The following article will guide you about how to determine metal forming problems. Also learn about the three methods used for solving the metal forming problems.
A simple tensile test is a case of uni-axial stress state, i.e. say σ11 ≠ 0 while all other components of stress state (or stress tensor) are equal to zero. For such a case we may write the yield condition as given below. The material is in plastic state if-
where σ0 is the yield strength of the material in tension or compression.
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However, in metal forming processes, more than one stresses are often involved, therefore, we must have a yield condition based on all the components of stress tensor. Here we are considering the condition when the metal body just comes to plastic state or yield point. Therefore, we may write the yield condition in multi-axial stress state as some function of stresses as given below-
Yield condition is a material property, therefore, the yield function for homogeneous and isotropic materials should in fact be a function of invariants of stresses. Secondly, according to Bridgeman, the hydrostatic pressure does not make a metal yield. It only produces elastic volume change. Therefore, we convert the stresses at a point as sum of two factors as given below-
Therefore, the function (4.7) can now be written as
Different yield criteria may be constructed depending upon the values of the constants A, B, C, D, etc. and the number of terms taken in the above expression. In this regard two yield functions are well known. These are given below.
(i) Von Mises’ hypothesis of yielding.
(ii) Tresca’s hypothesis of yielding.
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However, there are other hypotheses of yielding as well such as maximum shear strain energy theory, this and some others can be related to von Mises’ yield condition.
(i) Von Mises’ Hypothesis of Yielding:
Von Mises’ yield criterion takes into account the first two terms on the R.H.S. of the Eqn. (4.9). The Eqn. 4.9 becomes
Value of K may also be determined by experimenting with a different state of stress. Thus by subjecting a thin tube to torsion we may create a condition of pure shear. By increasing the torsion and hence shear stress the material of tube is brought to the yield point. Let σ12 be the shear stress due to the torsion and let τ0 be yield strength of the material in shear. By substituting σ12 = τ0 and all other stress components equal to zero, in Eqn. (4.11), we get the following.
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K = τ0 = yield strength of material in shear. (4.14)
From Eqn. (4.13) and Eqn. (4.14) it is obvious that,
Equation (4.15) gives a relationship between the yield strength of metal in shear to its yield strength in tension. This provides a test whether the metal obeys von Mises’ yield condition or not. One can determine the yield strengths in tension and shear and see if Eqn. (4.15) is satisfied.
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Example 1:
A metal body is in plastic state under the action of following stress state-
(ii) Tresca’s Hypothesis of Yielding:
Tresca was an engineer by profession. He carried out a large number of experiments and came to the following conclusion.
A metal body reaches the yield point when the maximum shear stress in the body reaches the value equal to the yield strength of the metal in shear.
If the stress state is given by three principal stresses σ1, σ2 and σ3 the three maximum shear stresses are as follows.
In a tension test we may take that σ1 ≠ 0, while all other stress components are zero. The yielding occurs when σ1 = σ0 = yield strength in tension. Also we know that in case of uni-axial tension the greatest value of shear stress is equal to half the tensile stress. Therefore, at the yield point,
If any of the three factors on the left hand side of Eqn. (4.18) becomes zero, the material reaches plastic state, and also, that factor will be the operating factor for the plastic flow. It should be noted that in Tresca’s yield condition, only the maximum and the minimum principal stresses take part in the yield condition.
The intermediate principal stress does not appear in the yield condition. In von Mises hypothesis, on the contrary, all the stresses contribute to the yield condition and are also placed symmetrically in the yield function.
The two yield conditions, however, give similar results in the following two cases:
(i) Two of the principal stresses are equal.
(ii) When deformation is in plane strain condition.
Methods for Solving the Metal Forming Problems:
1. Slip Line Method:
Theory of Slip Lines:
Slip lines are in fact directions of maximum shear stress in the body undergoing plastic deformation. For the plane strain deformation the equilibrium Eqns. (5.2) and (5.3) given below, along with yield condition (von Mises’ or Tresca’s) for plane strain case, are solvable for the three variables σxx, σyy and τxy. The partial differential equations along with yield condition in plane strain are in fact hyperbolic partial differential equations.
Their solution admits discontinuities. Directions of maximum shear are the characteristic directions of these equations. Therefore, slip may take place along the planes of maximum shear. In fact the slip takes place on the entire maximum shear stress surface, however, we in a figure show only a cross section of the body, so we see lines of maximum shear stress called slip lines.
If we take the directions of maximum shear as co-ordinate directions and transform the Eqns. (5.2 and 5.3) along these directions, these equations will get converted into algebraic equations. This is in tune with the general properties of hyperbolic partial differential equations and is discussed below-
In case of plane strain, the stresses σxx, σyy and τxy are functions of x and y only and ԑpzz = ԑpxz = ԑPyz = 0. According to plastic stress strain relations,
Both the yield conditions, i.e. von Mises’ and Tresca’s, reduce to same function in plain strain condition. This is illustrated below. From Eqn. (5.7) it is clear that the relative magnitudes of principal stresses are in the order σ1 > σ3 > σ2. The principal stress σ3 being the middle stress, Tresca’s yield condition becomes,
These equations are in fact the equilibrium equations (5.2 and 5.3) in which the coordinate are taken in the directions of maximum shear. The R.H.S. of Eqn. (5.17) is zero which means the expression (P + 2Kθ) is not a function of α., but it may be a function of β. Similarly from Eqn. (5.18) we get that (P – 2Kθ) is constant with respect to β. The Eqns. (5.17-5.18) may, therefore, be written as given below
The Eqns. (5.19) and (5.20) are algebraic equations obtained from the partial differential Eqns. (5.2) and (5.3) simply by changing co-ordinate system and incorporating yield condition.
Therefore, the solution of these algebraic equations along with the given boundary conditions is same as the solution of the partial differential Eqns. (5.2) and (5.3) along with the yield condition and the given boundary conditions. The Eqns. (5.19) and (5.20) are also called Hencky’s stress equations. Hencky first derived them in the year 1923.
Directions of α and β slip lines:
Figure 5.2 shows the directions of α and β-slip lines with respect to algebraically major principal stress σ1. The α-line is at 45° from σ1 in the clockwise direction, and β-line is at 90° with respect to α-line in the counter-clockwise direction. In other words the major principal stress lies in the first quadrant of (α, β) right hand coordinate system.
Geiringer Velocity Relations:
Figure 5.3 shows a quadrangular element bounded by slip lines. The normal stresses are all equal to the hydrostatic pressure. The stress normal to the plane of the slip lines is also equal to hydrostatic pressure. Thus we can write
This means that longitudinal strain-increments or strain rates along α and β lines are zero.
Now consider (Fig. 5.4) in which the particle velocity at the point D is divided into two components Vα and Vβ along the tangents to α and β-slip lines respectively. Consider an adjacent point E on α-slip line which has rotated by an angle dθ over the distance from point D to the point E.
Let the velocity components at the point E be Vα + dVα and Vβ + dVβ along the directions of α and β-lines respectively. Let us take the components of Vα + dVα and Vβ + dVβ along the directions α and β at the point D. Taking that there is no extension along α-slip line, we get the following relation for α-line.
Since the point E is very near to D, hence dθ is very small, thus we may take that cos (dθ) ≈ 1 and sin(dθ) ≈ dθ. Substituting these values in Eqn. (5.23) and neglecting the terms having more than one differential, we get a velocity relationship along α-line as given below.
Equations (5.24) and (5.25) are called Geiringer velocity equations. Slip lines are the directions of maximum shear stress. Therefore, if shear slip takes place it must be along a slip line. These slips give rise to velocity discontinuities which are explained below.
Velocity Discontinuities along Slip Lines:
Slip lines are the lines of maximum shear stress and hence, plastic shear deformation or slip may take place along a slip line. It is clear that slip can take place only along the direction of maximum shear stress and not along any other direction. When slip occurs along a slip line the velocities on the two sides of slip line become different.
There is a discontinuity in velocity along the tangent to the slip line, however, the velocity component normal to the slip line remains continuous, because any discontinuity in the normal component means fracture. The tangential discontinuity is illustrated in Fig. 5.5. The stresses are also continuous across slip lines.
It should be noted that if there is a velocity discontinuity on any point on a slip line, it will be there all along the slip line with the same magnitude but along tangent to the slip line.
When a body is undergoing plastic deformation, its various particles move with respect to one another under the action of external forces. Therefore, a velocity field is created in the deforming body. However, this actual velocity field is unknown to us. We assume a suitable velocity field which satisfies the boundary conditions and the incompressibility condition.
According to upper bound theorem the rate of energy dissipation calculated from the assumed velocity field is always higher than that by the actual velocity field. Therefore, the die load calculated from assumed velocity field is also higher than that due to the actual one. The statement of upper bound theorem is given below-
Of all the kinematically admissible velocity fields the actual one minimizes the function J where-
The R.H.S. of Eqn. (5.52) is the sum of different components of rate of work being done. For instance, the first term denotes the rate of work being done due to plastic strain rate. The second term gives the rate of energy or power being consumed on the frictional surfaces such as interface with tool as well as on surfaces of velocity discontinuity inside the body.
The third term signifies the rate of work being done against the external applied tractions. For instance, in cold rolling of sheets, back tension is often applied. The rolls pull the sheet against the back tension. The rate of energy being consumed due to back tension forms part of the third term.
Application of Upper Bound Theorem:
Compression of a Strip in Plane Strain:
In order to illustrate the application of upper bound theorem, we consider here a very simple problem that is compression of a strip in plane strain. When a long narrow strip is compressed, the material flow is mainly in the transverse (width) direction and very little flow takes place along the axis (length) of the strip, except near the ends of the strip. So this is a case of plane strain deformation.
Figure 5.16 shows a strip of length b, width 2L and height h being compressed between two flat dies. We take OX, OY and OZ as rectangular co-ordinates such that OZ passes through the center line of strip. Origin O of the co-ordinates lies on the bottom face of strip.
We take that the bottom die is stationary and the top die moves down with a velocity Ů. Therefore, the particles in the top layer of strip move with a velocity equal to — Ů. Particles on the bottom layer have zero down ward velocity because the bottom die is stationary. For other points in the strip, we take a linear distribution of- Ů over the height of the strip. The down ward velocity Ůz of a particle ‘A’ at a height z from bottom is given by,
The strain rates may be calculated by differentiating the above velocity field. Since the velocity in the y-direction is zero. It is taken that plane sections before the deformation remain plane during deformation, i.e. the velocity in x-direction is function only of x and that in the z-direction is function only of z, therefore, the shear strain rates are zero. All the strain rates are given below.
In the present case τ = frictional stress at the top and bottom interfaces between dies and work piece. Since we do not know the die pressure, we cannot use Coulomb’s law of friction, therefore, we take a constant value of frictional stress, i.e. τ = m.K where m is the friction factor with values ranging from 0 to 1. The value m = 0, corresponds to frictionless case and m = 1 stands for the sticking friction. Also here the magnitude of velocity discontinuity is given by,
The first term on the R.H.S. of Eqn. (5.73) denotes the contribution due to frictionless compression in plane strain condition and the 2nd term gives the effect of friction.
In the above example the velocity field is assumed to correspond to the case that plane sections remain plane during the deformation. We could also take bulging of sides by introducing a parameter in the velocity field and could optimize the expression given in Eqn. (5.52) to determine the optimum value of the parameter.
When a body is undergoing plastic deformation, a stress field and a velocity field are set up inside the body. The stress field so created satisfies the equilibrium equations and the yield condition and also balances the external tractions (tool forces). Generally we wish to determine the tool forces. But since the actual stress field is not known, the external tractions cannot be determined.
Therefore, we construct a stress field which satisfies the equilibrium equations and the yield condition. The rate of work done by such a stress field for the deformation with the prescribed velocities is always less than that due to actual stress field. This means that the calculated tool force or load on the equipment would be less than that due to the actual one. In other words it would give lower bound on the loads.
The constructed stress field may include an unknown parameter whose optimum value may be determined by optimizing the above expression.