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5. For a linear viscoelastic material, assume its = 3 seconds, T = 31 seconds, ER = 150 MPa, please (a) determine the Debye peak's magnitude and frequency in Hertz (Hz), (b) what is the magnitude of the complex modulus at the Debye frequency, (c) what is the relaxed modulus, (d) what is the instantaneous modulus?

5. For a linear viscoelastic material, assume its = 3 seconds, T = 31 seconds, ER = 150 MPa, please (a) determine the Debye peak's magnitude and frequency in Hertz (Hz), (b) what is the magnitude of the complex modulus at the Debye frequency, (c) what is the relaxed modulus, (d) what is the instantaneous modulus?

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### Problem Statement Given a linear viscoelastic material with the following properties: - \(\tau_{\epsilon} = 3\) seconds - \(\tau_{\sigma} = 31\) seconds - \(E_R = 150\) MPa Please answer the following questions: 1. Determine the Debye peak’s magnitude and frequency in Hertz (Hz). 2. What is the magnitude of the complex modulus at the Debye frequency? 3. What is the relaxed modulus (\(E_R\))? 4. What is the instantaneous modulus? ### Explanation For a linear viscoelastic material, \(\tau_{\epsilon}\) and \(\tau_{\sigma}\) are the relaxed and instantaneous time constants, respectively, and \(E_R\) is the relaxed modulus. These parameters are essential to understand the behavior of the material under different types of loading conditions, and the frequency response of the material. ### Solutions Steps (a) To determine the Debye peak’s magnitude and frequency: - The Debye peak frequency can be found using the time constants \(\tau_{\epsilon}\) and \(\tau_{\sigma}\). (b) To find the magnitude of the complex modulus at the Debye frequency: - This involves calculating the dynamic response of the material at the specific frequency corresponding to the Debye peak. (c) The relaxed modulus \(E_R\) is already given as 150 MPa. (d) The instantaneous modulus can be derived from the given material properties, often considering the relationship between \(\tau_{\epsilon}\) and \(\tau_{\sigma}\) and the given modulus \(E_R\). Note: Detailed calculations and formulas applicable to complex modulus, Debye peak frequency, and instantaneous modulus derivations should be included in the solved section to guide students through the problem.
Transcribed Image Text:### Problem Statement Given a linear viscoelastic material with the following properties: - \(\tau_{\epsilon} = 3\) seconds - \(\tau_{\sigma} = 31\) seconds - \(E_R = 150\) MPa Please answer the following questions: 1. Determine the Debye peak’s magnitude and frequency in Hertz (Hz). 2. What is the magnitude of the complex modulus at the Debye frequency? 3. What is the relaxed modulus (\(E_R\))? 4. What is the instantaneous modulus? ### Explanation For a linear viscoelastic material, \(\tau_{\epsilon}\) and \(\tau_{\sigma}\) are the relaxed and instantaneous time constants, respectively, and \(E_R\) is the relaxed modulus. These parameters are essential to understand the behavior of the material under different types of loading conditions, and the frequency response of the material. ### Solutions Steps (a) To determine the Debye peak’s magnitude and frequency: - The Debye peak frequency can be found using the time constants \(\tau_{\epsilon}\) and \(\tau_{\sigma}\). (b) To find the magnitude of the complex modulus at the Debye frequency: - This involves calculating the dynamic response of the material at the specific frequency corresponding to the Debye peak. (c) The relaxed modulus \(E_R\) is already given as 150 MPa. (d) The instantaneous modulus can be derived from the given material properties, often considering the relationship between \(\tau_{\epsilon}\) and \(\tau_{\sigma}\) and the given modulus \(E_R\). Note: Detailed calculations and formulas applicable to complex modulus, Debye peak frequency, and instantaneous modulus derivations should be included in the solved section to guide students through the problem.
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