Nonlinear Mechanics of Thin-Walled Structures

As climate change is one of the big challenges of the 21st century and beyond, energy efficiency in every regard is desired and sought for. One of the contributions, that we as scientists and engineers can make, is designing and building modern engineering structures and machines, which make use of thin-walled members. In contrast to conventional massive parts, properly designed thin-walled structures perform the same tasks and carry the same mechanical loads at significantly reduced mass. Hence, it is both cost-effective and resource-conserving to strive for an extension of the field of application of thin-walled structures.

Cross section of an airplane wing

This proves challenging in many situations, since thin-walled structures are liable to display a complex mechanical behavior (buckling, geometric / physical nonlinearity). In this respect, the Research Group Mechanics of Solids focusses on the development of novel analytical and numerical techniques to analyze the mechanics of thin-walled structures subject to static or dynamic loading conditions.

Though the recent increase of computing capacity makes it tempting to approach the full 3D continuum problem, it is necessary to limit the computational effort without sacrificing accuracy in certain situations (very thin structures, real-time requirement). The theories of structural mechanics, deduced with asymptotic methods or via application of certain principles of Lagrangian mechanics, achieve a dimensional reduction to deformable 1D or 2D continua. These structural models of rods, plates or shells are applicable to a wide variety of purely academic as well as application-oriented problems.

Three-dimensional rods

A generalization of the well-known theory of bending of beams to arbitrarily large deformations of elastic rods with initial curvature requires a solid theoretical foundation, based on the principle of virtual work and tensor calculus. Coupling between bending and torsion as well as absence of shear or inextensibility demand a specific treatment.

Additional effects like warping of cross-sections and reduction of the torsional stiffness at high tension may need to be accounted for in certain practically relevant applications; a prominent example being the analysis of the lateral run-off in a belt drive with geometric imperfections.

xyz coordinate system with two curves
Bending of a sagging belt as a simulation result

Plates and Shells

Structural elements, which are thin in one dimension compared to other two, may be considered using plate or shell models. Being initially curved, a shell structure is often beneficial for practical needs because it responds to external loads by in-plane membrane stresses in addition to bending moments, which essentially increases the load-carrying capacity.

Two-dimensional theories of thin-walled structures as material surfaces comprise different models of planar plates and shells. Being initially curved, a shell structure is often beneficial for practical needs because it responds to external loads by in-plane membrane stresses in addition to bending moments, which essentially increases the load-carrying capacity. At the same time,  shells tend to demonstrate essentially nonlinear behaviour already at small deformations, when deflections are of the order of smallness of the thickness of the shell. This makes the behaviour less intuitive and increases the importance of mathematical modelling.

Within our Research Group, analytical and numerical methods are developed and implemented to account for particular sources of nonlinearity and certain physical phenomena. In this regard, we focus on unique application-oriented problems, which cannot be tackled with commercial software. Novel solution algorithms are implemented in form of problem-specific in-house simulation tools. This enables easy adaptation, further extension and code re-use. Nevertheless, commercial software and physical experiments are used for validation purposes.

three half tubes cut in length as simulation result

Plasticity in Structural Mechanics

Irreversible strains because of inelastic material behaviour make the structural response essentially nonlinear in industrial processes like  sheet metal roll forming, which is currently investigated by the Research Group of Mechanics of Solids. The incremental bending of the initially flat sheet is carried out by consecutively arranged pairs of rolls, which are at a spatially fixed position and in contact with the axially moving plate. During this process irreversible strains are accumulated, as the plate is plastically deformed to obtain the shape of the final product.

Conventional approaches based on continuum models are not applicable for designing the roll forming process and developing model-based control algorithms due to excessive computational cost. We aim at resolving this issue by using a nonlinear shell model for the sheet metal in the framework of the Eulerian-Lagrangian kinematic description. In an effort to further increase simulation speed, we treat plasticity at the structural level and, thus, avoid the inefficient through-the-thickness integration of the evolution laws of inelastic variables (plastic strains, hardening parameters). This requires the yield surface and the constitutive laws to be expressed in terms of stress resultants of the shell model. Using special case solutions of three-dimensional equations as a reference, we strive for an efficient and accurate formulation, which would allow the stress-resultant plasticity to make its way into practice.

five different rolls on one roll station

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two times two rolls, which stand against each other on a flat surface

Publications

Schmidrathner, Christian, Yury Vetyukov, and Jakob Scheidl. "Non‐material finite element rod model for the lateral run‐off in a two‐pulley belt drive, opens an external URL in a new window." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 102, no. 1 (2022): e202100135.

Eliseev, Vladimir V., and Yury M. Vetyukov. "Finite deformation of thin shells in the context of analytical mechanics of material surfaces, opens an external URL in a new window." Acta Mechanica 209, no. 1 (2010): 43-57.

Scheidl, Jakob, Yury Vetyukov, Christian Schmidrathner, Klemens Schulmeister, and Michael Proschek. "Mixed Eulerian–Lagrangian shell model for lateral run-off in a steel belt drive and its experimental validation, opens an external URL in a new window." International Journal of Mechanical Sciences 204 (2021): 106572.

Kocbay, Emin, and Yury Vetyukov. "Stress resultant plasticity for plate bending in the context of roll forming of sheet metal, opens an external URL in a new window." International Journal for Numerical Methods in Engineering 122, no. 18 (2021): 5144-5168.

Literature Recommendation

Text Nonlinear Mechanics of Thin-Walled Structures
various mathematical formulas and equations

Contact

Univ.Prof. Mag. Dr. Yury Vetyukov

Send email to Yury Vetyukov