tearing, reinforcement and damage accumulation

Computer model reveals effective reinforcement strategies for fabric


Clothes are primarily designed to shield us from extreme environments. But we rarely see a knight's armor in boutiques - if we can find one at all. There is clearly a trade-off between rigidity and comfort in garments design and in our own personal choice of outfit - clothes can never be too tight and stiff lest it should compromise user's comfort. Striking a balance between these two takes its toll on fabric, the main clothing material.

Softer, lighter fabric, say satin, will definitely tear faster when subjected to shearing forces compared to tougher ones, like jeans. The more one goes to the comfort end of the spectrum, the more this becomes true, highlighting the need for structural reinforcement procedures that prolong the lifetime of clothing without sacrificing comfort.

This is perhaps the motivation behind the research endeavor conducted by George Allan P. Esleta and Christopher P. Monterola [Complex Systems Researchers]. Their results, published in the journal Computer Physics Communications [1], prove useful in two aspects: (1) from a computational point of view, it provides an elegant model of tearing and cracking; and (2) it has a practical aspect to it, particularly its direct application to fabric.

Physical model

Stress propagation across a square grid network of interconnected springs is simply no trivial matter (at least for the introductory physics student) albeit the simplicity and linearity of Hooke's law: the force F on a spring is proportional to the amount of displacement x it has elongated relative to its unstretched length, F = -kx, with the spring constant k measuring the stiffness of the spring. For a network of springs, one has to consider a set of different stiffness values k. And as real springs have limitations as to the extent up to which they can be stretched, a model of such network must also incorporate a set of breaking lengths xmax, proportional to the spring constants.

Such model, while rather computationally challenging, is best for modeling fabric: since springs have k values, the interconnected springs naturally incorporates stiffness found in real fabric. The breaking lengths xmax, on the other hand, is a relative measure of stability. In a way, the researchers found a way to computationally model the trade-off discussed earlier. Think of satin, as previously described, as having low k translating into its being comfortable, while correspondingly having low xmax value resulting into its being easily torn apart. In the same manner, jeans will be modeled as having a high k for its being stiff and high xmax for its being durable.

Esleta, the main proponent of the paper, solved the problem numerically, using different force profiles along the sides of the fabric and counting for the number of spring breakings (i.e. fabric tearings). Upon knowing the critical locations where the network is expected to tear apart, reinforcement strategies can be developed, stiffening springs on a particular region subject to a given spatial distribution.

Strategies: Random vs. targeted vs. distributed

Randomly choosing springs to strengthen do not resolve the problem. Of course, targeted reinforcement (i.e. finding the site to collapse and strengthening it) is best, but it is tedious and nearly impractical in reality. For fabric, this means actual testing where actual points on the weave are weakest, an impossible job for a yard of fabric, much more a skirt or gown.

The remedy is still to have reinforcement distributed over an area. Of the distributions tested, the centered Gaussian area distribution of reinforcement proved efficient, followed by the doughnut.

But why not uniform distribution? Again, in the context of fabric, this means neglecting the balance and tipping the scales to the strength side at the expense of comfort, a scenario not really advantageous for the user.

What doesn't kill you will make you stronger?

In addition to these published results, the group presented a conference paper on the effect of accumulated stress on the system [2].

The effect of accumulated stress on the system is described by an empirical law attributed to Basquin: the plot of the lifetime of the material versus the applied force obey's a power-law relation with a negative exponent roughly equal to 2.

Esleta and Monterola’s result seem to support the dictum advocated by these earlier authors: What doesn’t kill you will make you weaker. Yes, they are not as optimistic as Nietzsche, for whom the words in the title of this section are credited.

Not really into it

The irony here is not as much in the results as it is in the authors.

Esleta claims to have just ‘ended up solving’ the problem, it being assigned earlier to an undergraduate student he co-advises. He still refers to the liquid-solid transition in granular media as his ‘original work’ in complexity science.

Monterola, on the other hand, is a neural networks expert, and one active complex systems researcher interested in a whole wide range of natural and social phenomena. The work with Esleta is one of his earliest publications upon return from post-doctoral work.

As with the trade-off between comfort and durability, Esleta adds another facet: that of style. In and outside the lab, prior to and after this research, this seems to be the aspect of clothing he’s more into. ■

1. Esleta, G.A.P. and Monterola, C.P. (2008). Structural reinforcement in a spring-block model of stress-induced fracture propagation. Computer Physics Communications, doi:10.1016/j.cpc.2007.12.003.
2. Esleta, G.A.P. and Monterola, C.P. (2008). Proceedings of the 25th Samahang Pisika ng Pilipinas Congress.