Computational Mechanics

, Volume 58, Issue 3, pp 511–532 | Cite as

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

  • M. CerveraEmail author
  • N. Lafontaine
  • R. Rossi
  • M. Chiumenti
Original Paper


This paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales. A displacement sub-scale is introduced in order to stabilize the mean-stress field. Compared to the standard irreducible formulation, the proposed mixed formulation yields improved strain and stress fields. The paper investigates the effect of this enhancement on the accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P1P1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr–Coulomb and Drucker–Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.


Explicit mixed finite elements Stabilization Incompressibility Plasticity Strain softening Strain localization Mesh independence 



Financial support from the Spanish Ministry of Economy and Competitiveness under the project EACY—Enhanced accuracy computational and experimental framework for strain localization and failure mechanisms, ref. MAT2013-48624-C2-1-P, the Spanish Ministry of Foreign Affairs and Cooperation under the MAEC-AECID grants and the the European Commission under project NUMEXAS—Numerical Methods and Tools for Key Exascale Computing Challenges in Engineering and Applied Sciences, ref. FP7-ICT 611636, is gratefully acknowledged.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • M. Cervera
    • 1
    Email author
  • N. Lafontaine
    • 1
  • R. Rossi
    • 1
  • M. Chiumenti
    • 1
  1. 1.International Center for Numerical Methods in Engineering (CIMNE)Technical University of Catalonia (UPC)BarcelonaSpain

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