Strain rate intensity factors for a plastic mass flow between two conical surfaces

The concept of strain rate intensity factor was introduced in [1], where the asymptotic expansion of the velocity field in a perfectly rigid-plastic material was obtained near the maximum friction surface, which is determined by the condition that the specific friction forces on this surface are equal to the simple shear yield strength. In particular, it was shown in this paper that near the maximum friction surface the equivalent strain rate (the second invariant of the strain rate tensor) tends to infinity inversely proportional to the square root of the distance to this surface. We note that the same result was obtained in the case of plane flow in [2]. The strain rate intensity factor is defined to be the coefficient of the leading singular number in the series expansion of the equivalent strain rate near the maximum friction surface. It was shown in [3] that there is a sufficiently complete formal analogy between the strain rate intensity factor and the stress intensity factor in mechanics of cracks [4]. In [5], it was suggested to use the concept of strain rate intensity factor to estimate the thickness of the layer near the friction surface where one should take into account viscosity effects. (Thus, this is an intensive strain layer formed as a result of a very large equivalent strain rate.) Therefore, the problem of calculating the strain rate intensity factor in specific processes is topical in the development of the general concept based on the use of the strain rate intensity factor and its applications in the theory of metal forming processes. These factors have already been calculated for several processes such as plane upsetting and drawing [3]. In the present paper, we calculate the distribution of the strain rate intensity factor in a plastic mass flow through an infinite converging channel formed by two conical surfaces on which the law of maximum friction acts (Fig. 1). A specific characteristic of this problem is the existence of two maximum friction surfaces and, accordingly, two distributions of the strain rate intensity factor. Since, according to the theory [5], the strain rate intensity factor is related to the thickness of the intensive strain layer near the friction surface, the solution of this problem may serve as a starting point for experimental confirmations of the theory. Note that the intensive strain layer thickness can be determined experimentally without any difficulties [6, 7] and the flow in an infinite channel of the shape under study can successfully model the tube drawing process [8].     

Prediction of fracture in the vicinity of friction surfaces in metal forming processes

It is shown that the use of a fracture criterion containing a characteristic length of the flow region makes it possible to further develop the theory of fracture in the vicinity of the maximum friction surfaces in metal-forming processes, with allowance for an infinite equivalent strain rate arising near such surfaces. A model of perfectly plastic rigid solids is considered in formulating the criterion. It is noted that the approach can be extended to more complicated models of plastic solids. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 169–174, September–October, 2006.     

Strain rate intensity factors for a plastic material layer compressed between cylindrical surfaces

For some models of rigid-plastic bodies, the strain rate fields turn out to be singular near the maximum friction surfaces. In particular, the equivalent strain rate (the second invariant of the strain rate tensor) tends to infinity when approaching such frictions surfaces. The coefficient multiplying the leading singular term in the series expansion of the equivalent strain rate near the maximum friction surfaces is called the strain rate intensity factor. This coefficient occurs in several models predicting the development of intensive plastic deformation layers near friction surfaces and in equations describing the change in the material structure in such layers. In the present paper, the solution is constructed for the compression of a layer of a plastic material obeying the double shear model between cylindrical surfaces on each of which the maximum friction law holds. The dependence of two strain rate intensity factors on the material and process parameters is calculated and analyzed.     

Singular rigid/plastic solutions in anisotropic plasticity under plane strain conditions

Assuming a rigid plastic material model with arbitrary smooth yield criterion, it is shown that the plane strain solutions are singular in the vicinity of maximum friction surfaces. In particular, some components of the strain rate tensor and thus the equivalent strain rate approach infinity. It is also shown that the exact asymptotic representation of the solution near maximum friction surfaces depends on the shape of the yield contour in the Mohr stress plane.     

Mixed mode (I and II) crack tip fields in bulk metallic glasses

Stationary crack tip fields in bulk metallic glasses under mixed mode (I and II) loading are studied through detailed finite element simulations assuming plane strain, small scale yielding conditions. The influence of internal friction or pressure sensitivity on the plastic zones, notch deformation, stress and plastic strain fields is examined for different mode mixities. Under mixed mode loading, the notch deforms into a shape such that one part of its surface sharpens while the other part blunts. Increase in mode II component of loading dramatically enhances the normalized plastic zone size, lowers the stresses but significantly elevates the plastic strain levels near the notch tip. Higher internal friction reduces the peak tangential stress but increases the plastic strain and stretching near the blunted part of the notch. The simulated shear bands are straight and extend over a long distance ahead of the notch tip under mode II dominant loading. The possible variations of fracture toughness with mode mixity corresponding to failure by brittle micro-cracking and ductile shear banding are predicted employing two simple fracture criteria. The salient results from finite element simulations are validated by comparison with those from mixed mode (I and II) fracture experiments on a Zr-based bulk metallic glass.     

Compression of a viscoelastic layer between rough parallel plates

If the maximal friction law is applied, then some generalizations of the Prandtl solution for the compression of a plastic layer between rough plates do not exist. In particular, this pertains to the viscoplastic solutions obtained earlier. In the present paper, we show that these solutions do not exist because of the properties of the model material and introduce a model for which this solution can be constructed. The obtained solution is singular. In particular, the equivalent strain rate tends to infinity as the friction surface is approached, and its asymptotic behavior exactly coincides with that arising in the classical solution. The obtained solution is illustrated by numerical examples, which, in particular, show that an extremely thin boundary layer may arise near the friction surfaces.     

EVALUATION OF DUCTILE FRACTURE CRITERIA IN A GENERAL THREE-DIMENSIONAL STRESS STATE CONSIDERING THE STRESS TRIAXIALITY AND THE LODE PARAMETER

This paper is concerned with evaluation of various ductile fracture criteria in a general three-dimensional stress state of stress triaxiality, the Lode parameter and the equiva- lent plastic strain to fracture. Evaluation is carried out by comparing fracture loci constructed by fracture criteria to experimental results of A12024-T351. Comparison demonstrates that the Modified Mohr-Coulomb criterion and a newly proposed criterion provide sufficient predictabil- ity of fracture strain. Moreover, evaluation is emphasized on the predicted cut-off value for stress triaxiality. The evaluation demonstrates that the Cockcroft-Latham, Brozzo, Oh, Ko-Huh and the new criteria coupled a reasonable cut-off value for ductile materials.     

An efficient method for the identification of the modified Cockroft–Latham fracture criterion at elevated temperature

The paper presents the theoretical part of a method for the identification of the modified Cockroft–Latham ductile fracture criterion at elevated temperature. Quite a general viscoplastic model is adopted to describe material behavior. The original criterion is path-dependent and involves stresses. Therefore, the identification of constitutive parameters of this criterion, as well as many other ductile fracture criteria, is rather a difficult task that usually includes experimental research and numerical simulation. The latter is impossible without a precisely specified material model and boundary conditions. It is shown in the present paper that for a wide class of material models usually used to describe the behavior of materials at elevated temperatures, the criterion is significantly simplified when the site of fracture initiation is located on traction-free surfaces. In particular, this reduced criterion solely depends on two in-surface logarithmic strains. Since there are well-established experimental procedures to measure surface strains, the result obtained can be considered as a theoretical basis for the efficient method for the identification of the modified Cockroft–Latham ductile fracture criterion at elevated temperature.     

Pitting formation under rolling contact

The mechanism of pitting caused by rolling contact is analyzed using the fracture mechanics approach. The governing factors are the initial crack length, crack angle, contact force, friction, strain hardened layer, and the hydraulic pressure of trapped fluid acting on the crack surface. Mode I and II stress intensity and the strain energy density factors are calculated by application of the two-dimensional finite element method. The strain energy density criterion is applied to show that shallow angle crack under small rolling contact force and friction enhances the probability of pitting under the roller’s running surface. The presence of a strain hardened surface layer also tends to affect the fracture behavior. The analytical results agree well with the experimental observations.     

Elastic-plastic mechanics of steady crack growth under anti-plane shear

For a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.     

Extension of a V-notched bar and failure of plastic bodies

The effect of plastic strain localization near the domains of sharp variation in shape and transverse cross-section of bodies is well known. But such processes have not yet been studied analytically well enough. On the basis of the model of an ideally rigid-plastic body, we propose an approach for determining the strain fields near the concentrators on the basis the motion of the displacement velocity field (near surfaces or discontinuity lines in the form of rigid-plastic boundaries and centers of the fan of slip lines under plane strain). We consider the problem on plastic flow with failure for a V-notched bar. We show that the plastic flow is not unique (in the framework of the solution completeness).We propose to use the strain criterion for choosing the preferable plastic flow. On the basis of the solutions thus obtained, we state an approach to studying failure processes for more complicated models of bodies.     

Modelling of plastic flow localisation and damage development in friction stir welded 6005A aluminium alloy using physics based strain hardening law

Plastic flow localisation and ductile failure during tensile testing of friction stir welded aluminium specimens are investigated with a specific focus on modelling the local, finite strain, hardening response. In the experimental part, friction stir welds in a 6005A-T6 aluminium alloy were prepared and analysed using digital image correlation (DIC) during tensile testing as well as scanning electron microscopy (SEM) on polished samples and on fracture surfaces. The locations of the various regions of the weld were determined based on hardness measurements, while the flow behaviour of these zones was extracted from micro-tensile specimens cut parallel to the welding direction. The measured material properties and weld topology were introduced into a 3D finite element model, fully coupled with the damage model. A Voce law hardening model involving a constant stage IV is used within an enhanced Gurson type micro-mechanical damage model, accounting for void nucleation, growth and coalescence, as well as void shape evolution. The stage IV hardening, observed in Simar et al. (2010), was found to increase the stiffness during plastic flow localisation as well as to postpone the onset of fracture as determined by the void coalescence criterion. Furthermore, the presence of a second population of voids was concluded to strongly affect the fracture strain of the high strength regions of the welds. This modelling effort links the microstructure and process parameters to macroscopic parameters relevant to the optimisation of the welds.     

Potential flow model of cavitation-induced interfacial fracture in a confined ductile layer

Fracture of a thin ductile layer sandwiched between stiff substrates often results from growth and coalescence of microscopic cavities ahead of an extending crack. Cavitation induced by plastic flow in a confined, ductile layer is analyzed here to evaluate the interfacial fracture toughness of such sandwich structures. For rigid-plastic materials, a new method is proposed in which the potential flow field of a fluid is used to approximate the plastic deformation. The principle of virtual work rate is applied to determine the equivalent traction-separation law. The method is demonstrated and validated for spherically symmetric cavity growth, for which an exact solution exists. We then study in detail the growth of an initially spherical cavity in a cylindrical bar of finite length subject to uniform traction at its ends. The results show that the stress-separation curves depend strongly on initial cavity size and the strain-hardening exponent, and weakly on the nominal strain. The method has clear advantages over numerical methods, such as finite-element analysis, for parametric study of cavity growth with large plastic deformation.     

A finite element investigation of the effect of crack tip constraint on hole growth under mode i and mixed mode loading

In this work, the effect of constraint on hole growth near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, a 2-D plane strain, modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. A finite element procedure that accounts for finite deformations and rotations is used along with an appropriate version of J₂ flow theory of plasticity with small elastic strains. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. The interaction between the notch and hole is studied by examining the distribution of hydrostatic stress and equivalent plastic strain in the ligament between the notch tip and the hole, as well as the growth of the hole. The implications of the above results on ductile fracture initiation due to micro-void coalescence are discussed.     

Thermo-mechanical contact behavior of a finite graded layer under a sliding punch with heat generation

The problem of a rigid punch contacting with a finite graded layer on a rigid substrate is investigated within the framework of steady-state plane strain thermoelasticity, in which heat generated by contact friction is considered with a constant friction coefficient and inertia effects are neglected. The material properties of the graded layer vary according to an exponential function in the thickness direction. Fourier integral transform method and transform matrix approach are employed to reduce the current thermocontact problem to the second kind of Cauchy-type singular integral equation. Distributions of the contact pressure and the in-plane stress under the prescribed thermoelastic environment with different parameter combinations, including ratio of shear moduli, relative sliding speed, friction coefficient and thermal parameters are obtained and analyzed, as well as the stress singularity and the stress intensity factors near the contact edges. The results should be helpful for the design of surfaces with strong wear resistance and novel graded materials for real applications.     

Cracking mechanisms in lithiated silicon thin film electrodes

The massive cracking of silicon thin film electrodes in lithium ion batteries is associated with the colossal volume changes that occur during lithiation and delithiation cycles. However, the underlying cracking mechanism or even whether fracture initiates during lithiation or delithiation is still unknown. Here, we model the stress generation in amorphous silicon thin films during lithium insertion, fully accounting for the effects of finite strains, plastic flow, and pressure-gradients on the diffusion of lithium. Our finite element analyses demonstrate that the fracture of lithiated silicon films occurs by a sequential cracking mechanism which is distinct from fracture induced by residual tension in conventional thin films. During early-stage lithiation, the expansion of the lithium-silicon subsurface layer bends the film near the edges, and generates a high tensile stress zone at a critical distance away within the lithium-free silicon. Fracture initiates at this high tension zone and creates new film edges, which in turn bend and generate high tensile stresses a further critical distance away. Under repeated lithiation and delithiation cycles, this sequential cracking mechanism creates silicon islands of uniform diameter, which scales with the film thickness. The predicted island sizes, as well as the abrupt mitigation of fracture below a critical film thickness due to the diminishing tensile stress zone, is quantitatively in good agreement with experiments.     

The strain rate intensity factor in the plane strain compression of thin anisotropic metal strip

Using an available analytic solution for instantaneous plane strain compression of a plastically anisotropic strip between two parallel plates the strain rate intensity factor is found assuming Hill’s quadratic yield criterion. The distribution of material properties is uniform. The effect of parameters characterizing plastic anisotropy of the strip on the magnitude of the strain rate intensity factor is demonstrated. A possibility to replace the strain rate intensity factor with the plastic work rate intensity factor is discussed. Singular behavior of the plastic spin in the vicinity of the friction surface is revealed and discussed.     

A numerical method for determining the strain rate intensity factor under plane strain conditions

Using the classical model of rigid perfectly plastic solids, the strain rate intensity factor has been previously introduced as the coefficient of the leading singular term in a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. Since then, many strain rate intensity factors have been determined by means of analytical and semi-analytical solutions. However, no attempt has been made to develop a numerical method for calculating the strain rate intensity factor. This paper presents such a method for planar flow. The method is based on the theory of characteristics. First, the strain rate intensity factor is derived in characteristic coordinates. Then, a standard numerical slip-line technique is supplemented with a procedure to calculate the strain rate intensity factor. The distribution of the strain rate intensity factor along the friction surface in compression of a layer between two parallel plates is determined. A high accuracy of this numerical solution for the strain rate intensity factor is confirmed by comparison with an analytic solution. It is shown that the distribution of the strain rate intensity factor is in general discontinuous.     

The effect of crack orientation on fracture behavior of tantalum by multiscale simulation

The quasicontinuum (QC) multiscale method is used to investigate anisotropic fracture behaviors of body-centered cubic (BCC) rare metal tantalum (Ta) loaded in Mode I and different fracture mechanisms are discussed from nanoscopic to continuum perspectives to have a deep understanding of brittle and ductile fracture. Initial crack deflection, brittle fracture by cleaving along low surface energy plane, ductile fracture as a result of dislocation emission and fracture accompanied by deformation twinning are all observed near crack tips of different crystal orientations. Particularly, some of these fracture mechanisms are found to be consistent with the latest experimental results. By examining different fracture behaviors, we find the surface energy and the available slip planes play a combined role in determining the fracture mechanisms near a crack tip. Both isotropic and anisotropic critical stress intensity factors are derived and compared for different crack orientations. A straightforward criterion that is proved to be applicable is used to distinguish brittle fracture from ductile fracture.     

粗糙面在梯度表面层上滑动接触的应力分布

对粗糙面在梯度表面层上的滑动过程进行应力分布研究,以模拟实际摩擦过程中,考虑塑性变形情况下,梯度覆层体中的应力分布规律,同时与均质体及单覆层体进行比较研究,分析了在表面载荷相同时滑动接触的应力分布。结果表明覆层体出现塑性变形后,在接触表面上的压力分布与弹性变形时有很大变化,在界面处梯度层的应力分布比单层膜更为理想,其应变梯度也较小;受相同表面载荷作用下产生塑性变形时,梯度层膜在基体产生塑性变形较小