arc-length continuation in the nonlinear finite element method (FEM), to trace the branch solution curve at the pitchfork bifurcation point in parameter space. By this approach, branch direction can be determined without derivatives of tangent stiffness matrix. This approach is proved mathematically and is inserted into a general FEM code. Stability of equilibrium corresponding to each solution curve is estimated by the Lagrange-Dirichlet criteria. As an example, the whole process of the elastic mode jumping for rectangular thin plates, in which there are many pitchfork bifurcations in parameter space in loading and unloading, is simulated by the improved FEM code.The numerical results are identical with that of an experiment done by previous researchers.

Consistent Atomic-Scale Finite Element: A Unified Framework for the Multiscale Modeling of Carbon Nanostructures

Abstract: A new multiscale modeling technique called the Consistent Atomic Finite Element (CAFE) is introduced. Unlike traditional approaches for linking the atomic structure to its equivalent continuum, this method directly connects the atomic degrees of freedom to a reduced set of finite element degrees of freedom without passing through an intermediate homogenized continuum. As a result, there is no need to introduce stress and strain measures at the atomic level. The Tersoff-Brenner interatomic potential is used to calculate the consistent tangent stiffness matrix of the structure. In this finite element formulation, all local and non-local interactions between carbon atoms are taken into account using overlapping finite elements. In addition, a consistent hierarchical finite element modeling technique is developed for adaptively coarsening and refining the mesh over different parts of the model. This process is consistent with the underlying atomic structure and, by refining the mesh, molecular dynamic results will be recovered. This method is valid across the scales and can be used to concurrently model atomistic and continuum phenomena. Applicability of the method is shown with several examples of deformation of a graphite sheet under different loadings and boundary conditions.Copyright © 2004 by ASME

Design of lateral braces for columns considering critical imperfection of buckling

Abstract: The present paper discusses the design of column-type structures, which are composed of columns and lateral braces attached perpendicular to the columns. Buckling of the braces of this kind of structures directly leads to global buckling of the columns. The brace-buckling modes are successfully detected by considering higher-order geometrically nonlinear relations and by introducing Green's strain into the total potential energy of the structure. Sensitivity analysis of the eigenvalues of the tangent stiffness matrix under fixed load condition is carried out with respect to imperfections of the nodal locations. Furthermore, the critical imperfection that most drastically reduces the eigenvalues are calculated and buckling loads of the imperfect systems are evaluated. The numerical results show that the second or higher eigenmode of the tangent stiffness matrix of the perfect system should be sometimes used for estimating the buckling load of the imperfect system. Design examples are presented using the proposed method, and they are compared with those in accordance with an allowable-stress design standard. The results show a possibility of reducing the sizes of the brace sections.

Derivation of the geometric stiffness matrix of a truss element from a simple physical concept

Abstract: The derivation of the geometric stiffness matrix of a truss element based on simple physical way is presented in this study. It is derived based on the extension of the physical concept of rigid body motions. The derived element has abilities to simulate physical properties when it undergoes rigid body motions. Then it is combined with the linear stiffness matrix of a truss to form the tangent stiffness matrix of the truss element. This stiffness matrix was used in the prediction stage of geometrical non-linear analysis of truss structures or linear buckling analysis of truss structures. The nonlinearly incremental equations were solved iteratively by using an effective numerical method, the General Displacement Control (GDC) method. A truss dome structure was analysed by the proposed elements. Reliable results are sure through comparisons with studies done in the literature.

Solution of the Lyapunov matrix equation for a system with a time-dependent stiffness matrix

Abstract: The stability of the linearized model of a rotor system with non-symmetric strain and axial loads is investigated. Since we are using a fixed reference system, the differential equations have the advantage to be free of Coriolis and centrifugal forces. A disadvantage is nevertheless the occurrence of time-dependent periodic terms in the stiffness matrix. However, by solving the Lyapunov matrix equation we can formulate several stability conditions for the rotor system. Hereby the positive definiteness of a certain averaged stiffness matrix plays a crucial role.

Static stiffness study onparallel wire driven manipulators

Abstract: In this paper the static stiffness of a parallel wire driven robots (PWDRs) is theoretically studied. At first, based on the differential motion principle, a proposition about the small change of the Wire Matrix is presented and proved. And then the stiffness matrix of the manipulator consisting of structural-parameter-related stiffness and wire-tension-related stiffness is produced. Finally, the numerical simulation is done for verifying the conclusion. It is shown that the stiffness of this kind of manipulator depends not only on its structural parameters including the physical dimensions, the stiffness of wires and motors and the position and pose of the end effecter, but also on the tension of the wires. Those works lead to the conclusion that the stiffness of the PWDRs can be regulated by changing the internal tension of wires, and a variable stiffness mechanism can be realized.

Approximation of discontinuous curves and surfaces by discrete splines with tangent conditions

Abstract: This paper concerns the construction of a discontinuous paramet- ric curve or surface from a finite set of points and tangent conditions. The method is adapted from the theory of the discrete smoothing variational splines to introduce a discontinuity set and some tangent conditions. Such method is justified by a convergence result.