Deformation and Strain
The overall displacement of the particles in such body is known as deformation. By deforming, the body converts work done to elastic strain energy and kinetic energy. If there is no more external force exerted on it, the body will stop deform after such energy conversion is completed. When external forces reduced, the elastic strain energy will exert elastic restoring force, which will have its work done converted to kinetic energy and moves the particles back to their original position, if the bond between particles has not been break. There are two types of deformation: rigid body motion and non-rigid body deformation.
Rigid body motion occurs when the exerted force is enough to change the motion state of a body but insufficient stress is developed to overcome its attraction force. Rigid body motion includes translation and rotation. In translation, all points on the body have displacement of same magnitude and in the same direction. In rotation, all points on the body have angular displacement of same angle and in the same direction, except the points that lie along the axis of rotation.
Non-rigid body deformation occurs when stress developed is enough to overcome the attraction force, regardless the ability of force to change a body’s motion state. This type of deformation includes distortion and dilation, which is in fact the result of differential translation and rotation. Distortion is a process where a body change its shape, while dilation is a process where a body change its volume. Fig. 3.1 shows some examples of rigid and non-rigid body deformation in a girder.
As an observable and measurable quantity, deformation is an aspect that material scientists look up to identify the material’s mechanical properties. However, direct measurement of a body’s deformation is not reliable: a very short rubber stick deforms less than a very long steel rod. One can wrongfully conclude that rubber is more rigid
FIGURE 3.1 Examples of rigid body and non-rigid body deformations in girder.
FIGURE 3.2 Normal strain in a 1-D element.
than steel if he inteipreted test result this way. Also, when a solid is experiencing rigid body deformation, no differential deformation takes place throughout the body. Differential deformation only takes place during non-rigid body deformation, which is an indication of development of stress inside within the solid. For these reasons, strain, i.e a way to determine the severity of deformation using ratio of solid’s deformation to its original length along a specific direction, is looked up.
Assuming there is a 1-D element subjected to normal force and exhibited both rigid and non-rigid body deformation: translation and elongation (distortion). Let the distance between two nodes of the element, namely a and b be dx. Also, let the deformation along x and у axes be и and v, respectively. The correlations between the defined parameters are as shown in Fig. 3.2.
Translation caused the element to move и unit along positive x direction, and elongation added on extra deformation along x direction. This additional deformation is proportional to the length of element, which can be expressed as For element with
length of dx, the total deformation caused by elongation is ^ x dx.
The equation of normal strain is given as follows:
where e is normal strain, AL is the change in longitudinal length and L is the original length of body.
The following equation yielded by applying the case as illustrated in Fig. 3.2 into
This equation proves that rigid body deformation, i.e translation, does not influence the strain of body. Similarly, the strain expression for another two axes can be written in terms of v and tv, which is deformation along z direction as the following:
Assuming a 2-D element subjected to shear force and exhibited both rigid and non- rigid body deformation: translation and distortion. Let the nodes of the element be А, В, C and D, and the dimension of such element along x and у axes be dx and dy respectively. Similarly, deformation along .v and у axes will be и and v respectively, as shown in Fig. 3.3.
Translation caused the element to move и unit along positive x direction and v unit along positive у direction. Distortion is caused by the rotation of edge AB and AD, which now becomes A B' and A D’. Shear strain is defined as the total angular deformation of a body. In this case, it is defined as below:
If infinitesimal deformations are considered, then by small angle approximation,
FIGURE 3.3 Shear strain in a 2-D element.
Therefore, Eq. (3.5) can be expressed as follows:
Similarly, we can write the following shear strain expressions for another two axes:
By combining all the aforementioned cases of deformation, total deformation along each axis is defined as below:
In matrix form, the equations above are expressed as follows:
The relationship in Eq. (3.1) can be rewrite as this:
The following is obtained when comparison of equation above with Eq. (3.10) is made:
Shear strain is the product of coupled shear stress.
In Fig. 3.3, /3, and |32 are the effect of coupled shear stress on the body. Since the applied stresses are the same, and the variation of material properties over infinitesimal element is negligible, it can be concluded that:
Therefore, from Eq. (3.6), shear strain is defined as below:
For other two shear strain components we can write their expressions in the similar fashion, as shown below:
By substituting relationships in Eqs. (3.2), (3.3), (3.4), (3.12) and (3.13) into Eq. (3.11) yields the following: