Helical Structures

Structural performance in closed-form, analytical expressions


Fig. 1: Helix geometry and inner local forces (left) and effective 3D normal and radial structural response (right). 

Helical arrangements are encountered in a wide range of structures either natural or artificial. Tendons in our bodies are a typical example of naturally formed multiscale composite helical structures.  Helical formations have been used from the very early civilizations to transfer loads through ropes, cables or more recently to transfer electricity in power conductors.

A thorough analysis of the force-deformation relation of axially, torsionally and radially loaded helical structures can be found in [1]. What is more, the structural effect of thermal changes is quantified in [5]. In each case, simple formulas are provided in the form of parametric analytical expressions that relate loading and strains; formulas appropriate for use in everyday praxis.

Numerical Simulations

The computational modeling of helical structures commonly comes along with considerable numerical analysis costs, for their geometry to be adequately described. This can result in simulation limitations. Low order, two-dimensional finite element models allow for the computation of the mechanical behavior of helical constructions under axial, torsional, as well as radial and thermal loads in a rigorous manner, exploiting the geometric symmetry of the helical arrangement [3, 4]. The incorporation of symmetry substantially reduces the numerical cost, retaining important information on the local stress distribution (Fig. 2), quantities that cannot be captured by analytical or 1D formulations.

What is more, numerical simulations can provide insights in the effect of kinematic constraints that are hard to carry out experimentally, but whose effects are well known in the long term behavior of structures, such as cables [2].

Fig. 2: Normal stress profile of a simple strand with six helical  bodies subject to traction loads.

Optimal structural design: Torsionally equilibrated structural patterns

Multilayer helical assemblies as the ones commonly encountered in cable structures and electricity power conductors are commonly not self-equilibrated to torsion. The inherent creation of torsional loads induces rotational motions that increase wear and reduce the life expectancy of the structures. There are however braiding patterns that minimize the structure’s torsional propensity, and maximize its axial resistance (Fig. 3). A database of such patterns for assemblies of up to five layers is provided in [6], providing a design basis that covers a wide range of practical applications (cables, electricity power conductors).

Fig. 3: Axially loaded multilayer helical assemblies (left) with a structural pattern (right) that minimizes torsion.


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Advanced Metastructures

Architected beam structures: tuning deflections through inner couplings

Beam structures made of metamaterial that exhibit inner normal to shear strain couplings can balance bending loads through tension. Tetrachiral unit-cell designs are representative lattice patterns with an inherent coupled normal to shear strain behavior. Beam structures with an inner bending to normal loading coupling can deflect when axially loaded. The normal force required to equilibrate the effect of bending loads on beam structures can be analytically computed, providing relevant closed-form parametric expressions. The validity of the analytical formulas, is both numerically and experimentally verified for the case of cantilever beams. Results indicate that the coupling of normal and shear deformations can be used as a primal load-balancing mechanism, providing new possibilities in the control of the artificial structure’s kinematics and overall mechanics


Fig: Microarchitected beam structure with an inner tetrachiral lattice pattern exhibiting coupled axial and bending loading performance

Mechanically joined Aluminum and Steel Sheet Structures: Self-Piercing-Riveting Feasibility and strength

The effect of different rivet and die design parameters on the feasibility and quality of the self-piercing riveting (SPR) joining of metallic sheet stacks has been investigated (see Publication [31]). In particular, the riveting of similar aluminum-to‑aluminum sheet stacks (AA2019/AA7075-F) and dissimilar aluminum-to-steel sheet stacks (AA7075-F/BA0270) is analyzed, both experimentally and numerically. Towards this, the different rivet and sheet materials are experimentally characterized to obtain their elastoplastic and fracture attributes. Thereafter, SPR process finite element models are developed. Initially, the joint feasibility is experimentally probed for specific available rivet and die configurations, thus identifying SPR process parameters that allow for successful joining. Thereafter, a wider range of SPR process parameters is investigated numerically. It is observed that a combination of successful joint formation and high interlock values is obtained over specific ranges of rivet leg thickness and die depth values. Moreover, low rivet leg thickness values and near-unity normalized die depths – with the sum of the die depth and total sheet thickness normalized with respect to the rivet leg length – yield high quality SPR joints for both types of stacks considered. Further, it is concluded that the mean von Mises stress induced in the sheets after the SPR process is primarily affected by the selection of the die depth, with the induced stress being less sensitive to other design parameters, such as the rivet leg thickness or die shape for both joining cases.

Self-Piercing-Joining of aluminum-to-steel and steel-to-steel sheets as a function of different rivet and sheet thickness joining parameters

The mechanical failure of self-piercing rivet (SPR) joints connecting seven series aluminum and high strength steel sheets is investigated, both numerically and experimentally (see Publication [33]). The joint strength and failure mechanisms are characterized for a total of four distinct loading modes, including the lap-shear, cross-tension, inclined cross tension and coach peel. For the analysis of the underlying influential parameters in each loading case, joint designs with equal total sheet thickness and equal rivet head diameters are considered. The highest strength values are obtained in lap-shear loading for all joint types, while high rivet interlock joints are observed to pair with increased cross-tension strength values. Moreover, the loading mode in which the highest energy is absorbed directly relates to the joint type. Depending on the material combination to be joined, either the cross-tension or the coach-peel cases yielded the highest energy absorption. The experimental results indicate that high rivet hardness and bottom sheet strength values have a favorable impact on the coach-peel strength and the associated deformation response. For all failure modes, the lowest rivet hardness employed (H4) was proven sufficient to prevent rivet failure in the joint types employed, despite the substantial equivalent plastic strains developed in it. Furthermore, high interlocks were noted to primarily affect the lap shear failure mode, inducing significant bottom sheet damage upon fracture, with the failure response observed in all other loading cases to remain practically insensitive to the interlock magnitude.

Deformation and stress distribution profile of the half, 3D finite element SPR model upon cross-tension (a) and coach peel loading (b). The equivalent plastic strain distribution (PEEQ) upon coach peel and cross tension loading is provided in (c) and (d) respectively for all joint types. The maximum PEEQ value recorded is joint and loading-type specific.



Tailored effective static properties

        Architectured materials can be coined to exhibit static properties that well differ from the ones typically encountered in common engineering materials. Certain unit-cell arrangements of high material anisotropy can be used to tune the wave propagation attributes of the effective material in a systematic manne. In [8, B1], we demonstrate how the combination of high anisotropy and high element slenderness can be empoyed as mechanisms for wave propagation isolation characteristics to arise in a systematic manner along the weak material modulus direction of artificial materials (Fig. 4).


Fig. 4: The unit-cell anisotropy level constitutes a metric for both the static and the wave propagation attributes of the artificial material.



The inner material design allows for effective static properties properties that cannot be achieved by common engineering materials. Shear soft and shear still inner material architectures along with high and low volumetric resistance designs are analyzed in [13]. The yield limits of different metamaterial designs are summarized in [20], providing direct insights in the axial and shear yield performance of architectured material designs as a function of their relative density and unit-cell design.

Fig. 6: Metamaterial designs of low high and moderate bulk and shear laoding resistance

Large strain effects and wave propagation tuning

Large strains affect the wave propagation attributes af artificial materials. Their nonlinear dispersion characteristics can well differ from the linear ones for certain deformation modes. In [12], we demonstrate that the dispersion characteristics of periodic structures are affected by their nonlinear inner element kinematics, particularly for their lower modes. The unit-cell design and the amplitude and wavenumber of the propagating waves play a significant role in the magnitude of the nonlinear corrections.

Large strains can considerably modify the static properties of artificial materials, depending on the loading direction of the applied strain and the initial degree of the materials’ anisotropy. In [10], we show that for highly anisotropic material architectures, instabilities can arise at rather low strain magnitudes upon the application of normal strains that diminish the weak modulus direction. The latter can be used as wave propagation isolation mechanisms. What is more, we demonstrate that the non-reciprocity of the mechanical response of highly anisotropic artificial materials can be used to enhance the tunability of propagating longitudinal and shear waves. Moreover, we provide evidence that such a tuning requires the application of normal rather than shear strains, as volumetric changes are required for effects of the kind to be observed.


Fig. 8: The level of initial material anisotropy (left) dictates not only its large strain static behavior (up right), but also its wave propagation attributes so that it acts as

Menger Sponge-Like Structures

  • To appear

Multiscale Metamaterial designs

  • To appear



Machine Learning in Advanced Structural Design

Neural network modeling of the stress-strain response of polymers made through lithography

Photopolymerization is the governing chemical mechanism in two-photon lithography, a multi-step additive manufacturing process (see Publication [32]).  Negative-tone photoresist materials are widely used in this process, enabling the fabrication of structures with nano- and micro-sized features. The present work establishes the relationship among the process parameters, the degree of polymerization, and the nonlinear stress-strain response of polymer structures obtained through two-photon polymerization. Honeycomb structures are fabricated on a direct laser writing system (Nanoscribe) making use of different laser powers for two widely applicable, commercially available resins (IP-S and IP-Dip). The structures are then tested under uniaxial compression to obtain the corresponding stress-strain curves up to 30% strain. Raman spectroscopy is used to correlate the degree of conversion achieved upon different laser exposures of the base photoresist material with the selected mechanical properties (Young’s modulus, tangent modulus, deformation resistance) after polymerization. Significant differences are recorded in the observed constitutive responses. Higher degrees of conversion result in higher elastic moduli and strength at large strains. Moreover, it is found that the IP-Dip resin yields higher degrees of conversion for the same laser power compared to the IP-S resin. A neural network model is developed for each resin that predicts the stress-strain response as a function of the degree of conversion. For each material, an analytical form of the identified constitutive response is provided, furnishing basic formulas for engineering practice.