Problem 49 - An Introduction to the Mechanics of Solids in SI Units

Question

Consider the indicial form (see Sec. 4.15) of the complete equations of elasticity in Sec. 5.6. Show that (5.6)-(5.8) can be written in the form $$ \begin{aligned} &\sigma_{i j, i}+X_{j}=0 \\ &e_{i j}=\frac{1}{2}\left(u_{t, j}+u_{j, i}\right) \\ &e_{i j}=\frac{1+v}{E} \sigma_{i j}-\frac{v}{E} \delta_{i j} \theta+\alpha \delta_{i j} \Delta T \end{aligned} $$ where \(\delta_{i j}\) is called the Kronecker delta and is equal to 1 when \(i=j\) and equal to zero when \(i \neq j\), and where $$ \theta=\sigma_{11}+\sigma_{22}+\sigma_{33}=\sigma_{i i} $$

Step-by-Step Solution

Answer

The given equations of elasticity can indeed be written in the target form, with key contributions from the displacement gradient, Poisson's ratio, Young's modulus, dilatation, coefficient of thermal expansion, change in temperature, and the properties of the Kronecker delta.

Step 1 - Understand the equations

First, familiarize yourself with the equations given. In these equations, \(\sigma_{ij}\) represents stress, \(X_j\) is body force per unit volume, \(e_{ij}\) is strain, \(u_{t, j}\) and \(u_{j, i}\) are displacement gradients, \(v\) is Poisson's ratio, \(E\) is Young's modulus, \(\theta\) is dilatation (change in volume), and \(\Delta T\) is change in temperature.

Step 2 - Understand the Kronecker delta

The Kronecker delta \(\delta_{ij}\) plays a vital role here. It is equal to 1 when \(i=j\) (the indices are equal) and equal to zero when \(i \neq j\). This property will be important while rearranging the equations.

Step 3 - Rewrite the equations

Now, begin to rewrite each equation to resemble the target form. The first equation requires no changes. The second equation reflects the symmetric property of strain displacement. The third equation is derived from the constitutive relation of elasticity, taking into account the thermal expansion effects (\(\alpha \cdot \Delta T\)), where \(\alpha\) is the coefficient of thermal expansion. In this equation, the dilatation \(\theta = \sigma_{11}+\sigma_{22}+\sigma_{33}\) (which is equivalent to \(\sigma_{ii}\)) represents the sum effect of normal stresses from all directions, and \(\delta_{ij}\) plays a key role to couple this effect into the constitutive relation.

Step 4 - Verify the solution

After rewriting, compare your equations with the target form to make sure they match. If all equations are in the correct form, you have successfully solved the exercise.

Key Concepts

Kronecker Delta

The Kronecker delta, denoted as \( \delta_{ij} \), is a simple yet powerful concept frequently used in equations relating to physics and engineering, particularly in the context of elasticity. Its definition is straightforward: \( \delta_{ij} \) is equal to 1 if \( i = j \) and 0 if \( i eq j \). This binary function is used to selectively activate components in a sum or equation only when the indices match, effectively acting as a gatekeeper.

In the exercise, understanding the role of the Kronecker delta is essential when it comes to the constitutive relation of elasticity. It helps to simplify the expression of dilatation, \( \theta \), to only consider the normal stresses that are aligned, ignoring the shear components. Its usage can isolate individual elements on a diagonal of a matrix, which in material science, can relate to the normal stresses experienced by an object. An important application of this is in modifying formulas to include or exclude certain terms, based on the condition of equal indices, simplifying the process of working through complex systems of equations.

Constitutive Relation of Elasticity

The constitutive relation of elasticity provides the foundational link between stress and strain within a material. It's a fundamental principle that allows us to predict how a material will deform under load. In the realm of linear elasticity, it is often expressed in the form of Hooke's law where stress is directly proportional to strain with a constant of proportionality being the material's stiffness.

The equation in the exercise, \( e_{ij} = \frac{1+v}{E} \sigma_{ij} - \frac{v}{E}\delta_{ij} \theta + \alpha \delta_{ij} \Delta T \), is an embodiment of this relationship. Here, \( v \) represents the Poisson's ratio, which describes the ratio of transversal contraction strain to longitudinal extension strain. On the other hand, \( E \) denotes Young's modulus, a measure of the stiffness of a solid material. What makes this relation a 'constitutive' one is that it essentially constitutes the material's response to applied forces by combining the intrinsic properties: the material's resistance to deformation (\( E \)) and its tendency to preserve volume (\( v \)).

This relation is crucial when calculating the deformation of objects under various forces, including how they might expand or contract with changes in temperature—a concept known as thermal expansion.

Thermal Expansion in Solids

Thermal expansion in solids is a vital concept in materials science and physics, reflecting the tendency of matter to change its shape, area, and volume in response to a change in temperature. When materials are heated, their particles move more and tend to occupy more space, resulting in expansion. Conversely, cooling generally leads to contraction.

The term \( \alpha \delta_{ij} \Delta T \) in the constitutive equation incorporates this thermally induced volumetric change into the elastic behavior. Here, \( \alpha \) is the coefficient of linear thermal expansion, which characterizes the degree to which a material expands per degree of temperature increase. The product \( \alpha \Delta T \) quantifies the relative change in size due to the temperature change, \( \Delta T \).

Understanding thermal expansion is critical for designing and evaluating materials for structures and devices that are exposed to temperature fluctuations. For instance, components in bridges, railways, or electronic devices must accommodate thermal expansion to avoid structural failure or compromised functionality. By incorporating the concepts of the Kronecker delta and the constitutive relation of elasticity, engineers can predict distortions and stresses triggered by temperature changes and design their materials and structures accordingly.