Date: 11.2008

My role: modeling analyst and experimenter

The in-plane orthotropic material model works quite well in capturing the elastic behavior of spunbond nonwoven under small deformation^{1}. Although the model is a basic one in elasticity, some material constants such as the Poisson’s ratio and shear modulus are difficult to measure accurately for nonwoven. We proposed a practical material calibration way that is based on fitting the deformed shape of samples in uniaxial tensile tests with an inverse FEA procedure, to avoid the direct measurements of those parameters. This method transfers some of the experimental effort into analysis and provides an accurate material calibration for modeling of nonwovens.

The in-plane orthotropic material model has 4 independent constants, which can take the form of two Young’s moduli \(E_1\) and \(E_2\) in two respective principle directions—machine direction (MD) and cross machine direction (CMD), one Poisson’s ratio \(\nu_{12}\) (called major Poisson’s ratio) relating the induced deformation in CMD to the loaded deformation in MD, and one in-plane shear modulus \(G_{12}\). The strain-stress relation connected by the material compliance matrix is

$$\begin{bmatrix}\epsilon_{11} \\ \epsilon_{22} \\ \gamma_{12} \end{bmatrix}=\begin{bmatrix} \frac{1}{E_1} & -\frac{v_{12}}{E_1} & 0 \\ -\frac{v_{12}}{E_1} & \frac{1}{E_2} & 0 \\ 0 & 0 & \frac{1}{G_{12}}\end{bmatrix} \begin{bmatrix}\sigma_{11}\\\sigma_{22}\\\tau_{12}\end{bmatrix}.$$

Among the 4 material constants in the model, the two Young’s moduli can be measured by uniaxial tensile tests and have the best accuracy and precision. The in-plane major Poisson’s ratio of a typical nonwoven web is usually grater than 1 due to their composed microstructure. The severe necking deformation in a traditional uniaxial Poisson’s ratio test makes the calculation of transverse strain confusing. Thus the value of major Poisson’s ratio obtained may not be reliable. Furthermore, an accurate shear modulus test for nonwoven is even harder because of its extremely low resistance to shear buckling; thus, it’s difficult to gather enough meaningful data in a pure shear test of nonwoven before it buckles. A shear test using the picture frame may be viable but it causes more effort in machining clamps and preparation of samples.

To avoid the uncertainties of measurements on Poisson’s ratio and shear modulus, one more uniaxial test in some other angle—\(\theta\) besides the two principle directions is conducted. The orthotropic theory gives the expression of modulus in the \(\theta\) direction

$$\frac{1}{E_\theta}=\frac{\cos^4(\theta)}{E_1}+\frac{\sin^4(\theta)}{E_2}+\frac{1}{4}\bigg[ \frac{1}{G_{12}}-\frac{2\nu_{12}}{E_1} \bigg]\sin^2(2\theta).$$

Therefore, knowing the measured values of \(E_1\), \(E_2\), and \(E_\theta\) in \(\theta\), the shear modulus \(G_{12}\) and major Poisson’s ratio \(\nu_{12}\) are related by the above equation. There is only one variable left need to be determined. This allows us to transform the direct measurement problem into a single variable optimization problem—with the aid of finite element analysis used in an inverse and iterated fashion. The optimization objective is set to get an agreed deformed shape of test samples and FEA. The specific procedure is as follows:

- Three uniaxial tensile tests up to 1% strain are conducted in MD, CMD, and \(\theta\) direction. The three Young’s moduli were measured. And the deformed shape of sample in the MD test were recorded.
- A finite element model which simulates the MD tensile test of the same dimension sample is created. The material model in simulation is set as orthotropic elastic model with inputs of measured Young’s moduli \(E_1\) and \(E_2\), an assumed major Poisson’s ratio \(\nu_{12}\), and the calculated shear modulus \(G_{12}\) according to the above equation.
- A python script was developed to run multiple simulations, iterating Poisson’s ratio and corresponded shear modulus until the deformed shape of FEA matches that of the test. Then the complete material model is calibrated.

In seeking an agreed deformed shape between FEA and experiments, we’ve practiced using the lateral contraction only at the middle of the sample as the criterion. A more accurate calibration would use digital image correlation (DIC) to have a full field deformation measurement of nonwoven sample and minimize an average measure of displacement difference between FEA and test over multiple selected control points.

In summary we proposed and verified a practical material calibration way to fit parameters of orthotropic model for nonwoven. The method requires only three uniaxial tensile tests and avoids the uncertainty in measuring Poisson’s ratio and shear modulus. Since the method is based on an inverse FEA procedure of which the objective is to match the test deformation, the calibrated material model is especially accurate for future finite element analyses of nonwoven under similar tensile loadings. Nevertheless, it should be noted the proposed method is based on the validity of orthotropic model on depicting the elastic behavior of nonwoven, which is only validated in small deformation. If one aims to model behavior of large deformation where the material nonlinearity exhibits, more advanced constitutive models need to be developed. The orthotropic framework may still be valid, but material parameters must be made state dependent.