Finite elements for materials with strain gradient effects (original) (raw)
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This paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales (VMS). A displacement sub-scale is introduced in order to stabilize the pressure field. Compared to standard irreducible formulation, the proposed formulation yields improved strain and stress fields. The paper investigates the effect of this enhacement in accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P 1P 1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr-Coulomb and Drucker-Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.
Mixed finite elements for elasticity in the stress-displacement formulation
Contemporary Mathematics, 2003
We present a family of pairs of finite element spaces for the unaltered Hellinger-Reissner variational principle using polynomial shape functions on a single triangular mesh for stress and displacement. There is a member of the family for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and each is stable and affords optimal order approximation. The simplest element pair involves 24 local degrees of freedom for the stress and 6 for the displacement. We also construct a lower order element involving 21 stress degrees of freedom and 3 displacement degrees of freedom which is, we believe, likely to be the simplest possible conforming stable element pair with polynomial shape functions. For all these conforming elements the approximate stress not only belongs to H(div), but is also continuous at element vertices, which is more continuity than may be desired. We show that for conforming finite elements with polynomial shape functions, this additional continuity is unavoidable. To overcome this obstruction, we construct as well some non-conforming stable mixed finite elements, which we show converge with optimal order as well. The simplest of these involves only 12 stress and 6 displacement degrees of freedom on each triangle.
Finite Elements in Analysis and Design, 2015
The higher-order gradient plasticity theory is successful in explaining the size effects encountered at the micron and submicron length scale. Due to the incorporation of spatial gradients of one or more internal variables in these theories and the associated non-classical boundary conditions, special types of elements in the finite element method maybe necessary. This makes the numerical implementation of this higher-order theory not straightforward. In this paper, a robust but straightforward numerical implementation of higher-order gradient-dependent plasticity theories is presented. The novelty of this paper is in (1) the application of the meshless methods, particularly the moving weighted least square method, combined with the finite element method for the numerical computation of plastic strain gradients, and (2) the numerical implementation of different types of higher-order microscopic boundary conditions at internal/external surfaces, interfaces, and moving elastic-plastic boundaries. The proposed numerical implementation algorithms can be easily adapted in the implementation of any form of higher-order gradient-dependent constitutive models. Examples of applying the current numerical approach is demonstrated for capturing mesh-objective shear band formation and size effect and boundary layer formation in thin films on elastic substrates and metal matrix composites with embedded elastic inclusions through the consideration of stiff, intermediate, and soft interfaces.
The displacement-type finite element approach—From art to science
Progress in Aerospace Sciences, 1994
The finite element method originated in the aerospace industry in the mid-1950s to solve practical stress analysis problems associated with the structural design of aerospace vehicles. It is today the most overwhelmingly popular analysis and design tool in structural and solid mechanics and is being extensively applied to a very wide spectrum of engineering science, e.g. fluid mechanics, heat transfer and electromagnetics. In its formative years, the development of the method was guided mostly by engineering intuition, heuristic judgment and trial and error experimentation and validation. Its achievements have been remarkable and there are now very powerful general-purpose software codes that make a variety of analyses and design tasks routinely simple, that were once considered to be intractable. This is not to say that the progress of the method has been free of hurdles, especially in finding a complete scientific basis for it. Based on the recent studies of this author and his colleagues, this review attempts to provide a more complete paradigmatic understanding of the issues involved. Concepts such as consistency and (variational) correctness are introduced. These together with the more familiar completeness and continuity requirements are then employed to guide the construction of error-free robust finite elements and also provide procedures to perform a priori error estimates for the quality of approximation. These C-concepts, as we shall call them, are elucidated and their relevance to the design of several key elements commonly found in general-purpose packages used by the aerospace, automobile and mechanical engineering industries is briefly covered. The article reviews what has been achieved in areas where the C-concepts can be applied fruitfully in the study of the displacement type finite element method.
Finite Elements in Analysis and Design, 2012
The size effects exhibited in the structural behaviors of micro-sized loading components cannot be described with classical plasticity theory alone. Thus, strain gradient plasticity together with appropriate experiments has been used to account for this size effect. In previous implementations of strain gradient plasticity into finite element code, low order displacement elements with reduced integration, despite their versatility for solving various structural problems, have been excluded because of their inability to yield the strain gradient inside the element. In this work, a new method of evaluating the plastic strain gradient with linear displacement elements via an isoparametric interpolation of the averaged-at-nodal plastic strain is proposed. Rate-independent yield conditions are satisfied accurately by the Taylor dislocation hardening model with Abaqus UHARD subroutine. To verify the suggested approach, the structural behaviors of micro-sized specimens subjected to bending, twisting, and nanoindentation tests were modeled and analyzed. The predicted size effects are generally in good agreement with previously published experimental results. Computational efforts are minimized and user versatilities are maximized by the proposed implementation.
Numerical techniques for plasticity computations in finite element analysis
Computers & Structures, 1987
Common numerical techniques for plasticity computations in finite element analysis are examined. The plasticity theory considered is the simple rate-independent van Mises criterion for small strains. Work hardening is represented by a general isotropic model or by a linear, isotropic-kinematic mixed model. Algorithms to integrate the rate equations, strategies for stress updating over a time (load) step in implicit codes, and tangent operators consistent with the integration algorithm are discussed. The elastic predictor-radial return algorithm and a consistent tangent operator satisfy the requirements for a stable, accurate and efficient numerical procedure. An extension of this model for plane stress with mixed hardening is described. Two numerical examples are given to demonstrate the accuracy and efficiency for plane stress analyses.
Computer Methods in Applied Mechanics and Engineering, 1984
Finite element models for elasto-plastic incremental analysis are derived from a three-field variational principle. The Newton-Raphson method is applied to solve the nonlinear system of equations which is obtained from the stationarity condition of this principle. The iterative schemes are discussed in detail for pure displacement and for pure equilibrium models from which iterative schemes for hybrid models folfow directly. In the displacement model, the compatibility of the strains and the plasticity criterium are satisfied during the whole iterative process, while the equilibrium of the stresses is restored only in the mean after convergence. In the equilibrium model, the plasticity criterium and the compatibility of the strains are verified in the mean during the iterative process; when convergence is achieved, the stresses are locally in equilibrium with the applied external loads. In both cases, a tangential stiffness matrix can be constructed, even for perfectly plastic materials and it allowsone to obtain always very good convergence properties. Examples are shown for plane stress and axisymmetric cases.
A Finite Element approach with patch projection for strain gradient plasticity formulations
Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.
한국전산구조공학회 논문집, 2024
This paper presents two-dimensional boundary value problems of the stress-based gradient elasticity within the smoothed finite element method (S-FEM) framework. Gradient elasticity is introduced to address the limitations of classical elasticity, particularly its struggle to capture size-dependent mechanical behavior at the micro/nano scale. The Ru-Aifantis theorem is employed to overcome the challenges of high-order differential equations in gradient elasticity. This theorem effectively splits the original equation into two solvable second-order differential equations, enabling its incorporation into the S-FEM framework. The present method utilizes a staggered scheme to solve the boundary value problems. This approach efficiently separates the calculation of the local displacement field (obtained over each smoothing domain) from the non-local stress field (computed element-wise). A series of numerical tests are conducted to investigate the influence of the internal length scale, a key parameter in gradient elasticity. The results demonstrate the effectiveness of the proposed approach in smoothing stress concentrations typically observed at crack tips and dislocation lines.