Helical Constructions

Structural behavior in simple, analytical formulas


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.


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 correction.


Fig. 7: Dispersion diagram of hexagonal (left) and re-entrant hexagonal periodic structures accounting for the effect of their higher order inner material kinematics

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

Non-centrosymmetric and non-classical inner unit-cell material designs

To appear! Under construction!

Biomechanical artificial design

Inferring the inner structural composition of tendons

Tendons are biomechanical constructions with a multiscale inner structure (Fig. 7). Their restoration requires the design and use of biomimetic artificial materials that can emulate the experimentally observed tendon mechanical behavior.


Fig. 9: Tendon multiscale inner structure

In [7], biocompatible inner structural designs are obtained for the tendon fascicle scale. In particular,  dedicated fascicle finite element models are coupled to a Bayesian uncertainty quantification framework in order to infer optimal geometric and elastic material parameters for the design of artificial tendons. The method is extended in [14], to analyse the viscosity of the embedding matrix substance, providing for the first time estimates for the matrix relaxation contribution at the inner scales. The viscosity of tendon fibers is analysed in [16], while aging effects are addressed in [17].


Layered Higher Gradient Mechanics

In [22] we develop a higher gradient dynamic homogenization method with micro-inertia effects. To that scope, we compute the macroscopic constitutive parameters up to the second gradient, using two distinct approaches, namely the Hamilton’s principle and the total internal energy formulation. Thereupon, we analyze the sensitivity of the second gradient constitutive terms on the inner material and geometric parameters for the case of composite materials with a periodic, layered microstructure. The results suggest that the significance of the second gradient terms highly depends on the differences in the geometric and material properties of the underlying microstructural phases, as well as on the deformation mode of interest. What is more, the wave propagation study indicates that different higher gradient results are obtained, depending on the formulation used and on the wavenumber range of interest. In particular, Hamilton based, second gradient models perform poorly in the low wavenumber range, compared to first gradient or to analytic, exact solutions. Contrariwise, second gradient macroscopic constitutive formulations derived using the total higher gradient internal energy outperform first gradient approaches, better approximating the wave propagation characteristics of the effective medium, as the corresponding comparison in between the frequency diagrams and phase and group velocities suggest.

Fig. 10: Second gradient, energy-based homogenization of heterogeneous materials