Question

At a certain point \(P\) (which you can choose to be your origin) in a continuous solid, the strain tensor is \(\mathbf{E}\). Assume for simplicity that whatever displacements have occurred left \(P\) fixed and the neighborhood of \(P\) unrotated. (a) Show that the \(x\) axis near \(P\) is stretched by a factor of \(\left(1+\epsilon_{11}\right)\) (b) Hence show that any small volume around \(P\) has changed by \(d V / V=\operatorname{tr}\) E. This shows that any two strains that have the same trace dilate volumes by the same amount. In the decomposition \(\mathbf{E}=e \mathbf{1}+\mathbf{E}^{\prime}\) \((16.88),\) the spherical part \(e 1\) changes volumes by the same amount as \(\mathbf{E}\) itself, while the deviatoric part \(\mathbf{E}^{\prime}\) doesn't change volumes at all.

Short answer

(a) The x-axis is stretched by a factor of \(1+\epsilon_{11}\). (b) Volume change is \(dV/V=\operatorname{tr}\,\mathbf{E}\).

Step-by-step solution

Analyze the Strain Tensor

The strain tensor \( \mathbf{E} \) is a three-dimensional symmetric matrix that describes how a body deforms near a point \( P \). It consists of components \( \epsilon_{ij} \) that denote change per unit length.

Understand Stretching Factor

When a body is subjected to a strain, each axis can experience a stretch. The factor by which the \( x \)-axis is stretched is \( 1 + \epsilon_{11} \), where \( \epsilon_{11} \) is the normal strain along the \( x \)-axis.

Establish Volume Change Formula

The change in volume \( \frac{dV}{V} \) near point \( P \) can be related to the trace of the strain tensor: \( \operatorname{tr} \mathbf{E} = \epsilon_{11} + \epsilon_{22} + \epsilon_{33} \). This represents the sum of normal strains in all directions.

Connect to Trace of Strain Tensor

Since \( \operatorname{tr} \mathbf{E} = \epsilon_{11} + \epsilon_{22} + \epsilon_{33} \) determines the volumetric strain, any small volume around \( P \) changes by this value, illustrating that strains with the same trace result in the same volumetric change.

Decompose Strain Tensor

The strain tensor \( \mathbf{E} \) can be decomposed into a spherical part \( e \mathbf{1} \) and a deviatoric part \( \mathbf{E}^{\prime} \). The spherical part \( e \mathbf{1} = \frac{1}{3} \operatorname{tr} \mathbf{E} \cdot \mathbf{1} \), where \( \mathbf{1} \) is the identity matrix.

Analyze Volume Change Due to Decomposition

The spherical component \( e \mathbf{1} \) affects only the volume, preserving the trace of \( \mathbf{E} \), while the deviatoric component \( \mathbf{E}^{\prime} \) has no contribution to volume change as it doesn't alter the trace.

Key concepts

Volumetric Strain: The Bulk Response of a Material

Volumetric strain is a fundamental concept that describes how the volume of a material changes when it is subjected to stress. This change is crucial as it indicates how a material compresses or expands in response to external forces.
The concept encompasses the transformation of a differential volume at a point, characterized by the ratio of the volume change \((dV)\) to the original volume \((V)\). Mathematically, it is given by \(rac{dV}{V}\).
In the context of the strain tensor \(\mathbf{E}\), the volumetric strain is represented by the trace of the strain tensor. The trace \(\text{tr} \, \mathbf{E}\) is the sum of the normal strain components \(\epsilon_{11} + \epsilon_{22} + \epsilon_{33}\). This sum quantifies the total expansion or compression along the principal axes.
It is impressive to note that any two strain tensors with the same trace indicate equal volumetric changes, even if the detailed deformation patterns differ. This equivalence simplifies the analysis of material response under complex loading by focusing on volumetric changes.

Spherical and Deviatoric Decomposition: Understanding Strain Components

When dealing with strain tensors like \(\mathbf{E}\), it's helpful to break them down into more interpretable parts. This is where spherical and deviatoric decomposition comes in handy.
Spherical decomposition involves isolating the part of the strain that causes volume changes. The spherical part is denoted as \(e\,\mathbf{1}\), where \(e = \frac{1}{3} \text{tr} \, \mathbf{E} \) and \(\mathbf{1}\) is the identity matrix. This component affects only the volumetric strain, since it modifies the overall size but not the shape.
The deviatoric component \(\mathbf{E}^{\prime}\), on the other hand, describes shape changes without altering the material's volume. It's calculated as the remainder of the strain tensor after subtracting the spherical part, \(\mathbf{E}^{\prime} = \mathbf{E} - e\,\mathbf{1}\).
By utilizing spherical and deviatoric decomposition, engineers and scientists can separately assess how forces alter the size and shape of materials, contributing to our understanding of material behavior under different loading conditions.

Trace of the Strain Tensor: A Single Value that Tells a Comprehensive Story

The trace of a strain tensor \(\mathbf{E}\) is a significant quantity because it encapsulates the total volumetric strain experienced by a small volume at a point in a continuum. Essentially, it's a simple measure of how much the material has expanded or contracted.
This trace is calculated as the sum of the diagonal components of the strain matrix, \( \text{tr} \, \mathbf{E} = \epsilon_{11} + \epsilon_{22} + \epsilon_{33} \). These components are the normal strains along the principal axes and reflect the change in length along each direction.
Understanding the trace helps in evaluating whether a body is undergoing a simple volume change without any complex deformations like shearing. This is why it's often used to determine the volumetric strain, which is critical in applications where volume change dictates the material performance, such as in hydraulics and structural engineering.
In summary, the trace of the strain tensor provides a quick yet comprehensive insight into the volumetric aspect of deformation without delving into directional details.