Experimental verification of finite element failure criteria with respect to strain state and element size

P. Hogström1,*, J.W. Ringsberg1, E. Johnson2 1Department of Shipping and Marine Technology,

Chalmers University of Technology, Göteborg, Sweden 2SP, Technical Research Institute of Sweden, Borås, Sweden *Corresponding author. E-mail: Per.Hogstrom@chalmers.se

Abstract Within sheet metal forming, crashworthiness analysis in the automotive industry and ship research on collision and grounding, modelling of the material failure/fracture, including the behaviour at large plastic deformations, is critical for accurate failure predictions. In order to validate existing failure models used in finite element (FE) simulations, in terms of dependence on length scale and strain state, tests recorded with the optical strain measuring system ARAMIS have been carried out. With this system, the stress-strain behaviour of uniaxial tensile tests was examined locally, and from this information true stress-strain relations were calculated on different length scales across the necking region. Forming limit tests were carried out to study the multiaxial failure behaviour of the material in terms of necking and fracture. The failure criteria that were verified against the tests were chosen among those available in the FE software Abaqus and the BWH criterion proposed by Alsos et al. 2008. The results from the tensile tests confirmed that Barba’s law is valid for handling length scale dependence after necking. Also, the evolution of damage in the FE simulations was related to the processes ultimately leading to initiation and propagation of a macroscopic crack in the final phase of the tensile tests. Furthermore, the BWH criterion showed good agreement with the forming limit test results. The effect of pre-straining in the forming limit tests and the FE simulations of them is discussed. 1. Introduction Metal forming processes, crashworthiness in the automotive industry and ship collision and grounding are examples of areas where numerical models play an important role in process development, in design of full-scale tests/prototypes and more and more as a substitution for full-scale tests. The finite element (FE) method has been an established numerical tool suitable for these purposes for many years. Still, however, there is a need to continuously study, improve and validate the influence of many parameters in the numerical simulations and models used due to ever increasing demands for more accurate predictions of e.g. material failure or for mimicking the responses recorded in tests that have been carried out. Often, both the accuracy and reliability of numerical models and what they aim to represent is discussed, together with if their applicability is general or limited to specific conditions or types of analyses; see Alsos et al. [1]. This investigation focuses on the analysis of structures made of thin steel sheets considering their structural resistance to fracture as a result of large

plastic deformations caused by e.g. a collision between two cars or two ships. The objective was to analyse and compare models for material failure and fracture by means of numerical simulations using an explicit FE software and experimental results obtained from uniaxial tensile tests and forming limit tests. Specific topics are addressed such as element size of the mesh and length scale dependence on the failure limit (see Lehmann and Peschmann [2]), influence of multiaxial strain state on failure (see Alsos et al. [1]), and the applicability and reliability of the criteria studied in numerical investigations of ship-ship collision events (see Karlsson et al. [3] and Hong et al. [4]). In the experiments, the ARAMIS [5] optical strain measuring system was used which enabled accurate monitoring of displacements on the specimen surfaces. A brief description of the ARAMIS system is given in Section 2. In addition, when a material reaches the limit of its capacity to carry further loading, deformations localize into necking and become highly dependent on the length scale. In FE simulations, the length scale dependence must be taken into account when using elements of different sizes in the mesh. This is discussed together with the applicability of Barba’s law in Section 3. The influence of strain axiality on necking and fracture of the material can be investigated with information from forming limit tests, see e.g. Ramaekers [6] and Ragab [7]. This type of test is described in Section 4. In Section 5, an analytical solution of the BWH criterion [1] is presented and applied on the forming limit test results. The capacity of some failure criteria available in the FE software Abaqus/Explicit are investigated in Section 6. This section also presents the results from the FE simulations of the tests. Note that, throughout the current investigation, stresses and strains are always the true stresses and strains if nothing else is stated. 2. The ARAMIS system and the material tested The optical strain measuring system ARAMIS [5] was used in the tests to make precise measurements of the displacements on the surface of the specimens tested. The measuring system consists of two digital cameras connected to a computer with an image recognizing software matching the images from the cameras. The ARAMIS system divides the specimen surface into facets (pixels) whose distortion in three-dimensional space is monitored at certain time intervals throughout a test. The resulting facet displacement shows, for instance, regions of localized strains. The cameras used in the present study allow for a sampling frequency of 1 Hz. In addition, the material used in the tests was mild steel, denoted NVA by Det Norske Veritas (DNV) classification society [8], which is a commonly used material in shipbuilding. The thickness of the specimens used in the tests was 4 mm and the experimental setup fulfilled the requirements in the ISO 12004-2 standard [9] and the DNV rules [8]. 3. Tensile tests The tensile tests were carried out in accordance with the DNV rules [8], but slightly modified with regard to the loading speed of the tensile test machine. Because the elastic behaviour of the material was not in focus in the current

study, the speed of the tensile test machine crosshead stipulated for this region in the specification [8] was disregarded. Instead, the speed was set to achieve a high resolution in terms of output points from the ARAMIS system. Trends in tensile behaviour of metallic materials show that typical mild steel, like the NVA grade used here, has an engineering failure strain around 40% [11]. Based on this, a speed of 5 mm/min was used which gives an engineering strain of approximately 50% at the end of the test. Figure 1 shows the test setup for the tensile test. Two reference plates were introduced in the experimental setup: one was fixed to the machine foundation and the other plate followed the moving crosshead of the tensile test machine. The specimens were manufactured according to the DNV rules [8] with a width of 25 mm and a thickness of 4 mm. They were oriented with the loading direction aligned with the rolling direction of the sheet panel they were manufactured from; thus, the specimens were slightly curved prior to the test. This effect cannot be recorded by the ARAMIS system because it only measures the displacements on one surface of the test specimen; hence, also the position out of the plane had to be considered. By means of the recorded data, the last time step in the elastic region was used in the calculation of the initial area of the specimen. At this time step, the specimen had been straightened by the machine and the area contraction was negligible. In Fig. 2a, the stress-strain relationship obtained from the tests is presented. The “Aramis” stress was calculated using the force from the test machine divided by the actual area of the cross-section where failure (eventually) occurred.

Figure 1: Test setup for the tensile test.

The data obtained from the ARAMIS recordings allow for an analysis of the displacements over length scales defined by the engineer. As a result, the “Aramis” strain of the test rod can be measured using “virtual extensometers” (VE) which represent various length scales. A VE is defined as the distance between two points along the length of the test rod which are positioned on equal distance from the point of failure. A long VE corresponds to a strain value measured over the entire length of the test rod (see the bold line in Fig. 2a) while a small value of the VE corresponds to more local strain behaviour (see the dashed lines in Fig. 2a). In addition, necking is normally identified as the point in the engineering stress-strain diagram where the stress reaches its maximum; see the vertical line in Fig. 2a for its corresponding true strain value. Another definition which is used here is the point in the true stress-strain diagram where the different VE:s diverge, i.e. when the strains localize. Note that the two definitions of necking show good resemblance.

Reference plate 1 Test specimen Reference plate 2

The digital cameras

Figure 2: (a) Results from tensile tests with stresses calculated over the actual area of the test specimens and strains measured over virtual extensometers of different lengths (given in mm). (b) The failure strain as a function of the virtual extensometer length together with Barba’s relation fitted to the measured points. The curve fitted to the failure points in Fig. 2b is a relation with one asymptote in the global failure strain for the whole test rod and one asymptote when the length of the virtual extensometer approaches zero. Such a relation was first suggested by Barba in 1880 and many formulations have since then been proposed. In this study, the formulation proposed by Yamada et al. [10] was used which expresses the failure strain as:

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

VEf L

Wtce nεε ln (1)

where e is the mathematical constant defined as the base of the natural logarithm, W is the width (25 mm), t the thickness (4 mm), εn the necking strain (0.22) and LVE is the VE length over which the failure strain is measured. The Barba parameter, c, was calibrated (c = 0.8) to obtain a curve that matches the trend in the experimental points in Fig. 2b. 4. Forming limit tests Forming limit tests were carried out to study the multiaxial necking and material failure behaviour. The tests were carried out in accordance with the ISO 12004-2 standard [9] and the ARAMIS system [5] was used to monitor the surface displacements of the test specimens. Six different geometries were tested, see Fig. 3a, corresponding to six data points in the forming limit diagram (FLD). Three samples of each geometry were tested. The specimens were manufactured from the same sheet panel as the tensile test specimens. The orientation of the rolling direction was perpendicular to the major principal strain direction of the test section of the specimen. The specimens were clamped by bolts in the fixture, see Fig. 3a, and deformed quasi-statically normal to their plane using a half spherical punch with a diameter of 100 mm, see the test setup in Fig. 3b. A lubrication layer consisting of three layers of silicon grease and two sheets of reinforced Teflon was added between the test specimen and the punch in order to reduce the influence of friction. According

(a) (b)

to [9], the velocity of the punch should be 1.5 ± 0.5 mm/s. However, a velocity of 0.1 mm/s was used instead since it gives a higher resolution in time of the results; this is determined by the sampling frequency of the cameras of the ARAMIS system, see Section 2. Furthermore, a low velocity is also beneficial in order to reduce the influence from strain rate effects (note that this was not incorporated in the current study).

Figure 3: (a) The six test geometries: each of them corresponds to one strain state in the forming limit diagram. Geometry 1 is the circular plate (upper left) and geometry 6 is the narrowest of the geometries (lower right). (b) Test setup: the punch goes into the hold to which the specimen is clamped. The results from the forming limit tests are presented in the principal strain space in Fig. 4; here, ε1 is the major principal strain and ε2 is the minor principal strain. The evaluation of the points representing fracture was carried out in accordance with the ISO 12004-2 [9]. The points representing necking were evaluated analogously in the time step where the displacements on the surface of the test specimens started to localize. Moreover, the fracture points in Fig. 4 show a larger standard deviation of the results in the ε1 direction in comparison with the ε2 direction. The reason is that the scatter in the recorded data was larger in the major principal direction in contrast to the minor principal strain direction. Even though the evaluation procedure used to obtain these values is designed to reduce this scatter some remains [9].

Figure 4: Results from forming limit tests of the six geometries: mean values and standard deviations (denoted by the error bars) in the ε1 and ε2 directions, for necking and fracture.

(a) (b)

1 2

3 4 5 6

Note that the test specimens were manufactured so that the minor principal direction coincided with the rolling direction which has induced pre-straining of the material in this direction. This might be one explanation to why the necking points 2 to 6 are close to each other in the ε2 direction; the deformations in this direction have to overcome the pre-strain in the material. It also manifests in a slightly curved strain path: compare the points of necking and fracture for geometry 3 where the strain value for necking is positive for ε2 while the strain value for fracture is negative for ε2. At the point of fracture, the pre-strain has been redistributed and the deformations have started to grow in the ε2 direction. Geometry 1 is unaffected by this because it is rigidly fixed around its circumference. 5. Description of a forming limit curve using the BWH criterion Recently, Alsos et al. [1] proposed a failure criterion referred to as the BWH criterion. It is described in Eq. (2) on a stress-based form.

( )( )⎪

⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++

≤⎟⎟⎠

⎞⎜⎜⎝

⎛++

+++

+

=

otherwise)2/(1

3/23

2

0 if,1)1(3

21

5.013

2

21

212

1

ββ

ε

ββββ

εβββ

σn

c

nc

K

K

(2)

Here, 12 /εεβ &&= , K is the material strength coefficient, n is the strain hardening exponent [14] and ε1c is the local necking in an uniaxial strain state. The ε1c parameter can, when adapting the criterion to test values according to the procedure in [1], be interpreted as the point in the diagram where the forming limit curve (FLC) crosses the major principal axis. Studying the points of necking in the FLD in Fig. 4, a curve fitted to these points crosses the major principal axis at ε1 ≈ 0.27. The strain hardening exponent, n, was in the current study obtained by curve fit of the power stress-strain relationship to the tensile test results to n = 0.22. With this value and ε1c = 0.27, the BWH curve (on strain-based form) in Fig. 5 matches the test results.

Figure 5: Results (necking) from forming limit tests together with the analytical solution of the BWH criterion fitted to the experimental values.

6. Finite element simulations and results The finite element model of the test specimens used in the tensile and forming limit tests was built up with shell elements with reduced integration (S4R in Abaqus). Generally, shell elements which are thick in relation to their side lengths give poor results in bending since this type of element has a plane stress formulation. As a result, they are unable to resolve the stress gradients in their thickness direction. In the uniaxial tensile tests, no bending is present and in the forming limit tests, membrane and bending stresses are present but the membrane stresses are regarded to be dominant making the use of shell elements feasible. Element sizes between 1 and 6 mm were used throughout the study and only half of the test specimens were modelled due to symmetry conditions. The material was represented by an isotropic hardening model with piecewise linear isotropic hardening characteristics for the plastic behaviour. In the simulation of the forming limit tests, zero friction was assumed and the contact conditions between the punch and the specimen were modelled using the “general contact condition” in Abaqus/Explicit. For the modelling of different physical phenomena leading to failure of a material, Abaqus/Explicit offers several models that handle damage initiation and damage evolution. For damage initiation (DI) in ductile metals [12], either the ductile criterion, a phenomenological model for the nucleation, growth and coalescence of voids, or the shear criterion, which models shear band localization, may be used. In the current investigation, the ductile criterion was used to model the damage initiation in the FE simulation of the tensile tests using the necking strain from the tests, εn = 0.22, as the plastic strain at the onset of damage. In addition to these models, several criteria for simulating instability of sheet metal are available such as the “FLD criterion” which was used in the simulation of the forming limit tests [12]. In Abaqus/Explicit, criteria for sheet steel can only be used with elements that have a plane stress formulation, i.e. shell, beam or truss elements. After the damage initiation, a damage evolution model describes the degradation of the material up to the point of fracture. In Abaqus/Explicit, the evolution is defined either through the displacement at failure, uf, or the energy dissipated during the failure process, Gf. The former alternative was used in the current study. The displacement at failure is defined as uf = L×εf where L is a characteristic element length and εf is the plastic strain at failure taking into account the influence by the length scale. In the post necking region, the element size of the mesh has great influence on the solution; see the discussion in Section 2. Consequently, this dependency has to be accounted for when the damage evolution parameter is defined. In Abaqus/Explicit, damage evolution may be defined as linear, piecewise linear, or following an exponential behaviour. 6.1 Finite element simulations of the tensile tests The FE simulations of the tensile tests were carried out to study the damage evolution law in terms of mesh (element size or length scale) dependency and choice of damage evolution function. The stress-strain curve in Fig. 6a that represents the experiments is drawn up to the point where a macroscopic crack

started to propagate in the test specimen. This can be compared with Fig. 2a where two distinct phases of the true stress-strain relationship can be seen after necking; first a slow increase of the stress which then accelerates towards the final fracture when the macroscopic crack occurs and propagates. The two curves “linear” and “bilinear” damage evolution (DE) in Fig. 6a were obtained by FE simulations; see Fig. 6b for their definitions. The FE simulations were interrupted at a point which had a similar definition as the experimental point of interruption of a test, i.e. when the gradient of the curve rapidly increases and macroscopic failure is a fact. The displacement at failure, uf, used in the FE simulations was calculated using εf according to Barba’s law, see Eq. (1).

Figure 6: (a) Simulated tensile tests using a linear and a piecewise linear law for damage evolution (DE). (b) Illustration of linear and piecewise linear damage evolution (DE) relationships. Concerning the model with linear damage evolution, it deviates from the test curve at the point of necking. It also shows a too soft behaviour and underestimates the energy released during the damage process. This could be remedied if the damage variable, D, is allowed to evolve with a bilinear relation, see Fig. 6b. However, when defining such a law, the D - uf relation has to be corrected so that it results in the same amount of dissipated energy in order to reach zero stiffness at the same strain as the linear damage formulation; see the illustration of the damage evolution in Fig. 6b. Using a bilinear damage evolution relationship, a value of the x-parameter of 0.2 was found to give a damage evolution in good agreement with the test curves. A slower development in damage in the early phase and a more rapid development towards final fracture is, as discussed above, more in resemblance with the two phases of the experimental observations leading to macroscopic crack initiation and propagation. It is also in better agreement with the physical behaviour of the material. This behaviour of acceleration towards final fracture can be seen in the FE simulation with linear DE but in this particular case, it needs to be enhanced using a bilinear law. 6.2 Finite element simulations of forming limit tests In the FE simulations of the forming limit tests all six geometries in Fig. 3a were assessed. The FLD criterion in Abaqus/Explicit was used with tabular input values of ε1 and ε2 taken from the BWH necking curve presented in Fig. 5. The results from the FE simulations were evaluated in accordance with

(a) (b)

standards for evaluation of tests. It can be seen in the results presented in Fig. 7 that the trend of the FLC is captured even though the simulated points seem to slightly overestimate the major principal strain. Furthermore, the points representing the experimental results of geometries 2 to 6 in Fig. 5 are collected around the ε1 axis in contrast to the simulated points in Fig. 7 which are more separated in the ε1-ε2 space. One reason is that the material in the FE simulations was represented by isotropic material characteristics while the material of the specimens was known to be anisotropic and pre-strained from the rolling operation of the sheet panel they were manufactured from.

Figure 7: Results from the FE simulations of the forming limit tests. The analytical BWH criterion is shown for the same setup of material parameters. 7. Conclusions The results from the tensile tests showed a consistency in the comparison of different definitions of necking. Concerning the post-necking behaviour of the material used in the tests, this could, through theories and discussions of the initiation and propagation of a macroscopic crack, be related to the use of a bilinear law for damage evolution in the FE simulations. The method presented in Alsos et al. [1] to apply the BWH criterion to experimental values of forming limit tests showed good agreement between test values and the analytical solution. In terms of using the FLD criterion in Abaqus/Explicit, the results from the simulations capture the trend of the FLC used as input, but the major principal strain was slightly overestimated. In the comparison of the results from the tests and the FE simulations of the forming limit test geometries 2 to 6 (see Sections 4 and 6.2), the results from the FE simulations are more separated in the ε1-ε2 space in contrast to the test results. This difference can be explained by the pre-strain and anisotropy which is caused by the manufacturing process of the material tested. This effect was not incorporated in the description of the material characteristics of the FE model.

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[3] U. Karlsson, J.W. Ringsberg, E. Johnson, M. Hosseini, A. Ulfvarson, Experimental and numerical investigation of bulb impact with a ship side-shell structure, Accepted for publication in Marine Technology (2008) (In press)

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[8] Det Norske Veritas (DNV), Rules for classification of ships/high speed, light craft and naval surface craft, (2007)

[9] ISO 12004-2 Metallic materials – sheet and strip – Determination of forming limit curves – Part 2: Determination of forming limit curves in laboratory (2007)

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[12] Dassault Systèmes, Abaqus version 6.7 documentation (2007)