Large Deformation 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.

An extended Hamilton functional for the thermodynamic topology optimization of hyperelastic structures

AbstractWe present our work on a new variational approach for thermodynamic topology optimization of hyperelastic structures: building upon our previous works, we follow a thermodynamic approach for deriving a field equation that describes the evolution of the density. The problem of topology optimization is consequently solved without the need of expensive optimization routines. Furthermore, our new formulation can also be applied to hyperelastic structures which show a remarkable difference to structures optimized for small deformations. Important aspects like tension/compression asymmetry and buckling are inherently included in the topology optimization approach due to the large deformation formulation.

Gradient-extended damage analysis with reduced integration-based solid-shells at large deformations

The present contribution is concerned with the extension of a recently presented non-local damage analysis of shells to the geometrically non-linear regime. For this purpose, a gradient-extended two-surface damage-plasticity model for finite strains is incorporated into a low-order solid-shell finite element formulation for large deformations which is based on reduced integration with hourglass stabilization. Due to a specific combination of the assumed natural strain (ANS) and the enhanced assumed strain (EAS) method, the most relevant locking pathologies which stem from the linear interpolation of the displacement field are eliminated. Furthermore, an additional micromorphic nodal degree of freedom which comes from the gradient extension is considered. A polynomial approximation of the kinematic as well as the constitutively dependent quantities within the weak forms provides a suitable hourglass stabilization which is computationally highly efficient, since the corresponding element residual vectors and stiffness matrices can be determined analytically. Three numerical examples of different elastic as well as elasto-plastic plate and shell configurations, reveal the ability of the present framework to efficiently and accurately predict the damage processes within both geometrically non-linear thin and thick-walled structures.

A continuum mechanical model for asphalt based on the particle size distribution: Numerical formulation for large deformations and experimental validation

In this contribution, a physically motivated continuum mechanical model for asphalt concrete and stone mastic asphalt (asphalt material) at large deformations is proposed. The rheological model consists of a short-term and a long-term part connected in series governing the description of the short-term and the long-term behavior (volume preserving) of the asphalt material, respectively. The rheological element assembly of the short-term part is physically motivated by the particle size distribution of the asphalt material under investigation (choice of several networks formed by different size fractions of the aggregate mixture) and represents the asphalt material’s mainly viscoelastic behavior to short-term loading, e.g. during contact with a rolling tire. The long-term part captures large inelastic (accumulated) deformations due to repetitive loading and enables to represent e.g. rut formation during a structural pavement analysis. An experimental testing program including digital image correlation (DIC) for strain measurements on cylindrical samples was set up and enabled the identification of the model parameters. Furthermore, specific model parameters have been identified from the information provided by the manufacturer’s data sheet of the asphalt material based on the particle size distribution of aggregates, the binder content and the void content.

Designing 2D auxetic structures using multi-objective topology optimization

The behaviour of the auxetic structure under external load was regarded as the behaviour of the compliant (flexible) mechanism. The multi-objective topological optimization, based on genetic algorithms and the finite element method, was used to find the optimal shape of such two-dimensional compliant mechanism in this study. The optimization was performed on a quarter of a double-symmetric representative unit cell, which is a building block of the symmetrical auxetic structure. Static linear computational simulations were performed to determine the mechanical response of cell topologies. The proposed method leads to a set of best solutions positioned on the Pareto front with different topologies and gives a broad overview of possible designs of new auxetic structures. The method is highly effective and can be easily extended to large deformation formulations, nonlinear elasticity or elasto-plasticity.

Droplets on an elastic membrane: Configurational energy balance and modified Young equation

We study wetting of an elastic membrane whose surface can resist stretch by surface stress. Through energy minimization, we obtain the general equilibrium equations and a configurational energy balance equation at the contact line, i.e., the Young equation, modified for an elastic membrane. We use a 3D large deformation formulation with general elastic constitutive behavior for the membrane and its surfaces. Our results allow for non-flat reference membrane shape, non-uniform and anisotropic tension and dependence of surface stress on surface strain – the generalized Shuttleworth effects. We use two examples to demonstrate the application of our theory: plane strain and axisymmetric wetting of an initially flat elastic membrane. A large deformation solution for an axisymmetric droplet on a deformable membrane is obtained numerically. We show that when surface elastic moduli for solid–vapor and solid–liquid interfaces are different, continuity of membrane strain, a key assumption used in many previous studies, no longer holds at the contact line.

Modeling and simulation of nonlinear electro-thermo-mechanical continua with application to shape memory polymeric medical devices

Shape memory materials have gained considerable attention thanks to their ability to change physical properties when subjected to external stimuli such as temperature, pH, humidity, electromagnetic fields, etc. These materials are increasingly used for a large number of biomedical applications. For applications inside the human body, contactless control can be achieved by the addition of electric and/or magnetic particles that can react to electromagnetic fields, thus leading to a composite biomaterial. The difficulty of developing accurate numerical models for smart materials results from their multiscale nature and from the multiphysics coupling of involved phenomena. This coupling involves electromagnetic, thermal and mechanical problems. This paper contributes to the multiphysics modeling of a shape memory polymer material used as a medical stent. The stent is excited by electromagnetic fields produced by a coil which can be wrapped around a failing organ. In this paper we develop large deformation formulations for the coupled electro-thermo-mechanical problem using the electric potential to solve the electric problem. The formulations are then discretized and solved using the finite element method. Results are validated by comparison with results in the literature.

A low-order locking-free hybrid discontinuous Galerkin element formulation for large deformations

In this work, a hybrid discontinuous Galerkin (dG) quadrilateral element formulation is presented. As usual in hybrid formulations, the interior of the elements is treated separately from the skeleton, which represents the element boundaries. These are kinematically decoupled, i.e., displacement jumps can occur between the skeleton and the interior of the elements. The global degrees of freedom (dofs) are defined on the skeleton as the displacements at the corners, which allows the implementation into existing finite element codes. As usual in hybrid dG-formulations, the degrees of freedom in the interior are condensed out on the element level, leading to the same number of global degrees of freedom as for continuous bilinear elements.Instead of using conventional shape functions in the interior, the deformation gradient F is assumed constant within the element. Furthermore, F is connected to the skeleton degrees of freedom via the weak form. This leads to a very simple formulation and implementation. The element is tested for several computational examples from the literature. Special choices of the penalty parameter are investigated, which are partially derived analytically. It is found that the element is free of volumetric and shear locking. Moreover, the convergence is similar to that of other well-known locking-free finite element formulations.

Design of flat pneumatic artificial muscles

Pneumatic artificial muscles (PAMs) have gained wide use in the field of robotics due to their ability to generate linear forces and motions with a simple mechanism, while remaining lightweight and compact. However, PAMs are limited by their traditional cylindrical form factors, which must increase radially to improve contraction force generation. Additionally, this form factor results in overly complicated fabrication processes when embedded fibers and sensor elements are required to provide efficient actuation and control of the PAMs while minimizing the bulkiness of the overall robotic system. In order to overcome these limitations, a flat two-dimensional PAM capable of being fabricated using a simple layered manufacturing process was created. Furthermore, a theoretical model was developed using Von Karman’s formulation for large deformations and the energy methods. Experimental characterizations of two different types of PAMs, a single-cell unit and a multi-cell unit, were performed to measure the maximum contraction lengths and forces at input pressures ranging from 0 to 150 kPa. Experimental data were then used to verify the fidelity of the theoretical model.

Instability Analysis of Sand under Undrained Biaxial Loading with Rigid and Flexible Boundary

AbstractThe emergence of possible instability modes under undrained biaxial shear of sand was investigated to develop better understanding about the onset of localized and liquefaction-type solid-fluid instability modes using a generalized three-dimensional (3D) material model. These two modes are the primary instability modes that can exist in most cases while conducting undrained biaxial testing of sand at different densities, confining pressures, and boundary conditions. Using a 3D rate-independent and nonassociative constitutive model, the instability analysis was performed as a plane-strain bifurcation problem from a uniform stress-strain state. Large deformation formulation was used to simulate the biaxial test configuration with both rigid and flexible boundaries in the lateral direction. The existing theoretical framework of solid-fluid instability analysis under rigid boundaries was extended to the flexible lateral boundary condition. Interestingly, the onset of solid-fluid instability modes is s...

An Eulerian formulation for large deformations of elastically isotropic elastic-viscoplastic membranes

An Eulerian formulation for large deformations of elastically isotropic elastic-viscoplastic membranes

An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors mi that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors si are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of mi, which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.

Simulation-based investigations on the drape behavior of 3D woven fabrics made of commingled yarns

Composites based on 3D woven fabrics offer excellent mechanical properties due to the nearly non-crimp reinforcement fibers. Additionally, a reduced number of process steps results from the 3D nature of the reinforcement. Their in-plane and out-of-plane behaviors are purposefully adjustable to the expected loading conditions. A previously introduced modelling approach is extended to the simulation of 3D woven fabrics. The tensile, shear and bending behaviors of the fabric are considered in a material formulation for large deformations using shell elements. The model is successfully validated and parameter studies show the significant influence of shear and bending parameters on forming results. Furthermore, it is shown that also the forming process parameters have a significant influence on the draping results. It is concluded that the shearing and wrinkling can be reduced to a minimum with adapted material and process parameters.

An alternative approach for quasi‐static large deformation analysis of saturated porous media using meshfree method

SummaryAn alternative coupled large deformation formulation combined with a meshfree approach is proposed for flow–deformation analysis of saturated porous media. The formulation proposed is based on the Updated Lagrangian (UL) approach, except that the spatial derivatives are defined with respect to the configuration of the medium at the last time step rather than the configuration at the last iteration. In this way, the Cauchy stresses are calculated directly, rendering the second Piola–Kirchhoff stress tensor not necessary for the numerical solution of the equilibrium equations. Moreover, in contrast with the UL approach, the nodal shape function derivatives are calculated once in each time step and stored for use in subsequent iterations, which reduces the computational cost of the algorithm. Stress objectivity is satisfied using the Jaumann stress rate, and the spatial discretisation of the governing equations is achieved using the standard Galerkin method. The equations of equilibrium are satisfied directly, and the nonlinear parts of the system matrix are derived independent of the stresses of the medium resulting in a stable numerical algorithm. Temporal discretisation is effected based on a three‐point approximation technique that avoids spurious ripple effects and has second‐order accuracy. The radial point interpolation method is used to construct the shape functions. The application of the formulation and the significance of large deformation effects on the numerical results are demonstrated through several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

High Resolution, Large Deformation 3D Traction Force Microscopy

Traction force microscopy (TFM) is a powerful approach of quantifying cell-material interactions, which over the last two decades has contributed significantly to our understanding of cellular mechanosensing and mechanotransduction. In addition, recent advances in three-dimensional (3D) imaging and traction force analysis (3D TFM) have highlighted the significance of the third dimension in influencing various cellular processes. Yet irrespective of dimensionality almost all TFM approaches have relied on a linear elastic theory framework to calculate cell surface tractions.This talk presents a new high-resolution 3D TFM algorithm, which utilizes a large deformation formulation to quantify cellular displacement fields with unprecedented resolution. The results feature some of the first experimental evidence that cells are indeed capable of exerting large material deformations, which require the formulation of a new theoretical TFM framework to accurately calculate traction forces. Based on our previous 3D TFM technique we reformulate our approach to accurately account for large material deformation and quantitatively contrast and compare both linear and large deformation frameworks as a function of the applied cell deformation. Particular attention is paid in estimating the accuracy penalty associated with utilizing a traditional linear elastic approach in the presence of large deformation gradients.

High Resolution, Large Deformation 3D Traction Force Microscopy

Traction Force Microscopy (TFM) is a powerful approach for quantifying cell-material interactions that over the last two decades has contributed significantly to our understanding of cellular mechanosensing and mechanotransduction. In addition, recent advances in three-dimensional (3D) imaging and traction force analysis (3D TFM) have highlighted the significance of the third dimension in influencing various cellular processes. Yet irrespective of dimensionality, almost all TFM approaches have relied on a linear elastic theory framework to calculate cell surface tractions. Here we present a new high resolution 3D TFM algorithm which utilizes a large deformation formulation to quantify cellular displacement fields with unprecedented resolution. The results feature some of the first experimental evidence that cells are indeed capable of exerting large material deformations, which require the formulation of a new theoretical TFM framework to accurately calculate the traction forces. Based on our previous 3D TFM technique, we reformulate our approach to accurately account for large material deformation and quantitatively contrast and compare both linear and large deformation frameworks as a function of the applied cell deformation. Particular attention is paid in estimating the accuracy penalty associated with utilizing a traditional linear elastic approach in the presence of large deformation gradients.

Optimal Trajectory Planning for Flexible Link Manipulators with Large Deflection Using a New Displacements Approach

The main objective of the present paper is to determine the optimal trajectory of very flexible link manipulators in point-to-point motion using a new displacement approach. A new nonlinear finite element model for the dynamic analysis is employed to describe nonlinear modeling for three-dimensional flexible link manipulators, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. In comparison to other large deformation formulations, the motion equations contain constant stiffness matrix because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms. In this investigation, the computational method to solve the trajectory planning problem is based on the indirect solution of open-loop optimal control problem. The Pontryagin's minimum principle is used to obtain the optimality conditions, which is lead to a standard form of a two-point boundary value problem. The proposed approach has been implemented and tested on a single-link very flexible arm and optimal paths with minimum effort and minimum vibration are obtained. The results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the problem.

A large deformation formulation for fluid flow in a progressively fracturing porous material

A general numerical model has been developed for fluid flow in a progressively fracturing porous medium subject to large deformations. The fluid flow away from the crack is modelled in a standard manner using Darcy’s relation. In the discontinuity a similar relation is assumed for the fluid flow, but with a different permeability to take into account the higher porosity within the crack due to progressive damage evolution. The crack is described in a discrete manner by exploiting the partition-of-unity property of finite element shape functions. The nucleation and the opening of micro-cracks are modelled by a traction-separation relation. A heuristic approach is adopted to model the orientation of the cracks at the interfaces in the deformed configuration. A two-field formulation is derived, with the solid and the fluid velocities as unknowns. The weak formulation is obtained, assuming a Total Lagrangian formulation. This naturally leads to a set of coupled equations for the continuous and for the discontinuous parts of the mixture. The resulting discrete equations are nonlinear due to the cohesive-crack model, the large-deformation kinematic relations, and the coupling terms between the fine scale and the coarse scale. The capabilities of the model are shown at the hand of some example problems.

THE EFFECT OF ASYMMETRY AND AXIAL PRESTRAINING ON THE AMPLITUDE OF MECHANICAL STRESSES IN ABDOMINAL AORTIC ANEURYSM

In this work we study the effect of asymmetry and axial prestraining on the maximum effective mechanical stress in relatively small human abdominal aortic aneurysm (AAA) during a cardiac cycle. Our model is based on the fluid-structure interaction (FSI) methodology. The arterial wall is modeled using large strain and large deformation formulation, with hyperelastic material behavior, and isotropic nonlinear strain energy function (SEF) fitted to the averaged data set of biaxial tests of AAA tissue specimens. The results confirm that the magnitude of the maximum von Mises stress increases significantly with asymmetry and decreases with aneurysmal length. It is found that the amplitude in variation of the maximum effective stress changes in the same manner as the maximum stress. These results indicate that a short asymmetric AAA is under significant dynamic stress, which might be one of the important factors in a progressive AAA growth over time. It is also found that, in most considered cases, magnitude and amplitude of the maximum effective stress of an AAA is smaller when the artery is pre-stretched in axial direction, while a pre-stretched artery with aneurysm has slightly larger relative volume change of lumen during cardiac cycle than the unstretched AAA.

Nonlinear numerical analysis of the structural response of the intervertebral disc to impact loading

The analysis of intervertebral disc dynamics under impact loading, using computational simulation, is scarcely reported. In this study, the contribution of the characteristic structure of the disc to its dynamic response has been evaluated. The influence of several model features on the dynamic response was investigated. A hyperelastic large deformation formulation was used to describe the nonlinear behaviour of the soft tissues. The material parameters were determined by the fitting of experimental data from the literature. The model demonstrated pressure wave propagation and reflection through the disc, with a periodic oscillation of the system in response to a single impulse load, and highlighted a potential primary role played by the collagen fibre reinforcement. Their tensioning contributes to changing the stress propagation and oscillation, with a faster reduction in the internal pressure peak. The natural frequency of the disc was predicted to be approximately 9.8 Hz for the vertical oscillation.