Small Strain Problems Research Articles

A modular finite element approach to saturated poroelasticity dynamics: Fluid–solid coupling with Neo-Hookean material and incompressible flow

Several methods have been developed to model the dynamic behavior of saturated porous media. However, most of them are suitable only for small strain and small displacement problems and are built in a monolithic way, so that individual improvements in the solution of the solid or fluid phases can be difficult. This study shows a macroscopic approach through a partitioned fluid–solid coupling, in which the skeleton solid is considered to behave as a Neo-Hookean material and the interstitial flow is incompressible following the Stokes–Brinkman model. The porous solid is numerically modeled with a total Lagrangian position-based finite element formulation, while an Arbitrary Lagrangian-Eulerian stabilized finite element approach is employed for the porous medium flow dynamics. In both fields, an averaging procedure is applied to homogenize the problem, resulting in a macroscopic continuous phase. The solid and fluid homogenized domains are overlapped and strongly coupled, based on a block-iterative solution scheme. Two-dimensional simulations of wave propagation in saturated porous media are employed to validate the proposed formulation through a comprehensive comparison with analytical and numerical results from the literature. The analyses underscore the proposed formulation as a robust and precise modular approach for addressing dynamic problems in poroelasticity.

Dynamic response of floating piles subjected to axial loads considering three-dimensional soil deformations

This paper presents a systematic study on the effect of radial soil deformations on the dynamic response of floating piles subjected to axial loads. The study is facilitated by a novel mathematical model, that allows rigorously considering both vertical and radial soil displacements generated by axial pile vibration, under an extended Tajimi-type framework. Following the presentation of the mathematical model and the solution, we show that the model produces similar results compared to rigorous numerical boundary integral methods. Then we compare results of the proposed model against an earlier model that is formulated on the basis of confined elastodynamics i.e. on ignoring radial soil deformations. We demonstrate that the confined model overestimates stiffness and underestimates damping of the pile impedance in the low frequency range, regardless of the problem conditions, and we explain the mechanisms that lead to these discrepancies. Although the proposed model is based on the elastodynamic theory, findings of this study are not limited to small-strain problems, as the identified mechanisms are solely associated with the effect of radial soil deformations, and not with the constitutive behavior of soil.

A hyperelastic model for soils with stress-induced and inherent anisotropy

In this paper, modelling of the superposition of stress-induced and inherent anisotropy of soil small strain stiffness is presented in the framework of hyperelasticity. A simple hyperelastic model, capable of reproducing variable stress-induced anisotropy of stiffness, is extended by replacement of the stress invariant with mixed stress–microstructure invariant to introduce constant inherent cross-anisotropic component. A convenient feature of the new model is low number of material constants directly related to the parameters commonly used in the literature. The proposed description can be incorporated as a small strain elastic core in the development of some more sophisticated hyperelastic-plastic models of overconsolidated soils. It can also be used as an independent model in analyses involving small strain problems, such as dynamic simulations of the elastic wave propagation. Various options and features of the proposed anisotropic hyperelastic model are investigated. The directional model response is compared with experimental data available in the literature.

A variational flexible multibody formulation for partitioned fluid–structure interaction: Application to bat-inspired drones and unmanned air-vehicles

We present a three-dimensional (3D) partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow. While the incompressible Navier–Stokes system is discretized using a stabilized Petrov–Galerkin procedure, the multibody structural system consists of a generic interaction of multiple components such as rigid body, beams and flexible thin shells along with various types of joints and connections among them. A co-rotational framework is utilized for the category of small strain problems where the displacement of the body is decomposed into a rigid body rotation and a small strain component. This assumption simplifies the structural equations and allows for the incorporation of multiple bodies (rigid as well as flexible) in the system. The displacement and rotation constraints at the joints are imposed by a Lagrange multiplier method. The equilibrium conditions at the fluid–structure interface are satisfied by the transfer of tractions and structural displacements via the radial basis function approach, a scattered data interpolation technique, which is globally conservative. For the coupled stability in low structure-to-fluid mass ratio regimes, a nonlinear iterative force correction scheme is employed in the partitioned staggered predictor–corrector scheme. The convergence and generality of the radial basis function mapping are analyzed by carrying out systematic error analysis of the transfer of fluid traction across the non-matching fluid–structure interface where a third-order of convergence is observed. The proposed aeroelastic framework is then validated by considering a flow across a flexible pitching plate configuration with serration at the trailing edge. Finally, we demonstrate the flow across a flexible flapping wing of a bat modeling the bone fingers as beams and the flexible membrane as thin shells in the multibody system along with the joints.

Numerical integration for nonlinear problems of the finite cell method using an adaptive scheme based on moment fitting

Fictitious domain methods such as the finite cell method simplify the discretization process significantly as the mesh is decoupled from the geometrical description. However, this simplification in the mesh generation results in broken cells, which is why special integration methods are required. Usually, adaptive integration schemes are applied resulting in a large number of integration points and, thus, an expensive numerical integration — especially for nonlinear applications. To perform the numerical integration more efficiently, we propose an adaptive integration method using moment fitting. Thereby, we present a moment fitting approach based on Lagrange polynomials through Gauss–Legendre points to circumvent having to solve the moment fitting equation system. The performance of this integration method is shown by studying several numerical examples of the finite cell method for small and large strain problems in elastoplasticity.

Linear and nonlinear elastic analysis of closely spaced strip foundations using Pasternak model

In this study, an attempt is made to determine the interaction effect of two closely spaced strip footings using Pasternak model. The study considers small strain problem and has been performed using linear as well as nonlinear elastic analysis to determine the interaction effect of two nearby strip footings. The hyperbolic stress-strain relationship has been considered for the nonlinear elastic analysis. The linear elastic analysis has been carried out by deriving the equations for the interference effect of the footings in the framework of Pasternak model equation; whereas, the nonlinear elastic analysis has been performed using the finite difference method to solve the second order nonlinear differential equation evolved from Pasternak model with proper boundary conditions. Results obtained from the linear and the nonlinear elastic analysis are presented in terms of non-dimensional interaction factors by varying different parameters like width of the foundation, load on the foundation and the depth of the rigid base. Results are suitably compared with the existing values in the literature.

Large Strain and Small-Scale Biaxial Testing of Sheet Metals

Small-scale and multi-axial testing of sheet metals, particularly of lightweight alloys and advanced steels are becoming important as these materials exhibit forming behavior sensitive to their unique microstructural features and strain paths. As an alternative to large-scale standard tests, in this paper we introduce a novel biaxial tensile test apparatus utilizing miniature cruciform samples. The compact and portable apparatus includes a custom-built optical microscope and high-resolution digital image correlation (DIC) equipment for in-plane and in-situ strain measurements at the microstructure scale. The small strain and premature fracture problems common to the cruciform tests are solved by optimizing the sample design and by meticulously controlling the manufacturing steps and surface finish. Strain analyses reveal a key mechanism responsible for large strains and fracture at the center. This mechanism suppresses the local neck formation and allows uniform deformation under equibiaxial conditions until fracture. When normalized with the strain hardening exponent of the sample material (Al 6061-T6), the effective strain value before fracture, $$ \overline{\varepsilon}/n \sim 3 $$ , surpass the reported values for similar materials tested by cruciform and standard methods.

Homogeneous stress–strain states computed by 3D-stress algorithms of FE-codes: application to material parameter identification

In view of code verification of finite element implementations and for material parameter identification purposes it is of interest to make use of stress algorithms developed for three-dimensional finite element computations. In the case of homogeneous deformations various boundary-conditions for given displacements or stresses are possible and define a sub-problem of three-dimensional stress---strain states, which are either one-, two- or three-dimensional. Examples are uniaxial tension/compression, plane stress conditions or biaxial tensile problems. Caused by the fact that the stress algorithms are strain-driven, the constraints of zero stresses in a specific direction lead for elastic and inelastic constitutive models to a particular system of differential-algebraic equations. How to treat such stress algorithms and how to solve the resulting system of differential-algebraic equations, which are developed for finite element programs, for specific stress and displacement boundary conditions is discussed in this article. Additionally, it is worked out that the consistent tangent operator is required in the same manner as in 3D-FE computations. The second topic treats the extension of the entire procedure for material parameter identification procedure applied to test data for different materials such as steel, rubber material and powder. In this respect, uniaxial tensile, biaxial tensile tests, and laterally constrained loading paths are exemplarily investigated. These investigations and the proposed procedure are applied for small and finite strain problems. In this investigation measure of the quality of identification is discussed as well.

Thermodynamic aspects of Cottrell atmospheres

Using open system elastic constants, we can convert any of the many existing solved small-strain problems of stress fields of dislocation configurations to what the stresses and solute distribution equilibrium would be in alloys. We confine ourselves to small strain linear isotropic elasticity in which the solute atoms are centres of pure dilation and compute the inhomogeneous compositional distributions that are called Cottrell atmospheres, as well as the energy changes due to the segregation. We correct the misconception in the literature that the solute distribution does not affect the stress field of the dislocation and show that the atmosphere around an edge dislocation acts like an edge dislocation with a Burgers vector of opposite sign.

Some theoretical and computational aspects of single‐crystal strain‐gradient plasticity

AbstractVariational formulations are constructed for rate‐independent problems in single‐crystal strain‐gradient plasticity. The framework makes use of the flow rule expressed in terms of a dissipation function. The formulation extends to the finite‐deformation context earlier work on this problem. Provision is made for energetic and dissipative microstresses, and a range of defect energies is accounted for. The minimization problem corresponding to the time‐discrete formulation is derived. Two special cases are then treated: first, for the small‐strain problem with energetic microstresses, results on well‐posedness, convergence of finite element approximations, the associated algorithms, and computational examples, are discussed. Secondly, recent computational work on the the related large‐deformation viscoplastic problem, with both energetic and dissipative effects, is presented and discussed for problems involving ensembles of grains.

A coupled two‐scale shell model with applications to layered structures

SUMMARYIn this paper a coupled two‐scale shell model is presented. A variational formulation and associated linearization for the coupled global–local boundary value problem is derived. For small strain problems, various numerical solutions are computed within the so‐called FE 2 method. The discretization of the shell is performed with quadrilaterals, whereas the local boundary value problems at the integration points of the shell are discretized using 8‐noded or 27‐noded brick elements or so‐called solid shell elements. At the bottom and top surface of the representative volume element stress boundary conditions are applied, whereas at the lateral surfaces the in‐plane displacements are prescribed. For the out‐of‐plane displacements link conditions are applied. The coupled nonlinear boundary value problems are simultaneously solved within a Newton iteration scheme. With an important test, the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified.Copyright © 2013 John Wiley & Sons, Ltd.

Coupled multiscale finite element analysis of shell structures

AbstractIn this paper, a coupled two‐scale shell model is presented. A variational formulation and associated linearisation for the coupled global‐local boundary value problem is derived. The discretisation of the shell is performed with quadrilaterals, whereas the local boundary value problems at the integration points of the shell are discretised using 8‐noded or 27‐noded brick elements, or solid shell elements. The coupled boundary value problem is simultaneously solved within a Newton iteration scheme. Solutions for small strain problems are computed within the so‐called FE2 method. In an important test, the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified. Examples show that the developed two‐scale model is able to analyse the global and local mechanical behaviour of heterogeneous shell structures. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Linear Complementarity formulation of rate-independent finite-strain elastoplasticity. Part I: Algorithm for numerical integration

Abstract A methodology for the numerical integration of rate-independent, elastic–plastic finite-strain models is developed. The methodology is based on the idea of local linearization of the yield surface that was proposed in Maier (1969) , adopted as the basis for an integration scheme in Hodge (1977) , and developed further in Franchi and Genna, 1984 , Franchi and Genna, 1987 , so far for small-strain problems only. The proposed algorithm is based on the solution of a local Linear Complementarity Problem and is suited particularly for plasticity models that involve yield surfaces with singular points (corners, edges, apexes, etc.).

Analysis of spatial interpolation in the material-point method

This paper analyses different types of spatial interpolation for the material-point method. The interpolations include quadratic elements and cubic splines in addition to the standard linear shape functions usually applied. For the small-strain problem of a vibrating bar, the best results are obtained using quadratic elements. It is shown that for more complex problems, the use of partially negative shape functions is inconsistent with the material-point method in its current form, necessitating other types of interpolation such as cubic splines in order to obtain smoother representations of field quantities. The properties of different interpolation functions are analysed using numerical examples, including the classical cantilevered beam problem.

A corotational interpolatory model for fabric drape simulation

AbstractFabric drapes are typical large displacement, large rotation but small strain problems. In particle models for fabric drape simulation, the fabric deformation is characterized by the displacements of the particles distributed over the fabric. In this paper, a new particle model based on the corotational concept is formulated. Under the small membrane strain assumption, the bending energy can be approximated as a quadratic function of the particle displacements that are finite. In other words, the tangential bending stiffness matrix is a constant and only the tangential membrane stiffness matrix needs to be updated after each iteration or step. On the other hand, the requirement on the particle alignment is relaxed by interpolating the particle displacement in a patch of nine particles. To account for the membrane energy, a simple and efficient method similar to the three‐node membrane triangular element employing the Green strain measure is adopted. With the present model, the predicted drapes appear to be natural and match our daily perception. In particular, circular clothes and circular pedestal that can only be treated laboriously by most particle models can be conveniently considered. Copyright © 2008 John Wiley & Sons, Ltd.

A new skeletal model for fabric drapes

Fabric drapes are typical large displacement, large rotation and small strain problems. Compared with continuum shell finite element methods, methods that skeletonize the fabric sheet into a set of interconnected nodes appear to be more popular in drape problems with extensive wrinkles. These skeletal methods may resort to particle mechanics and formulate the elastic energy or the equations of motion in terms of the node-to-node distance and the angles between the straight lines joining adjacent nodes. Alternatively, beam elements can be employed to skeletonize the fabric sheet at the expense of using rotational in addition to translational nodal DOFs. In this paper, a new skeletal model based on the small strain and curvature assumptions is devised. In contrast to most, if not all, skeletal models for fabric drape simulation, all the stretching, shearing and bending energies in the model are simple polynomials of the grid-point␣displacement. Fabric drape problems with extensive wrinkles are examined. The presence of sharp fold, seam and cut in the undeformed fabric sheet as well as fabric-to-solid contact are considered. The predicted appearances are pleasant and conform to real life observations.

A co‐rotational grid‐based model for fabric drapes

AbstractFabric drapes are typical large displacement, large rotation and small strain problems. Compared to conventional geometric non‐linear shell analyses, computational fabric drape analysis is particularly challenging due to the extremely weak bending rigidities of fabrics. Compared to continuum shell finite element methods, grid‐ or particle‐based methods appear to be more successful in high drapeability problems. The latter methods often resort to simple particle mechanics and formulate the elastic energy in terms of the inter‐particle distances and trigonometrical functions of the angles between the straight lines joining adjacent particles. In this paper, the co‐rotational approach and commonly employed assumptions for small strain problems in finite element analysis will be adopted to formulate the elastic energy. It will be seen that the internal force vector and the stiffness matrix are considerably simpler than other grid‐based models, yet the sparsity of the tangential stiffness matrix remains unchanged. A number of examples are considered and the predicted results are promising. Copyright © 2003 John Wiley & Sons, Ltd.

SOLUTION OF FINITE-DISPLACEMENT SMALL-STRAIN ELASTICITY PROBLEMS BY REMOVAL OF RIGID BODY DISPLACEMENTS

If rigid body displacements are properly separated from a finite-displacement small-strain problem, the remaining deformation is within the scope of the small displacement theory. For this purpose, the rigid body displacement of each sufficiently small subdomain of an elastic body must be removed independently from other subdomains before the actual deformation occurs. This paper presents a rigorous and straightforward theoretical formulation for a numerical solution procedure based on this concept. The simplicity of the formulation is due to the fact that it is developed for a general three-dimensional solid rather than structural elements as in the past works.

A further study on the postbuckling of extensible elastic rods

The postbuckling of extensible elastic rods is studied using non-linear geometric models. Accordingly the kinematics and equilibrium are stated. Nine different strain–stress relationships are analyzed. The classical Strength of Materials approach is compared and discussed with other eight constitutive laws stated with Lagrangian and Eulerian descriptions. The well-known Cauchy and Green methods in Continuum Mechanics are alternatively employed. Four of the approaches are worked out until an explicit solution of the secondary equilibrium path is obtained. The analysis is applicable to small strain problems. The linearized problem is presented for all the laws together with numerical results for rods with various values of the extensibility parameter. The secondary equilibrium paths are numerically evaluated to illustrate the degree of discrepancy. A specific example that displays unexpected unstable behavior is shown. Both critical loads and postbuckling curves are coincident when the theoretical problem of an inextensible rod is solved. It is shown that even when small strains are addressed, the extensibility influence gives rise to disagreement of the postbuckling response when using the different alternatives.

Statistical properties of nonlinear mode spectrum for large amplitude forced vibration systems

A method to estimate the statistical properties of a geometrically nonlinear mode spectrum for large amplitude forced vibration systems is presented. It is shown here that this technique is readily applicable to geometrically nonlinear small strain problems in which the equations of motion of structures are derived using the finite element method. A general formulation for problems of this type is presented. The efficiency of the method is illustrated by its application to a nonlinear vibration of an elastic beam.