allc (b) (tr C)I – CT
Percentage
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Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
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Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
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Addition
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Use indicial notation to show that
![### Mathematical Expression Explanation
This section explains the partial derivative of a given function in the context of tensor calculus, specifically related to strain energy functions in continuum mechanics.
#### Expression:
\[
(b) \quad \frac{\partial I_1^c}{\partial \mathbf{C}} = (\text{tr} \, \mathbf{C}) \mathbf{I} - \mathbf{C}^T
\]
#### Explanation:
- **\(\frac{\partial I_1^c}{\partial \mathbf{C}}\)**: This represents the partial derivative of the first invariant \(I_1^c\) with respect to the tensor \(\mathbf{C}\).
- **\((\text{tr} \, \mathbf{C})\)**: This denotes the trace of the tensor \(\mathbf{C}\), i.e., the sum of the elements on its main diagonal.
- **\(\mathbf{I}\)**: Symbolizing the identity tensor, which is a matrix with ones on the diagonal and zeros elsewhere.
- **\(\mathbf{C}^T\)**: The transpose of tensor \(\mathbf{C}\), which flips the tensor over its diagonal.
In continuum mechanics, this equation can be used to relate the deformation gradient tensor \(\mathbf{C}\) to changes in strain energy, providing insights into the material behavior under various loading conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedadeb2e-cdac-43d1-98ad-015a8ab931d0%2Fc986be81-769d-4413-bbd5-da41034122f8%2Fruv2l0b_processed.jpeg&w=3840&q=75)
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