# Conveniently improving the mechanical properties of fibrous and discrete materials

A significant number of materials made by nature are fibre networks. Nature and evolution have designed them as fibre structures because they are light-weight, strong and relatively easy to construct. Based on the work of nature, numerous man-made materials also consist of fibres and other discrete constituents. Some examples are electronic textiles, cardboard, fibre-reinforced composites, electrospun scaffolds for tissue engineering and metal foams.

## Saving industry’s time and money

Manufacturers of products (containing fibrous materials) desire to design and optimise their products and materials such that certain mechanical specifications are met, e.g. ultimate load, failure strength. The design and optimisation of products can in general follow two routes: a physical one and a virtual one. The physical route starts with the design of a real prototype which is consequently mechanically tested. The test results are then used to optimise materials and product design and a new prototype is manufactured and again tested. If the new prototype does not meet the mechanical requirements, the test results are again used to optimise the materials and design of the prototype. This process is repeated until the mechanical requirements are met.

The virtual route follows exactly the same path as the physical one, but all steps are performed in a virtual manner. Not a real prototype is built, but a virtual one. This prototype is then virtually tested and the test results are subsequently used to optimise the materials and design of the prototype. This process is similarly repeated until the mechanical requirements are met in the virtual world. Only when that is the case, a real prototype based on the virtual one is manufactured and tested.

The virtual route is desired by industry because it saves a significant amount of TIME and MONEY. The reasons for this are that:

(i) the manufacture of a physical prototype costs significantly more time and money than constructing a virtual one because of several possibilities:

- new equipment has to be purchased in order to manufacture the prototype,
- manufacturing each prototype is timely as the design is different each time,
- existing manufacturing processes have to interrupted to produce a small batch of new material for the prototype, compromising the standard manufacture,
- existing machines have to be recalibrated to produce a small batch of new material for the prototype, and again when normal production restarts,
- small batches of new base materials have to be purchased which are substantially more expensive than large batches of base materials, and
- the planning of the standard production is undermined by production of small and new batches of materials.

(ii) the testing of a physical prototype costs significantly more time and money than virtual testing because of several possible reasons:

- new equipment has to be purchased in order to test the prototype, record and analyse the results,
- physical testing costs significantly more time than virtual testing, especially when fatigue is of interest,
- testing conditions are significantly more difficult to control and apply in physical experiments than in virtual experiments and sometimes are not possible to control and apply at all, and
- an enormous amount of detailed test results can easily be obtained from virtual tests which is not possible from physical tests.

Since the design process involves the manufacture and testing of several prototypes (the process is related several times), a huge amount of money and time can be saved by taking the virtual route.

# Aims

The virtual route is substantially more advantageous than the physical route. However, this is only the case if two conditions are met: (i) the models to test the virtual prototypes must be accurate enough to capture all relevant mechanics and (ii) the models must be fast enough to obtain test results in a timely manner. The aims are therefore to develop (i) highly accurate models for fibrous and discrete materials and (ii) numerical strategies that solve the highly accurate models in a substantially fast manner.

# Approach

## Discrete models of fibrous and discrete materials

The mechanical models for fibrous and discrete materials we have developed and will be developing represent each fibre/yarn segment using an individual spring or beam. The reason for this is that the mechanical response of fibrous materials is governed by the interplay of fibres at small length scales. Discrete models in which each fibre segment is individually modelled are therefore extremely accurate. As a result, individual mechanics typically occurring in fibrous materials and important for the global material response can be captured in detail and in a relatively straightforward manner (e.g. inter-fibre bond failure, inter-fibre sliding, inter-fibre contact and fibre failure). Even global material responses such as large re-orientations of fibres between each other (e.g. occurring in textiles) can be incorporated straightforwardly, whereas this is not the case if constitutive descriptions are used. So far we have developed several models for (electronic) textiles and paper materials including experimental validation, which amongst others include inter-fibre bond failure and the consecutive frictional fibre sliding.

## Multi-scale methods for discrete models

Hence, discrete models in which each fibre segment is individually represented are clearly highly detailed. However, an important disadvantage is that they cannot efficiently be used for engineering-scale applications/prototypes. Consequently, numerical strategies must be developed and employed to use the small-scale discrete models for large-scale computations. The numerical strategies we develop in the context for discrete models are *nested* multi-scale methodologies. This means that they directly couple small-scale mechanics, obtained from the small-scale discrete models, to the engineering-scale response. The term *nested* means that the small-scale discrete models are intrinsically present in the methodology and not indirectly coupled to the engineering-scale response via the use and calibration of a constitutive model. This has the advantage that the engineer using the methodology does not have to apply the coupling between the small-scale discrete model and the constitutive model himself, an issue which is often inaccurate and far from trivial.

The multi-scale method we have reformulated for the discrete models of fibrous materials is the quasicontinuum (QC) method. The QC method was originally developed for discrete atomistic models and is therefore specifically aimed at discrete models. One of the advantages is that it can easily be formulated for *concurrent* problems, which means that the discrete model is fully resolved in regions with large field fluctuations and coarse-grains the discrete model elsewhere. In this way, individual mechanisms can be captured in small domains whereas a substantial computational gain is made elsewhere. Another advantage of the method is that, in the right formulation, it is *equation-free*. This means that no additional equations are required to pass information from the small-scale model to the engineering-scale model and vice versa. As a result, the method is relatively easy to implement and understand. Also, scale-separation as required in numerous multi-scale frameworks is not required and the use of periodic boundary conditions can be avoided. Finally, the *equation-free* aspect of the method ensures that higher-order variants are as simple, transparent and easy to implement as the linear method.

As the QC method was originally developed for discrete atomistics which are intrinsically conservative, the method could not directly be applied to the non-conservative discrete models for fibrous materials (i.e. they include dissipation, e.g. by plasticity). A virtual-power-based QC method was therefore formulated that can deal with dissipative models. This has introduced a wide field of applications for QC frameworks. A QC variant was also formulated that can deal with beams instead of only spring networks. As beams use higher-order interpolation internally, the QC method for beams also requires higher-order interpolation; an issue that was not investigated in the past.

# Future work

The group has the knowledge and availability of QC methods that can deal with several discrete models of fibrous materials. These include local and non-local dissipation and beam models. An important drawback however is that the QC method cannot deal with discrete, irregular models. Currently, the group focuses on a *multi-field *QC framework that can deal with irregular models. Results on this method can be expected in due course. Consequently, we will have the availability of a *multi-field*, *virtual-power-based*, *higher-order**, equation-free* QC methods that can deal with the most complex discrete models for fibrous materials. From that point onwards, we can start to employ it to even more detailed discrete models than we have focused on in the past such as geometrically non-linear, materially non-linear beam networks including inter-fibre contact and inter-fiber bond failure for textiles, electronic textiles, paper materials and collagen networks. Another interesting development we aim to focus on in the future is to incorporate goal-oriented adaptivity. This would give us the availability of a QC framework that adaptively changes its interpolation depending on the desired error of a quantity the engineer is interested in.

A last important development related to the use of the virtual-power-based, higher-order QC framework is that it will incorporated in the DIGIMAT Software of E-Xstream Engineering (part of MSc Software). This will be the first time that engineers working in industry have the availability to use the QC method for the optimisation of several process steps in the development of new fibre-based composites.

# Summary of past achievements

Here we summarise the most important achievements of the group related to the aforementioned research.

## Discrete models of fibrous and discrete materials

- the formulation of a discrete model for electronic textile including the experimental calibration and validation,
- the formulation of a general discrete model that incorporates inter-fibre bond failure and subsequent fibre sliding.

## Multi-scale methods for discrete models

- the formulation of the
*virtual-power-based*QC method so the method can be used for discrete models including dissipation, - the formulation of a
*multi-field*,*virtual-power-based*QC framework that can deal with nonlocal dissipative mechanisms, e.g. for when inter-fibre bond failure and subsequent fibre sliding is included, - the formulation of a
*higher-order*QC framework that can deal with beam networks instead of only spring networks, - the formulation of novel sampling rules (one of the two steps that form the basis of QC methods) for different QC frameworks (amongst others those for atomistics), and
- the use of the virtual-power-based QC method to efficiently optimise an electronic textile with respect to a number of geometrical and material parameters.

Figure: Comparisons of the results of the direct computation and its *virtual-power-based, third-order *QC method for an X-braced lattice model of 200 x 100 unit cells in pure bending with an inclusion. Top-left: results of the direct lattice computation, top-right: results of the QC computation. Centre-left: results of the direct computation around the inclusion, centre-right: results of the QC computation around the inclusion, bottom: moment-angle curves for the direct computation (blue, solid) and the third-order QC computation (magenta squares).