Large Deformation Finite Element Formulation Research Articles

Coupled Large Deformation Finite Element Formulations for the Dynamics of Unsaturated Soil and Their Application

Unsaturated soil is a three-phase medium with three interfaces, and the mathematical equations that represent its behavior must be developed in a fully coupled manner for accurately predicting its hydromechanical behavior. In this paper, a set of fully coupled governing equations was developed for the dynamics of unsaturated soil, considering the interaction among the bulk phases and interfaces. In addition to implementing the complete governing equations, a simplified formulation was developed for practical applications. The derivation of the finite element formulation considering all the terms in the partial differential equations resulted in a formulation called complete formulation and was solved for the first time in this paper. Another formulation called reduced formulation was derived by neglecting the relative accelerations and velocities of water and air in the governing equations. In addition, small and large deformation theories were developed and implemented for both formulations. To show the applicability of the proposed models, the dynamic behavior of an unsaturated soil embankment was simulated using both small and large deformation formulations by applying minor and severe earthquakes. The reduced formulation was found to be computationally efficient and numerically stable. The smaller displacements predicted by large deformation theories show that the results are consistent with the expected behavior. Large deformation theories are considered suitable when the geotechnical system undergoes large deformation and may lead to accurate prediction.

Synthesis of C0 Path-Generating Contact-Aided Compliant Mechanisms Using the Material Mask Overlay Method

Contact-aided compliant mechanisms (CCMs) are synthesized via the material mask overlay strategy (MMOS) to trace desired nonsmooth paths. MMOS employs hexagonal cells to discretize the design region and engages negative circular masks to designate material states. To synthesize CCMs, the modified MMOS presented herein involves systematic mutation of five mask parameters through a hill climber search to evolve not only the continuum topology but also to position the rigid, interacting surfaces within some masks. To facilitate analysis with contact, boundary smoothing is performed by shifting boundary nodes of the evolving continuum. Various geometric singularities are subdued via hexagonal cells, and the V-notches at the continuum boundaries are alleviated. Numerous hexagonal cells get morphed into concave subregions as a consequence. Large deformation finite-element formulation with mean-value coordinates based shape functions is used to cater to the generic hexagonal shapes. Contact analysis is accomplished via the Newton–Raphson (NR) iteration with load incrementing in conjunction with the augmented Lagrange multiplier method and active set constraints. An objective function based on Fourier shape descriptors (FSDs) is minimized subject to suitable design constraints. Two examples of path-generating CCMs are presented, their performance compared with a commercial software and fabricated to establish the efficacy of the proposed synthesis method.

An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods

The objective of this work is the development of a new finite element formulation for beams according to the Kirchhoff theory of thin rods, which includes the deformation states of axial tension, torsion and bending. The proposed formulation accounts for large deformations in three-dimensional problem settings consisting of slender, prismatic beams with arbitrarily curved initial geometries and arbitrary cross-section shapes. Like in geometrically exact Reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations, rotations and strains. A novel orthogonal interpolation strategy is applied to the triad field representing the cross-section orientation in order to fulfill the Kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem. The continuity requirements resulting from the corresponding weak form are fulfilled by a cubic Hermite interpolation, thus leading to a C1-continuous representation of the beam centerline. Fundamental properties such as objectivity, path-independence, consistency and accuracy of the developed beam element are verified by means of suitable numerical examples.

A nonlinear visco-elastic constitutive model for large rotation finite element formulations

Although all known materials have internal damping that leads to energy dissipation, most existing large deformation visco-elastic finite element formulations are based on linear constitutive models or on nonlinear constitutive models that can be used in the framework of an incremental co-rotational finite element solution procedure. In this investigation, a new nonlinear objective visco-elastic constitutive model that can be implemented in non-incremental large rotation and large deformation finite element formulations is developed. This new model is based on developing a simple linear relationship between the damping forces and the rates of deformation vector gradients. The deformation vector gradients can be defined using the decomposition of the matrix of position vector gradients. In this paper, the decomposition associated with the use of the tangent frame that is equivalent to the QR decomposition is employed to define the matrix of deformation gradients that enter into the formulation of the viso-elastic constitutive model developed in this investigation. Using the relationship between the deformation gradients and the components of the Green–Lagrange strain tensor, it is shown that the damping forces depend nonlinearly on the strains and linearly on the classical strain rates. The relationship between the damping forces and strains and their rates is used to develop a new visco-elastic model that satisfies the objectivity requirements and leads to zero strain rates under an arbitrary rigid body displacement. The linear visco-elastic Kelvin–Voigt model frequently used in the literature can be obtained as a special case of the proposed nonlinear model when only two visco-elastic coefficients are used. As demonstrated in this paper, the use of two visco-elastic coefficients only leads to viscous coupling between the deformation gradients. The model developed in this investigation can be used in the framework of large deformation and large rotation non-incremental solution procedure without the need for using existing co-rotational finite element formulations. The finite element absolute nodal coordinate formulation (ANCF) that allows for straightforward implementation of general constitutive material models is used in the validation of the proposed visco-elastic model. A comparison with the linear visco-elastic model is also made in this study. The results obtained in this investigation show that there is a good agreement between the solutions obtained using the proposed nonlinear model and the linear model in the case of small deformations.

Modeling and simulation of the coupled mechanical–electrical response of soft solids

Dielectric elastomer (DE) is one type of electro-active polymers (EAP) that responds to electrical stimulation with a significant shape and size change. As EAPs, dielectric elastomers are lightweight, inexpensive, pliable and can be fabricated into various shapes, all of which are attractive properties to justify the intense research in the field. This paper presents a nonlinear, electrical and mechanical coupled, large deformation finite element formulation for DEAs. Maxwell’s equations for the electroquasistatic fields were solved simultaneously with equation of linear momentum. The hyperelastic Ogden model and total Maxwell stress method were combined to describe the material. The formulation was based on the weak forms of Maxwell’s equation and linear momentum expressed in the reference configuration. The closed form consistent tangent moduli for dielectric elastomers were derived. The results of the simulation compared with the experiments have demonstrated the validity of the method from the computational aspect.

Finite element modeling of tire/terrain interaction: Application to predicting soil compaction and tire mobility

Tire/terrain interaction has been an important research topic in terramechanics. For off-road vehicle design, good tire mobility and little compaction on terrain are always strongly desired. These two issues were always investigated based on empirical approaches or testing methods. Finite element modeling of tire/terrain interaction seems a good approach, but the capability of the finite element has not well demonstrated. In this paper, the fundamental formulations on modeling soil compaction and tire mobility issues are further introduced. The Drucker–Prager/Cap model implemented in ABAQUS is used to model the soil compaction. A user subroutine for finite strain hyperelasticity model is developed to model nearly incompressible rubber material for tire. In order to predict transient spatial density, large deformation finite element formulation is used to capture the configuration change, which combines with soil elastoplastic model to calculate the transient spatial density due to tire compaction on terrain. Representative simulations are provided to demonstrate how the tire/terrain interaction model can be used to predict soil compaction and tire mobility in the field of terramechanics.

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.

Modeling of pile installation using contact mechanics and quadratic elements

In geotechnical engineering, numerical analysis of pile capacity is often performed in such a way that piles are modeled using only the geometry of their final position in the ground and simply loaded to failure. In these analyses, the stress changes caused by the pile installation are neglected, irrespective of the installation method. For displacement piles, which are either pushed or hammered into the ground, such an approach is a very crude simplification. To model the entire installation process of displacement piles a number of additional nonlinear effects need to be considered. As the soil adjacent to the pile is displaced significantly, small deformation theory is no longer applicable and a large deformation finite element formulation is required. In addition, the continuously changing interface between the pile and the soil has to be considered. Recently, large deformation frictional contact has been used to model the pile installation and cone penetration processes. However, one significant limitation of the analysis was the use of linear elements, which have proven to be less accurate than higher order elements for nonlinear materials such as soils. This paper presents a large deformation frictional contact formulation which can be coupled consistently with quadratic solid elements. The formulation uses the so-called mortar-type discretisation of the contact surfaces. The performance of this contact discretisation technique is demonstrated by accurately predicting the stress transfer between the pile and the soil surfaces.

Contact friction simulation in powder compaction process based on the penalty approach

In this paper, the influence of powder–tool friction on the mechanical properties of the final product is investigated in pressing metal powders. A computational algorithm is presented for simulation of frictional contact in the compaction process of powder. The large deformation finite element (FE) formulation is characterized by the use of penalty approach in which a plasticity theory of friction is incorporated to simulate sliding resistance at the powder–tool interface. The constitutive relations for friction are derived from a Coulomb friction law. A double-surface cap plasticity model is employed together with the nonlinear contact friction behavior in numerical simulation of powder material. The contact friction algorithm together with the cap plasticity model is incorporated within the framework of large FE deformation in order to predict the non-uniform relative density distribution during large deformation of powder die pressing. Finally, the numerical schemes are examined for efficiency in modeling of a set of powder components.

Integration of Large Deformation Finite Element and Multibody System Algorithms

This paper presents an overview of research and development efforts that are currently being devoted to integrate large deformation finite element formulations with flexible multibody system algorithms. The goal is to develop computer simulation capabilities for the analysis of physics and engineering models with significant details. The successful development of such new and integrated algorithms will also allow modeling and simulation of systems that cannot be solved using existing computer algorithms and codes. One of the main difficulties encountered in this integration process is attributed to the fact that the solution procedures used in finite element codes differ significantly from those used in general-purpose flexible multibody system codes. Finite element methods employ the corotational formulations that are often used with incremental solution procedures. Flexible multibody computer codes, on the other hand, do not, in general, use incremental solution procedures. Three approaches are currently being explored by academic institutions and the software industry. In the first approach, gluing algorithms that aim at performing successful simulations by establishing an interface between existing codes are used. Using different coordinates and synchronizing the time stepping are among several challenging problems that are encountered when gluing algorithms are used. In the second approach, multibody system capabilities are implemented in existing finite element algorithms that are based on large rotation vector formulations. For the most part, corotational formulations and incremental solution procedures are used in this case. In the third approach, a new large deformation finite element formulation that can be successfully implemented in flexible multibody system computer algorithms that employ nonincremental solution procedures is introduced. The approach that is now being developed in several institutions is based on the finite element absolute nodal coordinate formulation. Such a formulation can be systematically implemented in general-purpose flexible multibody system computer algorithms. Nonlinear constraint equations that describe mechanical joints between different bodies can be formulated in terms of the absolute coordinates in a straightforward manner. The coupling between the motion of rigid, flexible, and very flexible bodies can also be accurately described. The successful integration of large deformation finite element and multibody system algorithms will lead to a new generation of computer codes that can be systematical and efficiently used in the analysis of many engineering applications.

Deformation modes in the finite element absolute nodal coordinate formulation

The objective of this study is to provide interpretation of the deformation modes in the finite element absolute nodal coordinate formulation using several strain definitions. In this finite element formulation, the nodal coordinates consist of absolute position coordinates and gradients that can be used to define a unique rotation and deformation fields within the element as well as at the nodal points. The results obtained in this study clearly show cross-section deformation modes eliminated when the number of the finite element nodal coordinates is systematically and consistently reduced. Using the procedure discussed in this paper one can obtain a reduced order dynamic model, eliminate position vector gradients that introduce high frequencies to the solution of some problems, achieve the continuity of the remaining gradients at the nodal points, and obtain a formulation that automatically satisfies the principle of work and energy. Furthermore, the resulting dynamic model, unlike large rotation finite element formulations, leads to a unique rotation field, and as a consequence, the obtained formulation does not suffer from the problem of coordinate redundancy that characterizes existing large deformation finite element formulations. In order to accurately define strain components that can have easy physical interpretation, a material coordinate system is introduced to define the material element rotation and deformation. Using the material coordinate system, the Timoshenko–Reissner and Euler –Bernoulli beam models can be systematically obtained as special cases of the absolute nodal coordinate formulation beam models. While a constraint approach is used in this study to eliminate the cross-section deformation modes, it is important to point out as mentioned in this paper that lower-order finite elements, some of which already presented in previous investigations, can be efficiently used in thin and stiff structure applications.

A finite element analysis of the flange earrings of strong anisotropic sheet metals in deep-drawing processes

Flange earrings of strong anisotropic sheet metals in deep-drawing process are numerically analyzed by the elastic-plastic large deformation finite element formulation based on a discrete Kirchhoff triangle plate shell element model. A Barlat-Lian anisotropic yield function and a quasi-flow corner theory are used in the present formulation. The numerical results are compared with the experimental ones of cylindrical cup drawing process. The focus of the present researches is on the numerical analysis and the constraining scheme of the flange earring of circular sheets with strong anisotropy in square cup drawing process.

The numerical analysis of anisotropic sheet metals in deep-drawing processes

Flange earing of strong anisotropic sheet metals in the deep-drawing process is analyzed numerically by the elastic–plastic large deformation finite element formulation based on the discrete Kirchhoff triangle plate shell element model. A Barlat–Lian anisotropy yield function and a quasi-flow corner theory are introduced in the present formulation. The numerical results are compared with the experimental results of the cylindrical cup drawing process. The focus of the present research is on the numerical analysis and the constraining scheme of the flange earing of circular sheets with strong anisotropy in the square cup drawing process.

Numerical study of the flange earring of deep-drawing sheets with stronger anisotropy

A Barlat–Lian anisotropy yield function is introduced into a quasi-flow corner theory of elastic–plastic finite deformation and the elastic–plastic large deformation finite element formulation based on the principle of virtual velocity and the discrete Kirchhoff triangle plate shell element model. The focus of the present researches is on the numerical simulation of the flange earring of deep-drawing process of circular sheets with stronger anisotropy, based on which, the schemes for controlling the flange earring are proposed.

Equivalence of the floating frame of reference approach and finite element formulations

The floating frame of reference formulation which is widely used in flexible multibody simulations leads to a highly non-linear inertia matrix. This matrix which exhibits a strong inertia coupling between the reference motion and the elastic deformation can be expressed in terms of a unique set of inertia shape integrals that depend on the assumed displacement field. In this paper, we demonstrate that the floating frame of reference formulation leads to the same dynamic relationship obtained using large deformation finite element formulations. This result clearly demonstrates that the floating frame of reference formulation can also be used in the large deformation analysis of constrained flexible bodies provided that no infinitesimal or finite rotations are used as nodal coordinates; instead, displacements and slopes are used to describe the element configuration. The relationship between the local and the global slopes is defined and used to establish a coordinate transformation which is used to demonstrate the equivalence of the dynamic relationships and the inertia forces obtained using two different finite element formulations. Using this equivalence relationship, a simple and systematic procedure for evaluating all the inertia shape integrals that appear in the floating frame of reference formulation from the constant mass matrix that appears in linear structural dynamics is developed. In order to develop this procedure, the concepts of the local and global shape functions are introduced. The local shape function, which is used in the floating frame of reference formulation, does not have any rigid body modes and is defined using an appropriate set of reference conditions. The global shape function, on the other hand, has a complete set of rigid body modes that can describe an arbitrary rigid body displacement. The analysis presented in this paper demonstrates that the floating frame of reference formulation does not lead to a separation between the rigid body motion and the elastic deformation, and in such a formulation, the reference motion cannot be considered as the rigid body motion of the deformable body.

Large displacement transient analysis of flexible structures

AbstractA new approach for numerical analysis of flexible space frames subjected to impact forces is presented. The method is particularly effective in the analysis of large‐amplitude vibration and dynamic stability found in extremely flexible structures.The proposed method of analysis is based on a concept of updated material frame, which allows the algorithm of handling large rotations exceeding 360° and extensions longer than 100 per cent without numerical difficulty and the need for iterations. Thus, an efficient explicit integration finite element algorithm can be adopted. The method is convenient for non‐linear and inelastic material models.The basic coding procedure is similar to the popular co‐rotational approach. Existing codes can be easily modified to solve transient dynamic problems. The modification steps necessary for the implementation are outlined in the paper.Several numerical examples are presented. Comparison with the standard co‐rotational approach and large deformation finite element formulations, such as the updated Lagrangian approach, are presented.