Deformation (engineering)
In engineering, deformation may be elastic or plastic.
If the deformation is negligible, the object is said to be rigid.
Main concepts
Occurrence of deformation in engineering applications is based on the following background concepts:- Displacements are any change in position of a point on the object, including whole-body translations and rotations.
- Deformation are changes in the relative position between internals points on the object, excluding rigid transformations, causing the body to change shape or size.
- Strain is the relative ''internal deformation, the dimensionless change in shape of an infinitesimal cube of material relative to a reference configuration. Mechanical strains are caused by mechanical stress, see stress-strain curve''.
In the above figure, it can be seen that the compressive loading has caused deformation in the cylinder so that the original shape has changed into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change. As a result, the material is forced out laterally. Internal forces resist the applied load.
Types of deformation
Depending on the type of material, size and geometry of the object, and the forces applied, various types of deformation may result. The image to the right shows the engineering stress vs. strain diagram for a typical ductile material such as steel. Different deformation modes may occur under different conditions, as can be depicted using a deformation mechanism map.Permanent deformation is irreversible; the deformation stays even after removal of the applied forces, while the temporary deformation is recoverable as it disappears after the removal of applied forces.
Temporary deformation is also called elastic deformation, while the permanent deformation is called plastic deformation.
Elastic deformation
The study of temporary or elastic deformation in the case of engineering strain is applied to materials used in mechanical and structural engineering, such as concrete and steel, which are subjected to very small deformations. Engineering strain is modeled by infinitesimal strain theory, also called small strain theory, small deformation theory, small displacement theory, or small displacement-gradient theory where strains and rotations are both small.For some materials, e.g. elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%, thus other more complex definitions of strain are required, such as stretch, logarithmic strain, Green strain, and Almansi strain. 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:
where
- is the applied stress;
- is a material constant called Young's modulus or elastic modulus;
- is the resulting strain.
Note that not all elastic materials undergo linear elastic deformation; some, such as concrete, gray cast iron, and many polymers, respond in a nonlinear fashion. For these materials Hooke's law is inapplicable.
Plastic deformation
This type of deformation is not undone simply by removing the applied force. An object in the plastic deformation range, however, will first have undergone elastic deformation, which is undone simply by removing the applied force, 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. An example of a material with a large plastic deformation range is wet chewing gum, which can be stretched to 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. 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.
Failure
Compressive failure
Usually, compressive stress applied to bars, columns, etc. leads to shortening.Loading a structural element or specimen will increase the compressive stress until it reaches its compressive strength. According to the properties of the material, failure modes are yielding for materials with ductile behavior or rupturing for brittle behavior.
In long, slender structural elements — such as columns or truss bars — an increase of compressive force F leads to structural failure due to buckling at lower stress than the compressive strength.
Fracture
A break occurs after the material has reached the end of the elastic, and then plastic, deformation ranges. At this point forces accumulate until they are sufficient to cause a fracture. All materials will eventually fracture, if sufficient forces are applied.Types of stress and strain
Engineering stress and engineering strain are approximations to the internal state that may be determined from the external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the true stress and true strain can be derived from the instantaneous size of the object.Engineering stress and strain
Consider a bar of original cross sectional area being subjected to equal and opposite forces pulling at the ends so the bar is under tension. The material is experiencing a stress defined to be the ratio of the force to the cross sectional area of the bar, as well as an axial elongation:Subscript 0 denotes the original dimensions of the sample. The SI derived unit for stress is newtons per square metre, or pascals, and strain is unitless. The stress–strain curve for this material is plotted by elongating the sample and recording the stress variation with strain until the sample fractures. By convention, the strain is set to the horizontal axis and stress is set to vertical axis. Note that for engineering purposes we often assume the cross-section area of the material does not change during the whole deformation process. This is not true since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on the original cross-section and gauge length is called the engineering stress–strain curve, while the curve based on the instantaneous cross-section area and length is called the true stress–strain curve. Unless stated otherwise, engineering stress–strain is generally used.
True stress and strain
In the above definitions of engineering stress and strain, two behaviors of materials in tensile tests are ignored:- the shrinking of section area
- compounding development of elongation
Here the dimensions are instantaneous values. Assuming volume of the sample conserves and deformation happens uniformly,
The true stress and strain can be expressed by engineering stress and strain. For true stress,
For the strain,
Integrate both sides and apply the boundary condition,
So in a tension test, true stress is larger than engineering stress and true strain is less than engineering strain. Thus, a point defining true stress–strain curve is displaced upwards and to the left to define the equivalent engineering stress–strain curve. The difference between the true and engineering stresses and strains will increase with plastic deformation. At low strains, the differences between the two is negligible. As for the tensile strength point, it is the maximal point in engineering stress–strain curve but is not a special point in true stress–strain curve. Because engineering stress is proportional to the force applied along the sample, the criterion for necking formation can be set as
This analysis suggests nature of the ultimate tensile strength point. The work strengthening effect is exactly balanced by the shrinking of section area at UTS point.
After the formation of necking, the sample undergoes heterogeneous deformation, so equations above are not valid. The stress and strain at the necking can be expressed as:
An empirical equation is commonly used to describe the relationship between true stress and true strain.
Here, is the strain-hardening exponent and is the strength coefficient. is a measure of a material's work hardening behavior. Materials with a higher have a greater resistance to necking. Typically, metals at room temperature have ranging from 0.02 to 0.5.