TMM4175 Polymer Composites

Home About Python Links Table of Contents

Questions and problems¶

Basic concepts¶

1) Explain the potential difference between plies and layers

2) What is a symmetric laminate?

3) What is a balanced laminate?

4) What is the meaning of homogeneous layers?

5) Can a laminate be homogeneous?

6) Explain the difference between plane stress and plane strain

7) Is it possible to be in a state of plane stress and plane strain simultaneously? If so, how?

8) Kirchoff assumptions implies that: A) Normals to the midplane remain straight and normal to the deformed midplane after deformation and B) Normals to the midplane do not change length. Make sketches that illustrate this assumption.

9) The Kirchoff assumtion (b) suggests plane strain while one of the other assumptions is plane stress. What is going on?

10) The transformed 2D (or plane stress) stiffnes matrix is shown below. Can the components $Q_{xs}$ and $Q_{ys}$ be zero, and if so, how? $$ \mathbf{Q}'= \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix} $$ 11) What is the meaning (illustrate) of a positive value of $\kappa_{x}$

12) What is the meaning (illustrate) of a positive value of $\kappa_{y}$

13) What is the meaning (illustrate) of a positive value of $\kappa_{xy}$

14) What is the unit of the curvatures parameters?

15) What is the unit of in-plane laminate forces?

16) What is the unit of laminate moments?

17) What is the unit of the laminate stiffness components A?

18) What is the unit of the laminate stiffness components B?

19) What is the unit of the laminate stiffness components D?

Computational problems and procedures¶

20) For plane stress, the compliance matrix is $$ \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \gamma_{12} \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} & 0 \\ S_{12} & S_{22} & 0 \\ 0 & 0 & S_{66} \end{bmatrix} \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \tau_{12} \end{bmatrix} $$
Show that the components of the stiffness matrix $\mathbf{Q} = \mathbf{S}^{-1}$ can be expressed by $$ Q_{11}=\frac{E_1}{1-v_{12}v_{21}}, \quad Q_{22}=\frac{E_2}{1-v_{12}v_{21}}, \quad Q_{12}=\frac{v_{12}E_2}{1-v_{12}v_{21}}, \quad Q_{66}=G_{12} $$

21) Use equation (3) in Plastic behavior to show that the 2D Tsai-Wu criterion with $f_{12}=-0.5$ and $X_T = Y_T = X_C = Y_C = \sigma_{yield}$ is equal to the von Mises yield criterion.

22) Show that $B = 0$ for a symmetric laminate.

23) Show the $A_{xs} = A_{ys} = 0$ for a balanced laminate.

24) Show that a laminate of type [0/-60/60]_s is a quazi-isotropic laminate using numerical examples.

25) Consider laminates made of layers of orthotropic plies having identical thicknesses. Identify the components of the laminate stiffness that are zero for the laminates with the following layups:

[0/90/0]
[0/90/0/90]
[45/-45/45/-45]
[90/30/90]
[30/-30/45/30/-30]
[45/-45/0/90]

26) Consider laminates made of layers of orthotropic plies having identical thicknesses. Identify the differences in the laminate stiffness matrix components for the pairs of layups:

[0/90/90] and [90/0/90]
[0/90/0/90] and [90/0/90/0]
[0/-45/45/0] and [0/45/-45/0]
[90/-45/90] and [90/45/90]
[0/0] and [0/0/0/0]
[0/0/90/90/0/0] and [0/90/0/0/90/0]

27) In Effective properties you find a study on the effective in-plane elastic constants for a [-θ/θ/θ/-θ] laminate as function of the angle θ where the material is the Carbon/Epoxy(a) Use the procedure in that page to compute the effective in-plane Poisson’s ratio as function of the angle when you change the material to a) E-glass/Epoxy, and b) Carbon/Epoxy(b). Based on the results, explain what governs the value of the in-plane Poisson’s ratio (and what it means)

28) Show that for a quazi-isotropic laminate, $$ A_{yy} = A_{xx}, \quad A_{ss}=\frac{A_{xx}-A_{xy}}{2} $$

Disclaimer:This site is about polymer composites, designed for educational purposes. Consumption and use of any sort & kind is solely at your own risk.
Fair use: I spent some time making all the pages, and even the figures and illustrations are my own creations. Obviously, you may steal whatever you find useful here, but please show decency and give some acknowledgment if or when copying. Thanks! Contact me: nils.p.vedvik@ntnu.no www.ntnu.edu/employees/nils.p.vedvik