1. Strain Matrix in Cylindrical Coordinate System

The strain tensor (ϵ) is defined as (1) Wecan recallits matrix form in Cartesian coordinate system: (2)

The strain tensor ϵ represents the deformation of a material. It is a mathematical object that relates the change in shape or size of an object to the forces acting on it. In this context, we first need to understand its representation in Cartesian coordinates, which serves as a foundation before we can translate this representation into cylindrical coordinates.

Imagine a rubber band being stretched. The strain tensor would help describe how the rubber band stretches in various directions as you pull on it.

We need to first express gradients in the cylindrical coordinate system. In Cartesian coordinate system, the gradient of a quantity is given by (3) whereas its definition in cylindrical coordinate system is (4). Notice that the partial derivative with respect to θ is divided by r as θ is non-dimensional and we are taking the gradient in space.

In the cylindrical coordinate system, gradients must be expressed differently than in Cartesian coordinates. The gradient helps to describe how a quantity changes in space. Specifically, in cylindrical coordinates, the partial derivative with respect to the angle θ is adjusted because this angle is a non-dimensional measure. This change allows us to correctly capture how changes in displacement occur within the cylindrical framework.

Think of standing in the center of a circular field. If you observe changes in how high grass grows at different angles from your position, the gradient helps you understand how that growth varies around you, accounting for the circular nature of your surroundings.

The displacement vector can be written in cylindrical coordinate system by decomposing it along cylindrical basis as follows: u = u_e + u_e + u_e (5) r r θ θ z z

The displacement vector, which describes how a point moves in space, can be represented in cylindrical coordinates by breaking it down into components along the radial (r), angular (θ), and axial (z) directions. Each of these components represents how much the displacement occurs in each respective direction, giving a complete picture of the object's deformation.

Imagine a balloon being inflated. The displacement vector helps you understand how the surface of the balloon stretches in three different directions: outward (radial), around the balloon (angular), and along the height (axial) as it expands.

Now, we can plug in the displacement vector given in equation (5) in the gradient definition (4) to obtain (6). For the terms here, the basis vectors act as a constant as they change only with θ.

By substituting the displacement vector into the gradient definition, we compute the gradient of the displacement, which captures how the displacement varies in space. We also observe that some aspects of the basis vectors remain constant except when changing direction around the cylindrical body.

Think of a hill. As you walk on the hill, certain aspects of the landscape (like the steepness in a particular direction) change while others may remain flat. The calculation described allows us to understand those changes in displacement over the landscape, capturing the 'gradient' of your movement.

Now, we can use equation (1) to obtain strain matrix which is the symmetric part of the displacement gradient matrix derived above. It turns out to be the following: (10)

The strain matrix is derived as the symmetric part of the displacement gradient matrix. The symmetric nature means it effectively captures how the material deforms under stress without redundantly incorporating rotational effects. Understanding this matrix is crucial for relating deformation to applied forces on the cylindrical body.

Think of a sponge. When you squeeze it, the way it compresses and expands represents its strain. The strain matrix allows us to quantify how each part of the sponge reacts differently when pressured, ignoring any twisting.

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