VFM

Pressure Reconstruction

The Virtual fields method can be applied to solve a broad range of problems in solid mechanics [Pierron and Grédiac [2012]]. This python package is limited to the particular case of load reconstruction during fast transient dynamics. The procedure for surface pressure reconstruction is based on the work of Pierron and Grédiac [2012] and Kaufmann et al. [2019].

The dynamic equilibrium equations for a thin plate can be written written on weak form using the principal of virtual work as [Dym and Shames, 1973]:

\begin{equation} \underbrace{\int\limits_{V} \rho ~ \boldsymbol{a} ~ \boldsymbol{u^*} dV}_{W_{inertial}^*} ~=~ \underbrace{ -\int\limits_{V} \boldsymbol{\sigma} : \boldsymbol{\varepsilon ^*} dV}_{W_{int}^*} + \underbrace{ \int\limits_{S} \overline{\boldsymbol{T}} \boldsymbol{u^*} ~dS + \int\limits_{V} \rho ~ \boldsymbol{F_{Vol}} ~ \boldsymbol{u^*} ~dV}_{W_{ext}^*} ~~~, \end{equation}

where \(W_{inertial}^*\), \(W_{int}^*\) and \(W_{ext}^*\) denotes the inertial virtual work, the internal virtual work and the external virtual work, respectively.

For the particular case of a thin plate represented by an isotropic linear elastic material, the principal of virtual work can be written using the Kirchoff-Love theory as:

\begin{equation} \begin{aligned} \int\limits_{S} p w^{*} dS ~ = ~& D_{xx} \int\limits_{S} \left( \kappa _{xx} \kappa ^* _{xx} +\kappa _{yy} \kappa ^* _{yy} + 2 \kappa _{xy} \kappa ^* _{xy} \right) dS \\ +~&D_{xy} \int\limits_{S} \left( \kappa _{xx} \kappa ^* _{yy} +\kappa _{yy} \kappa ^* _{xx} -2 \kappa _{xy} \kappa ^* _{xy} \right) dS +~\rho ~ t_S \int\limits_{S} a ~w^{*} dS ~~~. \end{aligned} \end{equation}

where \(S\) is the surface of the plate, \(p\) is the pressure acting on the surface of the plate. The deformation of the plate is given by the curvatures \(\kappa\) and the acceleration \(a\). The density of the plate material is denoted \(\rho\), the plate thickness is denoted \(t_S\), and \(D_{xx}\) and \(D_{xy}\) are the plate bending stiffness matrix components. Virtual quantities are marked with \(^*\).

As local pressure values are of interest, the surface is divided into subdomains. By assuming a constant pressure distribution within each subdomain, the integrals in the above equation is reformulated as discrete sums and the pressure \(p\) is solved for:

\begin{equation} \begin{aligned} p ~ = ~ \Biggl( ~&D_{xx} \sum\limits_{i = 1} ^{N} \kappa ^{i} _{xx} \kappa ^{*i} _{xx} +\kappa _{yy}^{i} \kappa ^{*i} _{yy} +2\kappa ^{i}_{xy} \kappa ^{*i} _{xy} \\ +~&D_{xy} \sum\limits_{i = 1} ^{N} \kappa ^{i}_{xx} \kappa ^{*i} _{yy} +\kappa ^{i}_{yy} \kappa ^{*i} _{xx} -2\kappa ^{i}_{xy} \kappa ^{*i} _{xy} +~\rho ~ t_S \sum\limits_{i = 1} ^{N} a^{i} ~w^{*i} \Biggr) ~ \left( \sum\limits_{i = 1} ^{N} w^{*i} \right) ^{-1} ~~~, \end{aligned} \end{equation}

where \(N\) is number of discrete surface elements.

The virtual fields based on 4-node Hermite 16 element shape functions Zienkiewicz [1977] are available for pressure reconstruction, see Pierron and Grédiac [2012] for more details.

Bibliography

1

C.L. Dym and I.H. Shames. Solid mechanics: a variational approach. Advanced engineering series. McGraw-Hill, 1973.

2

R. Kaufmann, B. Ganapathisubramani, and F. Pierron. Full-Field Surface Pressure Reconstruction Using the Virtual Fields Method. Experimental Mechanics, 59(8):1203–1221, 2019. doi:10.1007/s11340-019-00530-2.

3(1,2,3)

F. Pierron and M. Grédiac. The virtual fields method. Extracting constitutive mechanical parameters from full-field deformation measurements. Springer New-York, 2012. ISBN 978-1-4614-1824-5.

4

O.C. Zienkiewicz. The finite element method. McGraw-Hill, 1977. ISBN 9780070840720.