In this article we will discuss about the plastic deformation of metals with its diagram.
Referring to Fig. 6.2 the deformation of the specimen is elastic up to the yield point (or elastic limit) and it becomes plastic, i.e. the linear elastic region is followed by a nonlinear plastic region.
Although yield strength is an important design parameter for machine structures, but knowledge of plastic behaviour of metals too is essential for reasons:
(i) The more efficiently a given alloy is used for a machine component, the more closely its actual stress approaches the elastic limit. At times for reasons beyond control, the stresses may become very high, even above the factor of safety used.
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It is possible to make reasonably certain that the result of such misuse is only an inconvenience, and not a catastrophe. For example, a wheel of a car at high speed strikes unfortunately a barrier, the stress become higher than yield stress. The wheel may get bent if plastic deformation occurs (if the alloy used had ductility). The bending of the wheel causes inconvenience, far preferable to the alternative of fracture (if the alloy was brittle), when the breaking of the wheel could cause serious accident.
(ii) Plastic behaviour of metals and alloys are very important aspects to know as most of the metals are given controlled hot rolling or forging operations, etc.
(iii) Cold working operations like wire drawing, deep drawing, upsetting, etc. which are used for mass production of many metal parts, are restricted by plastic properties of metals.
The stress-strain curve has a positive slope beyond the elastic limit, i.e., in the plastic range the stress required to cause further deformation increases with increasing strain, a phenomenon called work-hardening or strain-hardening occurs.
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If the load is removed when the specimen is in the plastic range, it retraces a straight line path parallel to the initial line and reaches zero stress at a finite value of deformation after elastic part of deformation is recovered. At this stage, recovery occurs of the anelastic strain with the passage of time.
Ultimately, the strain left is the permanent plastic strain as shown in Fig. 6.13, which illustrates the initial part of the stress-strain curve. When the total deformation is small as in Fig. 6.13, the elastic and anelastic strains are significant, but can be neglected while studying large plastic deformations.
At least the anelastic strain may not be present. On reloading, !he plastic deformation starts again on reaching the stress level prior to Fig. 6.13 Initial part of a stress-strain curve showing the effect of suddenly unloading. The slope of the stress-strain curve decreases, as the strain increases. It becomes zero at some maximum stress.
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The engineering stress corresponding to this maximum is called the ultimate tensile strength (UTS), or simply the tensile strength of the material. Up to this point, the strain is uniformly distributed along the gauge length. Beyond this maximum, somewhere, normally near the middle of the specimen, a localised decrease in cross-section known as necking develops (Fig. 6.4 b).
Once the neck forms, further deformation is concentrated in the neck. The strain is no more uniform along the gauge length. As the strain increases, the cross-sectional area of the neck continuously decreases. This is called non-uniform plastic deformation. Pores nucleate (inside the metal) in the necked region at the interface of hard second-phase particles (or non-metallic inclusions) in the material.
These pores grow and coalesce as the strain increases further. The actual cross-section, bearing the applied load becomes very small as compared to the apparent cross-section, due to growth of these internal pores. At this stage, the specimen is no longer able to withstand the load. The load required to continue deformation tails to the point of fracture, when the specimen breaks into two pieces across the decreased cross-section at the neck.
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If a specimen does not contain second-phase particles or non-metallic inclusions to nucleate the pores, the specimen may neck down to a point before fracture occurs. The maximum point on the engineering stress-strain curve is called the point of plastic instability.
At this point, the fractional increase in stress, dσ/σ due to stain-hardening is exactly balanced by the fractional decrease in the cross-section, -dA/A due to the elongation of the specimen as:
Generally, ductility is measured in terms of % elongation (engineering strain at fracture, ef) or also called total elongation and % reduction in area at fracture.
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Both of these properties are obtained after fracture by putting the broken specimen together and then measuring Lf and Af as:
% E = {(Lf – L0) / L0 } x 100 …(6.27)
% R.A. = {(A0 – Af) / A0} x 100 …(6.28)
As appreciable fraction of the plastic deformation is concentrated in the necked region of the tensile test specimen, the value of % elongation, thus, depends on the gauge, length L0 over which the measurement has been done. Smaller the gauge length, higher is the value of % elongation and vice versa.
Thus, whenever, % elongation is reported, gauge length should be mentioned. Universally, if it is not mentioned, the gauge length is taken to be 5.00 cm. Reduction in area does not suffer from such a problem. It is, possible to change the values of reduction in area into an equivalent zero-gauge-length elongation, e0. As volume is supposed to remain constant during deformation, i.e., AL = A0L0 then,
True Stress-True strain Curve (Flow Curve):
The decrease in the engineering stress beyond UTS is not that the material softens, but actually the cross-sectional area of specimen decreases continuously up to the fracture, and the load required to continue the deformation falls, leading to decrease in the engineering stress, and consequently the fall-off in the engineering stress-strain curve, beyond the point of maximum load. Actually, the metal continues to strain-harden all the way up to fracture, and thus, the stress required to produce further deformation should also increase.
When the data is plotted as true stress (σ1 = F/A) versus the true strain [ɛi = In (L/L0)] as illustrated in Fig. 6.4 and 6.12, the curves have positive slopes right up to the point of fracture, illustrating that the strain-hardening occurs up to the point of fracture even though at a decreasing rate.
There is no maximum in the true stress-true strain curve, and the slope does not become zero before fracture. As the true stress is always higher than the engineering stress, and the true strain is smaller than the engineering strain, the true-stress-true strain curve lies above and to the left of the engineering stress-strain curve as illustrated in Fig. 6.4.
Relationship between True Strain and Engineering Strain:
The deformation behaviour of a metal are better understood with the help of true stress-true strain curve as it has been plotted based on changed dimensions of the specimen. True stress- true strain curve as based on instantaneous dimensions is more useful in metal-working processes, such as wire-drawing, where large changes in cross-sectional area are obtained.
This equation is applicable only up to the onset of necking as beyond that non-uniform plastic deformation occurs. Beyond the maximum load, the true strain is based on actual diameter measurements.
Relationship between True Stress and Engineering Stress:
It is assumed that volume remains constant even after deformation, and that the homogeneous plastic deformation occurs. If after deformation, the original area of cross section A0 and gauge length L0 change to A and L respectively. Then A0L0 = AL. The equation (6.33) can be written as-
If true stress is denoted by a and engineering stress by S, then
σ = F/A
S = F/A0
In true stress-true strain curve, Fig. 6.4 (a), three regions of deformation behaviour can be conveniently distinguished:
(i) Region of Elastic Deformation:
As the stress is gradually increased from zero, the metal can be considered to deform only elastically up to yield point, S0 (Elastic deformation continues to increase with increasing stress up to the fracture stress, σf, but in later stages, the amount of elastic deformation is negligibly small compared to plastic deformation).
(ii) Region of Uniform Plastic Deformation:
In the range of stresses between yield stress, S0, and the beginning of the necking (before nearly the straight line portion begins), the plastic deformation occurs uniformly.
(iii) Region of Non-Uniform Plastic Deformation:
When a neck begins to form in the specimen, it signals the beginning of non-uniform plastic deformation (the load begins to decrease in engineering stress-strain curve). As there is rapid decrease in the area A at the neck, the true stress, F/A continues to increase until finally the specimen breaks. This roughly linear curve illustrates that no work-softening occurs during this stage. The material strain-hardens continuously up to fracture.
True Tensile Strength:
It is the true stress at the maximum load. Let σu, ɛu and Au denote the true stress, true strain, cross-sectional area of the specimen at the maximum load.
True Uniform Strain:
The true uniform strain means true strain up to the maximum load. Thus, gauge length Lu, or cross-sectional area Au at the maximum load are needed to calculate it, or equation (6.35) could be used. Uniform strain is commonly used as a criterion of estimating the formability of the metals, such as obtained from a tensile test,
True Local Necking Strain, ɛn:
It is the strain produced when the specimen deforms from the maximum load to the fracture, i.e. is the strain of the non-uniform plastic deformation.
True Fracture Strain, ɛf:
It is the true strain based on the original area, A0 and the area after fracture, Af. This is the maximum true strain of the specimen before fracture.
The uniform plastic-deformation-region of the curve is generally expressed by the simple power curve relation-
σt = K ԑn …(6.44)
where, K is the strength coefficient, and n is the strain-hardening exponent. For some metals, n is constant throughout the uniform plastic region, but in other metals, n changes, generally decreases with the increasing strain. This strain-hardening exponent may have values from n = 0 (perfectly plastic solid) to n = 1 (elastic solid), but for most metals, n has values between 0.10 to 0.50.
Table 6.2 shows values of n and K of some metals and alloys. A high value of n refers to a high rate of strain-hardening and a high tensile strength as compared to the yield strength, i.e., the material exhibits more ductility as is true for copper and brass having n ≈ 0.5 as compared to heat treated steel (n ≈ 0.15).
Austenitic stainless steel has high value of n as it has low stacking-fault energy, because of which cross-slip of screw dislocations becomes difficult, and thus, dynamic recovery is difficult. The high value of n induces in the steel to have high tensile strength, more than twice of its yield strength.
This large gap produces high ductility in such steels (∼ 60% elongation). Addition of zinc to copper decreases the stacking- fault energy of alpha-brass, which has thus high value of n. This results in alpha-brass to be stronger and more ductile-than copper.
Problem:
Compare the engineering stress and-strain with true stress and true strain for an aluminium alloy at (a) the maximum load (35.6 kN) and (b) fracture (load is 33.8 kN at fracture) if the initial diameter is 12.5 mm and gauge length is 50 mm. The diameter at the maximum load is 12.30 mm and at fracture is 9.85 mm. What is the engineering strain at fracture and at fracture is 55.20 mm.
Solution:
(a) At the maximum load, using figures in meters-
(b) At fracture-
