Give definition of elastic deformation. Give definition of plastic deformation.

Question

Give definition of elastic deformation.

Give definition of plastic deformation.

What is the mechanism of elastic deformation in metals?

What is the mechanism of plastic deformation in metals?

Give definitions of true stress and true strain.

Write the formulas and explain all notations.

Sketch a typical engineering stress-strain curve for a ductile metal and mark the point at which noticeable necking begins.

Give definition of ductility.

What characteristic of a material is measured by the Charpy test, and what are its units?

What is slip in metals?

What is the relationship between critical resolved shear stress and yield strength?

Write the formula and explain all notations.

Explanation / Answer

Elastic deformation[edit]
For more details on this topic, see Elasticity (physics).
This type of deformation is reversible. Once the forces are no longer applied, the object returns to its original shape. Elastomers and shape memory metals such as Nitinol exhibit large elastic deformation ranges, as does rubber. However elasticity is nonlinear in these materials. Normal metals, ceramics and most crystals show linear elasticity and a smaller elastic range.

Linear elastic deformation is governed by Hooke's law, which states:

sigma = E arepsilon
Where sigma is the applied stress, E is a material constant called Young's modulus, and ? is the resulting strain. This relationship only applies in the elastic range and indicates that the slope of the stress vs. strain curve can be used to find Young's modulus. Engineers often use this calculation in tensile tests. The elastic range ends when the material reaches its yield strength. At this point plastic deformation begins.

Note that not all elastic materials undergo linear elastic deformation; some, such as concrete, gray cast iron, and many polymers, respond nonlinearly. For these materials Hooke's law is inapplicable.

Plastic deformation
See also: Plasticity (physics)
This type of deformation is irreversible. However, an object in the plastic deformation range will first have undergone elastic deformation, which is reversible, so the object will return part way to its original shape. Soft thermoplastics have a rather large plastic deformation range as do ductile metals such as copper, silver, and gold. Steel does, too, but not cast iron. Hard thermosetting plastics, rubber, crystals, and ceramics have minimal plastic deformation ranges. One material with a large plastic deformation range is wet chewing gum, which can be stretched dozens of times its original length.

Under tensile stress, plastic deformation is characterized by a strain hardening region and a necking region and finally, fracture (also called rupture). During strain hardening the material becomes stronger through the movement of atomic dislocations. The necking phase is indicated by a reduction in cross-sectional area of the specimen. Necking begins after the ultimate strength is reached. During necking, the material can no longer withstand the maximum stress and the strain in the specimen rapidly increases. Plastic deformation ends with the fracture of the material.

True Stress and True strain

Stress  
An applied force or system of forces that tends to strain or def orm a body.
Strain   A deformation produced by stress. Stress has units of a force measure divided by the square of a length measure, and the average stress on a cross-section in the tensile test is clearly the applied force divided by the cross-sectional area. Similarly, we may approximate the strain component along the long axis of the specimen as the change in length divided by the original, reference length.

It sounds simple enough, but you should realize that there are still some choices to make. Specifically, what area should be used for the cross-sectional area? Should you use the original area or the current area as the load is applied? By the same token, should changes in length always be compared to the original length of the specimen?

The answer is that we will define different types of stress and strain measures according to the way we perform the calculations. Engineering stress and strain measures are distinguished by the use of fixed reference quantities, typically the original cross-sectional area or original length. More precisely,

egin{displaymath}egin{array}{c} oldsymbol{sigma}_E = rac{P}{A_0}, oldsymbol{epsilon}_E = rac{Delta l}{l_0}. end{array}end{displaymath}   (1)

Engineering vs. True
Engineering stress and strain measures incorporate fixed reference quantities. In this case, undeformed cross-sectional area is used. True stress and strain measures account for changes in cross-sectional area by using the instantaneous values for area, giving more accurate measurements for events such as the tensile test. In most engineering applications, these definitions are accurate enough, because the cross-sectional area and length of the specimen do not change substantially while loads are applied. In other situations (such as the tensile test), the cross-sectional area and the length of the specimen can change substantially. In such cases, the engineering stress calculated using the above definition (as the ratio of the applied load to the undeformed cross-sectional area) ceases to be an accurate measure. To overcome this issue alternative stress and strain measures are available. Below we discuss true stress and true strain.
Figure 3: Engineering stress measures vs. true stress measures. The latter accounts for the change in cross-sectional area as the loads are applied.
egin{figure}hfil
epsfxsize =2.5in
epsfbox{Fig/crossection.eps} hfil
end{figure}
Stress: The true stress is defined as the ratio of the applied load to the instantaneous cross-se ctional area;
$displaystyle {oldsymbolsigma}_T = rac{P}{A}.$   (2)

True stress can be related to the engineering stress if we assume that there is no volume change in the specimen. Under this assumption,

$displaystyle Acdot l = A_0 cdot l_0,$  

which leads to
$displaystyle {oldsymbolsigma}_T = rac{P}{A} = rac{P}{A_0} cdot rac{l}{l_0} = {oldsymbolsigma}_E (1+{oldsymbolepsilon}_E ).$   (3)

True Strain: The true strain is defined as the sum of all the instantaneous engineering strains. Letting

$displaystyle depsilon = rac{dl}{l},$   (4)

the true strain is then
$displaystyle {oldsymbolepsilon}_T = int depsilon = intlimits_{l_0}^{l_f} rac{dl}{l} =$ ln$displaystyle rac{l_f}{l_0}.$   (5)

where $ l_f$ is the final length when the loading process is terminated. True strain can also be related back to the engineering strain, through the manipulation
$displaystyle {oldsymbolepsilon}_T =$ ln$displaystyle rac{l_f}{l_0} =$ ln$displaystyle rac{l_0 + Delta l}{l_0} =$ ln$displaystyle (1 + {oldsymbolepsilon}_E)$   (6)

In closing, you should note that the true stress and strain are practically indistinguishable from the engineering stress and strain at small deformations, as shown in Figure 4. You should also note that as the strain becomes large and the cross-sectional area of the specimen decreases, the true stress can be much larger than the engineering stress.

Key note:
As the strain becomes large and the cross-sectional area of the specimen decreases, the true stress can be much larger than the engineering stress.

Figure 4: Engineering stress-strain curve vs. a true stress, true strain curve.

Fig 2. A stress

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