Vicinity Of Maximum Friction Surfaces Research Articles

An accurate numerical solution for the singular velocity field near the maximum friction surface in plane strain extrusion

In the case of rigid perfectly plastic material, the velocity field is singular in the vicinity of maximum friction surfaces (the equivalent strain rate approaches infinity in the vicinity of such surfaces). This causes significant difficulties with the convergence of finite element solutions. On the other hand, an accurate description of the velocity field near frictional interfaces is crucial for predicting the generation of a narrow fine grain layer that frequently appears near such interfaces. The present paper provides an accurate numerical solution for the singular velocity field in the vicinity of the maximum friction surface in plane strain extrusion. The singularity in the velocity field is represented by means of the strain rate intensity factor. The numerical approach is based on the method of characteristics. The strain rate intensity factor is determined by means of simple formulae once the solution for the radii of curvature of characteristic lines and velocities has been found. It is shown that the distribution of the strain rate intensity factor along the friction surface is discontinuous. Comparison with an analytic solution for material flow through an infinite wedge-shaped die is made and a high accuracy of the numerical solution is demonstrated. On the other hand, it is shown that the analytic solution is not accurate enough to find the strain rate intensity factor even for rather long dies.

A numerical solution for the axisymmetric flow of a plastic tube on a rigid fibre

In the case of rigid perfectly plastic material the strain rate intensity factor appears in the vicinity of maximum friction surfaces. This factor controls the magnitude of the equivalent strain rate in a narrow region near the surface. On the other hand, the equivalent strain rate controls the evolution of material surfaces. Therefore, the existence of the strain rate intensity factor in theoretical solutions are in qualitative agreement with experimental observations that demonstrate that a narrow layer with drastically modified microstructure is often generated in the vicinity of frictional interfaces in machining and deformation processes. In order to use the strain rate intensity factor for describing such material behaviour, it is necessary to develop an efficient numerical method for calculating the strain rate intensity factor. Since this factor is involved in a singular series expansion, it is evident that commercial finite element packages are not capable of calculating the strain rate intensity factor. In the case of Tresca’s solids the theory of characteristics can be used for the development of the method in question. This paper presents a first step in this direction

A general kinematically admissible velocity field for axisymmetric forging and its application to hollow disk forging

An approach for selecting general kinematically admissible velocity fields for axisymmetric forging problems is outlined. The approach accounts for the existence of a rigid zone at frictional interfaces and for the singular behavior of real velocity fields in the vicinity of maximum friction surfaces. The plastic work rate for a material obeying the von Mises yield criterion and its associated flow rule is expressed in terms of one arbitrary function of a single argument, its derivative, and anti-derivative. An upper bound solution for constrained forging is given. Comparison with an available upper bound solution is made and it is shown that the new kinematically admissible velocity field results in a more accurate solution at high friction. Other problems that can be treated are briefly discussed.

On the nonexistence of certain solutions for damage mechanics models

The objective of this paper is to demonstrate that elastic and rigid plastic boundary value problems in damage mechanics may not have a solution. Two classes of damage mechanics models are considered. The constitutive equations of one class of models consist of a pressure-independent yield criterion, its associated flow rule, Hooke’s law and a damage evolution equation. The damage parameter enters the yield criterion. Therefore, these models are partly coupled. The constitutive equations of the other class are the constitutive equations of the classical rigid perfectly plastic model (or a rigid viscoplastic model) supplemented with an empirical ductile fracture criterion. The viscoplastic model contains a saturation stress. These models are uncoupled. In the case of partly coupled models, a simple boundary value problem is formulated and solved. It is shown that the solution breaks down for certain values of input parameters. In the case of uncoupled models, it is shown that empirical ductile fracture criteria are not compatible with solution behavior in the vicinity of maximum friction surfaces. An approach to formulate a new type of empirical ductile damage models is outlined.

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.

Effect of Plastic Anisotropy on the Strain Rate Intensity Factor: A Simple Analytic Solution

Solutions for many rigid/plastic models are singular in the vicinity of maximum friction surfaces. In particular, the magnitude of the equivalent strain rate near such surfaces is controlled by the strain rate intensity factor. This factor is the coefficient of the leading singular term is a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. Since the equivalent strain rate has a great effect of material properties, it is of important to reveal the dependence of the strain rate intensity factor on parameters characterizing material models. In the present paper, quite a general model of anisotropic plasticity under plane strain conditions is adopted. Then, using an analytic solution for instantaneous compression of a layer of plastic material between two parallel plates the effect of the shape of the yield locus on the asymptotic behavior of the equivalent strain rate in the vicinity of the friction surface is demonstrated.

An accurate upper bound solution for plane strain extrusion through a wedge-shaped die.

An upper bound method for the process of plane strain extrusion through a wedge-shaped die is derived. A technique for constructing a kinematically admissible velocity field satisfying the exact asymptotic singular behavior of real velocity fields in the vicinity of maximum friction surfaces (the friction stress at sliding is equal to the shear yield stress on such surfaces) is described. Two specific upper bound solutions are found using the method derived. The solutions are compared to an accurate slip-line solution and it is shown that the accuracy of the new method is very high.

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.

Singular solutions in viscoplasticity under plane strain conditions

The paper deals with asymptotic behavior of viscoplastic solutions in the vicinity of maximum friction surfaces under plane strain conditions. The definition of maximum friction surfaces is that the friction stress is equal to the shear yield stress at sliding. The constitutive equations of the viscoplastic model adopted include a saturation stress. It is shown that it is possible to choose parameters of the viscoplastic model such that the regime of sliding is possible at maximum friction surfaces. In this case solutions are singular in the vicinity of such surfaces. Because of this feature of solutions, the viscoplastic model chosen possesses a smooth transition of qualitative behavior between rigid perfectly plastic and viscoplastic solutions, and this may prove to be advantageous for some applications.

Ductile Fracture in Metal Forming: A Review of Selected Issues

The paper reviews several theoretical and experimental methods for the assessment of ductile fracture criteria and for their application to the fracture prediction in metal forming processes. In particular, distinguished features of two widely used ductile fracture criteria are demonstrated in the case of free surface fracture. Conventional empirical ductile fracture criteria are not compatible with behaviour of plastic solutions in the vicinity of maximum friction surfaces. An approach to overcome this difficulty is discussed. Finally, a theoretical/experimental method to reveal a possible effect of geometric singularities on the applicability of ductile fracture criteria is reviewed.

Modelling of Damage Evolution in the Vicinity of Frictional Interfaces in Metal Forming

Conventional ductile fracture criteria are not applicable in the vicinity of maximum friction surfaces for several rigid plastic material models because the equivalent strain rate (second invariant of the strain rate tensor) approaches infinity near such surfaces. In the present paper, a non-local ductile fracture criterion generalizing the modified Cockroft-Latham ductile fracture criterion is proposed to overcome this difficulty with the use of conventional local ductile fracture criteria. The final form of the new ductile fracture criterion involves the strain rate intensity factor which is the coefficient of the principal singular term in a series expansion of the equivalent strain rate in the vicinity of maximum friction surfaces. When the velocity field is not singular, the new ductile fracture criterion reduces to the modified Cockroft-Latham criterion. The strain rate intensity factor cannot be found by means of commercial finite element packages since the corresponding velocity field is singular. In the present paper, the new fracture criterion is illustrated with the use of an approximate semi-analytical solution for plane strain drawing. It is shown that the prediction is in qualitative agreement with physical expectations.

Velocity Gradients and the Evolution of Material Properties in a Narrow Layer near Surfaces of High Friction in Metal Forming Processes

Theoretical solutions for several rigid plastic models used to describe plastic flow in metal forming processes are singular in the vicinity of maximum friction surfaces. In particular, velocity gradients and the equivalent strain rate approach infinity near such surfaces. Such singular behavior can be excluded from consideration by choosing another friction law or material model. However, a different approach is proposed in the present paper. The starting point of this approach is that many experiments show that velocity gradients are very high in the vicinity of surfaces of high friction and that a narrow material layer is formed near such surfaces whose properties are very different from the properties in the bulk. Taking into account that the equivalent strain rate has a significant effect on the evolution of material properties, this experimental fact suggests that a theory based on the singular plastic solutions can be developed to describe the formation of the aforementioned material layer. In the present paper such a theory is proposed to describe the evolution of grain size. It is assumed that, in addition to the equivalent strain rate, the material spin has an effect of the evolution of grain size. It is then shown that the solutions for the material spin are singular as well. The interrelation between the present theory and strain gradient theories of plasticity is discussed. It is shown that it is necessary to account for the strain rate gradient to propose a more adequate theory to deal with the material flow near surfaces of high friction. Some experimental results on the formation of the narrow layer of ultra-fine grains in the vicinity of the fraction surface in extrusion are presented. An illustrative example to relate these experimental results and the new theory is given.

Behavior of anisotropic plastic solutions in the vicinity of maximum-friction surfaces

The instantaneous flow produced by the expansion and twisting of a rigid-plastic orthotropic material between coaxial cylindrical rough surfaces subjected to the maximum friction law is considered under the assumption that the major axes of anisotropy are inclined at an arbitrary angle to the friction surfaces. The solution is reduced to quadratures and analyzed. The behavior of the solution in the vicinity of friction surfaces is compared with the behavior of known solutions for other models of rigid-plastic materials.

Influence of pressure dependence of the yield criterion on the strain-rate-intensity factor

The strain-rate-intensity factor is the coefficient of the principal singular term in an expansion of the equivalent strain rate in a series in the vicinity of maximum-friction surfaces. Its value controls the physical processes in a thin material layer near the frictional interfaces. The objective of the present paper is to study the effects of the pressure dependence of the yield criterion on the strain-rate-intensity factor. The results can be used to develop new methods for predicting the evolution of material properties in the vicinity of surfaces with high friction in metal-forming processes.

Exact Solutions for Powder Plastic Materials

A comparative study of two analytic solutions for the classical plasticity of incompressible materials and a plasticity model for powder materials is made to reveal an effect of plastic compressibility on qualitative behaviour of the velocity field in the vicinity of maximum friction surfaces.

The Strain Rate Intensity Factor and its Applications: A Review

The present paper concerns with the concept of the strain rate intensity factor in rigid plastic solids. The strain rate intensity factor is the coefficient of the principal singular term in the expansion of the equivalent strain rate in a series in the vicinity of maximum friction surfaces. Such singular velocity fields appear in solutions based on several rigid plastic models. Because of this singularity in the velocity field, many conventional evolution equations for material properties are not compatible with such rigid plastic solutions. On the other hand, qualitative behaviour of the singular rigid plastic solutions in the vicinity of maximum friction surfaces is in agreement with a number of experimental results. Therefore, the primary objective of research in this direction is to develop an approach to relate parameters of the singular velocity fields and parameters characterizing material properties. The approaches proposed in previous works are based on the strain rate intensity factor. In the case of analytical and semi-analytical solutions the strain rate intensity factor can be found by means of an asymptotic analysis of the solutions. A number of such solutions obtained by inverse methods are reviewed in the present paper and the strain rate intensity factor is found. An effect of process parameters on its magnitude is shown and discussed.

Plane-Strain Compression of a Three Layer Strip Containing Viscoplastic Material with Saturation Stress

The plane strain compression of a long symmetric strip consisted of a three layer material between rigid, parallel, rough plates is under consideration. Two possible geometrical configurations of the layers are examined (a) a viscoplastic material is situated between two layers consisting of a rigid/perfectly plastic material, (b) a rigid/perfectly plastic material lies between two viscoplastic layers. It is assumed throughout the paper that the viscoplastic law is bounded in that sense that it reaches its critical value (saturation stress) as the strain rate tends to infinity. Exploiting closed form solutions obtained, qualitative differences between them and the known from literature solutions for three layer material structure with classic viscoplastic material are discussed. Asymptotic behaviour of solutions in the vicinity of maximum friction surfaces is analysed for any configuration.

Qualitative behaviour of viscoplastic solutions in the vicinity of maximum-friction surfaces

The maximum-friction surface is a source of singular solution behaviour for several rate-independent plasticity models. Solutions based on conventional viscoplastic models do not show such behaviour. For a class of materials, there is a range of temperatures and/or strain rates where a necessity of the consideration of rate effects depends on the area of application of the final result. Hence, the same material under the same conditions can be represented by either rate-independent or rate-dependent models. In this case, a reasonable requirement is that viscous effects should not be very significant and, in particular, the qualitative behaviour of viscoplastic solutions should be similar to that of solutions based on rate-independent models. The present paper deals with this issue by means of the solution for simultaneous shearing and expansion of a hollow cylinder under plane-strain deformation. One of the goals of the paper is to show that there is a class of viscoplastic models satisfying the requirement formulated. The other goal is to find an asymptotic representation of the solution in the vicinity of the maximum-friction surface and compare it with the rigid perfectly plastic solution.

Maximum friction law in plasticity

AbstractThe maximum friction law postulates that the friction stress is equal to the maximum possible shear stress admissible by the constitutive equations. The boundary value problems including the maximum friction stress as a boundary condition reveal special mathematical features which are of interest in the development of theories and models, numerical simulation and engineering applications. The present paper shortly reviews the singularity in velocity fields that can occur in the vicinity of maximum friction surfaces. A large class of rigid plastic (in a broad sense that the elastic portion of the strain rate tensor is neglected) is considered. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Viscoplasticity with a saturation stress: distinguishing features of the model

Viscoplastic models including a saturation stress are considered. The existence of the saturation stress significantly changes the mathematical structure of solutions near maximum friction surfaces (surfaces where the friction stress is equal to the local shear yield stress). The main features of solutions based on such theories are: (a) sliding must occur at the maximum friction surfaces under certain conditions, (b) the velocity field may be singular in the vicinity of maximum friction surfaces. The objective of the present paper is to study these features of solutions. The mathematical structure obtained is considered to be advantageous for a class of materials and may lead to a convergence of viscoplastic solutions to the corresponding rigid perfectly plastic solutions. It seems that the latter is of importance for the construction of a unified theory that could describe the material behavior in the range from rate-independent plasticity to viscoplasticity. In the present paper, the study of the main features of the model is based on the exact closed-form solution to the problem of flow between two coaxial rotating cylinders. In the case of sliding, in addition to the aforementioned features, the asymptotic behavior of the velocity field in the vicinity of the maximum friction surface is found for a class of constitutive laws.