In this article we will discuss about how to analyze strain in metal forming process.
If on the application of forces, the particles of a deformable body change their relative positions with respect to one another, the body is said to be deformed. However, if there is no change in the relative positions of the particles with respect to each other, the body is said to be rigid and any motion which may consist of translation or rotation of its particles is governed by laws of rigid body motion.
Let a deformable body occupy a region G in space (Fig. 3.11) designated by Cartesian orthogonal system of co-ordinates. Any particle P of the body is assigned a set of three coordinates (X1, X2, X3). After deformation the body occupies a new region G’ and the particle P gets a new position P’ given by another set of co-ordinates (x1, x2, x3). Similarly any other particle of the undeformed body occupies a unique position in the deformed state.
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There is one to one correspondence between positions of particles in the undeformed state and the deformed state. No two particles of undeformed body can occupy the same position in the deformed state. Similarly a particle of undeformed body cannot occupy two positions in the deformed state.
Therefore, there is unique relationship between the positions of particles in the deformed state and in the undeformed state. Hence, we can write-
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Definitions:
Consider the neighboring particles P, Q and R in the undeformed body (Fig. 3.12) such that PQ is parallel to X2 and Q is at a distance dX2 form P. Similarly PR is parallel to X3 and R is at a distance dX3 from P. The point P has co-ordinates (X1, X2, X3). The co-ordinates of points Q and R are (X1, X2 + dX2, X3) and (X1, X2, X3 + dX3) respectively. After deformation the three particles occupy the positions P’, Q’ and R’ respectively.
The longitudinal strain is defined as the relative change in length or change in length per unit length.
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The shear strain is defined as the change in the angle between two straight lines which were originally at right angles.
Now the determination of relative change in length or change in angle requires a reference state. We can take either (i) the original positions of particles in the undeformed state as our reference or (ii) we may take the co-ordinates in the deformed state as our reference. The strain measure in the first case is called Lagrangian measure of strain and the in second case is called Eulerian measure of strain.
Strains in Lagrangian Measure:
For Eulerian measure, let us consider the points L, M and N in the deformed state of the body (Fig. 3.13) and find the positions of these particles in the undeformed state. Let their positions in the undeformed state be L’, M and N” respectively. The strain measures are defined below.
Strain-Displacement Relations:
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Finite Strain:
We shall first analyze the longitudinal finite strains in Lagrangian measure. Let us consider two neighboring points P and Q (Fig. 3.12), such that PQ is parallel to X2. The co-ordinates of P and Q are (X1, X2, X3) and (X1, X2 + dX2, X3) respectively. After deformation the point P gets a displacement U and Q gets the displacement U + dU.
The magnitudes of the three components (dUi) of the vector dU are given below-
In the expression under the square-root sign, the differential terms are very small compared to unity, and hence we may approximate the above expression as given below.
The R.H.S. of the above expression (3.71) consists of two parts. The first part is linear while the second part is non-linear. In this text we shall be generally dealing with linear strain displacement relations. Such strains are also called infinitesimal strains.
The strain-displacement relations for infinitesimal strains are given below:
Equations of Compatibility:
A close look on Eqns. (3.73a-f) reveals that all the six components of strain are function of three components of displacement. If we know the displacement functions (Ui), we can determine strains uniquely. But if we know the strains, we may not get unique solution for displacements for arbitrarily prescribed strains.
We have six equations for the three unknowns i.e. U1, U2 and U3, which makes it an over determined system. Therefore, the strains have to be compatible. There must exist relations among the strains. These relations can be obtained by eliminating displacements as illustrated below. These relations are called equations of compatibility.
For derivation of these relations let us first consider Eqn. (3.73d) which is written below:
Equivalent Strain:
Like equivalent stress, equivalent strain is also often used in analysis of forming processes. It is defined as-
where γoct is the shear strain on octahedral plane.
In terms of general components of strain the effective strain is given below:
Equilibrium Equations in Curvilinear Co-Ordinates:
Often we need the equilibrium equations in cylindrical and spherical co-ordinates. These are given below for ready reference.
(i) Equilibrium Equations in Cylindrical Co-ordinates:
The cylindrical co-ordinates are r, θ, z (Fig. 3.13) and the stresses acting on an element are σrr, σθθ, σzz, σrθ, σθz and σrz.
(ii) Equilibrium Equations in Spherical Co-Ordinates:
The spherical co-ordinates are r, θ and ϕ (Fig. 3.14) and the stresses acting on an elements are σrr, σθθ, σϕϕ, σrθ, σrϕ and σθϕ.
Strain-Displacement Relations in Curvilinear Co-Ordinates:
In many real problems we need the strain displacement relation in cylindrical and spherical co-ordinate.
(i) The strain displacement relations in cylindrical co-ordinates are given below-
(ii) The strain displacement relations in spherical co-ordinates are given below-
