12. The Solution Methods Used i n Metal Forming Area

1. ME 612 Metal Forming and Theory of Plasticity 12. The Solution Methods Used in Metal Forming Area Assoc.Prof.Dr. Ahmet Zafer Şenalpe-mail: azsenalp@gmail.com Mechanical Engineering Department Gebze Technical University

2. 12. The Solution Methods Used in Metal FormingArea In metal forming area there are several methods that are used for the solution of problems. The most important of them can be listed as • limit theorems • slab method • slip line method • stress line method • finite difference method • the finite element method. In the below text the description of these methods are presented. Mechanical Engineering Department, GTU

3. 12.1. Limit Theorems 12. The Solution Methods Used in Metal FormingArea Limit theorems permit force calculations which provide values that are known to be either lower or higher than the actual forces. These calculations provide lower or upper bounds. A lower bound solution will yield a load prediction that is less than or equal to the exact load needed to cause a body to experience full plastic deformation. In metal forming operations, the prediction of a force that will surely cause the body to deform plastically to produce the desired shape change is more important. As a result of upper bound analysis a load prediction that is at least equal to or greater than the exact load needed to cause plastic flow is obtained. In upper bound analyses a yield criterion is satisfied and it is assumed that shape changes are geometrically self consistent. No attention is paid to stress equilibrium. Mechanical Engineering Department, GTU

4. 12.1. Limit Theorems 12. The Solution Methods Used in Metal FormingArea By applying upper bound theorem any estimate of the collapse load of a structure made by equating the internal rate of energy dissipation to the rate at which external forces do work in some assumed pattern of deformation will be obtained greater than or equal to the correct load [1]. For the upper bound solution an internal flow field is assumed that must account for the required shape change, and also be geometrically self consistent. The energy consumed internally in this deformation field is calculated using the strength properties of the work material. Besides the external forces or stresses are determined by equating the external work with the internal energy consumption. After obtaining solutions in the desired way, the assumed field can be checked for complete consistency by drawing a velocity vector diagram, namely hodograph. Several simplifying assumptions are made in applying the upper bound technique to metal forming operations. For the most of the cases that upper bound solution is applied, the flow is two dimensional [2]. Mechanical Engineering Department, GTU

5. 12.1. Limit Theorems 12. The Solution Methods Used in Metal FormingArea Upper bound solution can be applied to extrusion [3], closed die forging [4], forging [5], rolling of strip [6]. Although the upper bound method can be used for obtaining working loads in a short time and can be performed incrementally, it is not possible to predict either the details of the distribution of flow or any elastic deformation due to the prior assumptions mentioned above. Mechanical Engineering Department, GTU

6. 12.2. Slab Method 12. The Solution Methods Used in Metal FormingArea In slab method, also called the free body equilibrium approach, a force balance on a slab of metal of differential thickness is entailed. With this procedure a differential equation in one dimension is achieved. Using related boundary conditions, the solution is obtained by integrating this equation. The assumptions involved are; The direction of the applied load and planes perpendicular to this direction define principal directions, and the principal stresses do not vary on these planes. Although effects of surface friction are included in the force balance, these do not affect the internal distortion of the metal or the orientation of principal directions. Finally it is assumed that plane sections remain plane, thus the deformation is homogeneous [7]. The slab method is particularly used for processes where the material deformation mode is rigid plastic and the geometry of the workpiece is largely predetermined, such as sheet drawing [7], flat rolling [7], close die forging [8]. Mechanical Engineering Department, GTU

7. 12.3. Slip Line Method 12. The Solution Methods Used in Metal FormingArea For the application of the upper bound method, the deformation is assumed to occur along lines of intense shear. Metal inside any polygon moved as a solid block as deformation takes place. To consider a more realistic approach to metal forming operation, a type of field theory has been developed. This can be viewed as a graphical approach, which portrays the flow pattern from point to point in the deforming metal, and is named as slip line field analysis. Slip lines refer to planes of maximum shear which are oriented at 45° to principal planes. It should be noted that slip lines does not correspond to the planes of maximum shear stress. Mechanical Engineering Department, GTU

8. 12.3. Slip Line Method 12. The Solution Methods Used in Metal FormingArea  Similar to upper bound approach this analysis is based upon a deformation field that is geometrically consistent with the shape change. Furthermore, the stresses within the field are statically admissible. However, as the stress state outside the field is not considered, some proposed fields, which are otherwise acceptable, may violate equilibrium outside the deformation zone; in this case, such solutions are upper bounds. The commonapproach to this subject usually involves the following assumptions. The metal is isotropic homogenous and also rigid perfectly plastic implying that no work hardening is present. Deformation state is by plane strain. Moreover possible effects of temperature, strain rate, and time are not considered. As there is a constant shear stress at the interfacial boundary. usually, either a frictionless condition or sticking friction is assumed [9]. There are several works performed by this method on various metal forming subjects [10]. Mechanical Engineering Department, GTU

9. 12.4. Stress Line Method 12. The Solution Methods Used in Metal FormingArea Besides the wellknown methods mentioned above a new method has been developed lately for analysis of metal forming processes, namely stress line method (SLM). Extended slip line method and generative simulation is the starting point of stress line method [11]. The method of principal stress lines are applied to 2.5D geometry to determine the principal stress lines. This method has a range of application from the ideal rigid plastic material model based on plain strain to the linear elastic plastic and anisotropic strain hardening material model based on plain stress or strain or differentiated loads. Stress line method can be used to determine the stress variation and deformed shapes for the metal forming processes of bending, drawing, inward flanging process using blank holder, drawbead or stop bulge in one or more steps or stages with time scheduling. Mechanical Engineering Department, GTU

10. 12.4. Stress Line Method 12. The Solution Methods Used in Metal FormingArea Different concepts and models of SLM are present due to various application fields. Slip Line Method based on plastic plain strain which is the startingpoint of stress line method is used for the analytic solution of the 2D forming model based on the principles of plastic plain strain theory using the slip lines. The initial speed and stress rates and the final strain rates can be computed along the slip lines in the points of the blank sheet independently from each other. One application field for SLM is the rotational symmetric pieces with cylindrical wall and constant height without flange. Principal Stress Line Method is also used for the analytic solution of the 2.5D forming process model based on plastic plain strain theory using the principal stress lines, neglecting the changes of thickness, speed, stress and strain rates can be computed at any place on the deformed surface and in any time during the process independently from each others. As the method provides the exact plain strain history of any points, rigid plastic isotropic strain hardening material model can be applied both in theoretical and practical way. Moreover SLM is used to obtain the approximated solution of the 2.5D forming process model based on plastic plain stress theory using the estimated principal stress lines. The speed, stress and strain rates can not be computed and expressed independently from each others. Mechanical Engineering Department, GTU

11. 12.4. Stress Line Method 12. The Solution Methods Used in Metal FormingArea Thickness changing can be evaluated as the consequence of the medium stress. Normal anisotropy and strain hardening can be considered in the rigid plastic material model. Also hybrid models based on principal stress line method can be used as numerical methods, where the SLM provides the first principal strain lines for the model as the flowing paths along which curves the material segments are supposed to deform. Computing of the speed, stress and strain rates can only be made numerically step by step from the initial moment to the last using rigid or elastic plastic material models and supposing planar anisotropy [12]. The main advantage of using stress line method especially over the finite element method is the comparatively less solution time spent to obtain the stress variation and deformed shape. Over this advantage, stress line method cannot be used to determine the wrinkling behavior occurring in the flange part in sheet metal forming operations where no blank holder force is applied, due to the complex behavior of deformation composed of bending and stretching of the sheet metal. Mechanical Engineering Department, GTU

12. 12.5. Finite Difference Method 12. The Solution Methods Used in Metal FormingArea In finite difference method a numerical solution of the differential equation for displacement  or stress resultant is obtained for chosen points on the structure namely nodes or pivotal points, or just points of division. Thus the numerical solution is obtained from the differential equations which are applicable to the actual continuous structure unlike finite element method in which the actual continuous structure is idealized into an assembly of discrete elements, forwhich force displacement relations and stress distributions are determined and the complete solutionis obtained by combining the individual elements into an idealized structure. Mechanical Engineering Department, GTU

13. 12.5. Finite Difference Method 12. The Solution Methods Used in Metal FormingArea The numerical solution by finite differences generally requires replacing the derivatives of a function by difference expressions of the function at the nodes. At each node the differential equation governing the displacement or stress is applied in a difference form, relating the displacement to the external load applied at the given node and nodes in its vicinity. By this way a sufficient number of simultaneous equations for the displacements or stresses are achieved [13]. The finite difference coefficients of the equations applied at nodes on or close to, the boundary have to be modified, compared with the coefficients used at interior points, for satisfying the boundary conditions of the problem. This is one of the main difficulties and disadvantage in its use compared with the finite element method. This method is widely applied to many metal forming problems [14,15] until the improvement and extensive use of finite element method. Mechanical Engineering Department, GTU

14. 12.6. Finite Element Method 12. The Solution Methods Used in Metal FormingArea The finite element method (FEM) is the method whereby the difficulty of mathematically solving large complex problems is transformed from a differential or integral equation approach to an algebraic problem. This method can be applied to two or three dimensional continuum problems. First the continuum is discretized by dividing into a finite number of discrete regions; called elements which are connected at certain points; called nodes on the boundaries. In some cases internal nodes can also be used. The number of degrees of freedom at each node, normally refers to the displacements and their first partial derivatives (rotations) at the nodes. A displacement or velocity function is terms of nodal values are assumed to represent the corresponding field within each element. By using the strain displacement relations which relate the values at the nodes, and constitutive relations that relate stress and strain values at the integration points within an element, a virtual work or a variational formulation can be build up for the whole body. As a result the stiffness matrix relating the nodal displacement to nodal force is obtained. The stiffness matrices obtained for each element is assembled to obtain the global stiffness matrix. By the imposition of boundary conditions the corresponding solution to the problem is achieved. Mechanical Engineering Department, GTU

15. 12.6. Finite Element Method 12. The Solution Methods Used in Metal FormingArea With the improvement of computer technology and need for better designs, finite element method is also improved and extensively used worldwide. Today there are several commercial finite element programs available in the market. Some of them can be listed as, Ansys, Abaqus, Aska, Marc, Nastran. the development and application to several fields can be found in detail in Reference [15]. In the linear formulation of finite element method, it assumed that the displacements of the finite element assemblage are infinitesimally small and that the material is linearly elastic. There areseveral facts that distinguish nonlinear analysis from linear analysis. In nonlinear analysis the strain displacement matrix for each element is not constant and dependent to the element displacements. Material model and so the stress strain matrix is not linear. Also the boundary conditions may change related with the type of the problem. The analysis performed in metal forming area lays within the boundaries of nonlinear finite element analysis. Mechanical Engineering Department, GTU

16. 12.6. Finite Element Method 12. The Solution Methods Used in Metal FormingArea Nonlinear analysis can be classified into material and kinematic nonlinear effects. As a first kind in materially nonlinear only analysis, the nonlinear effects lies only in the nonlinear stress strain relation. The displacements and strains are infinitesimally small, as a result the usual engineering stress and strain measures can be used in the response description. Another type is composed of large displacements, large rotations but small strains. In this type of analysis the material is subjected to infinitesimally small strains that is measured in a body attached coordinate frame, while this frame undergoes large rigid body displacements and rotations. The most general analysis case is the one in which the material is subjected to large displacements, large rotations and large strains. For this case the stress strain relation is usually nonlinear. Another type of nonlinear analysis contains boundary conditions that change during the motion of the body. This is the case for the analysis of contact problems. Mechanical Engineering Department, GTU

17. 12.6. Finite Element Method 12. The Solution Methods Used in Metal FormingArea However for metal forming processes nonlinear finite element analysis nonlinear analysis subject can additionally be classified according to their material behavior. These are inelastic material behavior namely, rigid plasticity, elastoplasticity, creep and visco plasticity, and large strain elastoplasticity. For the metal forming operations where time effects on material behavior are important in the form of strain rate, a viscoplactic material model can be more appropriate to characterize the material response. A viscoplastic material model that is widely used is proposed by Perzyna [16]. This model relates the stress to strain rate. Several formulations and applications on time dependent viscoplasticity are presented by Owen and Hinton [17] Waszezyszyn [18] and Zienkiewicz and Cormeau [19]. Also viscoplastic approaches have been used with a pseudo time to analyze time independent elastoplasticity as presented in the works of Zienkiewicz and Cormeau [19] and Owen and Hinton [17]. Other applications on extrusion [20], and forging [21] have been presented. Mechanical Engineering Department, GTU

18. 12.6. Finite Element Method 12. The Solution Methods Used in Metal FormingArea Moreover elastic strains can be added to the viscoplastic strains to determine the elastic strains occurring during the deformation process. Then analysis type becomes elasto viscoplastic. In the work presented by Zienkiewicz and Cormeau [19] this analysis type is applied. Work presented by Chandra and Mukherje [22] considers elasto viscoplastic analysis with the inclusion of large strains and large deformations.  Rigid plastic method of analysis is applicable to metal forming problems in which the elastic recovery of the material is negligible. For the analysis of rigid plastic problems, Euler formulation consisting of small strains of increments are used. Some examples of application of this method are deformation of multipress bar drawing and extrusion by Mori, Osakada, and Fukuda, and simulation of plane strain rolling by Mori, Osakada, and Oda, simulation of three dimensional rolling by Mori and Osakada, calculation of forging by Briani, Berti, D'angelo, and Guggia. One of the latest works by Kiuchi, Yanagimoto and Victor presents a rigid plastic FEM code and describes its application to flow simulation during extrusion with aims to obtain knowledges and information usable for eliminating geometrical defects of products. The references for the above works are given in Reference [23] Mechanical Engineering Department, GTU

19. 12.6. Finite Element Method 12. The Solution Methods Used in Metal FormingArea Rigid plastic method of analysis can be used for operations where the material in consideration is in the form of bulk. Viscoplastic plastic method can be used also for the material in bulk form, but strain rate effects are also considered in this case. Therefore it can be deduced that rigid plastic and visco plastic methods can be used for metal forming processes where elastic response of the material is negligible. In order to determine the response of especially sheet metal forming operations elasto plastic analysis should be applied. For applying this method of analysis, either one of the solution methods which are previously described can be used taking the degree of displacement and rotations occurring during the deformation process into account. Marcal and King [24] studied on the elastic plastic analysis of the plane stress, plane strain and axisymmetrically loaded body of revolution. Yamada and Yoshimura improved Marcal and King's work by defining a scale factor for the elastic plastic transition. The detailed explanation for these works are presented by Darendeliler [23]. Mechanical Engineering Department, GTU