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Deformation Gradient (deformation + gradient)

Distribution by Scientific Domains
Engineering100%

Selected Abstracts

An objective incremental formulation for the solution of anisotropic elastoplastic problems at finite strain

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2001
S. Chatti
Abstract This paper presents an objective formulation for the anisotropic elastic,plastic problems at large strain plasticity. The constitutive equations are written in a rotating frame. The multiplicative decomposition of the deformation gradient is adopted and the formulation is hyperelastic based. Since no stress rates are present and the incremental constitutive law was formulated in a rotating frame, the formulation is numerically objective in the time integration. Explicit algorithm was proposed and has been optimized with regard to stability and accuracy. The incremental law was integrated in fast Lagrangian analysis of continua (FLAC) method to model anisotropic elastic,plastic problems at finite strain. Structural tests are carried out for isotropic and orthotropic materials. Copyright © 2001 John Wiley & Sons, Ltd. [source]

A uniform nodal strain tetrahedron with isochoric stabilization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
M. W. Gee
Abstract A stabilized node-based uniform strain tetrahedral element is presented and analyzed for finite deformation elasticity. The element is based on linear interpolation of a classical displacement-based tetrahedral element formulation but applies nodal averaging of the deformation gradient to improve mechanical behavior, especially in the regime of near-incompressibility where classical linear tetrahedral elements perform very poorly. This uniform strain approach adopted here exhibits spurious modes as has been previously reported in the literature. We present a new type of stabilization exploiting the circumstance that the instability in the formulation is related to the isochoric strain energy contribution only and we therefore present a stabilization based on an isochoric,volumetric splitting of the stress tensor. We demonstrate that by stabilizing the isochoric energy contributions only, reintroduction of volumetric locking through the stabilization can be avoided. The isochoric,volumetric splitting can be applied for all types of materials with only minor restrictions and leads to a formulation that demonstrates impressive performance in examples provided. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2007
Ragnar Larsson
Abstract The paper presents a higher order homogenization scheme based on non-linear micropolar kinematics representing the macroscopic variation within a representative volume element (RVE) of the material. On the microstructural level the micro,macro kinematical coupling is introduced as a second-order Taylor series expansion of the macro displacement field, and the microstructural displacement variation is gathered in a fluctuation term. This approach relates strongly to second gradient continuum formulations, presented by, e.g. Kouznetsova et al. (Int. J. Numer. Meth. Engng 2002; 54:1235,1260), thus establishing a link between second gradient and micropolar theories. The major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch. The driving vehicle for the derivation of the homogenized macroscopic stress measures is the Hill,Mandel condition, postulating the equivalence of microscopic and macroscopic (homogenized) virtual work. Thereby, the resulting homogenization procedure yields not only a stress tensor, conjugated to the micropolar stretch tensor, but also the couple stress tensor, conjugated to the micropolar curvature tensor. The paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics. Copyright © 2006 John Wiley & Sons, Ltd. [source]

F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005
Part I: formulation, benchmarking
Abstract This paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F-bar method proposed by de Souza Neto et al. (Int. J. Solids and Struct. 1996; 33: 3277,3296) and is conceptually very simple: It relies on the enforcement of (near-) incompressibility over a patch of simplex elements (rather than the point-wise enforcement of conventional displacement-based finite elements). Within the framework of the F-bar method, this is achieved by assuming, for each element of a mesh, a modified (F-bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre-defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near-incompressibility. As the original F-bar procedure, the present methodology preserves the displacement-based structure of the finite element equations as well as the strain-driven format of standard algorithms for numerical integration of path-dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi-static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton,Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two- and three-dimensional numerical examples. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Analysis of 3D problems using a new enhanced strain hexahedral element

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2003
P. M. A. Areias
Abstract The now classical enhanced strain technique, employed with success for more than 10 years in solid, both 2D and 3D and shell finite elements, is here explored in a versatile 3D low-order element which is identified as HIS. The quest for accurate results in a wide range of problems, from solid analysis including near-incompressibility to the analysis of locking-prone beam and shell bending problems leads to a general 3D element. This element, put here to test in various contexts, is found to be suitable in the analysis of both linear problems and general non-linear problems including finite strain plasticity. The formulation is based on the enrichment of the deformation gradient and approximations to the shape function material derivatives. Both the equilibrium equations and their variation are completely exposed and deduced, from which internal forces and consistent tangent stiffness follow. A stabilizing term is included, in a simple and natural form. Two sets of examples are detailed: the accuracy tests in the linear elastic regime and several finite strain tests. Some examples involve finite strain plasticity. In both sets the element behaves very well, as is illustrated in numerous examples. Copyright © 2003 John Wiley & Sons, Ltd. [source]

An arbitrary Lagrangian,Eulerian finite element method for finite strain plasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2003
Francisco Armero
Abstract This paper presents a new arbitrary Lagrangian,Eulerian (ALE) finite element formulation for finite strain plasticity in non-linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part (F = FeFp), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi-static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Geometrically non-linear damage interface based on regularized strong discontinuity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2002
Ragnar Larsson
Abstract The contribution of this paper concerns the fracture modelling of an interface with a fixed internal material surface in the context of geometrically non-linear kinematics. Typical applications are composite laminates and adhesive/frictional joints in general. In the model development, a key feature is the concept of regularized strong discontinuity, which provides a regular deformation gradient within the interface. The deformation gradient within the interface is formulated in a multiplicative structure with a continuous part and a discontinuous part, whereby the interface deformation is interpreted as a transformation between the material damaged configuration and the actual spatial configuration. In analogy with the continuum formulation of hyper-inelasticity, a constitutive framework is defined for the relation between the induced material traction and the displacement jump vector, which are defined on the material damaged interface configuration. Within this framework, a simple, but yet still representative, model for the delamination problem is proposed on the basis of a damage,plasticity coupling for the interface. The model is calibrated analytically in the large deformation context with respect to energy dissipation in mode I so that a predefined amount of fracture energy is dissipated. The paper is concluded with a couple of numerical examples that display the properties of the interface. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002
Jiun-Shyan Chen
Abstract A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh-free approximation, is generalized for non-linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh-free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non-uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Energy driven crack propagation at finite strains based on the embedded strong discontinuity approach

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009
Radan Radulovic
New advances in three-dimensional finite element modeling of crack propagation at finite strains are presented. The proposed numerical model is based on the Enhanced Assumed Strain concept. The enhanced part of the deformation gradient is associated with a displacement discontinuity. In contrast to previous works, a new, energy based criterion for crack propagation is presented. The necessity for a tracking algorithm for the crack path is avoided by using more than one discontinuity within each finite element. This leads to a strictly local formulation, i.e., no information about the neighboring elements are required. Further advantages of such a formulation are a symmetric tangent stiffness matrix and the reduction of locking effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Classes of Anisotropic Finite Plasticity Models and their Implementation in a Brick-Type Shell Element

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
N. Apel
We discuss two constitutive models formulated in terms of logarithmic strains suitable for the description of elastoplastic material response. We consider two different approaches to the definition of the plastic deformation. The first is based on the introduction of a plastic map yielding a multiplicative decomposition of the deformation gradient into an elastic and plastic part. The second one uses an additive decomposition of the current metric. A quantitative analysis of both approaches by means of numerical examples of sheet metal forming processes are presented. [source]