Among the different metal manufacturing techniques, the ones that require a proper theory to be described are those that modify the material through strain localization, or severe plastic strain development or material removal, or a combination of those. During machining, for example, material is removed from the work piece by inducing large amounts of shear in localized areas called Shear Bands (SBs) (see Figure 1
). Due to the high speed of the process (cutting velocities up to 90 m/s) [12
], high strain rates (
) are induced in the SBs [16
]. The high levels of stresses and strains give rise to plastic deformations, therefore inducing softening in the material at this same location. In addition to this, since the SBs usually span an extremely narrow area if compared to the global scale of the work piece, the continuum at this location experiences high stress and high strain gradients as well. To complete the picture, it must be mentioned that plastic deformations usually give rise to heat productions, and considering the case in which the rate of heat production is much larger than the heat flow within the material (especially in metals characterized by low thermal conductivity), high temperature fields are retained, and the SBs are referred to as Adiabatic Shear Bands (ASBs). In Figure 2
, the crystallographic analysis during a FSW process highlights the modification of the grain size and orientation induced during manufacturing. Shear bands have been reported to appear also during a high-temperature compression test on Nickel super-alloy [15
On top of the already complex scenario, at the location where the applied forces are transferred from the tool to the workpiece, the material experiences complex behaviors as well. High stress and shear gradients develop in this zone, leading to the same conditions which are found within the SB. Furthermore, the mechanism with which the loads are transferred from the workpiece to the material must be accurately assessed as well. The extreme conditions in which the contact must be modeled are actually different from the ones for which the classical contact models have been developed.
Overall, the problem of simulating machining seems to be composed of several sub-problems, some of them interconnected with each others, and some independent from one another. This complex problem can be more easily assessed if divided into separate and independent blocks. These might be listed as:
2.1. Strain Localization and Mesh-Size Dependency
Strain localization in a narrow area whose order of magnitude is comparable to the material grain size compels the adoption of a non-classical CM framework that would be able to model the size effect. In literature, several authors highlighted the inadequacy of the classical CM framework whenever it was used to reproduce experimental data in which the size effects were noticeable [20
]. Overall, the inconsistencies are more pronounced whenever the dimensions of the specimens become comparable to the order of magnitude of the specimen grain size, or, equivalently, when the deformations localize in areas whose size is comparable to the specimen grain size.
Fleck and Hutchinson performed several static torsion and tensile tests on copper wires of different diameters (12–170
]. Their results, reported in Figure 3
, highlight that smaller diameter wires are characterized by a much stiffer torsional response (scaled with the wire radius), although tensile tests performed on the same specimen demonstrate a very small, thus negligible, size dependency. Similar tests were performed by Liu et al. who reached the same conclusion [37
]. Fleck and Hutchinson also reported that the size-effect is present in a Vickers micro-indentation test conducted on single crystal tungsten specimens [20
]. The performed tests demonstrated a strong size-dependency, as the material hardness doubles by using an indent whose diagonal is one order of magnitude smaller. Similar size-effects in a micro-indentation test have also been reported by many other authors [23
The size-effect has also been experimentally investigated by Stölken and Evans [21
]. They presented the results of micro-bending tests conducted on 12.5
m and 50
m thin Nickel foils (Figure 4
). The results of their tests highlighted that foils with a smaller thickness were behaving stronger. The applied bending moment (normalized with respect to the foil thickness) of the 50
m thick foil is twice the one recorded with a 12.5
m thick foil. Similar tests leading to similar conclusions were conducted by Ehrler et al. [39
In the same referenced papers, so as in a broad branch of literature [28
], the authors point out that the classical CM is not able to capture these localization phenomena due to the absence of a length scale in the model that would counteract the localization of the fields. Micro-torsion tests also highlighted the shortcoming of the classical CM framework [40
]. The SGPT, whose mathematical description includes one or more length scales, has been proposed as a valid candidate to capture phenomena of different natures.
The size-effect can be predicted thanks to the definition of an additional deformation measure, i.e., the strain gradient. The dimensional analysis of this quantity reveals that it is not dimensionless and, unlike strain, it should be adimensionalized by means of a specific length, and the sensitivity of non-classical CM theories to size-effects comes directly from the definition of this non-dimensionless deformation measure. Many researchers investigated over the physical nature of the length scales of the SGPTs. Depending on the degrees of complexity of the observed phenomena, more than one characteristic lengths can be identified. For example, Fleck and Hutchinson identified up to five distinct characteristic lengths, each associated to one of the five invariants of the strain gradient tensor [20
]. These characteristic lengths however can be reduced to three in case of incompressible isotropic medium. In a successive study, Fleck and Hutchinson identified a single characteristic length in an equivalent plastic strain gradient theory [42
]. Liu and Dunstan, based on the SGPT of Fleck and Hutchinson, gave a physical interpretation to the characteristic length by making a connection to physical quantities via critical thickness theory [33
]. Duan et al., based on the same SGPT, associated the additional characteristic length to geometrically necessary dislocations through three different dislocation models [35
]. Similarly, Dahlberg and Boåsen assumed an equivalence between the microstructural length scale and dislocation density and provided an evolution law of the characteristic length [43
]. Zhang and Aifantis gave a comprehensive review of the interpretations of the characteristic length associated with SGPTs [44
Several other researchers focused on the calibration procedure and the quantification of this length for metals. Yuan and Chen proposed to identify the characteristic length from micro- and nano- indentation tests [45
]. Stölken and Evans developed a micro-bending test to measure the characteristic length [21
]. Abu Al-Rub and Voyiadjis also proposed to adopt micro- and nano- indentation tests to calibrate the characteristic length and its evolution law [34
Thereby, it can be concluded that the problem of properly capturing the size-effect rising during manufacturing simulation due to strain localization can be solved by adopting a medium description that includes the gradient of the strain as a deformation measure. In addition, adding the strain gradient as a deformation measure would solve not only the problem related to size-effect, but also the one related to mesh hypersensitivity.
Mesh hypersensitivity is an issue which has its root in the loss of ellipticity of the partial differential equation governing the medium equilibrium [47
]. In case the non-linear plastic behavior of the material experiences a decreasing flow stress, i.e., softening, the nature of the set of partial differential equations governing the equilibrium changes, and a strong mesh sensitivity is experienced, in particular, the solution does not appear to converge toward an asymptotic value when the mesh size is decreased. The process of recovering the property of mesh independence in literature is referred to as regularization procedure.
Jirasek and Rolshoven provided an extensive comparison of the regularization properties of many SGPT by analyzing the response of a mono-dimensional bar under tension, whose plastic behavior is characterized purely by softening [50
]. They explored the regularization mechanics of the Toupin-Mindlin elastic SGT [51
], the plastic version of Toupin-Mindlin SGT developed by Chambon et al. [53
], the Fleck and Hutchinson plastic SGT [20
] and the Mechanism-based SGT [54
]. Nguyen et al. coupled a non-local plasticity model with damage to successfully capture the softening behavior experienced by the material during ductile failure in the post-peak regime [55
]. Lele and Anand demonstrated that SGPTs are able to provide mesh independent results in case of viscoplastic material as well [56
Several other researchers overcame the mesh hypersensitivity issue through other solutions besides the SGT. Mediavilla et al. used a damage-enriched material model, in which the gradient of the damage field would enter in the material model, thereby achieving the same regularization effect produced by the SGTs [57
]. Many other researchers included the gradient of the damage variable in the model to regularize the solution [55
]. Higher Order continuum descriptions (see Section 3
) are also well known equivalent solution to avoid the pathological mesh dependence [61
]. In Figure 5
, proof of the regularization property of a Higher Order theory (Cosserat medium) is reported. A special combination of specimen geometry and applied boundary conditions induce the appearance of SBs inside the material [65
], and it has been demonstrated that the localization of the equivalent plastic strain inside the SB is mesh-independent in case a Generalized CM (Cosserat) theory is used.
Therefore, the issues of mesh hypersensitivity due to localization during machining simulations can be simultaneously addressed through the adoption of non-local theories, to which the SGPT belong.
2.2. Material Characterization at High Temperatures and High Strain Rates
The high strain rate fields produced during machining are inevitably intertwined with thermal-related consequences. Materials experiencing high strain rate are likely to plasticize and subsequently to generate heat. If the rate of heat production is higher than the rate of heat conduction, high temperatures will be retained in the plasticized zone. The consequences of high temperatures on material behavior can be easily modeled and they might be independently analyzed and simply assessed by performing material tests at high temperatures; on the contrary, the reliability of the model relating high strain rates with mechanical behavior, due to unavoidable raises in temperature, is dependent on the fidelity of the research done on the thermal behaviors.
In general, it must be recognized that reproducing the same level of shear deformation (
) and strain rate (
) that takes place during manufacturing processing under controlled lab conditions is not an easy task. The thermal aspect involved in the development of high strain rates was reported by Marchand and Duffy, who collected experimental data of both temperatures and shear strain fields during a dynamic torsion test [66
]. They tested a thin hollow tube of HY-100 steel in a torsional Kolsky (split-Hopkinson) bar properly modified to provide torsional loading at high strain rate (1600 s
). The specimen was 0.38 mm thick and 2.5 mm long, with an inner diameter of 9.5 mm. The authors adopted such high strain rates so as to induce the formation of adiabatic shear bands and then measure the temperatures that developed in the ASB. This condition is equivalent to the one that can occur during machining processes. From the collected data, reported in Figure 6
, it can be inferred that the thermal contribution to the energy balance cannot be neglected when SBs develop during machining.
The influence of the thermal field and strain rates on the mechanical behavior was also recently investigated by Rahmaan et al. [67
] (Figure 7
). They performed dynamic shear tests on 2 mm thick AA7075-T6 sheets at different strain rates (from 0.01 to 10
) inducing increments in temperatures of up to 60 °C in the specimen due to plasticity. The authors distinguished between a strain hardening dominant region and thermal softening dominant region. From their results, it is evident that plastic strain at high strain rates (500 s
) induce a temperature raise causing material softening. Additionally, the temperatures profiles obtained using different strain rates were directly compared, and the analysis showed that the influence of the strain rate on the developed temperature is considerably lower when the strain rates are reduced.
The researches reported in this section proves the fact that in the sought of a complete theory to model machining, the thermal and dynamic contributions cannot absolutely be neglected. Therefore, besides the usual governing equations relative to force equilibrium, a thermodynamical approach must be adopted so that energetic equilibrium equations would be included in the CM model. Therefore, the simulations of manufacturing processes involving very high deformation rates require an approach that would encompass thermodynamical considerations. This can be achieved by properly characterizing the media behavior from a deeper thermodynamical standpoint (from which the material constitutive behavior naturally derives), while solving the energy equation (heat equation) in addition to the standard set of equations governing the force equilibrium. This approach requires the temperature to be included as an additional degree of freedom of the continuum whose field is the solution of the energy equation. However, if the heat dissipation rate is very small compared to the head production rate, the boundary value problem can be considered as adiabatic, thus demoting the temperature from a degree of freedom of the system to a state variable [9
Another crucial consideration is that thermal-induced effects could be isolated and separately analyzed, but the opposite cannot be done with high strain rates, which are unavoidably related with heat productions. High temperatures soften the material, whereas higher strain rate have an opposite hardening effect. However, if high levels of strains are rapidly obtained, the material plasticize and temperature developments are expected to occur, thus interconnecting the strain rate effect with the thermal effect. So the ideal approach would be to separate these two aspects: first developing a model of the mechanical behavior that delivers the correct effect of thermal variation on the medium response, and then to successively incorporate the effect of the strain rate on mechanical behavior in terms of hardening.
2.3. Tool-Workpiece Contact
During machining processes, the contacts conditions between the tool and the workpiece lie beyond the assumptions made for the contact models usually used in mechanics. Especially the high temperatures and the high pressures, in fact, induce the material to behave more liquid-like than solid-like (see Figure 8
). The simple Coulomb law is not a good model anymore [69
Zorev’s model is a well-known refined friction model that is widely used for machining simulation. It recognizes two different zones at the tool-workpiece contact zone, each characterized by two different frictional behavior [70
]. He individuates a sliding zone, where the material response is purely elastic, and a sticking zone, that is instead characterized by plastic flow.
This special condition does not only require a modification of the classical CM governing equations, but it also impels the adoption of a proper numerical contact model able to cope with a change in the friction conditions from a sliding form to a sticking form (transversely promoted by high temperatures and stresses also). Based on this model, further modifications have been reported in literature attempting at delivering a precise model characterizing the metal behavior in the sliding and sticking regime [71
A throughout review of experimental data on the topic of tool-workpiece interaction was collected by Astakov [69
]. In this book chapter he reported many experiments which attempt to extrapolate many features of the phenomena from a tribological point of view such as the friction coefficient (assuming Coulomb law), contact stress distribution, thermal distribution and more.
Towards the development of a more appropriate contact model, the FSW benchmark, as suggested by Cahuc et al. [7
], might result in a useful mean which could be used to test the validity of the model itself. However, being FSW characterized by some on the main above-cited typical problems affecting machining, other independent features must be assesses before being able to test a more complex model.
The enhancement of the governing equations of the continuum mechanics from a classical to a generalized framework will likely bring the introduction of more complex variables associated with status of the continuum, such as strain gradients. Therefore, further considerations must be made on how these additional unknowns will fit into the proposed contact model, or even more important, how the contact conditions affect material deformations in enhanced continua.