Large elastic-plastic deformation of square membranes subjected to localised pulse pressure loads


  • N. Mehreganian Imperial College London, Exhibition Road, London SW7 2AZ UK
  • A. S. Fallah Brunel University, Kingston Lane, London, Uxbridge, UB8 3PH UK
  • L. A. Louca Imperial College London, Exhibition Road, London SW7 2AZ UK



Ductile isotropic materials are widely used in protective systems against transient pulse pressure loads, such as those of localised blasts. This is due to the combined elastic-plastic response which contributes to dissipation of total impulse from extensive loading as the energy stored elastically limits deformation while the energy expended plastically limits the level of transferred forces in the structure. In the case of thin, modern armour graded steel plates, the tailored metallurgy helps the structure store energy within the bounds of elastic region, which may be dissipated at a later stage as damping kills it off in subsequent cycles. On the other hand, the plastic work is almost entirely converted to heat and dissipates.

The present work focuses on the elastic and plastic energies in the membrane and aims at deducing, from the minimization of Föppl-Von-Kármán (FVK) energy functional combined with enforcing the constitutive relations of limit analysis, the dynamic elastic-plastic response of localised blast loaded square membranes undergoing large deformations. The presumed blast load function is a multiplicative decomposition of a prescribed continuous piecewise smooth spatial function and an arbitrary temporal function which may assume various temporal shapes (e.g. rectangular, linear, exponential).

Considering the elastic response, a single-degree-of-freedom model was developed from the prescribed displacement field and associated stress tensor having clamped and simply supported boundary conditions. The explicit closed form solutions were sought by using the Ritz-Galerkin’s variational method as well as the Poincaré-Lindstedt perturbation method. The theoretical solutions of rigid-perfectly plastic square membranes subjected to the same blast scenarios were then discussed. From the combined effects we deduce the load displacement curves representing the trajectory of the nonlinear elastic-perfectly plastic structure.

Key words: localised blast, square membrane, Ritz-Galerkin method.

Pages of the article in the issue: 126-133

Language of the article: English


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How to Cite

Mehreganian, N., Fallah, A. S., & Louca, L. A. (2019). Large elastic-plastic deformation of square membranes subjected to localised pulse pressure loads. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, (1), 126–133.



Differential equations, mathematical physics and mechanics