Note that this is the same laminate considered in Sample Problem 6.6,and the midplane strains and curvatures, ply strains, and ply stresses that will be induced immediately upon cooldown by the change in temperature have already been calculated. These quantities will all be modified due to the slow diffusion of water molecules into the epoxy matrix.
a. Midplane strains and curvatures: The laminate has experienced a change in moisture content ΔM = +0.5%, and consequently is subjected moisture stress and moment resultants. The effective moisture expansion coefficients for each ply are calculated using Equation 5.28, repeated here for convenience:
\beta _{xx}=\beta _{11}\cos^2(\theta )+\beta _{22}\sin^2(\theta )
\beta _{yy}=\beta _{11}\sin^2(\theta )+\beta _{22}\cos^2(\theta ) (5.28) (repeated)
\beta _{xy}=2\cos(\theta )\sin(\theta )(\beta _{11}-\beta _{12})
From Table 3.1, the moisture expansion coefficients for graphite–epoxy (relative to the 1–2 coordinate system) are β_{11} = 150 μm/m−%M and β_{22} = 4800 μm/m−%M. Therefore,
For ply #1 (the 30°ply):
\beta _{xx}^{(1)}=(150μm/m-\%M)\cos^2(30^\circ )+(4800μm/m -\%M)\sin^2(30^\circ )=1310μm/m −\%M
\beta _{yy}^{(1)}=(150μm/m-\%M)\sin^2(30^\circ )+(4800μm/m -\%M)\cos^2(30^\circ )=3640μm/m −\%M
\beta _{xy}^{(1)}=2\cos(30)\sin(30)[(150 -4800)μm/m – \%M] = -4030μrad/\%M
For ply #2 (the 0°ply):
\beta _{xx}^{(2)}=(150μm/m-\%M)\cos^2(0^\circ )+(4800μm/m -\%M)\sin^2(0^\circ )=150μm/m −\%M
\beta _{yy}^{(2)}=(150μm/m-\%M)\sin^2(0^\circ )+(4800μm/m -\%M)\cos^2(0^\circ )=4800μm/m −\%M
\beta _{xy}^{(2)}=2\cos(0)\sin(0)[(150 -4800)μm/m – \%M] = 0μrad/\%M
For ply #3 (the 90°ply):
\beta _{xx}^{(3)}=(150μm/m-\%M)\cos^2(90^\circ )+(4800μm/m -\%M)\sin^2(90^\circ )=4800μm/m −\%M
\beta _{yy}^{(3)}=(150μm/m-\%M)\sin^2(90^\circ )+(4800μm/m -\%M)\cos^2(90^\circ )=150μm/m −\%M
\beta _{xy}^{(3)}=2\cos(90)\sin(0)[(150 -4800)μm/m – \%M] = 0μrad/\%M
Both the ply interface positions as well as the [\overline{Q} ] matrices for each ply were calculated as a part of Example Problem 6.4. Hence, we now have all the information needed to calculate the moisture stress and moment resultants, using Equations 6.42. For example, Equation 6.42a is evaluated as follows:
N_{xx}^M\equiv \Delta M\sum\limits_{k=1}^{n}{\left\{\left[\overline{Q}_{11}\beta _{xx}+\overline{Q}_{12}\beta _{yy}+\overline{Q}_{16}\beta _{xy}\right]_k[z_k-z_{k-1}] \right\} } (6.42a)
N_{xx}^M\equiv \Delta M\sum\limits_{k=1}^{3}{\left\{\left[\overline{Q}_{11}\beta _{xx}+\overline{Q}_{12}\beta _{yy}+\overline{Q}_{16}\beta _{xy}\right]_k[z_k-z_{k-1}] \right\} }
N_{xx}^M =\Delta M\left\{\left(\left[\overline{Q}_{11}\beta _{xx}+\overline{Q}_{12}\beta _{yy}+\overline{Q}_{16}\beta _{xy}\right]_1[z_1-z_{0}]\right)+\left([\overline{Q}_{11}\beta _{xx}+\overline{Q}_{12}\beta _{yy}+\overline{Q}_{16}\beta _{xy}]_2 \\ \times [z_2-z_1]+\left[\overline{Q}_{11}\beta _{xx}+\overline{Q}_{12}\beta _{yy}+\overline{Q}_{16}\alpha _{xy}\right]_3[z_3-z_2] \right) \right\}
N_{xx}^M = (+0.5)\left\{\left(\left[(107.6\times 10^9)(1312\times 10^{-6})+(26.06\times 10^9)(3638\times 10^{-6}) \\ +(48.13\times 10^9)(-4027\times 10^{-6})\right]\left[\left(-0.0625+0.1875\right) \times10^{-3} \right] \right)+\left(\left[(170.9\times 10^9) \\ \times (150\times 10^{-6})+(3.016\times 10^9)(4800\times 10^{-6})+(0)(0)\right]\left[(0.0625+0.0625)\times 10^{-3}\right] \right) \right\} \\ +\left(\left[(10.05 \times 10^9)(4800\times 10^{-6})+(3.016\times 10^9)(150\times 10^{-6})+(0)(0)\right] \\ \times \left[(0.1875 – 0.0625)\times 10^{-3}\right] \right)
N_{xx}^M=8190N/m
The remaining moisture stress and moment resultants are calculated in similar fashion, resulting in
\left \{ \begin{matrix}N_{xx}^M \\ N_{yy}^M \\ N_{xy}^M \\ M_{xx}^M \\ M_{yy}^M \\ M_{xy}^M \end{matrix} \right \}=\left \{ \begin{matrix}8190N/m\\ 8460N/m\\ -233N/m\\0.05N–m/m\\ -0.05N–m/m\\ 0.03N–m/m \end{matrix} \right \}
We can now calculate midplane strains and curvatures using Equation 6.45,
\left \{ \begin{matrix} \varepsilon _{xx}^o \\ \varepsilon _{yy}^o \\ \gamma _{xy}^o \\\kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right \}=\left [ \begin{matrix} a_{11} & a_{12} & a_{16} & b_{11} & b_{12} & b_{16} \\ a_{12} & a_{22} & a_{26} & b_{21} & b_{22} & b_{26} \\ a_{16} & a_{26} & a_{66} & b_{61} & b_{62} & b_{66} \\ b_{11} & b_{21} & b_{61} & d_{11} & d_{12} & d_{16} \\ b_{12} & b_{22} & b_{62} & d_{12} & d_{22} & d_{26} \\ b_{16} & b_{26} & b_{66} & d_{16} & d_{26} & d_{66} \end{matrix} \right ]\left \{ \begin{matrix} N_{xx} & N_{xx}^T & N_{xx}^M \\ N_{yy} &N_{yy}^T & N_{yy}^M \\ N_{xy} & N_{xy}^T & N_{xy}^M \\ M_{xx} & M_{xx}^T & M_{xx}^M \\ M_{yy} & M_{yy}^T & M_{yy}^M \\ M_{xy} & M_{xy}^T & M_{xy}^M \end{matrix} \right \} (6.45)
which becomes*
\left \{ \begin{matrix} \varepsilon _{xx}^o \\ \varepsilon _{yy}^o \\\gamma _{xy}^o \\ \kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right \}=\left [ \begin{matrix} 3.757\times 10^{-8} & -1.964\times 10^{-9} &-1.038\times 10^{-8} & 1.440\times 10^{-4} & 3.905\times 10^{-6} & 8.513\times 10^{-5} \\ -1.964\times 10^{-9}& 1.037\times 10^{-7} & -4.234\times 10^{-8} &-1.866\times 10^{-6} &-6.361\times 10^{-4} & 4.628\times 10^4 \\-1.038\times 10^{-8} & -4.234\times 10^{-8} & 2.004\times 10^{-7}& 3.661\times 10^{-4} & 3.251\times 10^{-4} &-1.851\times 10^{-5} \\ 1.440\times 10^{-4} &-1.866\times 10^{-5} &3.661\times 10^{-4} & 7.064 & -3.122\times 10^{-2} & -4.572 \\ 3.905\times 10^{-6} & -6.361\times 10^{-4} &3.251\times 10^{-4} & -3.122\times 10^{-2} & 6.429 & -3.620 \\8.513\times 10^{-5} & 4.628\times 10^4 & -1.851\times 10^{-5} & -4.572 &-3.620 & 17.41 \end{matrix} \right ] \\ \left \{ \begin{matrix} -4060+8190 \\-7360+8460 \\ 2860-233\\ -0.62+0.05 \\ 0.62-0.05 \\ -0.36+0.03 \end{matrix} \right \}
\left ( \begin{matrix} \varepsilon _{xx}^o \\ \varepsilon _{yy}^o \\\gamma _{xy}^o \\ \kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right )=\left \{ \begin{matrix}18μm/m \\ -509μm/m \\ 420μrad \\ -1.0m^{-1} \\ 5.0m^{-1} \\ -4.4m^{-1} \end{matrix} \right \}
b. Ply strains relative to the x−y coordinate system: Ply strains may now be calculated using Equation 6.12.
\left \{ \begin{matrix}\varepsilon _{xx} \\ \varepsilon _{yy} \\ \gamma _{xy} \end{matrix} \right \}=\left \{ \begin{matrix}\varepsilon _{xx}^o \\ \varepsilon _{yy}^o \\ \gamma _{xy}^o \end{matrix} \right \}+z\left \{ \begin{matrix}\kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right \} (6.12)
For example, strains
present at the outer surface of ply #1 (i.e., strains present at z_o = −0.0001875 m) are
\left . \left \{ \begin{matrix}\varepsilon _{xx} \\ \varepsilon _{yy} \\ \gamma _{xy} \end{matrix} \right \}\right |_{z=z_0}=\left \{ \begin{matrix}\varepsilon _{xx}^o \\ \varepsilon _{yy}^o \\ \gamma _{xy}^o \end{matrix} \right \}+z_0\left \{ \begin{matrix}\kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right \}=\left \{ \begin{matrix}18\times 10^{-6}m/m \\ -509\times 10^{-6}m/m \\ 420\times 10^{-6}m/m \end{matrix} \right \}+(-0.0001875m)\left \{ \begin{matrix} -1.0rad/m \\5.0rad/m \\ -4.4rad/m \end{matrix} \right \}
\left . \left \{ \begin{matrix}\varepsilon _{xx} \\ \varepsilon _{yy} \\ \gamma _{xy} \end{matrix} \right \}\right |_{z=z_0}=\left \{ \begin{matrix} 206μm/m \\ -1450μm/m \\ 1240μrad \end{matrix} \right \}
Strains calculated at the remaining ply interface positions are summarized in Table 6.9.
c. Ply stresses relative to the x−y coordinate system: Ply stresses may now be calculated using Equation 5.30. The stresses present at the outer surface of ply #1 are (i.e., at z = z_0):
\begin{Bmatrix} \sigma _{xx} \\ \sigma _{yy} \\ \tau _{xy} \end{Bmatrix} = \begin{bmatrix} \overline{Q}_{11} & \overline{Q}_{12} & \overline{Q}_{16} \\ \overline{Q}_{12} & \overline{Q}_{22} & \overline{Q}_{26} \\ \overline{Q}_{16} &\overline{Q}_{26}& \overline{Q}_{66} \end{bmatrix} \begin{Bmatrix} \varepsilon _{xx}-\Delta T\sigma _{xx} – \Delta M\beta _{xx} \\ \varepsilon _{yy}-\Delta T\sigma _{yy} -\Delta M\beta _{yy} \\ \gamma _{xy}-\Delta T\sigma _{xy} -\Delta M\beta _{xy} \end{Bmatrix} (5.30)
\left . \left \{ \begin{matrix}\sigma _{xx} \\ \sigma _{yy} \\ \tau _{xy} \end{matrix} \right \}\right |_{z=z_0}^{ply1}=\left . \left [ \begin{matrix} \overline{Q} _{11} & \overline{Q}_{12} & \overline{Q}_{16} \\ \overline{Q}_{12} & \overline{Q}_{22} & \overline{Q}_{26} \\ \overline{Q}_{16} & \overline{Q}_{26} & \overline{Q}_{66} \end{matrix} \right ]\right |_{z=z_0}^{ply1}\left . \left \{ \begin{matrix} \varepsilon _{xx}-\Delta T\alpha _{xx}-\Delta M\beta _{xx} \\ \varepsilon _{yy}-\Delta T\alpha _{yy}-\Delta M\beta _{yy} \\ \gamma _{xy}-\Delta T\alpha _{xy}-\Delta M\beta _{xy} \end{matrix} \right \}\right |_{z=z_0}
\left . \left \{ \begin{matrix}\sigma _{xx} \\ \sigma _{yy} \\ \tau _{xy} \end{matrix} \right \}\right |_{z=z_0}^{ply1}=\left [ \begin{matrix}107.6\times 10^9 & 26.06\times 10^9 & 48.13\times 10^9 \\ 26.06\times 10^9 & 27.22\times 10^9 & 21.52\times 10^9 \\ 48.13\times 10^9 & 21.52\times 10^9 & 36.05\times 10^9 \end{matrix} \right ]\left \{ \begin{matrix} \left[(206)-(-155)(6.08)-(0.5)(1312)\right]\times 10^{-6} \\ \left[(-1450)-(-155)(20.0)-(0.5)(3638)\right]\times 10^{-6} \\ \left[(1240)-(-155)(-24.2)-(0.5)(-4027)\right]\times 10^{-6} \end{matrix} \right \}
TABLE 6.9
Ply Interface Strains in a [30/0/90] Graphite−Epoxy Laminate Caused by the Combined Effects of Cooldown from 175°C to 20°C and an Increase in Moisture Content of +0.5%
| z-Coordinate (mm) |
\varepsilon _{xx} (μm/m) |
\varepsilon _{yy} (μm/m) |
\gamma _{xy} (μ Radians) |
| −0.1875 |
206 |
−1450 |
1240 |
| −0.0625 |
80 |
−820 |
690 |
| 0.0625 |
−44 |
−190 |
150 |
| 0.1875 |
−170 |
440 |
−400 |
Note: Strains are referenced to the x−y coordinate system.
TABLE 6.10
Ply Interface Stresses in a [30/0/90] Graphite–Epoxy Laminate Caused by the Combined Effects of Cooldown from 175°C to 20°C and an Increase in Moisture Content of +0.5%
| Ply Number |
z-Coordinate (mm) |
\sigma _{xx} (MPa) |
\sigma _{yy} (MPa) |
\tau _{xy} (MPa) |
| Ply |
−0.1875 |
25 |
-2.2 |
2.2 |
| 1 |
−0.0625 |
1.4 |
-0.24 |
-9.9 |
| Ply |
−0.0625 |
-20 |
9.3 |
9 |
| 2 |
0.0625 |
-40 |
15 |
1.9 |
| Ply |
0.0625 |
16 |
-65 |
1.9 |
| 3 |
0.1875 |
17 |
43 |
-5.2 |
Note: Stresses are referenced to the x–y coordinate system.
\left . \left \{ \begin{matrix} \sigma _{xx} \\ \sigma _{yy} \\\tau _{xy} \end{matrix} \right \}\right |_{z=z_0}^{ply1}=\left \{ \begin{matrix} 25 MPa \\ -2.4 MPa \\ 2.2MPa \end{matrix} \right \}
Stresses calculated at the remaining plies and ply interface positions are summarized in Table 6.10.
A comparison of the results obtained in Example Problems 6.6 and 6.7 leads to the following observation: the initial ply stresses and strains caused by cooldown from cure temperatures to room temperatures are partially relieved by the subsequent adsorption of moisture. Although the interaction between temperature and moisture effects obviously depends on the details of the situation (material properties involved, stacking sequence, magnitudes of ΔT and ΔM, etc.), this observation is often true. That is, the thermal stresses predicted to develop in a multiangle laminate during cooldown are usually predicted to be relieved somewhat by subsequent adsorption of moisture.
* The [abd] matrix for a [30 /o/90]_T graphite–epoxy laminate was calculated in Sample Problem 6.3, and the thermal stress and moment resultants were calculated in Sample Problem 6.5.