Keywords: dynamic, far-field, acoustic, interaction, NLFEA, concrete, structures.

Unstructured FE continuum modelling of reinforced and pre-stressed concrete engineering facilities has for some time been the preferred analytical method when simulating the deformation response of complex structural geometries under extreme loading. While non-linearity of the material behaviour is the result of fracture growth, pore collapse and sliding (all discontinuous, local processes), an equivalent continuum approach is often adopted. This rather uncomfortable idealisation is a consequence of the current limits of computational power, which prevents a detailed local tracing of the evolution of each crack in civil-engineering-scale structures.

Despite much time having been invested worldwide on the development of a generalised constitutive model for this heterogeneous duplex material (concrete), we remain quite some way from realising a robust, accurate model. Of the more advanced models that do exist, the issues of stability, uniqueness, and physical realism continue to provide significant challenges to the computational mechanics researcher. Outside of constitutive modelling, another area of engineering mechanics where further work is required is that of providing efficient solutions to modelling large coupled dynamic systems where far-field (infinite) boundaries exist. As ever, the problem is one of finding accurate methods that do not impose an excessive computational burden.

This paper presents a personal view of some recent tools advanced in both areas (constitutive modelling and the far-field treatment), together with a discussion on some structural integrity indicators linked to physical and numerical instability. The text is split into 4 sections. First, the constitutive modelling is addressed. Here, a description of a recently developed isothermal inviscid frameworks is given [1,2]. The need for regularisation is discussed and a viscous extension to the elasto-plasticity model sketched. At the end of section 4, some remarks on material complexity are made. Addressing the need for new laboratory data, some results from multi-axial compression tests conducted at elevated temperature are presented, together with examples of uniaxial tension tests designed to reveal the material characteristic length.

In section 3, the discrete FE matrix form of the incremental non-linear dynamic equation of equilibrium is presented. Here an implicit approach is adopted, which is solved using a Newton-Raphson scheme with an element-by-element (GMRES) stabilised Bi-Conjugate Gradient non-symmetric iterative solver [3]. The FE system includes terms coupling an inviscid compressible fluid to the deformable solid [4]. The background to the technique simulating the dynamic far-field is explained and its introduction into a general FE code described [5]. Section 4 of this paper makes use of a hierarchical approach to assessing FE instability [6].

Within sections 3 and 4, there are FE examples illustrating the issues described above.

1

B. Tahar, "C₂ continuous hardening/softening elasto-plasticity model for concrete", PhD Thesis, University of Sheffield, 2000.

2

G. Etse and K.J. Willam, Fracture energy formulation for inelastic behaviour of plain concrete, ASCE, "J Eng Mech", 120(9), 1983-2011, 1994. doi:10.1061/(ASCE)0733-9399(1994)120:9(1983)

3

I.M. Smith, "General purpose parallel finite element programming", 7th ACME conference, Durham, 1999.

4

H.S. Wu, "3D non-linear dynamic fluid-structure interaction analysis of reinforced concrete structures", PhD Thesis, Department of Civil & Structural Engineering, University of Sheffield, UK, 2000.

5

J.P. Wolf and C. Song, "Finite-Element modelling of unbounded media", John Wiley, 1996.

6

N. Bicanic, R. de Borst, W. Gerstle, D.W. Murray, G. Pijaudier-Cabot, V. Saouma, K.J. Willam and J. Yamazaki, Computational Aspects of Structures (Chapter 7), "Finite element analysis of reinforced concrete structures II", Proceedings of the International Workshop, J. Isenberg (Ed), ASCE, 1991.

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