About Frequency Domain Deformation Modes and Test Data

Deformation Modes in the Frequency Domain

Two deformation modes are available in the frequency domain: uniaxial, E, and shear, G. Selecting a deformation mode means that you are selecting a built-in simulation model that is available for calibration and specifies whether the test data specifies storage and loss modulus data in terms of Young's modulus or the shear modulus. Both deformation modes require test data that includes frequency and at least one response from the storage modulus, the loss modulus, or the tan delta data. You can also specify mean strain, dynamic strain, and temperature.

The simulations for both deformation modes are based on a homogenous deformation. Many Dynamic Mechanical Analysis (DMA) tests impart uniform shear or normal deformation in the specimen, and the built-in calibration models characterize these tests well. However, some DMA tests such as bending and torsion tests can impart nonuniform deformation states on the specimen. In these tests, the complex material's Young's modulus or shear modulus is usually determined based on the force-displacement or torque-twist behavior along with structural mechanics equations that assume linear elastic material behavior. For example, the equation for the torsion of a bar, $G=\frac{l}{J}\frac{T}{\theta }$ , relates the linear elastic shear modulus, $G$ , to the bar's torsion constant, $J$ ; length $l$ ; torque $T$ ; and the angle of twist, $\theta$ . A homogeneous deformation state might not represent the behavior for these tests, especially for nonlinear viscoelastic material models in which the viscous response exhibits dependence on the mean strain or dynamic strain amplitude.

Frequency Data

Frequency data points specify the number of cycles per unit of time. All imported test data sets in the frequency domain must include frequency data.

Storage Modulus Data

Storage modulus data, $E\prime$ , is optional when you import test data in the frequency domain. The selected deformation mode determines whether the storage modulus is the Young's ( $E$ ) or shear ( $G$ ) modulus. The storage modulus is the real part of the complex modulus, and it is defined based on the equation $E\prime =\frac{{\sigma }_0}{{\epsilon }_0}\cos \left(\delta \right)$ , where ${\epsilon }_0$ is the harmonic strain perturbation about the mean strain, ${\sigma }_0$ is the magnitude of the stress perturbation about the mean stress, and $\delta$ is the phase offset between the strain and stress.

Loss Modulus Data

Loss modulus data, $E^{\mathrm{..}}$ , is optional when you import test data in the frequency domain. The deformation mode you specify determines whether the loss modulus represents the Young's modulus ( $E$ ) or the shear ( $G$ ) modulus. The loss modulus is the imaginary part of the complex modulus, and it is defined by the equation $E\mathit{"}=\frac{{\sigma }_0}{{\epsilon }_0}\sin \left(\delta \right)$ , where ${\epsilon }_0$ is the harmonic strain perturbation about the mean strain; ${\sigma }_0$ is the magnitude of the stress perturbation about the mean stress, and $\delta$ is the phase offset between the strain and stress.

Tan Delta Data

The tangent of the phase offset $\delta$ between stress and strain, also known as tan delta data, is optional when you import test data in the frequency domain. Tan delta data is related to the storage and loss modulus according to the equation $\tan \left(\delta \right)=\frac{E"}{E\prime }$ .

Mean Strain

The mean strain describes the nominal strain value about which the strain is harmonically perturbed at a given frequency. You can specify this value as a column of test data for cases where the mean strain varies with frequency, or you can specify a single mean strain for the entire test. The app uses this value in the numerical execution mode only.

Dynamic Strain

The dynamic strain defines the peak-to-peak variation in nominal strain for a given frequency. You can specify this value as a column of test data for cases where the dynamic strain varies with frequency, or you can specify a single dynamic strain for the entire test. The app uses this value in the numerical execution mode only.

Temperature

You can specify varying temperatures explicitly as a column of test data, or you can specify a single test temperature for all test data.

Test Data Weights

For some test data sets imported for calibration, you might want to emphasize some parts of the data. In this case, you can define varying relative weights as a column of test data—larger weights put more emphasis on a row of data, smaller weights put less emphasis. Because the scaling is relative, the app internally normalizes the defined weights.