Kirchhoff–Love plate theory

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Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)

The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love[1] using assumptions proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form.

The following kinematic assumptions that are made in this theory:[2]

  • straight lines normal to the mid-surface remain straight after deformation
  • straight lines normal to the mid-surface remain normal to the mid-surface after deformation
  • the thickness of the plate does not change during a deformation.

Assumed displacement field[edit]

Let the position vector of a point in the undeformed plate be \mathbf{x}. Then


   \mathbf{x} = x_1\boldsymbol{e}_1+x_2\boldsymbol{e}_2+x_3\boldsymbol{e}_3 \equiv x_i\boldsymbol{e}_i\,.

The vectors \boldsymbol{e}_i form a Cartesian basis with origin on the mid-surface of the plate, x_1 and x_2 are the Cartesian coordinates on the mid-surface of the undeformed plate, and x_3 is the coordinate for the thickness direction.

Let the displacement of a point in the plate be \mathbf{u}(\mathbf{x}). Then


  \mathbf{u} = u_1\boldsymbol{e}_1+u_2\boldsymbol{e}_2+u_3\boldsymbol{e}_3 \equiv u_i\boldsymbol{e}_i

This displacement can be decomposed into a vector sum of the mid-surface displacement and an out-of-plane displacement w^0 in the x_3 direction. We can write the in-plane displacement of the mid-surface as


  \mathbf{u}^0 = u^0_1\boldsymbol{e}_1+u^0_2\boldsymbol{e}_2 \equiv u^0_\alpha\boldsymbol{e}_\alpha

Note that the index \alpha takes the values 1 and 2 but not 3.

Then the Kirchhoff hypothesis implies that


   \begin{align}
     u_\alpha(\mathbf{x}) & = u^0_\alpha(x_1,x_2) - x_3~\frac{\partial w^0}{\partial x_\alpha} \equiv u^0_\alpha - x_3~w^0_{,\alpha} ~;~~\alpha=1,2 \\
     u_3(\mathbf{x}) & = w^0(x_1, x_2)
   \end{align}

If \varphi_\alpha are the angles of rotation of the normal to the mid-surface, then in the Kirchhoff-Love theory


   \varphi_\alpha = w^0_{,\alpha}

Note that we can think of the expression for u_\alpha as the first order Taylor series expansion of the displacement around the mid-surface.

Displacement of the mid-surface (left) and of a normal (right)

Quasistatic Kirchhoff-Love plates[edit]

The original theory developed by Love was valid for infinitesimal strains and rotations. The theory was extended by von Kármán to situations where moderate rotations could be expected.

Strain-displacement relations[edit]

For the situation where the strains in the plate are infinitesimal and the rotations of the mid-surface normals are less than 10° the strain-displacement relations are[clarification needed]


  \begin{align}
    \varepsilon_{\alpha\beta} & = \frac{1}{2}\left(\frac{\partial u_\alpha}{\partial x_\beta} + 
      \frac{\partial u_\beta}{\partial x_\alpha}\right) \equiv \frac{1}{2}(u_{\alpha,\beta}+u_{\beta,\alpha})\\
    \varepsilon_{\alpha 3} & = \frac{1}{2}\left(\frac{\partial u_\alpha}{\partial x_3} + 
      \frac{\partial u_3}{\partial x_\alpha}\right) \equiv \frac{1}{2}(u_{\alpha,3}+u_{3,\alpha})\\
    \varepsilon_{33} & = \frac{\partial u_3}{\partial x_3} \equiv u_{3,3}
  \end{align}

Using the kinematic assumptions we have


  \begin{align}
    \varepsilon_{\alpha\beta} & = \tfrac{1}{2}(u^0_{\alpha,\beta}+u^0_{\beta,\alpha})
      - x_3~w^0_{,\alpha\beta} \\
    \varepsilon_{\alpha 3} & = - w^0_{,\alpha} + w^0_{,\alpha} = 0 \\
    \varepsilon_{33} & = 0
  \end{align}

Therefore the only non-zero strains are in the in-plane directions.

Equilibrium equations[edit]

The equilibrium equations for the plate can be derived from the principle of virtual work. For a thin plate under a quasistatic transverse load q(x) these equations are


   \begin{align}  
      &\cfrac{\partial N_{11}}{\partial x_1} + \cfrac{\partial N_{21}}{\partial x_2} = 0 \\
      &\cfrac{\partial N_{12}}{\partial x_1} + \cfrac{\partial N_{22}}{\partial x_2} = 0\\
      &\cfrac{\partial^2 M_{11}}{\partial x_1^2} + 2\cfrac{\partial^2 M_{12}}{\partial x_1 \partial x_2} +
      \cfrac{\partial^2 M_{22}}{\partial x_2^2} = q
   \end{align}

where the thickness of the plate is 2h. In index notation,


   \begin{align} 
     N_{\alpha\beta,\alpha} & = 0 \quad  \quad N_{\alpha\beta} := \int_{-h}^h \sigma_{\alpha\beta}~dx_3 \\
     M_{\alpha\beta,\alpha\beta} - q & = 0 \quad  \quad M_{\alpha\beta} := \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3
   \end{align}

where \sigma_{\alpha\beta} are the stresses.

Bending moments and normal stresses
Torques and shear stresses

Boundary conditions[edit]

The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtual work. In the absence of external forces on the boundary, the boundary conditions are


   \begin{align}
      n_\alpha~N_{\alpha\beta} & \quad \mathrm{or} \quad u^0_\beta \\
      n_\alpha~M_{\alpha\beta,\beta} & \quad \mathrm{or} \quad w^0 \\
      n_\beta~M_{\alpha\beta} & \quad \mathrm{or} \quad w^0_{,\alpha}
   \end{align}

Note that the quantity  n_\alpha~M_{\alpha\beta,\beta} is an effective shear force.

Constitutive relations[edit]

The stress-strain relations for a linear elastic Kirchhoff plate are given by


   \begin{align}
     \sigma_{\alpha\beta} & = C_{\alpha\beta\gamma\theta}~\varepsilon_{\gamma\theta} \\
     \sigma_{\alpha 3} & = C_{\alpha 3\gamma\theta}~\varepsilon_{\gamma\theta} \\
     \sigma_{33} & = C_{33\gamma\theta}~\varepsilon_{\gamma\theta} 
   \end{align}

Since \sigma_{\alpha 3} and \sigma_{33} do not appear in the equilibrium equations it is implicitly assumed that these quantities do not have any effect on the momentum balance and are neglected. The remaining stress-strain relations, in matrix form, can be written as


   \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} = 
   \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\
                   C_{13} & C_{23} & C_{33} \end{bmatrix}
   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix}

Then,


   \begin{bmatrix}N_{11} \\ N_{22} \\ N_{12} \end{bmatrix} = 
   \int_{-h}^h \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\
                   C_{13} & C_{23} & C_{33} \end{bmatrix}
   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix}
    dx_3 = \left\{
   \int_{-h}^h \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\
                   C_{13} & C_{23} & C_{33} \end{bmatrix}~dx_3 \right\}
   \begin{bmatrix} u^0_{1,1} \\ u^0_{2,2} \\ \frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \end{bmatrix}

and


   \begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = 
   \int_{-h}^h x_3~\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\
                   C_{13} & C_{23} & C_{33} \end{bmatrix}
   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix}
    dx_3 = -\left\{
   \int_{-h}^h x_3^2~\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\
                   C_{13} & C_{23} & C_{33} \end{bmatrix}~dx_3 \right\}
   \begin{bmatrix} w^0_{,11} \\ w^0_{,22} \\ w^0_{,12} \end{bmatrix}

The extensional stiffnesses are the quantities


   A_{\alpha\beta} := \int_{-h}^h C_{\alpha\beta}~dx_3

The bending stiffnesses (also called flexural rigidity) are the quantities


   D_{\alpha\beta} := \int_{-h}^h x_3^2~C_{\alpha\beta}~dx_3

The Kirchhoff-Love constitutive assumptions lead to zero shear forces. As a result, the equilibrium equations for the plate have to be used to determine the shear forces in thin Kirchhoff-Love plates. For isotropic plates, these equations lead to


   Q_\alpha = - D\frac{\partial}{\partial x_\alpha}(\nabla^2 w^0) \,.

Alternatively, these shear forces can be expressed as


   Q_\alpha = \mathcal{M}_{,\alpha}

where


   \mathcal{M} := -D\nabla^2 w^0 \,.

Small strains and moderate rotations[edit]

If the rotations of the normals to the mid-surface are in the range of 10^{\circ} to 15^\circ, the strain-displacement relations can be approximated as


  \begin{align}
    \varepsilon_{\alpha\beta} & =  \tfrac{1}{2}(u_{\alpha,\beta}+u_{\beta,\alpha}+u_{3,\alpha}~u_{3,\beta})\\
    \varepsilon_{\alpha 3} & = \tfrac{1}{2}(u_{\alpha,3}+u_{3,\alpha})\\
    \varepsilon_{33} & = u_{3,3}
  \end{align}

Then the kinematic assumptions of Kirchhoff-Love theory lead to the classical plate theory with von Kármán strains


  \begin{align}
    \varepsilon_{\alpha\beta} & = \frac{1}{2}(u^0_{\alpha,\beta}+u^0_{\beta,\alpha}+w^0_{,\alpha}~w^0_{,\beta})
      - x_3~w^0_{,\alpha\beta} \\
    \varepsilon_{\alpha 3} & = - w^0_{,\alpha} + w^0_{,\alpha} = 0 \\
    \varepsilon_{33} & = 0
  \end{align}

This theory is nonlinear because of the quadratic terms in the strain-displacement relations.

If the strain-displacement relations take the von Karman form, the equilibrium equations can be expressed as


   \begin{align} 
     N_{\alpha\beta,\alpha} & = 0 \\
     M_{\alpha\beta,\alpha\beta} + [N_{\alpha\beta}~w^0_{,\beta}]_{,\alpha} - q & = 0 
   \end{align}

Isotropic quasistatic Kirchhoff-Love plates[edit]

For an isotropic and homogeneous plate, the stress-strain relations are


   \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}
   = \cfrac{E}{1-\nu^2}
   \begin{bmatrix} 1 & \nu & 0 \\
                   \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix} \,.

The moments corresponding to these stresses are


   \begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = 
   -\cfrac{2h^3E}{3(1-\nu^2)}~\begin{bmatrix} 1 & \nu & 0 \\
                   \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
   \begin{bmatrix} w^0_{,11} \\ w^0_{,22} \\ w^0_{,12} \end{bmatrix}

In expanded form,


   \begin{align}
     M_{11} & = -D\left(\frac{\partial^2 w^0}{\partial x_1^2} + \nu \frac{\partial^2 w^0}{\partial x_2^2}\right) \\
     M_{22} & = -D\left(\frac{\partial^2 w^0}{\partial x_2^2} + \nu \frac{\partial^2 w^0}{\partial x_1^2}\right) \\
     M_{12} & = -D(1-\nu)\frac{\partial^2 w^0}{\partial x_1 \partial x_2}
   \end{align}

where  D = 2h^3E/[3(1-\nu^2)] = H^3E/[12(1-\nu^2)] for plates of thickness H = 2h. Using the stress-strain relations for the plates, we can show that the stresses and moments are related by


   \sigma_{11} = \frac{3x_3}{2h^3}\,M_{11} = \frac{12 x_3}{H^3}\,M_{11} \quad \text{and} \quad
   \sigma_{22} = \frac{3x_3}{2h^3}\,M_{22} = \frac{12 x_3}{H^3}\,M_{22} \,.

At the top of the plate where x_3 = h = H/2, the stresses are


   \sigma_{11} = \frac{3}{2h^2}\,M_{11} = \frac{6}{H^2}\,M_{11} \quad \text{and} \quad
   \sigma_{22} = \frac{3}{2h^2}\,M_{22} = \frac{6}{H^2}\,M_{22} \,.

Pure bending[edit]

For an isotropic and homogeneous plate under pure bending, the governing equations reduce to


   \frac{\partial^4 w^0}{\partial x_1^4} + 2\frac{\partial^4 w^0}{\partial x_1^2  \partial x_2^2} + \frac{\partial^4 w^0}{\partial x_2^4} = 0 \,.

Here we have assumed that the in-plane displacements do not vary with x_1 and x_2. In index notation,


   w^0_{,1111} + 2~w^0_{,1212} +  w^0_{,2222} = 0

and in direct notation


   \nabla^2\nabla^2 w = 0

The bending moments are given by


   \begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = 
   -\cfrac{2h^3E}{3(1-\nu^2)}~\begin{bmatrix} 1 & \nu & 0 \\
                   \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
   \begin{bmatrix} w^0_{,11} \\ w^0_{,22} \\ w^0_{,12} \end{bmatrix}

Bending under transverse load[edit]

If a distributed transverse load -q(x) is applied to the plate, the governing equation is M_{\alpha\beta,\alpha\beta} = -q. Following the procedure shown in the previous section we get[3]


   \nabla^2\nabla^2 w = \cfrac{q}{D} ~;~~ D := \cfrac{2h^3E}{3(1-\nu^2)}

In rectangular Cartesian coordinates, the governing equation is


  w^0_{,1111} + 2\,w^0_{,1212} + w^0_{,2222} = -\cfrac{q}{D}

and in cylindrical coordinates it takes the form


   \frac{1}{r}\cfrac{d }{d r}\left[r \cfrac{d }{d r}\left\{\frac{1}{r}\cfrac{d }{d r}\left(r \cfrac{d w}{d r}\right)\right\}\right] = - \frac{q}{D}\,.

Solutions of this equation for various geometries and boundary conditions can be found in the article on bending of plates.

Cylindrical bending[edit]

Under certain loading conditions a flat plate can be bent into the shape of the surface of a cylinder. This type of bending is called cylindrical bending and represents the special situation where u_1 = u_1(x_1), u_2 = 0, w = w(x_1). In that case


   \begin{bmatrix}N_{11} \\ N_{22} \\ N_{12} \end{bmatrix} = 
   \cfrac{2hE}{(1-\nu^2)}~\begin{bmatrix} 1 & \nu & 0 \\
                   \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
   \begin{bmatrix} u^0_{1,1} \\ 0 \\ 0 \end{bmatrix}

and


   \begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = 
   -\cfrac{2h^3E}{3(1-\nu^2)}~\begin{bmatrix} 1 & \nu & 0 \\
                   \nu & 1 & 0 \\
                   0 & 0 & 1-\nu \end{bmatrix}
   \begin{bmatrix} w^0_{,11} \\ 0 \\ 0 \end{bmatrix}

and the governing equations become[3]


   \begin{align}
   N_{11} & = A~\cfrac{\mathrm{d}u}{\mathrm{d} x_1} \quad \implies \quad
     \cfrac{\mathrm{d}^2 u}{\mathrm{d} x_1^2} = 0\\
   M_{11} & = -D~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x_1^2} \quad \implies \quad \cfrac{\mathrm{d}^4 w}{\mathrm{d} x_1^4} = \cfrac{q}{D} \\
   \end{align}

Dynamics of Kirchhoff-Love plates[edit]

The dynamic theory of thin plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes.

Governing equations[edit]

The governing equations for the dynamics of a Kirchhoff-Love plate are


   \begin{align}
     N_{\alpha\beta,\beta} & = J_1~\ddot{u}^0_\alpha \\
     M_{\alpha\beta,\alpha\beta} + q(x,t) & = J_1~\ddot{w}^0 - J_3~\ddot{w}^0_{,\alpha\alpha}
   \end{align}

where, for a plate with density \rho = \rho(x),


   J_1 := \int_{-h}^h \rho~dx_3 = 2~\rho~h ~;~~ 
   J_3 := \int_{-h}^h x_3^2~\rho~dx_3 = \frac{2}{3}~\rho~h^3

and


   \dot{u}_i = \frac{\partial u_i}{\partial t} ~;~~ \ddot{u}_i = \frac{\partial^2 u_i}{\partial t^2} ~;~~
   u_{i,\alpha} = \frac{\partial u_i}{\partial x_\alpha} ~;~~ u_{i,\alpha\beta} = \frac{\partial^2 u_i}{\partial x_\alpha \partial x_\beta}

Solutions of these equations for some special cases can be found in the article on vibrations of plates. The figures below show some vibrational modes of a circular plate.

Isotropic plates[edit]

The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected. In that case we are left with one equation of the following form (in rectangular Cartesian coordinates):


   D\,\left(\frac{\partial^4 w}{\partial x^4} + 2\frac{\partial^4 w}{\partial x^2\partial y^2} + \frac{\partial^4 w}{\partial y^4}\right) = -q(x, y, t) - 2\rho h\, \frac{\partial^2 w}{\partial t^2} \,.

where D is the bending stiffness of the plate. For a uniform plate of thickness 2h,


   D := \cfrac{2h^3E}{3(1-\nu^2)} \,.

In direct notation


   D\,\nabla^2\nabla^2 w = -q(x, y, t) - 2\rho h \, \ddot{w} \,.

For free vibrations, the governing equation becomes


   D\,\nabla^2\nabla^2 w = -2\rho h \, \ddot{w} \,.

References[edit]

  1. ^ A. E. H. Love, On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–549.
  2. ^ Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.
  3. ^ a b Timoshenko, S. and Woinowsky-Krieger, S., (1959), Theory of plates and shells, McGraw-Hill New York.

See also[edit]