All solids get deformed, i.e., show change in shape if an external load is applied on them.
The basic types of deformations could be broadly divided into two categories:
1. Time-Independent Deformation:
(i) The time independent deformation, which disappears (i.e., object returns to its original shape) on the release of load is called elastic deformation. Thus, elastic deformation is a reversible deformation. In the practical sense, it is true to assume that elastic deformation occurs instantaneously with the application of the external load.
Elastic strain is linear with crosshead stress in most materials particularly at low stresses as illustrated by stress- strain curves of some materials in Fig. 6.2 to 6.6. Brittle materials like cast irons, concrete, silicate glasses under tensile stress show elastic deformation right up to the point of fracture, such as for Al2O3 in Fig. 6.2.
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(ii) As the external load is increased, the elastic behaviour of most metals changes to a state when the material no longer recovers its original dimensions, but is deformed permanently. The type of deformation which occurs almost instantaneously and remains despite the removal of load is called plastic deformation. It is irreversible deformation. Ductile materials such as aluminium Fig. 6.3 and copper and even low carbon steel. Fig. 6.2 is elastic up to a certain stress called the elastic limit. Thereafter, they deform plastically.
2. Time-Dependent Deformation:
(i) For recoverable deformation to occur, the elastic strain need not necessarily be a simple function of stress, i.e., there can be recoverable deformation dependent on time. The attainment of maximum elastic strain can lag behind the attainment of maximum stress causing it.
This deviation from the unique simple relationship between stress and strain before the large scale plastic flow occurs is called “inelasticity”. Here the body regains its original dimensions only with the passage of time, as illustrated in Fig. 6.8.
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(ii) In the similar way, the time-dependent plastic deformation could occur under a constant stress. It is called as creep, as illustrated in Fig. 6.9.
(iii) Under certain conditions particularly at high temperatures, and in amorphous materials, viscous flow can occur. In such conditions, the material cannot support an applied stress, and the material deforms continuously, even at vanishingly small stress at a rate which is nearly proportional to the applied stress. This is illustrated in Fig. 6.10.
Normally, the critical parts and structures are made of metal and alloys, or materials derived from them. The very nature of metallic bond makes the metals to undergo plastic deformation, and as a result of which strengthening (strain-hardening) occurs. Even at points of stress-concentration, when the stress may become very high to even cause cracking, but as the metals have high plasticity they undergo plastic deformation at the tip of the crack, and thus get strengthened.
The process of fracture is thus arrested. The non-metallic materials are incapable of plastic deformation and the resulting strain-hardening, and thus get fractured as soon as the stress at the tip of the crack exceeds a definite value. Elastic deformation may be followed directly by fracture as shown by stress-strain curve for AI2O3, a ceramic material in Fig. 6.2.
Elastic Deformation:
The elastic deformation of metals, although may be as small as one hundredth of plastic deformation, but is of great interest to engineers for two main reasons:
(i) Normally, machine components are designed based on maximum stress level of yield strength of that metal, below which deformations are almost entirely elastic. The stresses must remain within elastic-range to avoid large plastic deformation.
(ii) In many structures, even the little elastic deformation must be controlled to keep the minimum clearances between the components otherwise stresses shall be induced.
A few terms can be best understood with reference to a tensile test of a specimen. A tensile test measures the resistance of a material to a static or slowly applied force. Fig. 6.1 illustrates a simple test set-up with a standard specimen which could be cylindrical or flat (such as 12.5 mm diameter and gauge length of 50 mm). The force F is the load. A strain gauge or extensometer measures the amount by which the specimen elongates between the gauge marks when the load is applied.
Stress:
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The application of a force on a body results in stress, which is the distribution of internal forces produced in the body. Stress is measured and defined as the force per unit sectional area on which it acts. A0 is the original area of cross section and L0 is the gauge length.
As some load F is applied, the gauge length increases slightly and there takes place slight decrease in diameter of the specimen. If the original area, A0 is used to calculate the stress, then the resultant stress is known as the engineering stress. Generally, the stress will not be uniform over the area A0, because of inherent anisotropy between grains in a polycrystalline material. The equation 6.1 gives the value of average stress.
Strain:
It is defined as the resulting deformation per unit-dimension. If with the application of external force F, the initial gauge length L0 changes to L, the engineering strain (average linear strain), e is the change in length per unit of original length.
Strain is a dimensionless quantity, since both the change in length and the original length are expressed in the units of length. The % elongation is obtained by multiplying the engineering strain by 100.
Engineering stress, σ = F / A0 … (6.1)
Engineering strain, e = (L – L0) / L0 … (6.2)
The engineering stress-strain curves are shown in Fig. 6.2 and 6.3.
Since, the external force deforms the body and thus, changes its cross-sectional area, the engineering stress does not give the correct indication of the resistance of the body to further deformation. True stress is defined in terms of the force F acting per unit of instantaneous area A.
In the same way, it is more useful to use true strain, which is defined as the change in linear dimension divided by the instantaneous value of length. This is also called natural strain. True strain finds more useful applications while dealing with plasticity and metal forming.
True stress, = F/A …(6.3)
True stress-strain curve is illustrated in Fig. 6.4 along with engineering stress-strain curve.
Elastic deformation may also result in a change in the initial angle between any two lines as illustrated in Fig. 6.7 (c). This change in a right angle is called shear strain. The original angle of 90° is reduced by an angle θ. The shear strain is given by,
Shear strain, ϒ = x / h …(6.5)
= tan θ
As for small angles, the tangent of an angle, and the angle (in radians) are equal, the shear strain is often given as angle of rotation
ϒ = θ …(6.6)
In both ductile as well as brittle metals, the stress-strain curve starts with the elastic deformation. When a force is first applied to a specimen, the bonds between the atoms are stretched and the specimen elongates.
The bonds return to their original length and the specimen returns to its initial size, when the load is removed. In this initial linear portion of the curve, the stress is proportional to the strain, i.e. Hooke’s law is obeyed up to point A in Fig. 6.5. This point A, called proportional limit is the stress at which the stress-strain curve deviates from linearity.
The point A’ (which is slightly higher or of almost equal value as of stress at A) is the elastic limit, which is defined as the greatest stress that the metal can withstand without getting a permanent strain when the load is removed.
The engineering stress corresponding to this transition is called the yield strength (YS). Its determination is important as it is an important design parameter. A large number of materials exhibit a continuous change from the elastic region to the plastic region such as shown in Fig. 6.3.
In such cases, it becomes difficult to determine the yield strength precisely. In such cases, yield strength is defined as the stress which produces a small amount of permanent deformation, such as 0.1 or 0.2% but commonly 0.2% is specified. This is normally called proof strength’ or ‘off-set yield strength’ as illustrated in Fig. 6.5.
In some materials, the onset of plastic deformation is denoted by a sudden drop in load indicating both an upper and a lower yield point, such as in mild steels, Fig. 6.2. It is easy to specify the yield strength. Below the proportionality limit, there is an important characteristic of the materials, the ratio between the stress and the strain, called Young’s modulus.
E = σ / ɛ = constant …(6.7)
This constant is also called modulus of elasticity. Fig. 6.6 illustrates that the Young’s modulus of steel is higher than that of aluminium.
When a tensile force in the x-direction produces an extension in the x-direction (as per Hooke’s law), it also produces a contraction in the transverse y and z directions. This decrease is known as Poisson contraction, and is determined quantitatively by Poisson’s ratio, v, i.e., the lateral strain has been found to be a constant fraction, v, of the longitudinal extension.
The relationship between transverse and axial strain is-
ɛy = ɛz = – v ɛx = – vσx / E …(6.8)
Poisson’s ratio is normally taken to be close to 0.33 for most metals (Fe = 0.28; W = 0.29; Cu = 0.33; Au = 0.42; Pb = 0.44, Ge = 0.28).
Suppose a cube is acted upon by normal stresses σx, σy and σz and shearing stresses xy, yz, zX and is an isotropic material, then the components of strains in x, y and z directions are:
Relationship between Original Gauge Length and Cross-Sectional Area of Test Piece:
Certain standards have been used to compare elongation values on different- sized test pieces, i.e.
Original gauge length,
l0 = k √A, … (6.23)
where, k is a constant, and A is the area of cross-section of test piece
ASTM recommends,
l0 = 4.5 √A, … (6.24)
which gives for round test pieces,
l0 = 4 x diameter of the test piece
The British Standard Institution recommends
l0 = 5.65 √A … (6.25)
which for round specimens is
l0 = 5 x diameter of lest piece