Prof. ERWIN STEIN Remarks on crucial problems of inelastic deformations, and on new ideas for the calculation of damaged materials Invited Lecture at the Appreciation Dinner of COMPLAS XIV on the Occasion of the 30th Anniversary of COMPLAS September 6, 2017

Dear Eugenio, dear Roger, dear Professor Peric, dear Professor Chiumenti, Ladies and Gentlemen,

Many thanks for the kind invitation to the memorable 30^th anniversary of COMPLAS. As a member of the Founding Committee of IACM, I felt the enthusiasm which swept over all continents in the mid-eighties to realize the challenging tasks and ideas of paying tribute to the fast development of computational methods in engineering mechanics and related subjects, which –of course– was strongly influenced by the incredibly fast growth of computer hard- and software.

The first COMPLAS Conference was organized in 1987 by Professor Eugenio Oñate and Professor Roger Owen. At this time COMPLAS was not yet related to IACM; this followed in the 1990’s and holds true until today. It has to be pointed out that COMPLAS is also a thematic conference of ECCOMAS.

The worldwide successful and active IACM could not be structured and guided from the top alone; such a widespread association must be mentally and practically supported from the basis by continuous motivation and support of the national associations. In this sense, IACM strongly encourages and promotes many periodical thematic conferences, of which COMPLAS was among the firsts. I would also like to mention my own continuous engagement in the special interest conferences “COMPLAS” and “Adaptive Modeling and Simulation”.

The COMPLAS conferences had and still have a strong mental and practical influence on the development of physical and mathematical modeling, especially at microscales, including experimental observations, and of course on appropriate mathematical modeling and ,of course, analytical and numerical solution strategies with direct variational approximation methods. It has to be pointed out that verification and validation are of equal importance, concerning solution and model adaptivity, based on proper error analysis and mathematically filtered experimental data at macro- and microscales with parameter identification.

Not to forget: COMPLAS also encouraged industrial cooperation and applications together with academic research.

It also should be remarked that COMPLAS is an important reference conference for all types of plastic forming with all their different aspects, and not to forget the important contributions to biomechanics.

Let me give a short review on the last decades:
In the last third of the 20th century, many research projects and computer programs for variational discretization methods were realized with mesh-based discretizations.
Many research articles were published on numerical stability with respect to mathematical, structural and material effects. Furthermore, fast and stable solvers and related computer programs for solutions of large dimensional algebraic systems had to be developed continuously in order to meet industrial expectations, and to use improved computer hardware.

In the second main period of Finite-Element-developments for inelastic deformations, the important problems of verification and validation came up, i.e. a priori and a posteriori bounded error analysis in space and time, because this concerns the main requirements of reliability, accuracy and efficiency of related for numerical approximation methods in computational mechanics.
With respect to the choice of direct variational discretizations, next to mesh-based ansatz- and test polynomials, also mesh-free methods (and more general various concepts for particle methods) became important because of their physical advantages, despite the fact that they additionally require the control of numerical integrations and some other nontrivial algorithmic and programmatic efforts, like the weak approximation of Dirichlet boundary conditions.
In this respect, also the Isogeometric Analysis has to be adduced. It opened an important access to geometrically consistent and efficient numerical methods, avoiding geometrical stiffening or instability effects, for instance in shell-buckling.

Another complicated present topic in engineering is the theory and numerical analysis of uncertain material and structural data and probabilistic loads, especially for the prediction of damage evolution, such as arising micro-voids and micro- cracks in brittle materials or micro-cracks with localizations in inelastic materials, leading to damage, fatigue and failure, strongly depending from the histories of static and dynamic loadings. This holds especially for complicated structures built from several high-strength materials and composites and designed for specified lifetimes, such as modern wide-spanned bridges.

Main challenges of today are: experimental observation and mathematical modeling of materials, especially the numerical computation of damage evolution at the microscale, including identification of internal variables. This is necessary for various types of static and dynamic loadings with the appearance of first micro- voids and micro-cracks and the transition to macro-cracks, according to the macroscopically detected damage, fatigue, and failure.
We have to distinguish between various types of essentially brittle and ductile materials with their specific microstructures, like mono-crystals, poly-crystals, and further mainly brittle composites with amorphous structure, especially steel- or carbon-fiber reinforced concrete, as well as glass- or carbon-fiber reinforced materials.

All the so far developed damage models are mathematically based on the Riemannian manifold and metric with differentiable displacements of C1- Boltzmann-continua, i.e. the continuous-differentiable point-continuum. But damage at microscales mostly causes local dissolving of connections at a crucial scale, e.g. as voids and micro-cracks in various directions. The most common geometrical approximation for damage is still the volumetric Kachanov-model. But it looks to be disadvantageous and not reasonable to map discontinuously deformed structural subdomains due to damage and fatigue back to the continuous C¹-metric of the point-continuum, both from physical and mathematical points of view.

Even generalized Riemannian manifolds and related metrics, like the Finsler- manifold, -e.g. with distributions instead of classical derivatives- cannot be used for discontinuous subdomains, mainly for micro-voids and discontinuous, structured micro-crack patterns. An important reason is the existence of stress singularities at the boundaries of voids and cracks.
Therefore, I propose to define damaged macro-elements in time and space, which I call damage chips, in the first step as mesh-based finite element discretizations. The damaged material has patterns of voids or/and cracks with evolution equations according to experiments and thermodynamic; they can be pre-calculated for triangular, rectangular, cubic, or other macro-elements. The reverse motivation of this methodology comes from homogenization processes. The transition of kinematical assembling to the non-damaged macro-elements has to be performed with nodal kinematical quantities. However, generalizations, for instance with mixed variational methods, are possible.

Up to thousands of those pre-calculated damage chips might be necessary for all reasonable damaged microstructures. In total, physically realistic numerical calculations of micro-damaged deformation processes will be possible by this staggered two-step modelling and calculation, and it will be computationally efficient.

By this combination of regularly elasto-plastic deformations in non-damaged subdomains of a structure with discontinuously deforming “damage chips” it will be possible from an engineering point of view to analyze micro-damage and failure processes via their true topologies and metrics in order to get reliable results for large scale structures in a computationally efficient way.

Ladies and gentlemen, this closes my remarks on crucial problems of inelastic deformations, and on new ideas for the calculation of damaged materials.

This night, we celebrate the merits and achievements of COMPLAS and especially of its eminent organizers.
Dear Eugenio, dear Roger, once more I cordially thank you for thirty years of stimulating and creative organization of COMPLAS conferences with excellent service and a special friendly and familiar atmosphere. You are, so to say, the absolute cream of the crop of IACM, and therefore, we sing together:

For they are jolly good fellows,
For they are jolly good fellows,
For they are jolly good fellows,
and so say all of us!

Ladies and gentlemen, thank you very much for your attention!