2011 Taylor & Francis Group, London, ISBN 978-0-415-67771-4. Different finite element refinement strategies for the computation of the strain energy density in ...
Advances in Marine Structures – Guedes Soares & Fricke (eds) © 2011 Taylor & Francis Group, London, ISBN 978-0-415-67771-4
Different finite element refinement strategies for the computation of the strain energy density in a welded joint C. Fischer, A. Düster & W. Fricke Hamburg University of Technology (TUHH), Hamburg, Germany
ABSTRACT: Structural failure due to a lack of fatigue strength and crack initiation is often caused by local stress concentrations at welded joints. By decreasing the weld notch radius to small values, the magnitude of stress concentration no longer affects the fatigue strength but a mean value of stress determined around the notch tip. Moreover, in linear elasticity the local stress distribution becomes singular with vanishing radius. Besides the common methods for fatigue assessment, the utilisation of an averaged value of strain energy, computed for a small volume of material, is an alternative approach. The method shows promise with respect to finite element analysis of complex problems. In most of the previous work the quality of the numerical approximation was controlled by mesh refinement. In this paper, alternative refinement techniques are investigated, accounting for the singular behaviour of the exact solution in the vicinity of the re-entrant corner. After the introduction of the concept based on the averaged strain energy density, different refinement strategies improving the quality of the finite element approximation are described and applied to a selected example. The results of the different approaches related to both h and p-refinement are presented and discussed also in view of application to more complex three-dimensional problems. 1
INTRODUCTION
In the theory of linear elasticity, a vanishing notch radius ρ results in a singular stress distribution. However, the fatigue strength is controlled by a mean stress value averaged in a small volume of material. The different common fatigue approaches, which have been summarised by Radaj et al. (2006), estimate the fatigue strength in various ways. For instance, the effective notch stress approach applies a fictitious notch radius of ρref = 1 mm to account for the effect of microstructural support at the notch root (Hobbacher, 2009). Alternatively, the stress intensity factors describe the local asymptotic stress distribution at the weld toe in a similar way as for crack tips (Lazzarin & Tovo, 1996), with an opening angle 2α differing from zero, Figure 1. The stress intensity factors have been proven to be suitable parameters concerning fatigue assessment of welds (Lazzarin & Tovo, 1998). However, the demand for very fine finite element meshes in highly stressed regions is one of the main disadvantages of the approach. Besides evaluation of stresses, the strain energy is utilized by different failure assessment methods. For the prediction of fatigue life under multiaxial loading conditions Glinka et al. (1995) defined a parameter, which is based on the strain energy occuring inside the critical plane. Alternatively, in
Figure 1.
Control sector surrounding the weld toe.
a small defined region close to the singular notch root the so-called Strain Energy Density (SED) is computed. The SED indicates fracture of brittle materials weakened by a sharp V-notch under static loads (e.g. Yosibash et al., 2004) and of cracked components under mixed mode loading by estimating the product of the SED and a critical distance from the crack tip (Sih, 1974). For high-cycle fatigue of welded joints assuming a V-notch with ρ = 0, Lazzarin & Zambardi (2001) used a geometrically based SED parameter, which is averaged over a small cylindrical volume.
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Based on the relationship between the SED and the notch stress intensity factors, a unified S-N curve independent of the welded detail has been developed. The approach shows promise because a relative coarse finite element mesh yields satisfactory results. When discretizing problems of linear elasticity with singular behaviour, the refinement strategy has a very important influence on the quality and efficiency of the computation. In this paper the influence of different refinement techniques computing the averaged SED has been investigated for a doubler plate with fillet-welded joints. After a brief introduction of the SED approach the different refinement strategies will be described and the numerical results will be presented. 2
STRAIN ENERGY DENSITY APPROACH
The application of the SED approach for fatigue assessment of welded joints with a sharp V-shaped notch including an opening angle of 2α, has been recommended by Livieri & Lazzarin (2005) and Lazzarin et al. (2008). The local region, where the total strain energy is averaged, is defined as a cylindrical volume surrounding the notch root. The weld toe considered as re-entrant corner typically exhibits an opening angle of 2α = 135° and the weld root at slit ends 2α = 0°. In case of two-dimensional problems the volume degenerates to a sector with a characteristic radius R0, Figure 1. The size of the radius R0 over which the SED needs to be computed depends on the material and has been derived from Beltrami’s failure criterion by Lazzarin & Zambardi (2001) considering stress intensity factors for welded joints under mode I loading and high cycle fatigue data (N = 5 × 106). Choosing R0 = 0.28 mm results in a conservative approach for welds made of steel and arc welding technology with failure from weld toe and root (Livieri & Lazzarin, 2005). For aluminium alloys a value of 0.12 mm has been suggested. It should be noted that these considerations are solely based on continuum mechanics. The material effects are taken into account by the high cycle fatigue failure data. Based on an evaluation of about 650 fatigue tests, which cover both non-load carrying and load-carrying fillet cruciform welded joints with various plate thicknesses, a uniform S-N curve with reference to the range of the averaged SED ΔW has been obtained by Lazzarin et al. (2008). Considering probabilities of survival Ps = 2.3% and 97.7%, Figure 2 shows the corresponding scatter band involving fractures initiated from weld toe as well as weld root.
Figure 2. Fatigue strength depending on range of the averaged SED ΔW (Lazzarin et al., 2008).
Furthermore, a relationship between the range of the averaged SED ΔW and the equivalent local stress range Δσeq is given under plane strain conditions by: Δσ eq =
2 E ⋅ ΔW 1 − v2
(1)
where E is Young’s modulus and ν is Poisson’s ratio. 3
EXAMPLE
Within various types of welded joints, Fricke & Feltz (2009) investigated a partial-load carrying doubler plate on both sides of a continuous main plate. Here two different weld throat thicknesses a were considered resulting in two types of specimen, which are denoted as D.3 and D.7. Figure 3 shows the geometry of the specimens. Besides statistical evaluation of the fatigue tests, the specimens were numerically investigated by different approaches of fatigue assessment (Fricke & Feltz, 2009). The computations were conservative, but some restrictions and shortcomings concerning the estimation of the critical notch were discovered. Furthermore, the notch stress intensity factor approach was applied by Fischer et al. (2010) analysing the averaged SED in the control sector at both the weld toe and the weld root. In addition to the mesh layouts suggested by Lazzarin et al. (2008) two different uniform meshes, which use both elements with a quadratic shape function, were applied: a rather coarse mesh with triangular elements having a size of 0.28 mm and a finer mesh, see Figure 4. The computed values of the averaged SED ΔW for the two specimen types are conservative and are located within the proposed scatter band.
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Figure 3. 2009).
Geometry of the specimens (Fricke & Feltz,
Figure 5. Convergence rates of different refinement strategies applied to a 2D problem of linear elasticity with singularities.
Figure 4. Meshes used for the computation of the mean SED.
Moreover, the predicted crack locations, which are identified by means of a higher SED-value, agree with the tests. 4
CONTROL OF THE DISCRETIZATION ERROR
The quality of a finite element approximation can be controlled either by reducing the size of the elements (h-refinement) or by increasing the polynomial degree p of the shape functions of elements (p-refinement), see Szabo & Babuska (1991). A combination of both approaches yields the hp-version of the finite element method. A uniform h or p-refinement corresponds to a global reduction of the element size or an increase of the polynomial degree of all elements, respectively. The convergence rate of the refinement process depends on the strength of the singularity λ controlled by the angle of the V-notch and the chosen refinement strategy. In Figure 5 the error in energy norm is plotted in a double logarithmic
style against the number of degrees of freedom N for different refinement strategies applied to a two-dimensional linear elastic problem. In absence of singularities a uniform p-extension yields an exponential rate of convergence, which is characterized in Figure 5 by a curve with increasing (negative) slope resulting in a highly efficient discretization. Unfortunately, in almost all problems of linear elasticity, singularities are present and a p-extension on a uniform mesh will result in an algebraic rate of convergence, being still twice as high as the one of an h-extension based on uniform meshes. More efficient finite element refinements can be obtained when combining the increase of the polynomial degree with a local mesh refinement. In the hp-extension the mesh is refined in a geometric progression with a grading factor q = 0.15 towards the point of singularity and the polynomial degree is increased simultaneously. Small elements close to the corner are assigned to a low polynomial degree and with increasing distance from the singularity the polynomial degree is elevated linearly. The layout of these so-called geometric meshes is depicted in Figure 6 using R0 as reference parameter. In practice it is sufficient to carry out a uniform p-extension on a strongly graded geometric mesh. This extension process yields an exponential rate of convergence in the pre-asymptotic range, which asymptotically slows down to an algebraic rate still twice as high as the one of a uniform h-refinement, see Figure 5. When computations need to be performed with finite elements fixed in polynomial degree of low order (e.g. p = 2), an optimal convergence rate can be obtained applying radical meshes, see Figure 7, which are strongly refined towards the singularity (Szabo & Babuska, 2011). For the radical mesh the distance of the nodes with respect to the singular point are set to
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toe as well as the weld root are performed under plane strain conditions with Young’s modulus E = 2.06 × 105 MPa and Poisson’s ratio ν = 0.3. While considering frictionless contact conditions between doubler and main plate with zero gap, both p-version and h-version refinements were analysed. In the following, results are only described for the weld toe notch. Thereby, the convergence behaviour of the SED is studied based on the relative error e, which is defined similar to the error in energy norm by Figure 6. Geometric mesh with three layers at weld toe (not to scale).
e=
| Wex W | Wex
(3)
where Wex is obtained by the most accurate numerical approximation using finite elements with a polynomial degree of p = 8. It was decided to estimate the relative error on the basis of the averaged SED instead of the equivalent stress Δσeq mentioned in equation (1). 5.2
Figure 7. Nodal distances for radical mesh three layers applied (not to scale).
dk
⎛ p + 1⎞ λ ⎠
⎛ k ⎞⎝ R0 ⎜ ⎟ ⎝M⎠
,k
, ,
,M
(2)
where M is the number of layers of elements, k is the number of the current layer and R0 represents the reference length. For further details on h, p and hp-refinement we refer to Szabo & Bakuska (1991) and Szabo et al. (2004). A detailed description of radical meshes can be found in Szabo & Babuska (2011). 5 5.1
p-version refinement
Computations increasing the polynomial degree of the finite elements from five up to eight on a fixed mesh were carried out by Actis applying the finite element software StressCheck® (2010). A relatively coarse geometric mesh with 121 elements graded towards the notch root has been utilized, see Figure 8. Table 1 lists the number of degrees of freedom (DOF) for the global model and elements inside the sector as well as the computed mean SED ΔW and the error e according to equation (3). 5.3
h-version refinement
In this section the results obtained by an h-refinement will be considered. The corresponding computations were performed under plane strain conditions with ANSYS® 11.0 using element
NUMERICAL ANALYSIS OF THE DOUBLER PLATE Problem description
In order to study the influence of the different refinement strategies the example of the doubler plate with a small weld throat thickness a = 3 mm (D.3) is considered. Due to symmetry only a quarter of the doubler plate has to be discretized. The structure is subjected to a nominal axial stress range Δσn = 150 MPa. The computations of the SED within a sector of R0 = 0.28 mm at the weld
Figure 8. Finite element model used for p-version refinement computed with StressCheck®.
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Table 1. Computed ΔW and relative error e for R0 = 0.28 mm and Δσn = 150 MPa at weld toe using p-refinement.
Table 3. Computed ΔW and relative error e for R0 = 0.28 mm and Δσn = 150 MPa at weld toe for a geometric mesh.
Polynomial DOF DOF ΔW [Nmm/mm3] degree p (model) (sector) at weld toe e [%]
DOF (model)
DOF (sector)
ΔW [Nmm/mm3] at weld toe
e [%]
28,629 28,749 28,985 29,229
36 70 138 206
3.50250 × 10−1 3.51275 × 10−1 3.51424 × 10−1 3.51424 × 10−1
8.3 6.3 5.9 5.9
5 6 7 8
3,163 4,441 5,961 7,723
670 880 1,120 1,390
3.52732 × 10−1 3.52667 × 10−1 3.52656 × 10−1 3.52654 × 10−1
1.5 0.6 0.2 0.0
Table 2. Computed ΔW and relative error e for R0 = 0.28 mm and Δσn = 150 MPa at weld toe for a uniform mesh.
Table 4. Computed ΔW and relative error e for R0 = 0.28 mm and Δσn = 150 MPa at weld toe for a radical mesh.
DOF (model)
DOF (sector)
ΔW [Nmm/mm3] at weld toe
e [%]
DOF (model)
DOF (sector)
ΔW [Nmm/mm3] at weld toe
e [%]
28,629 32,619 122,451 477,611
36 120 418 1554
3.50250 × 10−1 3.50920 × 10−1 3.51468 × 10−1 3.51738 × 10−1
8.3 7.0 5.8 5.1
28,629 28,749 28,985 29,221
36 70 138 172
3.50250 × 10−1 3.51473 × 10−1 3.51460 × 10−1 3.51463 × 10−1
8.3 5.8 5.8 5.8
Figure 9. Coarsest and finest mesh utilised by uniform h-refinement at weld toe.
type PLANE82, which is an eight-node element utilizing quadratic shape functions. Applying a uniform mesh refinement within the sector results in a noticeable growth of the global number of degrees of freedom and therefore computational time, since the refinement spreads globally. The computed values of the SED ΔW for the four different levels of mesh refinement and the corresponding error e based on the most accurate approximation obtained by the p-version are shown in Table 2. Furthermore, the geometric and the radical meshes, which are geometrically graded towards the singular point, are considered. An example of each mesh with three layers of elements is shown in Figures 6 and 7, taking the size of parameter R0 as basis. Both refinement strategies increase the number of finite elements in the vicinity of the singular point, so that the surrounding mesh does not change. The corresponding results are shown in Tables 3 and 4, respectively.
Figure 10. model.
5.4
Error curves plotted against DOF of whole
Comparison of results
In order to compare the results of the different refinement strategies and to assess the convergence rate, the relative error e is plotted against the global degrees of freedom in a double logarithmic style, Figure 10. From this, it is evident that the p-extension on the geometric mesh yields the most efficient approach, which is due to its high convergence rate. Furthermore it can be observed that when being restricted to finite elements of low order, mesh refinement in a radical or geometric fashion improves the efficiency. In order to illustrate the effect of the geometric and radical mesh refinement more clearly, in Figure 11 the relative error e is plotted against the degrees of freedom related to the sector only. From this it can be seen that the application of geometric and radical meshes yields an improvement as
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out the p-version finite element analysis of the example presented in this paper applying the software StressCheck®. We would also like to thank both of them for fruitful discussions and access to their unpublished data and literature. REFERENCES
Figure 11. Error curves plotted against DOF inside sector.
compared to a uniform h-refinement, modifying the mesh only very locally. When comparing Figures 10 and 11 with respect to the efficiency of the computations, it should be kept in mind that the total number of degrees of freedom is relevant for the computational time. Hence, the p-refinement on a geometric mesh yields the best performance. 6
SUMMARY AND CONCLUSIONS
In estimating the fatigue strength of welded joints, the computation of the averaged strain energy in a cylindrical volume represents a promising approach. In order to compute the SED efficiently different finite element refinement strategies have been applied and compared to each other. To this end, a doubler plate has been chosen as a benchmark problem. The most efficient approach to compute the SED is to perform a p-extension on a geometric mesh. If finite element computations are limited to low-order elements, geometric or radical mesh increase the efficiency significantly. Although in two dimensions the SED can be also computed with h-refinement strategies, the efficiency of high-order finite elements is very important since the numerical effort in three dimensions increases dramatically, calling for efficient strategies such as the p-version of the finite element method. Further research will focus on the SED approach in three dimensions applied to welded joints which will be discretized with the efficient discretisation strategies studied in this paper for two-dimensional problems. ACKNOWLEDGEMENTS The authors would like to thank Prof. Szabo and Dr. Actis from Engineering Software Research and Development, Inc., St. Louis, U.S.A., who carried
Fischer, C., Feltz, O., Fricke, W. & Lazzarin, P. 2010. Application of the notch stress intensity and crack propagation approaches to weld toe and root fatigue. IIW-Doc. XIII-2337-10/XV-1354-10, Int. Inst. of Welding. Fricke, W. & Feltz, O. 2009. Fatigue tests and numerical analyses of partial-load and full-load carrying fillet welds at cover plates and lap joints. Welding in the World 54, No. 7/8, R225–233. Glinka, G., Shen, G. & Plumtree, A. 1995. A multiaxial fatigue strain energy parameter related to the critical fracture plane. Fatigue Fract. Eng. Mater. Struct. 18, 37–46. Hobbacher, A. 2009. Recommendations for Fatigue Design of Welded Joints and Components. IIW-Doc. 1823-07, New York: Welding Research Council Bulletin 520. Lazzarin, P., Berto, F., Gomez, F.J. & Zappalorto, M. 2008. Some advantages derived from the use of the strain energy density over a control volume in fatigue strength assessments of welded joints. Int. J. Fatigue 30, 1345–57. Lazzarin, P. & Tovo, R. 1996. A unified approach to the evaluation of linear elastic stress fields in the neighbourhood of cracks and notches. Int. J. Fract. 78, 3–19. Lazzarin, P. & Tovo, R. 1998. A notch intensity factor approach to the stress analysis of welds. Fatigue Fract. Eng. Mater. Struct. 21, 1089–103. Lazzarin, P. & Zambardi, R. 2001. A finite-volume-energybased approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. Int. J. Fract. 112, 275–298. Livieri, P. & Lazzarin, P. 2005. Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local strain energy values. Int. J. Fract. 133, 247–276. Radaj, D., Sonsino, C.M. & Fricke, W. 2006. Fatigue Assessment of Welded Joints by Local Approaches. Cambridge: Woodhead Publishing (2nd Edition). Sih, G.C. 1974. Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 10, 305–321. StressCheck 2010. http://www.esrd.com. Szabo, B. & Babuska, I. 1991. Finite Element Analysis. New York: John Wiley & Sons. Szabo, B. & Babuska, I. 2011. Introduction to Finite Element Analysis: Formulation, Verification and Validation, to be published by John Wiley & Sons. Szabo, B., Düster, A. & Rank, E. 2004. The p-version of the finite element method. In E. Stein, R. de Borst, T.J.R. Hughes (ed.), Encyclopedia of Computational Mechanics, Volume 1: Fundamentals, Chapter 5, pp. 119–139, New York: John Wiley & Sons. Yosibash, Z., Bussiba, A.R. & Gilad, I. 2004. Failure criteria for brittle materials. Int. J. Fract. 125, 307–333.
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