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A novel computational technique is presented for embedding mass-loss due to burning into the ANSYS finite element modelling code. The approaches employ a range of computational modelling methods in order to provide more complete theoretical treatment of thermoelasticity absent from the literature for over six decades. Techniques are employed to evaluate structural integrity (namely, elastic moduli, Poisson’s ratios, and compressive brittle strength) of honeycomb systems known to approximate three-dimensional cellular chars. That is, reducing the mass of diagonal ribs and both diagonal-plus-vertical ribs simultaneously show rapid decreases in the structural integrity of both conventional and reentrant (auxetic, i.e., possessing a negative Poisson’s ratio) honeycombs. On the other hand, reducing only the vertical ribs shows initially modest reductions in such properties, followed by catastrophic failure of the material system. Calculations of thermal stress distributions indicate that in all cases the total stress is reduced in reentrant (auxetic) cellular solids. This indicates that conventional cellular solids are expected to fail before their auxetic counterparts. Furthermore, both analytical and FE modelling predictions of the brittle crush strength of both auxteic and conventional cellular solids show a relationship with structural stiffness.

Composite materials consist of two chemically distinct component materials mechanically bonded such that the overall amalgamate has more superior properties (in some sense) than the individual constituents. For example fibre-reinforced polymeric composites have a high strength and stiff fibrous (or particulate) reinforcement phase embedded within a matrix resin [

An example where enhanced thermal effects may be beneficial is in the fire retardant (FR) materials field. Here, improved mechanical resilience, reduced oxygen permeability, and reduced tendency to oxidation of very fragile or brittle charred structures formed under heat or fire conditions are desirable. Research has developed a barrier fabric containing cellulosic fibres dispersed with an intumescent [

In order for materials to be developed with enhanced thermal and fire retardant properties, it is necessary to gain an improved understanding of the relationship between material structure and both the mechanical and thermal behaviour of the foamed charred structures formed by the aforementioned intumescent compounds. Since typically, such components are used in many structural applications, appreciation of the mechanical and failure properties when subjected to mass-loss is of particular interest especially from a design viewpoint. The residual strength of the foamed intumescent is a salient characteristic, since fire presents threats to the continued presence of the insulating and protective foam through live loading from deposits, expansion of the underlying material, flame, buoyant flows, and impact of moving debris.

Much of the recent literature is dedicated to the experimental examination of structural integrity of combustible composite materials during burning (e.g., [

One model of such a two-dimensional cellular solid structure is shown in Figure

(a) Reentrant (auxetic) honeycomb and (b) conventional honeycomb.

Materials which exhibit this phenomenon of becoming fatter when stretched and conversely reducing in cross section when compressed are termed auxetic materials [

The models developed in this paper take the form of honeycomb systems shown in Figure

Whence the standpoint with regard to the theoretical modelling of elasticity values has not really changed for over seven decades. This paper will therefore show that with the wide range of computational continuum (and cellular) modelling methods now available a more complete theoretical treatment of thermoelasticity can now be realistically attempted. It should be noted however it is not the intention of the work described herein to provide a detailed analysis of particular mass-loss process in relation to burning of polymers more to provide describe modelling procedures once particular empirical data are available and or modelling of such are available. That is, we will assume that such charring in flame retarded resin components within a composite always form an expanded or intumescent char. It is therefore understood throughout the work that follows that few polymers form intumescent chars when heated alone under nitrogen or in air; the polyphenolics and, to a lesser extent, the epoxies (both used as composite resin components) are examples of significant char, forming resins with some level of intumescent char formation occurring under certain circumstances. Moroever, it is not our intention to present detailed modeeling procedures of the chemical and physical processes taking place during charing, that is, to say reactions to consider when the IFR/polymer system is heated in air there being numerous treatments in the literature (e.g., [

A number of models have been developed to predict the mechanical properties of cellular solids, the most prevalent being analytical and finite element methods or a combination of these two approaches. Analytical models for the deformation of periodic regular honeycombs, model the honeycomb cell walls (ribs) as beams, the deformations of which are calculated using standard beam formulae [

Numerical approaches based on the finite element (FE) method have been developed to predict the properties of irregular and nonperiodic honeycombs [

As mentioned previously several analytical models are evident in the literature, ranging from single deformation mode models [

In general mass-loss is a complicated chemical process consisting of four stages of pyrolysis (

The first function was a linear inverted saturation and the other a hyperbolic tangent. In the latter case this can be expressed thus [

Equations (

The majority of the analytical approaches used to date have maintained the thickness of the ribs constant. However, Whitty et al. [

Here, the salient expressions in order to calculate the temperature dependent in-plane mechanical properties of honeycomb systems are derived. We apply expressions for elastic modulli, in the two principal orthogonal (

These expressions allow mass-loss to be incorporated into both the numerical and analytical modelling procedures via rib thickness variations, which in turn are functions of temperature. Poisson’s ratios and elastic moduli were calculated for the following three scenarios: mass-loss from (i) both vertical and diagonal ribs simultaneously; (ii) the vertical ribs only; (iii) diagonal ribs only, via application of (

It should be noted at all the elastic moduli data also enable inference as to the failure of these material systems. It is suggested in [

Numerical FE models were used initially to validate the analytical procedures detailed in the previous section since for the mass-loss scenarios considered throughout this work could not be compared directly with experiment. Homogenized FE methods are evident in the literature (e.g., [

Indeed, the more classical FE models described in the following subjections are able to predict other thermoelastic properties of periodic honeycomb systems, which as far as we are aware until now, analytical models do not exist in the literature. These thermoelastic property predictive FE models were solved using a two-stage modelling process in order to investigate the thermal stress distributions throughout these cellular systems. The details of the FE modelling procedures will now be considered in turn.

FE calculations were performed using two commercial preparatory finite element codes [

Honeycomb models: (a) mechanical and (b) thermal boundary conditions (temperature difference for most simulations being 100 K).

Strains were determined by calculating the fractional change in distance between pairs of nodes aligned along the direction of interest and offset from each of the opposing edges of the honeycomb by one-unit cell. This was done to minimize edge effects and is again consistent with the previous theoretical and experimental works for comparison. Forces were applied to the nodes of the free edge, opposite the fixed edge, and in a direction perpendicular to the fixed edge. The total force applied to the free edge was obtained by summing all the applied nodal forces, which then enabled the applied stress to be calculated by dividing the total applied force by the area (i.e., the tributary area) of the edge upon which the force was applied. Preliminary calculations demonstrated linear stress-strain and transverse strain-longitudinal strain behaviour, up to applied strains of at least 0.02%, as also observed in the experimental data and predicted by the analytical models [

The effects of mass-loss were incorporated into both the numerical and analytical modelling procedures via rib thickness variations, which in turn are functions of the temperature, that is, (

These models were also used to predict to the temperature dependent compressive strengths by conducting a further linear elastic bulking analysis and evaluating the maximum stress of structure at the first mode of bulking.

Thermal stress can be induced as a result of a uniform temperature change in the solid or as a result of a temperature gradient; in this work we investigated both phenomena via application of a structural analysis and a coupled thermal-structural analysis.

The modelling techniques were initially verified with other models evident in the literature [

Excellent agreement was shown between the empirical mass-loss data [

Comparisons between assumed mass-loss functions and empirical mass-loss data.

The hyperbolic tangent curve shows the classic trend. Initially there is very little change in the mass of the substance, and up until a temperature of about 400°C (the onset of combustion) the mass-loss is mainly due to off-gassing alone; then a rapid increase in the next 100°C as combustion generates charring and both convective and radiant heating of the material from hot surfaces and flame. It is not the purpose of this paper to consider complex ignition processes. The mass-loss continues at an almost constant highly increased rate until 10% of the mass of the substance is remaining, after which the rest of the mass dies away slowly until a temperature of 800°C at which the substance has completely decomposed. Note that this model is being applied to fabric material, rather than solid. Moreover, the complex insulating properties of the char and real-world external effects that might dislodge the char are neglected.

The calculated elastic moduli for the

Conventional honeycomb in-plane normalized elastic moduli values as functions of temperature.

Reentrant (auxetic) honeycomb in-plane normalized elastic moduli values as functions of temperature. Auxetic honeycomb rib reducion data.

These are given for the aforementioned three types of mass-loss scenarios (using geometrical and intrinsic material properties as the reference [

All three mass-loss scenarios lead to a decrease in

Initial decrease in

The variation of Poisson’s ratio

Conventional honeycomb in-plane Poisson’s ratio variations with temperature.

Reentrant (auxetic) honeycomb in-plane Poisson’s ratio variations with temperature.

Figures

It should be noted that in all cases the FE and analytical models are in excellent agreement for all calculations of these mechanical properties.

Preliminary FE calculations showed that the rib stresses were all negative (i.e., compressive) if a positive temperature gradient (heating) was applied; and for negative temperature gradients (cooling) the converse was observed.

Figure

Thermal stress distributions due to (a) constant temperature change of 100°C and (b) temperature gradient of 100°C.

Figure

Regression coefficient data.

Honeycomb | Model | Regression | Constant | Correlation coefficient |
---|---|---|---|---|

(MPa/K) | (MPa) | |||

Conventional | Diagonal | 0.0517 | −1.035 | 1.000 |

Reentrant | Diagonal | 0.0465 | −0.931 | 1.000 |

Conventional | Vertical | 0.0329 | −0.658 | 1.000 |

Reentrant | Vertical | 0.0151 | −0.3017 | 1.000 |

The linear regression here is most informative showing that conventional honeycomb increases with temperature at a greater rate than the reentrant. The negative values for the constant can be explained by extrapolation of the graph in Figure

The

Additionally, vertical rib stretching becomes dominate only at higher temperatures (and hence beams with extremely thin vertical ribs), in which case the use of elastic beam theory to describe the deformation may become inappropriate. Mass-loss of the diagonal rib leads to a rapid reduction in the elastic moduli (i.e.,

Figures

Thermal stress comparisons between the auxetic and conventional honeycombs subjected to a 100°C/m temperature gradient.

The FE and analytical models indicate that reentrant honeycombs will have enhanced thermal properties, such as reduced thermal stress by virtue of (

Strength predictions due to the temperature and hence mass-loss are shown in Figures

(a) Conventional and (b) auxetic honeycomb residual strength predictions.

Analytical and FE methods have been employed to investigate the effects of mass-loss on the structural integrity and thermal stress distributions of cellular solids known to take the form of a char [

The mechanical properties of polymeric honeycombs have been shown to be functions of temperature and hence the relative densities of these cellular systems. In principle this technique can be employed to predict the structural integrity and resulting thermal stress distributions for any foamed polymeric cellular solid undergoing mass-loss.

Mass-loss from the vertical ribs only leads to a reduction in the magnitude of

Mass-loss from only the diagonal ribs leads to a decrease in both

When exposed to constant temperature changes, cellular auxetic materials show less stress buildup than conventional solids. Thus it may be concluded that related properties such as thermal shock will be increased.

For all of the mass-loss scenarios investigated the thermal stress has been shown to rapidly increase as mass is lost from cellular systems. Moreover the brittle collapse of the structure appears related to the stiffness in that particular principal orthogonal direction under consideration, which differs significantly with what is observed in many continuum material analogues.

It is however noted that current empirical evidence from many sources suggests that typical polymer-based intumescent chars comprise spherical or distorted spherical voids commensurate with a normal foam cellular structure not an auextic one as investigated here. The paper therefore provides an appropriate computational models to determine the mechanical properties of char structures if and only if the variation in the underlining properties of the intrinsic materials is known from (as in this case) empirical data and/or other chemical/mathematical modelling methods [

The novelty of this research therfore is determination of a porous (intumescent) char has improved strength retention properties when exposed to an intense heat source if it has an auxetic property as opposed to one which has a conventional. However, the validity of the research hypothesis could be increased if there was evidence that actual intumescent chars had an auxetic structure or the potential to form one. Current empirical evidence from many sources suggests that typical polymer-based intumescent chars comprise spherical or distorted spherical voids commensurate with a normal foam cellular structure. With the advancement made in the production of auxetic composite materials [

Honeycomb breadth

Length of

Length of

Original mass of cellular solid

Mass-lost from cellular solid

Current thickness of vertical or diagonal rib

Original thickness of vertical or diagonal rib

Elastic modulus in the

Elastic modulus of

Flexural stiffness of a

Hinging stiffness of a

Stretching stiffness of

Stretching stiffness of

Number of unit cells parallel to the abscissa

Numbers of unit cells parallel to the ordinate

Temperature change

Thermal shock

Dimensionless temperature

Original (ambient) temperature

Final (burning) temperature

Linear coefficient of thermal expansion (CTE)

Fracture strain of honeycomb/intrinsic material

Orientation angle of

Poisson’s ratio of

Orthotropic longitudinal (

Solid intrinsic material density

Honeycomb material density

Normalized residual relative density

Principal stress

Fracture strength of honeycomb/intrinsic material.

The authors declare that there is no conflict of interests regarding the publication of this paper.