Mechanical Properties

Shell elements behave in two ways, either membrane or bending behavior. The Mindlin plate elements that are used by Radioss account for bending and transverse shear deformation. Hence, they can be used to model thick and thin plates.

Membrane Behavior

The membrane strain rates for Mindlin plate elements are defined as:

{\dot{e}}_{x x} = \frac{\partial v_{x}}{\partial x}

{\dot{e}}_{y y} = \frac{\partial v_{y}}{\partial y}

{\dot{e}}_{x y} = \frac{1}{2} (\frac{\partial v_{x}}{\partial y} + \frac{\partial v_{y}}{\partial x})

{\dot{e}}_{x z} = \frac{1}{2} (\frac{\partial v_{x}}{\partial z} + \frac{\partial v_{z}}{\partial x}) = \frac{1}{2} (ω_{y} + \frac{\partial v_{z}}{\partial x})

{\dot{e}}_{y z} = \frac{1}{2} (\frac{\partial v_{y}}{\partial z} + \frac{\partial v_{z}}{\partial y}) = \frac{1}{2} (- ω_{x} + \frac{\partial v_{z}}{\partial y})

Where, ${\dot{e}}_{i j}$ is the membrane strain rate.

Bending Behavior

The bending behavior in plate elements is described using the amount of curvature. The curvature rates of the Mindlin plate elements are defined as:

{\dot{χ}}_{x} = \frac{\partial ω_{y}}{\partial x}

{\dot{χ}}_{y} = - \frac{\partial ω_{x}}{\partial y}

{\dot{χ}}_{xy} = \frac{1}{2} (\frac{\partial ω_{y}}{\partial y} - \frac{\partial ω_{x}}{\partial x})

Where, ${\dot{χ}}_{i j}$ is the curvature rate.

Strain Rate Calculation

The calculation of the strain rate of an individual element is divided into two parts, membrane and bending strain rates.

Membrane Strain Rate: The vector defining the membrane strain rate is:
${\dot{e}}_{m} = {{\dot{e}}_{x^{'}} {\dot{e}}_{y^{'}} 2 {\dot{e}}_{x y}}$; This vector is computed from the velocity field vector ${v}_{m}$ and the shape function gradient ${B}_{m}$ :
${\dot{e}}_{m} = {B}_{m} {v}_{m}$; Where,
${v}_{m} = {v_{x^{'}}^{1} v_{y^{'}}^{1} v_{x^{'}}^{2} v_{y^{'}}^{2} v_{x^{'}}^{3} v_{y^{'}}^{3} v_{x^{'}}^{4} v_{y}^{4}}$
${[B]}_{m} = [\begin{matrix} \frac{\partial Φ_{1}}{\partial x} & 0 & \frac{\partial Φ_{2}}{\partial x} & 0 & \frac{\partial Φ_{3}}{\partial x} & 0 & \frac{\partial Φ_{4}}{\partial x} & 0 \\ 0 & \frac{\partial Φ_{1}}{\partial y} & 0 & \frac{\partial Φ_{2}}{\partial y} & 0 & \frac{\partial Φ_{3}}{\partial y} & 0 & \frac{\partial Φ_{4}}{\partial y} \\ \frac{\partial Φ_{1}}{\partial y} & \frac{\partial Φ_{1}}{\partial x} & \frac{\partial Φ_{2}}{\partial y} & \frac{\partial Φ_{2}}{\partial x} & \frac{\partial Φ_{3}}{\partial y} & \frac{\partial Φ_{3}}{\partial x} & \frac{\partial Φ_{4}}{\partial y} & \frac{\partial Φ_{4}}{\partial x} \end{matrix}]$
Bending Strain Rate: The vector defining the bending strain rate is:
${\dot{e}}_{b} = {{\dot{χ}}_{x^{'}} {\dot{χ}}_{y^{'}} 2 {\dot{χ}}_{x y^{'}} 2 {\dot{e}}_{z x^{'}} 2 {\dot{e}}_{y z}}$; As with the membrane strain rate, the bending strain rate is computed from the velocity field vector. However, the velocity field vector for the bending strain rate contains rotational velocities, as well as translations:
${\dot{e}}_{b} = {B}_{b} {v}_{b}$; Where,
${v}_{b} = {ω_{y}^{1}, - ω_{x}^{1}, ω_{y}^{2}, - ω_{x}^{2}, ω_{y}^{3}, - ω_{x}^{3}, ω_{y}^{4}, - ω_{x}^{4}, v_{z}^{1}, v_{z}^{2}, v_{z}^{3}, v_{z}^{4}}$
${[B]}_{b} = [\begin{matrix} \frac{\partial Φ_{1}}{\partial x} & 0 & \frac{\partial Φ_{2}}{\partial x} & 0 & \frac{\partial Φ_{3}}{\partial x} & 0 & \frac{\partial Φ_{4}}{\partial x} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\partial Φ_{1}}{\partial y} & 0 & \frac{\partial Φ_{2}}{\partial y} & 0 & \frac{\partial Φ_{3}}{\partial y} & 0 & \frac{\partial Φ_{4}}{\partial y} & 0 & 0 & 0 & 0 \\ \frac{\partial Φ_{1}}{\partial y} & \frac{\partial Φ_{1}}{\partial x} & \frac{\partial Φ_{2}}{\partial y} & \frac{\partial Φ_{2}}{\partial x} & \frac{\partial Φ_{3}}{\partial y} & \frac{\partial Φ_{3}}{\partial x} & \frac{\partial Φ_{4}}{\partial y} & \frac{\partial Φ_{4}}{\partial x} & 0 & 0 & 0 & 0 \\ Φ_{1} & 0 & Φ_{2} & 0 & Φ_{3} & 0 & Φ_{4} & 0 & \frac{\partial Φ_{1}}{\partial x} & \frac{\partial Φ_{2}}{\partial x} & \frac{\partial Φ_{3}}{\partial x} & \frac{\partial Φ_{4}}{\partial x} \\ 0 & Φ_{1} & 0 & Φ_{2} & 0 & Φ_{3} & 0 & Φ_{4} & \frac{\partial Φ_{1}}{\partial y} & \frac{\partial Φ_{2}}{\partial y} & \frac{\partial Φ_{3}}{\partial y} & \frac{\partial Φ_{4}}{\partial y} \end{matrix}]$; Figure 1. Strain Rate Calculation

Mass and Inertia

Consider a rectangular plate with sides of length

a

and

b

, surface area

A = a b

and thickness

t

, as shown in Figure 2.

Due to the lumped mass formulation used by Radioss, the lumped mass at a particular node is:

m = \frac{1}{4} ρ A t

The mass moments of inertia, with respect to local element reference frame, are calculated at node $i$ by:

I_{x x} = m (\frac{b^{2} + t^{2}}{12})

I_{y y} = m (\frac{a^{2} + t^{2}}{12})

I_{z z} = m (\frac{a^{2} + b^{2}}{12})

I_{x y} = - m \frac{a b}{16}

Inertia Stability

With the exact formula for inertia (Equation 18 to Equation 21), the solution tends to diverge for large rotation rates. Belytschko proposed a way to stabilize the solution by setting $I_{x x}$ = $I_{y y}$ , that is, to consider the rectangle as a square with respect to the inertia calculation only. This introduces an error into the formulation. However, if the aspect ratio is small the error will be minimal. In Radioss a better stabilization is obtained by:

I_{x x} = m (\frac{A}{f} + \frac{t^{2}}{12})

I_{z z} = I_{y y} = I_{x x}

I_{x y} = 0

Where, $f$ is a regulator factor with default value $f$ =12 for QBAT element and $f$ =9 for other quadrilateral elements.