European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2000 Barcelona, 11-14 September 2000 ECCOMAS
COMBINATION OF SMEARED AND DISCRETE APPROACHES WITH THE USE OF INTERFACE ELEMENTS Ricardo A. Einsfeld*, Luiz F. Martha†, and Túlio N. Bittencourt
§
*
Instituto Politécnico (IPRJ), Universidade do Estado do Rio de Janeiro (UERJ), Nova Friburgo, RJ, 28601-970, Brazil, e-mail:
[email protected] †
Departamento de Engenharia Civil, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, 22453-900, Brazil, e-mail:
[email protected] §
Departamento de Engenharia de Estruturas e Fundações, Universidade de São Paulo (EPUSP), São Paulo, SP, 05508-900, Brazil, e-mail:
[email protected]
Key words: Fracture Mechanics, Interface Elements, Non-Linear Analysis, Plasticity. Abstract. This paper presents a new procedure for combining the smeared and discrete approaches with the use of interface elements. The original methodology, developed by the first author, considers the discrete crack insertion in the continuum without any liaison between its faces. This procedure was tested with relative success but numerical instabilities arose during the analysis. The numerical problems can be attributed to the unbalanced forces that remain after the discrete crack insertion and the bad approximations obtained to the history-dependent variables mapped from the old mesh to the new mesh configuration that is obtained after the insertion of the discontinuity in the structural model. With the introduction of interface elements for the discrete crack representation, the two main aspects that affect the stability of the numerical solution can be eliminated. The great advantage of the original methodology, that is, the possibility of having a discrete crack representation without the need of a predefined crack path, still remains with the use of interface elements.
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
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INTRODUCTION
Finite element models for crack analysis are grouped according to crack representation into discrete and smeared crack models. The discrete approach deals with the fracture process introducing a discontinuity in the geometry of the structure, while the rest of the structure behaves in a linear elastic way. On the other hand, the smeared approach keeps the geometry fixed and the strain localization is simulated in a representative region of the structure through the material constitutive law. The smeared approach offers a number of advantages over the discrete approach. Remeshing techniques and continuous topological redefinition are not necessary in the smeared process. Nevertheless, a major drawback of the smeared analysis is its inability to model the surfaces of the crack. Also, the representation of cracked materials as a continuum induces locked-in stresses in the elements close to the localization zone1 and a too stiff response of the structure is frequently found in the analysis. Stress locking due to finite element compatibility can be gradually eliminated combining both approaches in a single model, inserting a discontinuity in the geometry of the structure when the strain localizes. Both approaches are combined in a method proposed by the first author2. The smeared process is used to predict the strain localization and the position and direction of the corresponding discrete crack. Based on an energy criterion, the discrete crack is inserted or propagated in the finite element model with the stresses totally released through its surfaces. After discrete crack insertion, the geometry of the model is modified and the region around the crack is remeshed. The nodal displacements and the history-dependent variables are mapped from the old mesh configuration to the new one. In this paper, it is presented a technique where the original methodology is modified with the use of interface elements. With this new technique, the energy is totally dissipated via discrete crack model instead of the smeared approach. It is expected an improvement of the mapping approximations after remeshing, and a more stable numerical convergence during the analysis. Numerical tests with interface elements are still in course by the time this paper is being written, and the results are expected to be presented at the Congress. Some examples of numerical instability obtained when using the original methodology are presented at the end of the paper. 2 THE COMBINATION OF SMEARED AND DISCRETE APPROACHES In the original methodology proposed, the discrete crack is inserted or propagated in the finite element model with the stresses totally released through its surfaces, that is, without the need of any interface element. The smeared process, with a plasticity-based constitutive model, is used to predict the strain localization and the position and direction of the corresponding discrete crack. The criterion to insert the discrete crack is based on the fracture energy Gf. The discrete crack is inserted or propagated in the finite element model when the dissipated energy during the post-peak behavior of the material at the integration points reaches Gf. Once the discrete crack is inserted with the stresses totally released through its surfaces, the geometry of the model is modified and the region around the crack is remeshed.
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
The nodal displacements and the history-dependent variables are mapped from the old mesh configuration to the new one. In a previous work3, it was shown that the sequence of the analysis depends strongly on the approximations obtained to these variables. The cumulative errors after a few remeshing procedures, which are necessary for crack propagation, do not permit the system to find the equilibrium configuration at a certain stage of the analysis. The criterion originally established for discrete crack insertion considers the total dissipation of the fracture energy at the Gauss point where strain localization occurs. This situation is difficult to achieve, since the smeared analysis spreads the plastic behavior over a relatively large region of the structure and causes difficulties in concentrating strain in only a few Gauss points. For this reason, the discrete crack is inserted as soon as the Gauss points of the finite elements close to the crack tip show a certain amount of decohesion. The discrete crack is inserted without any liaison between its faces and can be considered as a damage introduced in the structural model. This methodology was used with a relative success. The singularity introduced in the structural model after discrete crack insertion caused jumps in the load-deflection diagram and also caused elastic unloading. Although numerical convergence was improved using different strategies for updating the history-dependent variables, the same kind of behavior was obtained when a higher rate of decohesion was established for insertion of the discontinuity in the structural model. This behavior suggests that the approximations obtained for the variables in the softening regime are inadequate and can be considered responsible for the instability of the numerical solution. The magnitude of the remained unbalanced forces in the split region turns to be another factor of instability in the numerical process. This is a consequence of the fact that the discrete crack is inserted in the finite element model before the total dissipation of the fracture energy. The numerical instability, due to the abrupt transition from a continuum representation to a totally opened crack, causes difficulties in establishing an appropriate criterion for discrete crack insertion based on decohesion rate at the Gauss points, when trying to obtain the correct post-peak behavior of the structural system. In order to control the unbalanced forces and to permit a better representation of the cohesive soft behavior, the use of interface elements was suggested to be introduced in the discrete crack representation. In this adaptation from the original method, the smeared analysis is still used to predict the strain localization and the position and direction of the corresponding discrete crack. But the material post-peak behavior is solely represented within the interface elements that connect the opposite faces of the discrete crack. The crack evolution through this methodology is schematically shown in Figure 1. The discrete crack is inserted as soon as the stress at any Gauss point reaches the ultimate stress established by the constitutive model used in the analysis. 3
DISCRETE CRACK MODELED WITH INTERFACE ELEMENTS
In the discrete approach, the crack is constrained to follow a predefined path modeled with interface elements. The softening characteristics of the material are attributed to this elements,
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
while the rest of the structure behaves in a linear-elastic way. Different types of interface elements have been used since the pioneer work of Ngo and Scordelis4. These authors used lumped interface elements in order to evaluate the displacements and forces at the node sets. A distinction is made between two possible finite element formulation for interface elements. The first is called the nodal or lumped interface formulation, while the second is named the continuous or numerically integrated interface formulation. Although these formulations are equivalent, the resulting element stiffness matrices are different. Continuous interface elements were used by Bittencourt et al.5 for linear and non-linear crack analysis. For this type of element various integration schemes can be used (nodal lumping, Lobatto, Gauss and Newton-Cotes). For concrete and geotechnical applications the Gauss scheme leads to oscillatory results. For line interface Rots suggested a nodal lumping scheme which caused the oscillations to disappear (see reference 1).
a
b
c
d
Figure 1: Procedure for discrete crack insertion sing nodal interface elements: a) Identification of the element where the stress at the Gauss point reaches the ultimate stress. b) The identified element and all the surrounding elements are eliminated. c) Discrete crack propagation and remeshing of the region around the crack. d) Sequence of the analysis.
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
In performing the numerical analysis through the discrete approach, a high initial stiffness has to be supplied for the interface elements as no significant deformations should occur in the elastic stage of the loading process. In the method presented in this article, the analysis performs through the smeared approach and the interface elements are introduced within the original mesh only when the softening behavior occurs. In this way, there is no need of introducing high initial stiffness for the elements. Below, the formulation for the nodal interface elements that are being initially tested is presented. The implemented 2D two-node spring-like elements with two degrees of freedom per node are shown in Figure 2. Nodal interface elements Finite Element B 2
n
Direction of Interface t Finite Element A 1 Figure 2: 2D nodal interface elements
The element nodal displacement vector is defined as6 v=
{v
n
1
, vn2 ,vt 1, vt 2
T
}
(1)
where subscript n indicates the normal to the direction of the interface, and subscript t the tangent to the interface direction. Superscripts indicate the element nodes. It should be pointed that v is defined for the element local axes. The relative displacement vector is defined through the use of a matrix B that relates the relative displacements to nodal displacements ∆un ∆u = = Bv ∆ut
(2)
−1 +1 0 0 B= 0 0 −1 +1
(3)
where B is defined as
The traction-relative displacement constitutive relation is given by matrix D
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
tn dn t = = D ∆u = tt 0
0 ∆un dt ∆ut
(4)
The principle of minimum total potential energy can be invoked to obtain the linear elastic stiffness matrix K (5)
K = ∫ B T D B dA A
The application of equation 5 for the 2D case gives dn − d n = K A 0 0
− dn dn 0 0
0 0 dt − dt
0 0 − dt dt
(6)
where A is the surface contribution of the node set. This contribution factor depends on the type of interpolation and the dimension of the surrounding elements. For the case of 2D quadratic element used in this work, A = 1/6 for corner nodes, and A = 2/3 for mid-side nodes. A complete list of the values of the surface contribution factor A can be found in reference 6. 4
NUMERICAL INSTABILITY
The stability of the numerical simulation is affected by the introduction of the geometrical discontinuities in the finite element mesh. Two main aspects of this process should be considered. First, due to the nature of the smeared analysis, the representation of cracked material induces locked-in stresses in the elements close to the localization zone. For this reason, it was not possible to dissipate all the fracture energy in the corresponding Gauss point, and unbalanced forces remain at the cracked nodes as the analysis restart. Difficulties in obtaining the numerical equilibrium after the remeshing process can be attributed to these remained unbalanced forces. This problem was overcome by controlling the extension of the crack, introduced as a damage in the structural model. Second, as a consequence of the remeshing of the region around the discrete crack, the history-dependent variables need to be mapped from the old mesh to the new mesh configuration. It was shown (see reference 3) that the sequence of the analysis depends strongly on the approximations obtained for these variables, and some strategies were implemented and tested with the specific objective of improving these approximations. Numerical convergence improved as a better configuration for the updated stress was obtained. In order to illustrate these facts, a numerical simulation was carried out for the plain concrete three-point symmetric notched beam, solved analytically by Petersson7. The concrete behavior was modeled using the energy-based plasticity model of Pramono and Willam8. The results, considering the deflection of the upper-center of the beam are presented in Figure 3.
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
90 80
Load (kN/m)
70 60
Petersson Strategy 1 Strategy 2 Strategy 3
50 40 30 20 10 0 0
0.05
0.1
0.15
0.2
Deflection (mm) Figure 3: Numerical instability in non-linear analysis
The singularity introduced in the structural model after discrete crack insertion causes jumps in the diagram, as shown for strategies 1 and 2. Although numerical convergence was improved using strategy 3, the same kind of behavior was obtained when a higher rate of decohesion at the Gauss points was established for insertion of the discontinuity in the structural model. Strategy 3 allows a better representation of discrete crack propagation, but the load-deflection diagram does not represent the post-peak behavior of the structure, as can be seen in the figure. With the introduction of interface elements for the discrete crack representation it is possible to avoid the stability problems related above. This new procedure permits the controlling of the unbalanced forces at the discrete crack nodes. Also, since the softening behavior is uniquely represented within the interface elements, only the pre-peak historydependent variables need to be mapped to the new mesh, and a better approximation for the variables can be obtained. With this technique, the two main aspects that affect the stability of the numerical simulation are eliminated and the correct representation of the post-peak behavior of the structure can be expected. In this new procedure with the use of interface elements, the great advantage of the original methodology still remains: it is possible to have a discrete crack representation without the need of a predefined crack path. 5
CONCLUSIONS
In the last years, the discrete representation of the crack evolution in fracture analysis has found more supporters, since remeshing no longer represents a drawback in the process. Automatic remeshing techniques for finite element analysis have improved along with computer graphics capabilities and the development of sophisticated data structures that assure mesh consistency. Due to these recent advances, adaptive remeshing schemes have
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Ricardo A. Einsfeld, Luiz F. Martha, and Túlio N. Bittencourt.
been widely used to improve the efficiency of the finite element method, reducing the discretization errors. With the increasing use of non-linear finite element analysis applied to inelastic problems, the mapping of local variables from the old mesh to the new mesh has become a major issue of interest due to its influence in the sequence of the numerical analysis. Numerical problems were found in the analysis of concrete structures using a technique that combines the smeared and discrete approaches. These problems are attributed to the unbalanced forces that remain after the discrete crack insertion, and the bad approximations obtained in updating the history-dependent variables at the Gauss points of the new mesh configuration. This article suggests a new procedure, with the introduction of interface elements for discrete crack representation, in order to avoid the numerical problems reported. This technique shows to be promising although no objective conclusion can be made up to this moment. The numerical tests are still in course with the nodal interface elements formulated in item 3, and the results are expected to be presented at the Congress. Besides the nodal interface elements, the continuous interface elements should also be implemented and tested as a sequence of this work. REFERENCES [1] J.G. Rots, Computational Modeling of Concrete Fracture, Ph.D. Dissertation, Delft University (1988). [2] R.A. Einsfeld, Simulação Numérica de Fraturamento em Estruturas de Concreto Combinando os Processos Discreto e Distribuído, D.Sc. Thesis, Departamento de Engenharia Civil, PUC-Rio (1997). [3] R.A. Einsfeld, D. Roehl, T.N. Bittencourt, and L.F. Martha, “Mapping of Local Variables Due to Remeshing in Non-Linear Plasticity Fracture Problems”, Fourth World Congress on Computational mechanics, Vol.I, p.511 (Abstract), CD (Paper) (1998). [4] O. Ngo, and A.C. Scordelis, “Finite Element Analysis of Reinforced Concrete Beams”, ACI Journal, 64:3, 152-163 (1967). [5] T.N. Bittencourt, P.A. Wawrzynek, A.R. Ingraffea, and J.L. Souza, “Quasi-Automatic Simulation of Crack Propagation for 2D LEFM Problems, Engineering Fracture Mechanics, 52:2, 321-334 (1996). [6] J.C.J. Schellekens, Interface Elements in Finite Element Analysis, report nr. 25-2-90-517, TU-Delft (1990). [7] P.E. Petersson, Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials, report TVBM-1006, Division of Building Materials, Lund Institute of Technology (1981). [8] E. Pramono, and K. Willam, “Fracture Energy-Based Plasticity Formulation of Plain Concrete”, Journal of Engineering Mechanics, ASCE, 115:6, 1183-1204 (1989).
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