Loads and deformations¶
The laminate theory connects in-plane forces $\mathbf{N}$ and moments $\mathbf{M}$ to midplane strains $\boldsymbol{\varepsilon^0}$ and kurvatures $\boldsymbol{\kappa}$ by a set of linear relations:
$$ \begin{bmatrix} \mathbf{N} \\ \mathbf{M} \end{bmatrix}= \begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B} & \mathbf{D} \end{bmatrix} \begin{bmatrix} \boldsymbol{\varepsilon^0} \\ \boldsymbol{\kappa} \end{bmatrix} $$The relations will be derived from the previously established theories and assumptions.
Laminate deformation¶
The Kirchhoff assumption implies that the displacements at a position $z$ through the thickness are related to the midplane displacements and rotations expressed by:
\begin{equation} \begin{aligned} &u(x,y,z) = u_0(x,y)-z \frac{\partial w_0}{\partial x} \\ &v(x,y,z) = v_0(x,y)-z \frac{\partial w_0}{\partial y} \\ &w(x,y)=w_0(x,y) \end{aligned} \tag{1} \end{equation}where $u$, $v$ and $w$ are the displacements in $x$, $y$ and $z$ directions respectively, and $u_0$, $v_0$ and $w_0$ are the displacements of the reference plane.
From the strain-displacement relations;
\begin{equation} \begin{aligned} &\varepsilon_x = \frac{\partial u}{\partial x} =\frac{\partial u_0}{\partial x} - z \frac{\partial^2 w_0}{\partial x^2} = \varepsilon_x^0 + z \kappa_x \\ &\varepsilon_y = \frac{\partial v}{\partial y} =\frac{\partial v_0}{\partial y} - z \frac{\partial^2 w_0}{\partial y^2} = \varepsilon_y^0 + z \kappa_y \\ &\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} =\frac{\partial u_0}{\partial y} + \frac{\partial v_0}{\partial x} - 2z \frac{\partial^2 w_0}{\partial x \partial y} = \gamma_{xy}^0 + z \kappa_{xy} \end{aligned} \tag{2} \end{equation}where $\varepsilon_x^0$, $\varepsilon_y^0$ and $\gamma_{xy}^0$ are mid-plane strains and $\kappa_x$, $\kappa_y$ and $\kappa_{xy}$ are curvatures given by
\begin{equation} \kappa_x=-\frac{\partial^2 w}{\partial x^2}, \quad \kappa_y=-\frac{\partial^2 w}{\partial y^2}, \quad \kappa_{xy}=-2\frac{\partial^2 w}{\partial x \partial y} \tag{3} \end{equation}The relations can now be summarized using matrix form as
\begin{equation} \begin{bmatrix} \varepsilon_x \\ \varepsilon_y \\ \gamma_{xy} \end{bmatrix}= \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix}+z \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} \tag{4} \end{equation}or the short notation
\begin{equation} \boldsymbol{\varepsilon}' = \boldsymbol{\varepsilon}^0 + z \boldsymbol{\kappa} \tag{5} \end{equation}Note that
$$\varepsilon_{z} = \frac{\partial w}{\partial z} = \frac{\partial w_0}{\partial z} = 0$$$$\gamma_{xz} = \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} = \frac{\partial u_0}{\partial z} + \frac{\partial w_0}{\partial x} = 0$$$$\gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} = \frac{\partial v_0}{\partial z} + \frac{\partial w_0}{\partial y} = 0$$Illustrating the curvatures:
The curvatures can be visualized by superposing the resulting out-of-plane displacement $w$ given by the contributions from $\kappa_x$, $\kappa_y$ and $\kappa_{xy}$:
$$ \frac{\partial^2 w}{\partial x^2} = -\kappa_x \Rightarrow \frac{\partial w}{\partial x} = -\kappa_x x + c_1 \Rightarrow w=-\frac{1}{2}\kappa_x x^2 + c_1 x + c_2 $$The boundary conditions $w(x=0) = 0$ and $\frac{\partial}{\partial x}w(x=0)$ leads to
$$ w=-\frac{1}{2}\kappa_x x^2 $$Correspondingly, the contribution from the curvature $\kappa_y$ when we assume that $\frac{\partial}{\partial y}w(x=0)$ is
$$ w=-\frac{1}{2}\kappa_y y^2 $$For $\kappa_{xy}$:
$$ \frac{\partial^2 w}{\partial x \partial y} = -\frac{1}{2}\kappa_{xy} \Rightarrow \frac{\partial w}{\partial y} = -\frac{1}{2} \kappa_{xy} x + c_5 \Rightarrow w=-\frac{1}{2}\kappa_{xy} x y + c_5 y + c_4 $$The boundary conditions $w(x=0) = 0$, $\frac{\partial}{\partial x}w(x=0)$ and $\frac{\partial}{\partial y}w(x=0)$ gives
$$ w=-\frac{1}{2}\kappa_{xy} x y $$Finally;
\begin{equation} w(x,y) = -\frac{1}{2}\kappa_x x^2 -\frac{1}{2}\kappa_y y^2 -\frac{1}{2} \kappa_{xy} x y \tag{6} \end{equation}Implementation and examples:
def illustrateCurvatures(Kx,Ky,Kxy):
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
#ax = fig.gca(projection='3d')
ax = fig.gca()
# Make data.
X = np.arange(-0.5, 0.6, 0.1)
Y = np.arange(-0.5, 0.6, 0.1)
X, Y = np.meshgrid(X, Y)
Z1 = (- Kx*X**2 - Ky*Y**2 - Kxy*X*Y)/2
surf1 = ax.plot_surface(X, Y, Z1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
%matplotlib inline
illustrateCurvatures(Kx=0.5, Ky=0.0, Kxy=0.0)
%matplotlib inline
illustrateCurvatures(Kx=-0.5, Ky=0.5, Kxy=0.0)
%matplotlib inline
illustrateCurvatures(Kx=0.0, Ky=0.0, Kxy=-0.25)
%matplotlib inline
illustrateCurvatures(Kx=0.0, Ky=0.25, Kxy=0.5)
Laminate loads and constitutive relations¶
Due to the discontinuous variation of stresses from layer to layer, we will conveniently consider the integrated effect of the stresses on the laminate. The laminate loads acting on a laminate are the in-plane forces $N_x$, $N_y$ and $N_{xy}$, and the moments $M_x$, $M_y$ and $M_{xy}$ as illustrated in Figure-1.
Figure-1: Laminate loads
Force resultants¶
The in-plane forces per unit length on individual layers are \begin{equation} N_{x,k}=\int_{h_{k-1}}^{h_k} \sigma_x dz, \quad N_{y,k}=\int_{h_{k-1}}^{h_k} \sigma_y dz, \quad N_{xy,k}=\int_{h_{k-1}}^{h_k} \tau_{xy} dz \tag{7} \end{equation}
Figure-2: Force per unit length of a single layer
The resultant forces per unit length on the laminate is the sum of contributions from all layers:
\begin{equation} N_{x}=\sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \sigma_x dz, \quad N_{y}=\sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \sigma_y dz, \quad N_{xy}=\sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \tau_{xy} dz \tag{8} \end{equation}Equation (8) can be expressed in matrix form as
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \end{bmatrix} = \sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \begin{bmatrix} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{bmatrix} dz \tag{9} \end{equation}The stresses can now be expressed through Hooke's law for the plane stress case:
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \end{bmatrix} = \sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \varepsilon_x \\ \varepsilon_y \\ \gamma_{xy} \end{bmatrix} dz \tag{10} \end{equation}Recall that the strain at a position $z$ is
\begin{equation} \begin{bmatrix} \varepsilon_x \\ \varepsilon_y \\ \gamma_{xy} \end{bmatrix}= \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix}+z \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} \tag{11} \end{equation}Substituting (11) into (10) yields:
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \end{bmatrix} = \sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix} dz + \sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} z dz \tag{12} \end{equation}Since the stiffness matrix, mid-plane strains and curvatures do not vary through the thickness of a layer, equation (12) can be rearranged to
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \end{bmatrix} = \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix} \int_{h_{k-1}}^{h_k} dz + \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} \int_{h_{k-1}}^{h_k} z dz \tag{13} \end{equation}Performing the integrations,
\begin{equation} \int_{h_{k-1}}^{h_k} dz = h_k - h_{k-1}, \quad \int_{h_{k-1}}^{h_k}z dz = \frac{1}{2}(h_k^2 - h_{k-1}^2) \tag{14} \end{equation}such that equation (13) can written as
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \end{bmatrix} = \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix} (h_k - h_{k-1}) + \frac{1}{2} \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} (h_k^2 - h_{k-1}^2) \tag{15} \end{equation}Now defining
\begin{equation} \begin{bmatrix} A_{xx} & A_{xy} & A_{xs} \\ A_{xy} & A_{yy} & A_{ys} \\ A_{xs} & A_{ys} & A_{ss} \end{bmatrix}= \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k (h_k - h_{k-1}) \tag{16} \end{equation}and
\begin{equation} \begin{bmatrix} B_{xx} & B_{xy} & B_{xs} \\ B_{xy} & B_{yy} & B_{ys} \\ B_{xs} & B_{ys} & B_{ss} \end{bmatrix} = \frac{1}{2} \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k (h_k^2 - h_{k-1}^2) \tag{17} \end{equation}and finally equation (15) becomes
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \end{bmatrix} = \begin{bmatrix} A_{xx} & A_{xy} & A_{xs} \\ A_{xy} & A_{yy} & A_{ys} \\ A_{xs} & A_{ys} & A_{ss} \end{bmatrix} \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix}+ \begin{bmatrix} B_{xx} & B_{xy} & B_{xs} \\ B_{xy} & B_{yy} & B_{ys} \\ B_{xs} & B_{ys} & B_{ss} \end{bmatrix} \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} \tag{18} \end{equation}Moment resultants¶
Correspondingly, resultant moments per unit length are:
\begin{equation} \begin{bmatrix} M_x \\ M_y \\ M_{xy} \end{bmatrix} = \sum_{k=1}^{n} \int_{h_{k-1}}^{h_k} \begin{bmatrix} \sigma_x \\ \sigma_y \\ \tau_{xy} \end{bmatrix}z dz \tag{19} \end{equation}Substitution and integration leads to
\begin{equation} \begin{bmatrix} M_x \\ M_y \\ M_{xy} \end{bmatrix} = \begin{bmatrix} B_{xx} & B_{xy} & B_{xs} \\ B_{xy} & B_{yy} & B_{ys} \\ B_{xs} & B_{ys} & B_{ss} \end{bmatrix} \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \end{bmatrix}+ \begin{bmatrix} D_{xx} & D_{xy} & D_{xs} \\ D_{xy} & D_{yy} & D_{ys} \\ D_{xs} & D_{ys} & D_{ss} \end{bmatrix} \begin{bmatrix} \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} \tag{20} \end{equation}where
\begin{equation} \begin{bmatrix} D_{xx} & D_{xy} & D_{xs} \\ D_{xy} & D_{yy} & D_{ys} \\ D_{xs} & D_{ys} & D_{ss} \end{bmatrix} = \frac{1}{3} \sum_{k=1}^{n} \begin{bmatrix} Q_{xx} & Q_{xy} & Q_{xs} \\ Q_{xy} & Q_{yy} & Q_{ys} \\ Q_{xs} & Q_{ys} & Q_{ss} \end{bmatrix}_k (h_k^3 - h_{k-1}^3) \tag{21} \end{equation}Equations (18) and (20) can now be combined into
\begin{equation} \begin{bmatrix} N_x \\ N_y \\ N_{xy} \\ M_x \\ M_y \\ M_{xy} \end{bmatrix} = \begin{bmatrix} A_{xx} & A_{xy} & A_{xs} & B_{xx} & B_{xy} & B_{xs} \\ A_{xy} & A_{yy} & A_{ys} & B_{xy} & B_{yy} & B_{ys} \\ A_{xs} & A_{ys} & A_{ss} & B_{xs} & B_{ys} & B_{ss} \\ B_{xx} & B_{xy} & B_{xs} & D_{xx} & D_{xy} & D_{xs} \\ B_{xy} & B_{yy} & B_{ys} & D_{xy} & D_{yy} & D_{ys} \\ B_{xs} & B_{ys} & B_{ss} & D_{xs} & D_{ys} & D_{ss} \end{bmatrix} \begin{bmatrix} \varepsilon_x^0 \\ \varepsilon_y^0 \\ \gamma_{xy}^0 \\ \kappa_x \\ \kappa_y \\ \kappa_{xy} \end{bmatrix} \tag{22} \end{equation}Summary¶
Force resultants: $$ \mathbf{N} = \sum_{k=1}^{n} \int_{h_{k-1}}^h \boldsymbol{\sigma}'dz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\varepsilon}'dz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k(\boldsymbol{\varepsilon}^0 + z \boldsymbol{\kappa})dz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\varepsilon}^0 dz+ \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\kappa}zdz = \\ \Big[ \sum_{k=1}^{n} \mathbf{Q}'_k (h_k-h_{k-1}) \Big] \boldsymbol{\varepsilon}^0 + \Big[\frac{1}{2}\sum_{k=1}^{n} \mathbf{Q}'_k(h_k^2-h_{k-1}^2 \Big] \boldsymbol{\kappa} \Rightarrow \\ \mathbf{N} = \mathbf{A} \boldsymbol{\varepsilon}^0+\mathbf{B} \boldsymbol{\kappa} $$
Moment resultants:
$$ \mathbf{M} = \sum_{k=1}^{n} \int_{h_{k-1}}^h \boldsymbol{\sigma}'zdz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\varepsilon}'zdz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k(\boldsymbol{\varepsilon}^0 + z \boldsymbol{\kappa})zdz = \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\varepsilon}^0 z dz+ \sum_{k=1}^{n} \int_{h_{k-1}}^h \mathbf{Q}'_k\boldsymbol{\kappa}z^2 dz = \\ \Big[\frac{1}{2}\sum_{k=1}^{n} \mathbf{Q}'_k (h_k^2-h_{k-1}^2) \Big] \boldsymbol{\varepsilon}^0 + \Big[\frac{1}{3}\sum_{k=1}^{n} \mathbf{Q}'_k(h_k^3-h_{k-1}^3 )\Big] \boldsymbol{\kappa} \Rightarrow \\ \mathbf{M} = \mathbf{B} \boldsymbol{\varepsilon}^0+\mathbf{D} \boldsymbol{\kappa} $$where
\begin{equation} \begin{aligned} &\mathbf{A} = \sum_{k=1}^{n} \mathbf{Q}'_k (h_k-h_{k-1}) \\ &\mathbf{B} = \frac{1}{2}\sum_{k=1}^{n} \mathbf{Q}'_k (h_k^2-h_{k-1}^2) \\ &\mathbf{D} = \frac{1}{3}\sum_{k=1}^{n} \mathbf{Q}'_k (h_k^3-h_{k-1}^3) \end{aligned} \tag{23} \end{equation}Finally,
\begin{equation} \begin{bmatrix} \mathbf{N} \\ \mathbf{M} \end{bmatrix}= \begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B} & \mathbf{D} \end{bmatrix} \begin{bmatrix} \boldsymbol{\varepsilon^0} \\ \boldsymbol{\kappa} \end{bmatrix} \tag{24} \end{equation}
