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Mobile App Chapter 7: Problem 1
Show (a) that there are a maximum of 21 independent elastic constants, using the fact that both stress and strain are symmetric dyadics, and \((b)\) that there exists a symmetry among the constants arising from the reciprocity relations $$ \frac{\partial f_{i j}}{\partial e_{k !}}=\frac{\partial f_{k}}{\partial \epsilon_{i j}} $$ where \(i j\) and \(k l\) stand for pairs of values of \(x, y, z\).
Short Answer
Expert verified
Answer: There are a maximum of 21 independent elastic constants. The symmetry among these constants exists due to the fact that both the stress and strain tensors are symmetric, and also due to the reciprocity relations between the stress and strain tensors.
Step by step solution
01
Find the independent components of stress and strain tensors
The stress and strain tensors are symmetric, which means that each tensor has \(6\) independent components. We can represent these tensors as follows:
Stress tensor (\(\sigma\)):
$\begin{bmatrix}
\sigma_{11} & \sigma_{12} & \sigma_{13}\\
\sigma_{12} & \sigma_{22} & \sigma_{23}\\
\sigma_{13} & \sigma_{23} & \sigma_{33}
\end{bmatrix}$
Strain tensor (\(\epsilon\)):
$\begin{bmatrix}
\epsilon_{11} & \epsilon_{12} & \epsilon_{13}\\
\epsilon_{12} & \epsilon_{22} & \epsilon_{23}\\
\epsilon_{13} & \epsilon_{23} & \epsilon_{33}
\end{bmatrix}$
Both tensors have six independent components each.
02
Find the number of independent elastic constants
Two general forms of Hooke's law are:
\(S_{ijkl}=\sum_{i, j, k, l = 1}^3 C_{ijkl}E_{ij}E_{kl}\) (contracted form)
\(\sigma_{ij} = C_{ijkl}\epsilon_{kl}\) (expanding for each \(i\) and \(j\))
where \(S_{ijkl}\) is the stiffness tensor of rank 4, \(C_{ijkl}\) are the elastic constants and \(E_{ij}\) and \(E_{kl}\) are the linearized stress and strain tensors, respectively. Considering the contracted form of Hooke's law and since the stress (\(\sigma_{ij}\)) and strain (\(\epsilon_{kl}\)) tensors each have six independent components, we have a total of \((6)*(6)=36\) possible unique combinations for the indices \(i,j,k,l\).
However, the elastic constant tensor \(C_{ijkl}\) is symmetric with respect to the pairs of indices \((ij)\) and \((kl)\). This means that \(C_{ijkl}=C_{jikl}=C_{ijlk}=C_{jilk}\).
Using this symmetry property of the elastic constant tensor, we can conclude that there are only \(21\) independent elastic constants:
\(C_{1111}, C_{1122}, C_{1133}, C_{2222}, C_{2233}, C_{3333}\) - \(6\) constants (\(i \neq j, i=k=l\))
\(C_{1123}, C_{2231}, C_{3312}\) - \(3\) constants (\(i \neq j, j=k \neq l\))
\(C_{1212}, C_{1313}, C_{2323}\) - \(3\) constants (\(i \neq j=k\))
\(C_{1213}, C_{1312}, C_{2321}\) - \(3\) constants (\(i=k \neq j=l\))
\(C_{1112}, C_{1131}, C_{2213}, C_{2232}, C_{3311}, C_{3322}\) - \(6\) constants (\(i=k \neq j, l \neq i\))
Thus, there are a maximum of \(21\) independent elastic constants.
03
Show the symmetry among the elastic constants
We are given the reciprocity relations:
\(\frac{\partial f_{ij}}{\partial e_{kl}}=\frac{\partial f_{kl}}{\partial \epsilon_{ij}}\)
Using the expressions of Hooke's law, we have:
\(\frac{\partial (\sigma_{ij})}{\partial (\epsilon_{kl})}=\frac{\partial (\sigma_{kl})}{\partial (\epsilon_{ij})}\)
Since from Hooke's law, \(\sigma_{ij} = C_{ijkl} \epsilon_{kl}\), the differentiation gives:
\(C_{ijkl} = C_{ilkj}\)
This expression shows that the elastic constants have additional symmetry property across the pairs of indices \((ij)\) and \((kl)\). So the above relation, along with the other symmetries discussed in Step 2, proves that the elastic constants have the required symmetry and there is a maximum of 21 independent elastic constants.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Symmetric Tensors
When dealing with the mechanical properties of materials, we come across certain mathematical entities known as 'symmetric tensors.' Tensors, in general, are geometric objects that describe linear relations between different types of physical quantities, such as vectors, scalars, and even other tensors. Symmetric tensors have a unique property where their values remain unchanged when the indices are swapped. For instance, if we have a second-order tensor like the stress (\(\sigma\)) or strain (\(\epsilon\)) tensors, they are said to be symmetric if \(\sigma_{ij} = \sigma_{ji}\) and \(\epsilon_{ij} = \epsilon_{ji}\).
A key example is the strain tensor, which occurs in the study of elasticity and material deformation. It can be visualized as a 3x3 matrix whose elements represent the stretching or compression along and across different axes. Because of these symmetric properties, we can simplify our calculations. Instead of having to account for nine separate components for a 3D strain or stress tensor, we only need to consider six - reducing the complexity significantly. This simplification is fundamental when we attempt to establish relationships like Hooke's Law for elastic materials.
A key example is the strain tensor, which occurs in the study of elasticity and material deformation. It can be visualized as a 3x3 matrix whose elements represent the stretching or compression along and across different axes. Because of these symmetric properties, we can simplify our calculations. Instead of having to account for nine separate components for a 3D strain or stress tensor, we only need to consider six - reducing the complexity significantly. This simplification is fundamental when we attempt to establish relationships like Hooke's Law for elastic materials.
Delving into Hooke's Law
Hooke's law is a principle of physics that relates the mechanical stress applied to an object to the resulting strain (deformation) observed within the elastic limit of that object. The law is named after the 17th-century physicist Robert Hooke, who first formulated it in 1678. In its simplest form, the law states that the force \(F\) needed to extend or compress a spring is directly proportional to the distance 'x' that the spring is stretched or compressed; mathematically, it's written as \(F = kx\), where 'k' is a constant characteristic of the spring.
In the context of materials science and engineering, Hooke's law is extended to express the relationship between tensile (or compressive) stress \(\sigma\) and strain \(\epsilon\) in solid materials. The proportionality constant in this case is known as the modulus of elasticity or Young's modulus. When we examine this in three dimensions, we use tensors, and Hooke's law takes the form: \(\sigma_{ij} = C_{ijkl} \epsilon_{kl}\), connecting the stress tensor to the strain tensor with elastic constants \(C_{ijkl}\) serving as a bridge.
In the context of materials science and engineering, Hooke's law is extended to express the relationship between tensile (or compressive) stress \(\sigma\) and strain \(\epsilon\) in solid materials. The proportionality constant in this case is known as the modulus of elasticity or Young's modulus. When we examine this in three dimensions, we use tensors, and Hooke's law takes the form: \(\sigma_{ij} = C_{ijkl} \epsilon_{kl}\), connecting the stress tensor to the strain tensor with elastic constants \(C_{ijkl}\) serving as a bridge.
Stress and Strain: The Twin Pillars of Elasticity
The concepts of stress and strain form the foundation of the study of material deformation and are crucial in understanding elastic behavior. Stress is the force applied to a material divided by the area over which the force is distributed. It describes the internal forces within a material that arise in response to external loads. Strain, on the other hand, measures the deformation or displacement that results from these external loads. Specifically, it is a dimensionless measure of how much an object is stretched or compressed relative to its original length.
Stress and strain are related in that the former causes the latter. When materials are subjected to stress within their elastic limits, they deform proportionally to the stress applied, as per Hooke's law. It's this proportionality that allows engineers and scientists to predict how materials will behave under different loads, which is essential for designing safe and efficient structures. Moreover, the symmetric nature of stress and strain tensors simplifies the mathematical complexity of this relationship, allowing deeper insights into material properties.
Stress and strain are related in that the former causes the latter. When materials are subjected to stress within their elastic limits, they deform proportionally to the stress applied, as per Hooke's law. It's this proportionality that allows engineers and scientists to predict how materials will behave under different loads, which is essential for designing safe and efficient structures. Moreover, the symmetric nature of stress and strain tensors simplifies the mathematical complexity of this relationship, allowing deeper insights into material properties.
Reciprocity Relations in Elasticity
While discussing elastic constants, we cannot neglect the critical aspect of reciprocity relations. These are mathematical expressions that demonstrate symmetry in the partial derivatives of functions describing the relationship between stress and strain. To be specific, they reveal that the derivative of stress with respect to strain is identical to the derivative of strain with respect to stress when considering the interchanges of the relevant indices.
In elasticity, this means that the fundamental properties governing the behavior of materials are inherently balanced. Such relations don't just serve as a mathematical curiosity; they have profound implications in the material sciences field by ensuring the consistency and stability of the theoretical frameworks used to describe material behavior. These relations further limit the number of independent elastic constants we must consider when analyzing the stress-strain relationship, thus streamlining the study and practical application of elastic behavior in various materials.
In elasticity, this means that the fundamental properties governing the behavior of materials are inherently balanced. Such relations don't just serve as a mathematical curiosity; they have profound implications in the material sciences field by ensuring the consistency and stability of the theoretical frameworks used to describe material behavior. These relations further limit the number of independent elastic constants we must consider when analyzing the stress-strain relationship, thus streamlining the study and practical application of elastic behavior in various materials.
