Samer Adeeb

Displacement and Strain: Simple Examples of Displacement Fields

Rigid Body Displacement

A rigid body displacement is represented by a constant displacement vector at every point. The new (deformed) position x\in\mathbb{R}^3 of every point is related to the old (reference) position X\in\mathbb{R}^3 as follows:

    \[ x=X+c \]

where c\in\mathbb{R}^3 is a constant vector. The displacement field at every point is the difference between the deformed and reference positions and is constant:

    \[ u=x-X=c \]

Change the components of the vector c in the following tool to view its effect on the displacement of the cuboid.

Rigid Body Rotation

A rigid body rotation is represented by a rotation matrix Q\in\mathbb{M}^3 (see Orthogonal Tensors) such that the new (deformed) position x\in\mathbb{R}^3 of every point is equal to the rotation matrix Q multiplied by the old (reference) position X\in\mathbb{R}^3 as follows:

    \[ x=QX \]

The displacement field at every point is the difference between the deformed and reference positions:

    \[ u=x-X=QX-X=(Q-I)X \]

Recall that any rotation matrix can be viewed as consecutive rotations around each of the basis vectors of the coordinate system. Clockwise rotations with an angles \theta_a, \theta_b, \theta_c around the basis vectors e_1, e_2 and e_3 are given by the following matrices Q_a, Q_b and Q_c, respectively:

    \[ Q_a=\left(\begin{array}{ccc} 1& 0& 0\\ 0& \cos(\theta_a) &\sin(\theta_a)\\ 0&- \sin(\theta_a)&\cos(\theta_a)\end{array}\right) \]

    \[ Q_b=\left(\begin{array}{ccc} \cos(\theta_b)& 0& -\sin(\theta_b)\\ 0& 1&0\\ \sin(\theta_b) & 0 &\cos(\theta_b)\end{array}\right)\]

    \[Q_c=\left(\begin{array}{ccc} \cos(\theta_c)& \sin(\theta_c)&0\\ -\sin(\theta_c) &\cos(\theta_c)&0\\0&0&1\end{array}\right) \]

It is important to notice that the order of rotation changes the final position of the rotated object. The rotation matrix Q_1=Q_cQ_bQ_a describes a rotation of \theta_a around e_1 followed by a rotation of \theta_b around e_2 and finally a rotation of \theta_c around e_3. On the other hand, the rotation matrix Q_2=Q_aQ_bQ_c describes a rotation of \theta_c around e_3 followed by a rotation of \theta_b around e_2 and finally a rotation of \theta_a around e_1. In general: Q_1\neq Q_2. In the following example, the red box represents the original box after rotation around the basis vectors. Try it out: rotate the box 90 degrees around x and then slowly rotate it around y. This order is applied to the image on the left. The order of rotation applied to the one on the right is reversed! Compare the two orders of rotation. The overall matrix of transformation is displayed at the bottom of each image.

Rigid Body Motion

A rigid body motion is a combination of both a rigid body displacement and a rigid body rotation such that the deformed position x\in\mathbb{R}^3 is function of the reference position X\in\mathbb{R}^3 as follows:

    \[ x=QX+c \]

where Q\in\mathbb{M}^3 is a rotation matrix and c\in\mathbb{R}^3 is a vector representing the rigid body displacement. The displacement field can be expressed as:

    \[ u=x-X=(Q-I)X+c \]

In component form, the relationship between the vectors x and X can be written as follows:

    \[ \left(\begin{array}{c} x_1\\ x_2\\ x_3\end{array} \right)= \left(\begin{array}{ccc} Q_{11} & Q_{12} & Q_{13}\\ Q_{21} & Q_{22} & Q_{23}\\ Q_{31} & Q_{32} & Q_{33} \end{array} \right) \left(\begin{array}{c} X_1\\ X_2\\ X_3\end{array} \right) + \left(\begin{array}{c} c_1\\ c_2\\ c_3\end{array} \right) \]

Note that in some numerical analysis software and tools, the above relationship adopts the following form:

    \[ \left(\begin{array}{c} x_1\\ x_2\\ x_3\\ 1\end{array} \right)= \left(\begin{array}{cccc} Q_{11} & Q_{12} & Q_{13}&c_1\\ Q_{21} & Q_{22} & Q_{23} &c_2\\ Q_{31} & Q_{32} & Q_{33}&c_3\\ 0 & 0 & 0 & 1\ \end{array} \right) \left(\begin{array}{c} X_1\\ X_2\\ X_3\\ 1\end{array} \right) \]

Change the angles of rotation and the components of the vector c in the following tool to see the effect on the final position of the cube.

Uniform Extension and Contraction

A uniform extension or contraction can be characterized by three positive parameters k_1, k_2, and k_3\in\mathbb{R}^3 that represent the ratios between the three vector components in the deformed configuration to the components in the reference configuration:

    \[ \left(\begin{array}{c} x_1\\ x_2\\ x_3\end{array} \right)= \left(\begin{array}{ccc} k_1 & 0 & 0\\ 0 & k_2 & 0\\ 0 & 0 & k_3 \end{array} \right) \left(\begin{array}{c} X_1\\ X_2\\ X_3\end{array} \right)= \left(\begin{array}{c} k_1X_1\\ k_2X_2\\ k_3X_3\end{array} \right) \]

Note that the relationship can be written as a linear transformation x=MX where M\in\mathbb{M}^3 has the form:

    \[ M= \left(\begin{array}{ccc} k_1 & 0 & 0\\ 0 & k_2 & 0\\ 0 & 0 & k_3 \end{array} \right) \]

In the following example, you can vary the values of k_1,k_2 and k_3 to see the effect on the deformation of a cube. What values constitute compression and what values constitute tension? Also, what does it mean that the value of k_i is equal to 1 or 0?

Simple Shear

The simple shear motion is described by a shearing angle along a certain direction and perpendicular to another direction. The following relationship describes a simple shear motion in which the planes parallel to the basis vectors e_2 and e_3 are sheared in the direction of e_1:

    \[ \left(\begin{array}{c} x_1\\ x_2\\ x_3\end{array} \right)= \left(\begin{array}{ccc} 1 & \tan{(\theta)} & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right) \left(\begin{array}{c} X_1\\ X_2\\ X_3\end{array} \right)= \left(\begin{array}{c} X_1+\tan{(\theta)}X_2\\ X_2\\ X_3\end{array} \right) \]

Note that the relationship can be written as a linear transformation x=MX where M\in\mathbb{M}^3 has the form:

    \[ M= \left(\begin{array}{ccc} 1 & \tan{(\theta)} & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right) \]

In the following example, change the values of \theta_{xy}, \theta_{yz} and \theta_{xz} in the matrix M:

    \[ M= \left(\begin{array}{ccc} 1 & \tan{(\theta_{xy})} & \tan{(\theta_{xz})}\\ 0 & 1 & \tan{(\theta_{yz})}\\ 0 & 0 & 1 \end{array} \right) \]

and observe the effect on the deformation x=MX. The term simple shear applies to the deformations when only one of the angles \theta_{xy}, \theta_{yz} and \theta_{xz} is non-zero.

Pure Shear

The following relationship describes a pure shear motion with an angle \theta in the plane of e_1 and e_2:

    \[ \left(\begin{array}{c} x_1\\ x_2\\ x_3\end{array} \right)= \left(\begin{array}{ccc} 1 & \tan{(\theta/2)} & 0\\ \tan{(\theta/2)} & 1 & 0\\ 0 & 0 & 1 \end{array} \right) \left(\begin{array}{c} X_1\\ X_2\\ X_3\end{array} \right)= \left(\begin{array}{c} X_1+\tan{(\theta/2)}X_2\\ \tan{(\theta/2)}X_1+X_2\\ X_3\end{array} \right) \]

Note that the relationship can be written as a linear transformation x=MX where M\in\mathbb{M}^3 has the form:

    \[ M= \left(\begin{array}{ccc} 1 & \tan{(\theta/2)} & 0\\ \tan{(\theta/2)} & 1 & 0\\ 0 & 0 & 1 \end{array} \right) \]

The difference between pure shear and simple shear can be viewed in the following two dimensional example. Change the value of \theta_{xy} to see the deformation of a rectangle under pure shear (on the left) and under simple shear (on the right). The matrix M in each case is given underneath the figure:

In the following example, change the values of \theta_{xy}, \theta_{yz} and \theta_{xz} in the matrix M:

    \[ M= \left(\begin{array}{ccc} 1 & \tan{(\theta_{xy/2})} & \tan{(\theta_{xz/2})}\\ \tan{(\theta_{xy/2})} & 1 & \tan{(\theta_{yz/2})}\\ \tan{(\theta_{xz/2}) & \tan{(\theta_{yz/2}) & 1 \end{array} \right) \]

and observe the effect on the deformation x=MX. The term pure shear applies to the deformations when only one of the angles \theta_{xy}, \theta_{yz} and \theta_{xz} is non-zero.

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