General Continuum Mechanics Approach Research Articles

In-plane nonlinear postbuckling analysis of circular arches using absolute nodal coordinate formulation with arc-length method

In this study, Absolute Nodal Coordinate Formulation (ANCF) in conjunction with Crisfield’s arc-length method is utilized in order to predict the nonlinear postbuckling behaviour of circular arches. The whole primary equilibrium path in load-displacement space of circular arches under central concentrated load is obtained. Three ANCF based approaches, i.e., the conventional two-dimensional fully parameterized shear deformable ANCF beam element based on the General Continuum Mechanics (GCM) approach, the same element modified by the Strain Split Method (SSM) approach and the ANCF planar Higher Order Beam Element (HOBE) with GCM approach are used. Circular arches with various geometric configurations and boundary conditions such as clamped-clamped, hinged-hinged, clamped-hinged and three-hinged arches are studied which exhibit nonlinear response in the form of snap-through, snap-back and looping phenomenon. The obtained results are compared with the analytical solutions, experimental result (where available in the literature) and numerical approximations (by using the commercially available FEM package). In this paper, the recently proposed ANCF based approaches are successfully implemented which validate and verify the utility of ANCF in nonlinear postbuckling analysis. The characteristics of the three approaches with regard to the adoptability of arc-length method are compared and discussed.

Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.

A geometrically accurate deformable-body approach for the analysis of robotic and parallel-mechanism systems

Strength and durability investigations of parallel-mechanism (PM) and robotic systems, used as integral part in modern manufacturing, are necessary at the virtual design stage. Nonetheless, because of the limitations of existing finite element (FE) approaches, flexible-link geometries are often simplified or distorted when converting solid models to FE analysis meshes. This article introduces and demonstrates the use of a new unified geometry/analysis approach for the small-deformation analysis of robotic and parallel mechanism (RPM) systems with flexible links. The approach used in this investigation integrates the geometrically accurate absolute nodal coordinates formulation (ANCF) with the computationally efficient floating frame of reference (FFR) formulation that allows for systematic elimination of high-frequency and insignificant deformation modes. The proposed ANCF/FFR approach allows modeling accurately the stress-free reference configuration geometry, captures the dynamic coupling between the rigid-body and elastic displacements, is based on a unified geometry/analysis mesh that eliminates the need for geometry/analysis model conversion or coordinate transformation, and allows for efficient and accurate solution of the nonlinear dynamic equations that govern the RPM motion. The formulation of the spatial ANCF/FFR equations of motion including the elastic (stress) forces using a general continuum-mechanics approach is presented, and use of the strain split method (SSM) as an FE locking-alleviation technique is discussed. A frequency-convergence analysis of flexible links with tapered geometry is performed and the obtained numerical results are compared with solutions obtained using commercial FE software. The application of the new procedure in the nonlinear dynamics simulations of RPM systems is demonstrated using a spatial parallel mechanism that includes flexible links and prismatic, revolute, and spherical joints.

Development of Geometrically Accurate Continuum-Based Tire Models for Virtual Testing

AbstractIn this paper, an approach based on the integration of computer-aided design and analysis (I-CAD-A) is used to develop new continuum-based finite element (FE) tire models for the small and large deformation analyses. Based on given tire specifications, the mechanics-based geometry/analysis absolute nodal coordinate formulation (ANCF) is used to define the tire geometry with the same degree of accuracy as B-splines and nonuniform rational B-spline (NURBS), widely used in the computer-aided design (CAD) systems. In the case of large deformations, the ANCF geometry can be used directly as the analysis mesh without the need for conversion or adjustments. In order to define the material parameters that characterize the ANCF tire composite structure, a virtual test rig is developed, and the tire calibration process is performed according to the standards defined by the Society of Automotive Engineers (SAE). In order to develop small-deformation models that can be used in the prediction of the tire frequencies and mode shapes, the ANCF position vector gradients are consistently written in terms of rotation parameters, leading to geometrically accurate floating frame of reference (FFR) finite elements, referred to as ANCF/FFR elements. Using this mechanics-based geometry/analysis approach, new geometrically accurate reduced-order tire models are systematically developed and used to define vibration equations for the prediction of the tire frequencies, which are verified using a commercial FE software. The element stiffness matrix is calculated using the general continuum mechanics approach (GCM), and the effectiveness of the strain split method (SSM) for locking alleviation is tested. The results obtained in this investigation show that the I-CAD-A tire modeling approach can be used to develop geometrically accurate tire models suited for the large-deformation multibody system (MBS) problems as well as for the prediction of the tire frequencies and mode shapes.

Development and implementation of geometrically accurate reduced-order models: Convergence properties of planar beams

A geometrically accurate infinitesimal-rotation planar beam element is developed and implemented in this study. The performance of the element, which is suited for developing reduced-order models for both structural and multibody systems (MBS), is evaluated using an eigenvalue analysis. Unlike conventional infinitesimal-rotation finite elements (FE), the new element is compatible with the computer-aided design (CAD) B-spline and NURBS (Non-Uniform Rational B-Spline) representations and allows for a straightforward linear transformation of CAD solid models to FE analysis meshes. The absolute nodal coordinate formulation (ANCF) elements, which are related to B-splines and NURBS by linear mapping, are used as the basis for developing the planar beam element. The new element has a shape function matrix expressed in terms of geometric coefficients obtained using the ANCF position vector gradients in the reference configuration. The change in the position vector gradients is written in terms of infinitesimal rotation coordinates using a velocity transformation that defines constant element mass and stiffness matrices. Using this approach, initially straight and curved configurations can be modeled using the same displacement field. The eigenvalue analysis is used to evaluate the element performance and examine the effect of shear locking on the predicted frequencies. Several elastic force formulations are used to evaluate the convergence characteristics, including the direct displacement method (DDM), general continuum mechanics (GCM) approach, elastic line (EL) approach, and strain split method (SSM). The element performance is compared with the conventional Euler-Bernoulli and Timoshenko elements as well as the analytical solutions.

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

Prediction of dynamic stresses using flexible multibody system algorithms: Application to tracked vehicle upper structure

The objective of this investigation is to develop a general procedure for predicting the dynamic stresses when the nonlinear floating frame of reference formulation is used to model flexible components in multibody system applications. The procedure utilizes the concept of the floating frame of reference intermediate finite element coordinate system and the general continuum mechanics approach to develop accurate definitions of the floating frame of reference kinematic equations that enter into the formulations of the strains. These general equations are used to design a computational algorithm for the calculation of the dynamic stresses. In order to demonstrate the use of the procedure developed in this investigation for the stress prediction, a tracked vehicle system is used as the study model. Taking into consideration the actual endurance test, a model for a bulldozer upper structure is developed using computer aided engineering solution framework which includes flexible multibody system algorithms. In order to obtain the dynamic stresses, the flexible structure of the tracked vehicle is modeled using Mindlin shell finite elements. The equations of motion are developed using the finite element/floating frame of reference formulation that allows for effectively reducing the number of degrees of freedom by neglecting high-frequency modes of vibration using component mode synthesis techniques. The stresses are obtained using a procedure that takes into consideration the dynamic coupling between the rigid body motion and the elastic deformations. The use of the general continuum mechanics theory to formulate the strain and stress equations allows for applying the procedure developed in this paper to other element types. Kinematic relationships are used to formulate several components and elements of the undercarriage including the idler cushion system and the maximum reverse angle. Numerical results are presented in order to demonstrate the use of the general floating frame of reference procedure in the stress analysis of the tracked vehicle undercarriage system.

Three-Dimensional Fully Parameterized Triangular Plate Element Based on the Absolute Nodal Coordinate Formulation

In this work, a new three-dimensional fully parameterized triangular plate element based on the absolute nodal coordinate formulation (ANCF) is introduced. This plate element has 12 coordinates per node; therefore, it can be used in thick plate applications. The proposed 12 shape functions are obtained by adding three shape functions to the nine shape functions that were previously used with the ANCF thin triangular plate element. Unlike the existing ANCF thin triangular plate element, which allows only the use of classical Kirchoff's plate theory, the fully parameterized ANCF triangular plate element proposed in this work allows for the use of a general continuum mechanics approach and also allows for a straight forward implementation of general nonlinear constitutive equations. Moreover, all deformation modes including thickness deformation can be captured using the fully parameterized ANCF triangular plate element proposed in this paper. The numerical results obtained in this investigation show that in case of negligible deformation, the fully parameterized ANCF triangular plate element behaves like a rigid body. Moreover, it is found that there is a good agreement between the solutions obtained using the proposed fully parameterized ANCF triangular plate element and the theoretical model in the case of small deformations. Furthermore, it is shown that the results of the proposed element agree well with the results obtained using the existing fully parameterized ANCF rectangular plate element when large deformation conditions are applied. The twist behavior of the proposed element is verified by comparison with the results obtained using a conventional nonlinear rectangular plate element.

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.

Coupled Deformation Modes in the Large Deformation Finite Element Analysis: Generalization

The effect of the absolute nodal coordinate formulation (ANCF)–coupled deformation modes on the accuracy and efficiency when higher order three-dimensional beam and plate finite elements are used is investigated in this study. It is shown that while computational efficiency can be achieved in some applications by neglecting the effect of some of the ANCF-coupled deformation modes, such modes introduce geometric stiffening/softening effects that can be significant in the case of very flexible structures. As shown in previous publications, for stiff structures, the effect of the ANCF-coupled deformation modes can be neglected. For such stiff structures, the solution does not strongly depend on some of the ANCF-coupled deformation modes, and formulations that include these modes lead to numerical results that are in good agreement with formulations that exclude them. In the case of a very flexible structure, on the other hand, the inclusion of the ANCF-coupled deformation modes becomes necessary in order to obtain an accurate solution. In this case of very flexible structures, the use of the general continuum mechanics approach leads to an efficient solution algorithm and to more accurate numerical results. In order to examine the effect of the elastic force formulation on the efficiency and the coupling between different modes of deformation, three different models are used again to formulate the elastic forces in the absolute nodal coordinate formulation. These three methods are the general continuum mechanics approach, the elastic line (midsurface) approach, and the elastic line (midsurface) approach with the Hellinger–Reissner principle. Three-dimensional absolute nodal coordinate formulation beam and plate elements are used in this study. In the general continuum mechanics approach, the coupling between the cross section deformation and the beam centerline or plate midsurface displacement is considered, while in the approaches based on the elastic line and the Hellinger–Reissner principle, this coupling is neglected. In addition to the fully parametrized beam element used in this study, three different plate elements, two fully parametrized and one reduced order thin plate elements, are used. The numerical results obtained using different finite elements and elastic force formulations are compared in this study.

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.

Coupled Deformation Modes in the Large Deformation Finite-Element Analysis: Problem Definition

In the classical formulations of beam problems, the beam cross section is assumed to remain rigid when the beam deforms. In Euler–Bernoulli beam theory, the rigid cross section remains perpendicular to the beam centerline; while in the more general Timoshenko beam theory the rigid cross section is permitted to rotate due to the shear deformation, and as a result, the cross section can have an arbitrary rotation with respect to the beam centerline. In more general beam models as the ones based on the absolute nodal coordinate formulation (ANCF), the cross section is allowed to deform and it is no longer treated as a rigid surface. These more general models lead to new geometric terms that do not appear in the classical formulations of beams. Some of these geometric terms are the result of the coupling between the deformation of the cross section and other modes of deformations such as bending and they lead to a new set of modes referred to in this paper as the ANCF-coupled deformation modes. The effect of the ANCF-coupled deformation modes can be significant in the case of very flexible structures. In this investigation, three different large deformation dynamic beam models are discussed and compared in order to investigate the effect of the ANCF-coupled deformation modes. The three methods differ in the way the beam elastic forces are calculated. The first method is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second method is based on the elastic line approach that systematically eliminates these modes. The ANCF-coupled deformation modes eliminated in the elastic line approach are identified and the effect of such deformation modes on the efficiency and accuracy of the numerical solution is discussed. The third large deformation beam model discussed in this investigation is based on the Hellinger–Reissner principle that can be used to eliminate the shear locking encountered in some beam models. Numerical examples are presented in order to demonstrate the use and compare the results of the three different beam formulations. It is shown that while the effect of the ANCF-coupled deformation modes is not significant in very stiff and moderately stiff structures, the effect of these modes can not be neglected in the case of very flexible structures.

A Non-Incremental Nonlinear Finite Element Solution for Cable Problems

In this investigation, a nonlinear finite element method for the large deformation and rotation of cable problems is presented. This method is based on finite element absolute nodal coordinate formulation that guarantees the continuity of all displacement gradients and leads to a constant mass matrix. The classical cable theory is first reviewed and the assumptions used in this linear theory are defined in order to demonstrate the basic differences between the linear theory and the nonlinear finite element formulation proposed in this paper for cable applications. The elastic cable forces in the absolute nodal coordinate formulation are obtained using a general continuum mechanics approach that accounts for the effect of all geometric nonlinearities. It is shown in this investigation that the use of the general continuum mechanics approach leads to a simpler and more efficient formulation as compared to the use of the assumptions of the linear theory that employs a local finite element coordinate system. The results obtained using the absolute nodal coordinate formulation show a good agreement with the results obtained using the classical cable theory when linear cable problems are considered. In particular it is shown that the use of perturbation methods to linearize the finite element equations of motion leads to modal characteristics results that are in a good agreement with the linear theory. The results of this investigation obtained using explicit numerical integration also show the potential of the proposed finite element formulation in the nonlinear analysis of cables that experience large rotations and deformations. The generalization of the procedure presented in this paper to three-dimensional cable problems is demonstrated and the computer implementation in multibody algorithms is discussed.

A Non-Incremental Finite Element Procedure for the Analysis of Large Deformation of Plates and Shells in Mechanical System Applications

In this investigation, a non-incremental solution procedure for the finite rotationand large deformation analysis of plates is presented. The method, whichis based on the absolute nodal coordinate formulation, leads to plateelements capable of representing exact rigid body motion. In thismethod, continuity conditions on all the displacement gradients areimposed. Therefore, non-smoothness of the plate mid-surface at the nodalpoints is avoided. Unlike other existing finite element formulationsthat lead to a highly nonlinear inertial forces for three-dimensionalelements, the proposed formulation leads to a constant mass matrix, andas a result, the centrifugal and Coriolis inertia forces are identicallyequal to zero. Furthermore, the method relaxes some of the assumptionsused in the classical and Mindlin plate models and automatically satisfiesthe objectivity requirements. By using a generalcontinuum mechanics approach, a relatively simple expression for theelastic forces is obtained. Generalization of the formulation to thecase of shell elements is discussed. As examples of the implementationof the proposed method, two different plate elements are presented; oneplate element does not guarantee the continuity of the displacementgradients between the nodal points, while the other plate elementguarantees this continuity. Numerical results are presented in order todemonstrate the use of the proposed method in the large rotation anddeformation analysis of plates and shells. The numerical results, whichare compared with the results obtained using existing incrementalprocedures, show that the solution obtained using the proposed methodsatisfies the principle of work and energy. These results are obtainedusing explicit numerical integration method. Potential applications ofthe proposed method include high-speed metal forming, vehiclecrashworthiness, rotor blades, and tires.

Study of the Centrifugal Stiffening Effect Using the Finite Element Absolute Nodal Coordinate Formulation

The finite element absolute nodal coordinate formulation is usedin this investigation to study the centrifugal stiffening effect onrotating two-dimensional beams. It is demonstrated that the geometricstiffening effect can be automatically accounted for in the above mentionedfinite element formulation by using an expression for the elastic forcesobtained with a general continuum mechanics approach. The Hill equation thatgoverns the vibration of the rotating beam is obtained in terms of a set ofgeneralized coordinates that describe the beam displacements and slopes.Under the assumption of small deformation, the Hill equation is linearized,and the complete solution is obtained and used to demonstrate analyticallythat such a solution does not exhibit instabilities as the angular velocityof the beam increases. The results obtained using this finite elementprocedure are compared with the results reported in the literature.