Use of ductile fracture criteria to predict the failure in the formed plastic sheets

The modern and competitive industry requires a better knowledge and design of its processes and products. There is currently a widespread trend towards the simulation and optimization of industrial processes, including processes of forming of sheet metal. One of the basic tools to meet this demand is the use of numerical techniques and the development of reliable criteria for predicting the failure of sheet metal products. This paper examines the ability of different criteria of ductile fracture to predict the failure in the process of forming plastic sheet. Among the criteria explored are those proposed by Cockcroft and Latham, Brozzo et al., Oyane et to the. and Chaouadi et to the... Likewise, it is proposed a new approach that arises as a modification of the criterion of Freudenthal to include the effect of hydrostatic pressure on the ruling. To these criteria, discusses not only his goodness to predict the time and area of failure, but their ability to reproduce the physical process of judgement based on the experimental evidence.

In recent years the increasing demand of the industry of automotive and Aeronautics for the development of new products with increasingly complex geometries, as well as the use of ever more demanding materials (stronger, lighter, etc.), it has brought a renewed interest for a better understanding of the fundamentals of the process of forming of sheet metal. In general terms, the limiting factor of any process of formed sheet metal is the failure of this. Understanding any event that render for the failure of the sheet. They include located the sheet tear, the ductile fracture and the wrinkling by local buckling. The most commonly observed in practice is the failure by tear, which is a consequence of plastic instability in certain areas of the sheet in which occurs a thinning or incipient localized necking of the thickness of the same. Recently, to assess and predict the failure of the sheet, are being used different criteria of ductile fracture in conjunction with the use of the method of finite elements (FEM) to numerically simulate the process of forming [1,2].

This paper examines the ability of several criteria of ductile fracture, widely used in practice to predict and replicate the physical process of the judgment in processes of forming plastic sheets. Likewise proposed and analyzed an alternative criterion of judgement, which emerges the well-known criterion of Freudenthal be modified to include the effect of hydrostatic pressure on fracture process. To focus the study has been a model of finite elements (EF) of the process of deep drawing of sheet metal axilsimétrico, using the commercial package of ANSYS finite element. About this process, we have implemented the different criteria of fracture ductile, analyzing the place where they predict failure, the State of stresses and deformations in the area, the limit drawing ratio (RLE), etc. The results of the simulations have been compared with the experimental results obtained by Takuda et to the. [1] for sheets of different steels.

In the scientific literature, there are a wide variety of criteria or models to predict the failure by ductile fracture in metals. In General, these criteria can be grouped into three categories. The first consists of criteria based on purely experimental or semi-empíricas rules. The second category includes those methods obtained from modelling the physical process of fracture observed experimentally, that is, simulate nucleation, growth process and subsequent coalescence of voids that occurs during the ductile fracture of metals. Finally, the third group includes those criteria that predict the failure of the material as a result of the appearance of instability or fork in the plastic flow.

This paper will explore four ductile fracture criteria widely known: the classic criteria of Cockcroft and Latham [3] and Brozzo et to the. [4], including in the first category, and the criteria of Oyane et to the. [5] and Chaouadi et to the.[6], belonging to the second described group. These criteria require to know the history of load and deformation suffered material throughout the process, and can express themselves in their entirety in the following way:

Where  and^p represents the equivalent plastic deformation,  and^p_f is the equivalent plastic deformation in the instant of the fracture, s is the equivalent tension, s_h is the hydrostatic tension, s_I is the maximum main tension and the C_i  (i=...6) they are constant of the material to determine experimentally. In the here presented form, the exposed criteria predict that the failure of the material by ductile fracture produces when the value of the integral (left term of the equality) reaches an equal value to the unit.

Alternatively to the previous criteria presented, goes to propose in this work a new criterion semi-empirical based in a modification of the classical criterion Freudenthal. The criterion of Freudenthal establishes that the fracture initiates when the plastic work by unit of volume in a determinate point of the material reaches a true critical value. Said model has been employed successfully to predict the failure in processes of global plastic deformation [5]. Nevertheless, when it applies to processes of conformed of sheet his behaviour has not been at all satisfactory [2]. Between the causes argued to explain this fact, the accepted is that the original criterion of Freudenthal does not include the effect of the hydrostatic tension in the process of ductile fracture. It is very known that whereas the hydrostatic tension does not have an appreciable repercussion in the process of plastification, during the ductile fracture is all the contrary, favouring decisively the phase of growth of cavities and contributing with this to accelerate the process fracture. To take into account this effect, proposes here include in the expressesion of the plastic work an extra term that depend on the hydrostatic tension. The simplest form that poses next is a linear dependency, remaining the new criterion of the form:

                                                  (5)

where C₇ and C₈ and are constant of the material. Now, and by simplicity, will refer us to said criterion like criterion of the plastic Work Modified (criterion TPM).

A 2D model of finite elements of an axilsimétrico of sheet metal drawing process has been to analyse and compare the different criteria of judgement. The program of EOF's purpose has been used general ANSYS [8]. The proposed model focused on reproducing the experiments conducted by Takuda et to the. [2]. The figure 1 shows the geometry of the punch and matrix. They have modeled four different radii of punch: 2, 4, 8 and 20 mm (hemispherical). For successive simulations, preform them starting have been circular plates of various diameters.

Figure 1. Dimensions of the model of 2D drawing

The model of developed EF made a Lagrangian description, using a formula that has large deformations due to the strong non-linearity of the problem, mainly due to the plastic material behavior and contact between the different elements. The matrix, the awl and the prensa-chapa have been considered rigid elements. For the sheet metal behavior has been elasto-plastic with isotropic hardening strain. In first approximation has been used a model of associative plasticity isotropic and independent of the speed of deformation, taking the approach of von Mises yield criterion. This approach is appropriate where the characteristic of the sheet anisotropy is not very high.

Figure 2 shows the mesh used in a generic configuration. The elements used have been ring of 8 nodes (PLANE82). Depending on the thickness of the steel has been used 4 layers of elements for less than 1 mm thick sheets and 5 layers for thickness of 1 mm. The surfaces in contact have meshing using pairs of elements CONTA172 and TARGE169 of ANSYS. It is assumed that the friction between the surfaces is a Coulomb model, with a coefficient of friction of 0.1. The initial force exerted by the prensa-chapa on the sheet was obtained from the well-known expression of Siebel. Finally, it was considered the process of drawing as a quasi-static problem, solving incrementally. The established criterion of convergence implies this is reached when the module (Euclidean norm) of the vector of waste is less than 0.5% of the module of the vector of external loads.

Figure 2. Finite element mesh (deformed generic)

As mentioned in the preceding paragraph, the simulations try to reproduce Takuda drawing test et to the. [1]. Such tests are conducted using three different materials: 430 stainless steel with thickness of 0.78 mm (Material 1), sheet steel of high resistance with a thickness of 1 mm (Material 2) and zinc coated steel of 0.7 mm thick (3 Material). The mechanical properties of these materials are shown in table 1. As you can see in this table, the coefficient of normal anisotropy in the different badges is sufficiently close to the unit to consider acceptable, from a point engineering view, the assumption of isotropic behavior in simulations of EOF.

Material	And (GPa)	su (MPa)	K (MPa)	n	r	e(I), (f) (uniaxial)	e(I), (f) (def.plana)
1	210	488	829	0.20	0.81	0595	0.336
2		672	1020	0.14	0.91	0.634	0.285
3		748	858	0.03	0.87	0.303	0.130

Table 1. Mechanical properties of the materials (model s= Kandⁿ ,  r   coefficient of normal anisotropy )

For each one of the materials commented, has simulated the filling of circular sheets of different initial diameters, evaluating the failure or no of the same from the criteria of fracture described previously. The Table 2 sample the constants of the material for the distinct criteria. Said constants have calculated from the two values of and_I,f of the Table 1, which represent the maximum main deformation of fracture obtained respectively for an essay of traction uniaxial pure and for an essay of traction under conditions of flat deformation. Therefore, to determine the constants suffices with particularizar each one of the previous criteria for the conditions of both essays and clear his values. In the case that the criterion have an only constant to determine, the value of said constant corresponds with the half value of the obtained to the particularizar the criterion for the two essays.

	C1 (MPa)	C2	C3	C4	C5	C6(MPa)	C7	C8 (MPa)
Material 1	313.3	0.556	0.124	0.272	1.480	670.87	32.69	4407.54
Material 2	411.6	0.536	-0.070	0.167	5.240	2065.65	-8.78	-1026.06
Material 3	189.9	0.252	-0.094	0.073	4.430	836.24	-9.67	-541.13

Table 2. Constant of the material for the different criteria

Figures 3, 4 and 5 show the limit relationship of drawing (RLE) predicted the different criteria for three materials analyzed and tested four rays of punch. The value of the integral of each criterion corresponds to the average value of such an approach along the thickness. In this way when this value reaches a value equal to the unit said that the sheet has failed in this area. As you can see in the above figures, the obtained estimates are in reasonable agreement with the experimental results. In general terms, is observed that the numerical results slightly overestimate the experimental results, being less than 10% maximum differences. This trend was expected in this case, given that it is a direct result of not having considered the anisotropy of the material in the simulations. However, small differences were found between values predicted and observed for these materials, justify the use of an isotropic model, as it had already.

Once analyzed the predictive ability of the proposed models, the question that arises in a natural way is if those criteria adequately reproduce the physical process of judgement based on the experimental evidence. In the trials of Takuda et to the. the failure occurs in the area of the radius of the punch, for all situations and materials used. Notes that initially deformation is concentrated in a very close of the order of the thickness of the steel band, which is known as incipient localized necking, and finally there is the fracture of the material in that zone. This fact is in perfect agreement with the predictions obtained by all models analyzed for more ductile materials, i.e. materials 1 and 2 Material.

Figure 3. Connection limit of drawing (RLE) predicted by the criteria analyzed based on the ratio of the punch for Material 1

Figure 4. List drawing (RLE) limit predicted by criteria analyzed based on the radius of the punch for the 2 Material.

Figure 5. Connection limit of drawing (RLE) predicted by the criteria analyzed based on the radius of the punch for Material 3.

Figure 6a presents one mock case, showing the evolution of the value of integrals of the criteria (average thickness of the sheet) close to the bug instantly. As you can see, the failure is invariably predicted in the area where the localized thinning of the thickness of the steel is produced.

Figure 6a. evolution of the integrals of the criteria and deformation in the thickness to 2 material (results corresponding to a radius of the awl rp = 2 mm and initial radio of preform r0 = 42 mm)

Figure 6b represents the evolution of the relationship in the area of fault to go down the punch. You can see that at the beginning of the bug takes very close to 0.577 values in all the nodes, indicating that the tenso-deformacionales conditions in the incipient necking are approximately crush flat (zero deformation in circumferential direction). This fact has been widely observed in practice. Plastic instability, and elle is observed the failure of the material, appears when local conditions in the incipient necking deformation flat, regardless of the overall state of tension in the rest of the component (see for example reference [9]). This indicates that, for ductile materials with low anisotropy, ductile fracture models analyzed here appropriately predict the macroscopic physical process leading to the failure of the sheet.

Figure 6b. Distribution of the tensions in the area of incipient necking in each layer of material 2 nodes (results corresponding to a radius of the awl rp = 2 mm and initial radio of preform r0 = 42 mm)

For Material 3, whose ductility is quite low, the behavior depends on the used approach. The experimental results obtained by Takuda et to the. they are once again the failure in the incipient localized necking produced in the area of the radius of the punch. The criteria of Oyane et al., Chaouadi et to the. and the TPM predict failure in perfect agreement with these results. However, the models of Cockcroft and Latham and Brozzo et to the. They predict fair judgement at the end of the radius of agreement of the matrix. However, seen that in the area where there is the thinning of the sheet the integral of such criteria also takes very high values, and this is therefore an area of high probability of failure (see figure 7a). This occurs for the radii of punch 2, 4 and 8 mm. The 20 mm radius evolution is identical to the of other criteria, predicting the failure in the area of agreement the embedded piece base-pared.

Figure 7a. evolution of the criteria of Cockcroft and Latham and deformation in the thickness to 3 material (results corresponding to a radius of the awl rp = 2 mm and initial radio of preform r0 = 41 mm)

Figure 7b shows the evolution of deformation at the point where the failure is predicted. As you can see, the conditions of deformation at this point evolve maintain approximately constant thickness, that is. The failure is predicted, however, when deformation through thickness is homogenised, so in the case of a typical failure by ductile fracture, produced under conditions of pure shear in the plane of the sheet. This type and status of failure, although it is not strictly observed in the trials reviewed, appears frequently on little ductile material, as it is the case of some embuticiones with certain alloys of aluminium [2].

Figure 7b. Distribution of major deformations in the area of the material 3 failure prediction (results corresponding to a radius of the awl rp = 2 mm and initial radio of preform r0 = 41 mm)

This paper gives the following conclusions:

• The capacity has been analysed to predict the failure in different criteria of ductile fracture of sheet products. The predictions obtained very reasonably fit the experimental results obtained by Takuda et to the. for different steels.
• These criteria are predicting the failure in the area of incipient necking located when the local State of tension reaches the flat deformation conditions, based on the experimental evidence.
• For the less ductile material criteria of Cockcroft and Latham and Brozzo et to the. they have a slight tendency to predict the failure in areas of pure shear (constant thickness), fact experimentally observed also in some embuticiones made with aluminum alloys.
• The measure of working plastic modified proposed shows results in good agreement with experimental results and the rest of analyzed criteria.

[1] Takuda H., H. Fujimoto, Kuroda y. and Hatta N., Finite element analysis of formability of a few kinds of special steel sheets, Steel research 68, 9, 398-402, 1997.

[2] Takuda H., Mori k. and Hatta N. (1999) "The application of some criteria for ductile fracture to the prediction of the forming limit of sheet metals" j. Mater. Proc. Tech. 95, 116-121.

[3] Cockcroft M. g. and Latham D. j., Ductility and the workability of metals, j. Inst. Metals 96, 33, 1968.

[4] Brozzo p., De Luca B. and Rendina r., A New Method for the Prediction of the Formability Limits of Plastic Sheets, Proc 7th Biennial Congress of the IDDRG, 1972.

[5] M. Oyane, T. Sato, Okimoto k. and Shima S., Criteria for Ductile Fracture and Their Application, j. Mech. Working Tech., 4, 66-81, 1980.

[6] Chaouadi r., De Meester p. and Vandermeulen w., Damage work as ductile fracture criterion, Int. Journal of Fracture 66, 155-164, 1994.

[7] Clift S. e., Hartley p., Sturges C. e. b. and Rowe g. w., Fracture Prediction in Plastic Deformation Processes, Int. J. Mech. Sci., 32, 1-17, 1990.

[8] ANSYS User's guide, Release 5.5, ANSYS Inc., 1998.

[9] Forming Limit Diagram: Concepts, Methods and Application, Edited by r. H. Wagoner, k. S. Chan and S. p. Keeler, The Minerals, Metals & Materials Society, Warrendale, Pennsylvania, 1989.