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Mesoscopic-macroscopic concept of physical modelling and controlling shear fracture and processes determined by this fracture
E.S. Dzidowski
Wroclaw University of Technology, Faculty of Mechanical Engineering, Wroclaw, 50-372 Poland
Numerous reports indicate that the classical continuous medium mechanics no longer is able to reliably predict the behaviour of materials. This is due mainly to the macroscopic character of this theory and its incompatibility with the microscopic description of the behaviour of deformable materials. The above limitations hinder the development of both advanced materials and the theory and technology of mechanical working of materials. There are numerous indications that the problem can be solved as further advances are made in the newly established scientific discipline - physical mesomechanics. The main aim of this paper is to present a concept of physical mesoscopic-macroscopic modelling and controlling the shear fracture of materials. It is the shear fracture which determines most processes of mechanical working of materials and many tribological and other processes. The concept has been experimentally verified and applied to modelling and controlling a wide range of plastic forming and machine cutting processes. Examples of such applications are given and directions for further researches are suggested.
1. Introduction
Mesomechanics is a newly created scientific discipline [1, 2]. The establishment of this discipline provided a stimulus for innovative research and has made possible effective management of the previously existing interdisciplinary researches.
The main aim of this paper is to show the benefits stemming from the application of mesomechanics to the analysis of the development of strains and fracture in processes based on shear. The earlier methods of such analysis were based mainly on the slip line field theory and the theory of transition zones. Both theories have serious limitations.
According to the slip line field theory (Fig. 1(a)), the plastic sinking of the cutting tool in the sheared material first causes gradual widening and then narrowing of the plastic strain area [3]. Characteristically, this area finally assumes the shape and dimensions of a line (a plane) with zero thickness (Fig. 1(a, IV)). The line defines the location of slip velocity discontinuity and it is identified with the presumed trajectory of fracture. It is thought that the only way in which the shape of the fracture trajectory can be changed is by eliminating the rotation (bending) of the sheared material, which is usually done by pressing the sheared material against the cutting tool (Fig. 1 (a, V-VI)). With this the possibilities of the method are exhausted. The fact that shear strain determined by this method approaches infinity poses an additional problem. The problem has been partially solved by the development of the theory of transitional zones.
The introduction of the theory of transitional zones (Fig. 1(b)) made the values and distribution of strain in the final stage of shearing real [4]. This means that instead of a line (a surface) with zero thickness (Fig 1(a, IV)), an area having the shape of a biconvex lens is considered (Fig. 1(b, XI-XII)). The beginning of the formation of this area is identified with conditions corresponding to the action of an absolute stress concentrator (Fig. 1(c, VII-X)). But it is not known when and why such a significant change in the stress concentration conditions occurs. Moreover, it is assumed that once the lens is formed, it does not change its shape but only diminishes as the displacement of the cutting tool increases (Fig. 1(c, XIII-XIV)).
To sum up, the above macroscopic theories do not explain clearly enough the mechanism and causes of the fracture of a material during its shearing. This makes the control and optimisation of shear-based processes (machining, die shearing, etc.) difficult. In the author’s opinion, these limitations can be overcome through a recourse to the mesoscopic aspects of shearing.
2. Mesoscopic-macroscopic concept and model of shearing process
The mesoscopic-macroscopic concept of the shearing process, proposed by the author of this paper, is illustrated in Figs. 2 and 3. According to this concept, the onset (Fig. 2(b)) and then the development of strain localization in mesoscopic shear bands (Fig. 3(a)) are ofkey importance.
© E.S. Dzidowski, 2004
A concept based on slip line theory
Macroscopic concepts of shear process
A concept of shear with quasi-absolute stress concentrators (for comparison)
0 ____i_
A concept based on transition zone theory
Fig. 1. Macroscopic concepts of shear process: shear stages according to slip line theory (I-IV) and effect of clamp Q on shape of slip velocity discontinuity line as likely fracture trajectory (V-VI) (a); shear stages according to transition zone theory (XI-XIV) (b); distribution of slip lines for shear with flat (VII-VIII) and sharp-pointed (IX-X) stress concentrator (c). Based on [3] and [4]
Here a case of strain localization in quasi-isothermal, mesoscopic shear bands (SB) is considered. The development of shear bands manifests itself in the appearance of a lenticular strain localization zone (Fig. 2(b)).
The beginning of strain localization in the mesoscopic shear bands puts an end to the displacements of the free surface: Us = Us max (Fig. 2(a, b)). The moment when shear bands appear and the free surface displacements are inhibited can be easily predicted. It is enough to know the relationship between limiting strain Ul and strain-hardening coefficient n [5]. Strain (displacement) Ul is limiting from the strain localization point of view. The development and properties of the dislocation structure within shear bands (Fig. 2(h, i, j) determine the susceptibility of the material to
fracturing along the shear bands. The macroscopic course of fracture depends on the shape of the boundaries of the strain localization zone (Fig. 2(b), 3(b)) and the magnitude of the displacements of the material along the defective grain boundaries (Fig. 3(a)). The original grain boundaries become defective as a result of the interaction between them and the shear bands (Fig. 3(a, boundary GB2). This means that the course and effects of the shearing process depend here only on the synergy between the strain localization mechanism and the mechanism of fracture along mesoscopic shear bands.
As Fig. 3 shows, the fracture of the sheared material consists in its separation along mesoscopic shear bands SB (Fig. 3(a, separation AL)). Initially the fracture propagates
MACRO II Strain weakening due to localization of strains in mesoscopic, quasiisothermal shear bands
Displacement of sheared material, h
Fig. 2. Mesoscopic-macroscopic concept of shearing. Description in text
^ Complete opening
b
(I) Ai ti
A »/ bV
2'X
Boundaries
of layers q')(
\gB2 (grain and layer boundary BF-i or 2-3)
S-shape trajectory (surface Ar2-3-4) of final fracture of specimen on two pieces (I) and (II)
Fig. 3. Model of shear mechanism (b) and scanning electron microscopy results which validate it (a, c). Description in text
Classical shearing
Shearing with tension
Shearing with dynamic recrystallization
Shearing with compresion
Effect
Interpretation
Macroscopic course of shear with two active stress concentrators (2-ASCs)
Macroscopic shear model with 2-ASCs
Macroscopic course of shear with 1-ASC
Description
Complete fracture along shear zone boundary
Incomplete, cyclic fracture along boundaries of new shear zones
No fracture along boundaries of shear zones
Effect ^^LEMENTAL
Fig. 4. Examples of interpretation possibilities of the mesoscopic-macroscopic model of shear mechanism
along the shear bands (Fig. 3(b, trajectories A-B and E1-Dl)) and it consists in the loss of cohesion between the particular layers of the material. As the fracture reaches points 2 and 3, the fracture mechanism changes. From now on the fracture propagates only along the defective grain boundaries (Fig. 3(b, line 2-3) and Fig. 3(a, boundary GB2)).
Hence the final shape of fracture surface A-2-3-4' and that of surface A-2-3-4 A1 - 2 - 3 - 4 depend on the shape and width of the lenticular strain localization zone and on the magnitude of the displacements along the defective grain
boundaries (Fig. 3(a, b)). This means that the fracture initially propagates along the boundaries and then across the strain localization zone formed by the shear bands (Fig. 3(b, c)). Characteristically, the strain localization zone shrinks from top and bottom and eventually widens as a result of intercrystalline displacements of the material. Immediately before the total separation of the sheared material into two parts, the zone assumes a shape similar to parallelogram BCE1F1 (Fig. 3(b, blackened area)).
The above shear fracture mechanism model relates fracture not only to the properties (misorientation) of the mate-
rial substructure within shear bands (Fig. 2(i, j)), but also to transverse (acting transversely to the direction in which shear bands (SBs) develop) tensile stresses. Such stresses may arise naturally or be artificially generated as in shear with tension (Fig. 4). One should note here that the effective value of artificially generated tensile stresses amounts to about 0.25 of the yield point value (R02) [5].
Conventional shearing (Fig. 3) is an example of the natural generation of transverse tensile stresses. The stresses arise because of interaction between SBs and the original grain boundaries (see Fig. 3(a, GB2)). The development of shear bands results in strong flattening and rotation of the grains and in the formation of characteristic laminar lenticular strain localization zones (SLZ). This may be accompanied by the formation of wedge-shaped discontinuities along the original grain boundaries (see Fig. 3(a, GB2)). The tendency to form such discontinuities depends to a large degree on the condition of the original grain boundaries. One of the factors conducive to the lamination of the original grain boundaries may be adsorption of foreign atoms.
The defective grain boundaries become some kind of inclined planes whereby the sheared portions of material move and separate along the SBs (Fig. 3(a, shear bands SB1-SB4 and the next ones)). The separation is the most complete near the boundaries of SLZs, i.e. at the places where displacement (non-dilatational strain) gradients are the steepest (see Fig. 3(a and b)).
To sum up, the shape of slip fracture trajectories depends here on: the way in which shear bands develop, the properties of the shear bands, the condition of the original GBs and the shape of the SLZ formed by the mesoscopic SBs.
This means that by properly changing the properties and direction of development of shear bands and the way in which transverse tensile stresses are generated one can change the shape of the slip fracture trajectories or totally eliminate the fracture. Examples of such possibilities are given in the next subsection.
3. Mesoscopic-macroscopic criteria and principles of controlling shearing process
The presented mesoscopic-macroscopic concept of the physical modelling of the shearing process generates criteria for the effective control of processes based on the shearing of materials. The criteria include everything which affects strain localization in mesoscopic shear bands and the properties of the dislocation structure forming within the shear bands. Depending on the needs, stacking fault energy, the strain hardening ability of the material, the angle of misorien-tation of the substructure forming within the shear bands, the direction in which the shear bands propagate, the original condition of the grain boundaries, the location of the strain localization zone and so on can be such a criterion. Selected examples illustrating the possibilities of controlling the shea-
a m [c]
development development development
of two intersecting of shear bands of one family families of shear bands
Fig. 5. Mesoscopic concept of limit strain curve for sheet-metal forming processes: ways in which shear bands develop depending on state of strain (a-c)
ring process by acting on the properties and on how the mesoscopic shear bands develop are shown in Figs. 4 and 5.
Figure 4 shows mainly the influence of the length of the material cropped part, the additional state of stress and the properties of the shear bands on the course and effects of shear fracture.
According to Fig. 4, a change in the length of the cropped part leads to a change in the kind of process. In the considered case, the processes are cropping and orthogonal cutting (machining). They differ only in the shape of the macroscopic strain localization zone (MSLZ) and the way in which the latter develops. In the case of cropping, the zone assumes the shape of a biconvex lens whose axis is parallel to the direction of shearing. In the case of machining, the MSLZ assumes the shape of a half-lens whose axis is skew to the direction of shearing. Because of the skewness of the axis machining is a process of cyclic formation of countless MSLZs, owing to which a chip forms. The type of chip depends solely on the properties of the substructure forming within mesoscopic shear bands.
This means that:
- the same phenomenon—the localization of strains in mesoscopic shear bands — underlies both processes;
- the material tendency to slip fracture depends solely on the properties of the substructure which forms within
shear bands; the substructure susceptibility to transverse tension is a significant factor here (Fig. 2(i, j));
- the direction of development and shape of the fracture trajectory depend on the shape of the MSLZ and the way in which the transverse tensile stresses are generated.
Another example of the benefits stemming from taking the mesoscopic aspects of shear fracture into account is a new concept of the limit strain curve (LSC) for sheet-metal forming (Fig. 5).
It has been established and experimentally verified that the same phenomenon — the localization of strains in mesoscopic shear bands — underlies strain localization and sheet metal fracture in forming processes. On this basis a new mesoscopic concept of limit strain curve (Fig. 5) has been developed. The concept is derived from the synergy between the mechanism of localization of strains in mesoscopic shear bands and the mechanism of fracture along the shear bands.
As it follows from Fig. 5 the way in which shear bands develop in sheet-metal forming is determined by the state of strain in the given process or in the given point of the drawpiece. Depending on the state of stress the following occur:
- dispersed development of two intersecting families of shear bands (Fig. 5(a));
- concentrated development of a single family of shear bands (Fig. 5(b));
- dispersed development of a single family of shear bands (Fig. 5(c)).
The mesoscopic concept of the LSC makes it possible to avoid the paradoxes associated with the hitherto existing forms of the LSC, such as the paradox (pertaining to the areas denoted as a and c in Fig. 5) inherent in the classical continuous medium mechanics and the deficiencies of the current theories of fracture. The problem is discussed in more detail in [9].
The mesoscopic-macroscopic concepts of slip fracture modelling presented here have already found wide practical application. It should be mentioned, that owing to this concept of modelling [5-7], several economic methods of improving the quality of cutting have been developed. The methods do not require the use of complicated tools or machines. The precision of cutting is within the grinding allowance. In addition, new models and principles of controlling the type of chip which forms during the machining of materials [8] and a new concept of the limit strain curve for sheet-metal forming processes [9] have been developed.
4. Suggested direction of further research
The further development of the concept of modelling and controlling processes based on shearing requires a quan-
titative assessment of the properties of the structure forming within the mesoscopic shear bands. One should also take into account the sensitivity of this structure to transverse tension since tensile stresses facilitate material separation along the shear bands and straighten the trajectory of fracture. Moreover, there is a need to better understand and describe the mechanism of dynamic recrystallization which accompanies the shearing of materials at ambient temperature (Fig. 2(j)). Such recrystallization contributes to the elimination of fracture and in the considered case it made it possible to obtain a smooth cut surface.
5. Conclusions
The presented mesoscopic-macroscopic concept of shearing opens up new possibilities for:
1. The development of principles of modelling and controlling shear fracture.
2. The creation of a mesomechanical theory of processes of mechanical working of materials.
3. The development of new technologies of mechanical working of materials.
4. The development of new materials characterized by specified susceptibility or resistance to shear fracture and understood differently than before machinability, drawabi-lity, abrasive wear resistance and so on.
References
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[2] V.E. Panin (Ed.), Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, Cambridge Interscience Publishing, Cambridge, 1998.
[3] VA. Timoshchenko, Elementy Teorii i Tekhnologii Razdelitel’nykh Protsessov, Shtiintsa, Kishinev, 1979.
[4] V.I. Belyaev (Ed.), Inzhenernaya Teoriya Plastichnosti, Nauka i Tekhnika, Minsk, 1985.
[5] E.S. Dzidowski, Mechanism of Shear Fracture in the Aspect of Controlled Decohesion of Metals (in Polish), Scientific Papers of the Technical University of Wroclaw, Monographs No. 11, Wroclaw, 1990.
[6] E.S. Dzidowski, Shear fracture at liquid nitrogen temperature, Materials Science and Engineering, A168 (1993) 11.
[7] E.S. Dzidowski, Some Critical Remarks about Application of Know Fracture Criteria in Modelling Processes Determined by Shear Fracture, in the 7* International ESAFORM Conference on Material Forming, Trondheim, Norway (2004) 745.
[8] E.S. Dzidowski and G. Chruscielski, Mesoscopic-Macroscopic Model of Chip Formation during Machining in Quasi-Isothermal Conditions, in Proceedings of the 6th ESAFORM Conference on Material Forming, Salerno, Italy (2003) 523.
[9] E.S. Dzidowski, Advances of Fracture Mesomechanics in Sheet Metal Forming Processes, in the 4th International ESAFORM Conference on Material Forming, Vol. I, Liege, Belgium (2001) 241.
