Optimizing non-Newtonian fluids for impact protection of laminates

Significance Complex fluids that alter their mechanical response as the applied forces change enable smart materials. A prime example is flexible body armor infused with a shear-thickening suspension that hardens on impact. During impact, there is a complex interplay between solid deformation and fluid flow that complicates predictive design. We construct and experimentally validate a theoretical model for a fluid–solid laminate that describes display glass applications, such as in smartphones. Strikingly, we find that, now, sandwiching a fluid that becomes less viscous during impact between a top and a bottom layer protects both against impact. Our approach establishes design principles for smart fluid–solid composites.

Non-Newtonian fluids can be used for the protection of flexible laminates.Understanding the coupling between the flow of the protecting fluid and the deformation of the protected solids is necessary in order to optimise this functionality.We present a scaling analysis of the problem based on a single coupling variable, the effective width of a squeeze flow between flat rigid plates, and predict that impact protection for laminates is optimised by using shear-thinning, and not shear-thickening, fluids.The prediction is verified experimentally by measuring the velocity and pressure in impact experiments.Our scaling analysis should be generically applicable for non-Newtonian fluid-solid interactions in diverse applications.
Complex fluids that alter their mechanical response as the applied forces change enable smart materials.A prime example is flexible body armour infused with a shear thickening suspension that hardens on impact.During impact there is a complex interplay between solid deformation and fluid flow that complicates predictive design.We construct and experimentally validate a theoretical model for a fluidsolid laminate that describes display glass applications, such as in smartphones.Strikingly, we find that, now, sandwiching a fluid that becomes less viscous during impact between a top and a bottom layer protects both against impact.Our approach establishes new design principles for smart fluid-solid composites.
Woven fabrics impregnated with a shear-thickening colloidal fluid, whose viscosity increases suddenly at a critical shear rate, can function as body armour 1 .Perhaps surprisingly, the shear-thickening fluid does not provide protection in body armour because of their bulk rheology that allows, for example 'running on cornstarch' 2 .Instead, the fluid increases inter-fibre friction and so prevents fibres from being pulled apart 3 , so that they form a rigid layer to spread impact and protect the material underneath.
Partly inspired by this application, there is growing interest in smart materials that incorporate various non-Newtonian fluids in solid structures [4][5][6][7][8] .In particular, in direct analogy with body armours, it is envisaged that including shear-thickening fluids in laminates may provide impact protection.However, analysing the impact response of fluid-solid composites is challenging even in the case of Newtonian fluids 9 .Deformation of the solid drives fluid flow, which then generates a pressure, which in turn changes the solid deformation, creating feedback.For a non-Newtonian fluid, such fluid-solid interaction is even more challenging, because the fluid property changes as the flow develops throughout impact, and analyses to date are limited, e.g., to blood flow [10][11][12] , or stationary process such as blade coating 13 .
We consider fluid-solid interactions in a laminate consisting of a non-Newtonian fluid sandwiched between a flexible sheet above and a rigid base below, which is a model for various a) Electronic mail: james.a.richards@ed.ac.uk real-life applications, e.g., a display in which the base layer is an LCD panel and the top layer is a piece of glass, both of which must be protected from concentrated impacts at ≲ O (10 m s −1 ).The physics differs from that in shear thickening body armour.The requirement here is to protect both solid layers, while body armour is optimised for the protection of the single lower layer.
We perform a scaling analysis of the coupling between fluid flow, rheology and solid deformation in our geometry based on the idea of an 'effective squeeze flow width', and verify our analysis using controlled-velocity impact experiments.We find that the effective squeeze flow width varies weakly throughout the impact, so that the process can be approximated as a simple rigid squeeze flow.From this we find, surprisingly, that shear thinning, not thickening, is optimal for protection.

Modelling
Using a quasi-2D setup, we analyse the downward impact of a point mass  at the origin,  = 0, with speed  on a flexible plate initially at height ℎ  parallel to a rigid bottom plate, with the gap filled by a fluid, Fig. 1A.The width of the plate  ≫ ℎ  , and breadth of the plate (perpendicular to the page)  ≫ ℎ  .The upper plate is pushed down, leaving a gap ℎ 0 () at the impact point, and bending deformation Δℎ(, ) upwards.The net motion causes a fluid flow, .If the impact velocity is significantly sub-sonic,i.e. 0 ≪ O (1000 m s −1 ) for most solids and liquids, then incompressibility and mass conservation require with  the distance from the impact.The pressure gradient associated with the impact-driven flow is given by where the fluid viscosity  is constant for a Newtonian fluid.The pressure, (, ), which satisfies ( → ∞) = 0, pushes arXiv:2311.08280v1[physics.flu-dyn]14 Nov 2023 . back on the impacting mass , and bends the flexible layer, which has thickness ℎ  and rigidity  =  ℎ 3  /12 (with  its Young's modulus).The shape of the layer follows the Euler-Bernoulli equation 14 ,    4 Δℎ  4 = (, ), where we have neglected the laminate mass as ≪ .Self consistency requires that (4) solves to give small plate deflection, so that flow is essentially along , as is assumed in the 'lubrication approximation' 15 , (2).The coupled integro-differential equations, Eqs.(1-3) need to be supplemented by a form for the rate-dependent viscosity, ( ), if the fluid is non-Newtonian.The complex feedback between quantities, Fig. 1B, means that finite element or immersed boundary numerical methods are needed to solve specific fluidsolid interaction problems for Newtonian 9 and non-Newtonian fluids 16,17 ; but such solutions offer little physical insight into fluid-solid interactions, for which we turn to a different approach.

Simplified closure
To analyse the fluid-solid interactions in our geometry, note first that since the pressure gradient    ∝ ℎ −3 , we need only consider the region around the impact where deformation is small, Δℎ ≲ ℎ 0 . 18Within this region the surface is only weakly curved, and a calculation of the shear rate shows that it is adequate to treat it as a flat surface, ℎ() ≈ ℎ 0 (SI Appendix, Fig. S1).We therefore define an effective flat plate width,  eff , such that the pressure created by a rigid plate squeeze flow bends the flexible plate by Δℎ = ℎ 0 at  =  eff .The squeeze flow for || ≤  eff ≪  is solved analytically 19 , but we neglect fluid flow and deformation outside (|| >  eff ), Fig. 1C-D.Within this local approximation, boundary conditions can be neglected as volume conservation will be ensured by, e.g., the surface being pushed up further away from the impact zone.
We use a scaling analysis to determine  eff , which is not known a priori.The flux created by the rigid-plate squeeze flow  ≃  eff gives    ≃ 12 eff /ℎ 3 0 and  ≃ 12 2 eff /ℎ 3 0 .Equation (4) implies that the the deflection Δℎ ≃  4 eff × /.Self consistency demands that this Δℎ ≈ ℎ 0 , which combines with  to give .
(5) While higher , faster  and narrower ℎ 0 bend the plate more strongly and reduce  eff , the dependence is weak.The somewhat unusual 1 6 exponent is traceable to the dependence of plate deflection on  6 eff . 20The nearly-constant  eff means that the dynamics can be thought of as a modified fixed width squeeze flow that scales approximately as ℎ −3 0 .To capture the lowest order effects of a rate-dependent viscosity,  = ( ), in non-Newtonian fluids, a further approximation is made.We take the fluid to be an effectively Newtonian with a single viscosity,  eff = (   ), where   is the shear rate at the edge of the effective plate ( =  eff ) for a fluid of this viscosity.This again ensures self-consistency; it also recalls the use of the rim shear rate in calculating the viscosity in parallel-plate rheometry 21 .
We use a power-law model,  eff =   −1  , to explore the effect of thinning ( < 1) and thickening ( > 1) on impact protection.Now, (5) becomes (see SI Appendix) and which reduce to Newtonian results, (5), for  =  and  = 1.Equation ( 6) gives the force per unit length in terms of (/, , ) and a single dynamical variable ℎ 0 () with its derivative ℎ 0 = ; this then allows us to understand how a flexible solid-fluid laminate may be protected against impact.
At some optimal  ≈ 0.2,  max is minimised at  opt max , part D. The impact is absorbed over the whole gap with a near-constant  = ℎ 0 , but eventually slows before  diverges.As  eff is weakly dependent on ℎ 0 , reducing the divergence in  directly gives a flatter  ().This, however, still peaks as the gap narrows, part C [bold dashed line], increasing 50% from  = 0 before dropping rapidly to zero.To obtain a minimum  max with a flat  () profile, we turn to non-Newtonian fluids.
Consider first a constant-speed impact.We plot in Fig. 3A-B the ℎ 0 () dependence implied by (6): The force and bending moment in a shear-thickening fluid laminate ( = 1.5, 2) diverge more sharply as the gap narrows than the Newtonian case ( = 1).However, a shear thinning fluid ( = 0.5, 0.33, 0) leads to a weaker force divergence.For  = 0.5 the bending moment also diverges more weakly than the Newtonian case.Interestingly, decreasing  further brings a constant  ( = 0.33) and then a decreasing  ( = 0).These results suggests that for laminate protection a shear-thinning, not thickening, fluid is needed.
We next confirm and generalise our analysis with numerical solutions of the dynamical equation for ℎ 0 (): (9)   where the second term modifies the Newtonian equation, (7), and ,  and  have been set to unity.
For any value of  ≳ 0.4, we find an optimal  for which the maximum bending moment is minimised (comparable to Fig. 2D, but with  → ).Increasing  from the Newtonian value of unity, this optimal value  opt max increases, Fig. 4C, i.e., a shear-thickening fluid decreases protection.In contrast, decreasing  below unity, i.e., changing to progressively more shear-thinning fluids, lowers  opt max , thus offering increasing impact protection, consistent with our constant- analysis.
For  < 0.4, we find that decreasing  below its optimal value brings laminate failure, as ℎ 0 → 0. So, we predict that optimal impact protection is offered by a shear-thinning fluid with  = 0.4, somewhat higher than the 1 3 from the constant- analysis, but is insensitive to pre-factors in our scaling analysis.Physically, a shear-thinning fluid is optimal as it is harder to push out of large gaps (low , higher  eff , larger ) than for narrow gaps (high , lower  eff , smaller ), which smooths  () and hence  ().

Constant velocity experiments
We verify our analysis in an experimental realisation of our quasi-2D set up from Fig. 1A, using a universal testing machine to drive a wedge downwards at a laminate consisting of a fluid sandwiched between a 0.3 mm-thick flexible glass plate and a 10 mm-thick polydimethylsiloxane (PDMS) base, Fig. 4A, at low enough constant velocity, , to allow us to follow the force on the wedge, , as a function of time, or, equivalently, (downward) displacement, Δ.The gap height is ℎ 0 = ℎ  − Δ + /, where ℎ  is the initial gap height, and  is the (separately measured) stiffness of the system.We measured  (Δ) at different imposed , and monitored the pressure on the PDMS via photoelastic imaging.Experimental details are in Materials and Methods.

Newtonian fluids
We begin with a Newtonian fluid laminate with ℎ  = 0.7 mm, using glycerol as the 'sandwich filling', increasing  from 0.5 mm min −1 , Fig. 4B [dark (purple) lines], to 200 mm min −1 [light (yellow) lines].At low , the fluid can almost freely drain and  is low, only increasing as Δ → 0.8 mm and ℎ 0 → 0. With increasing ,  (Δ) takes on a sigmoidal shape.Converting Δ to ℎ 0 and normalising by √  collapses the data to within a factor of 1.5 over a 400-fold variation in , Fig. 4C.confirming the √  scaling of (5).Indeed, / = 12(/ℎ 2 0 ) 1/2 offers a credible account of the collapsed data (dashed line).That this is within an order-unity numerical factor ( √ 12 ≈ 3.5) of ( 5) validates the physics embodied in our scaling analysis: an effective squeeze flow that shrinks in extent as the viscous forces more strongly bend the flexible upper layer.
To illustrate this physics, we turn to photo-elastic measurements, where light intensity is a proxy for the pressure, so that we can visually distinguish between a point and a distributed load, Fig. 4A (ii) and (iii) respectively.At  = 20 mm min −1 , a bright region, evidencing high pressure, emerges at ℎ 0 ≲ 0.35 mm Fig. 4C, and grows in intensity as ℎ 0 decreases further.The half width of a Gaussian fitted to the measured intensity pattern decreases only weakly, from 9.9(2) to 6.19(3) mm as ℎ 0 decreases from 0.53 to 0.09 mm.The observation of a localised high pressure region is consistent with assumption of squeeze flow in a confined region of some effective width  eff .The weak dependence of  eff on ℎ 0 is also consistent with (5), from which we predict  eff ≃ (ℎ 4 0 /12) 1/6 = 9 mm at ℎ 0 = 0.35 mm down to  eff ≃ 4 mm at ℎ 0 = 0.09 mm, comparable to the observed widths and trends of the high-pressure region.Finally, these results are consistent with our assumptions of lubrication flow ( eff ≫ ℎ 0 ) and neglecting boundaries ( eff ≪  = 75 mm).Thus, the complex feedback between fluid flow and plate deformation can indeed be captured in an 'effective flat plate' treatment.

Non-Newtonian fluids
We next tested a laminate filled with an  = 0.4 shearthinning suspension, Fig. 5A (filled circles); this and the shearthickening suspension (see below) can be treated as continua, as the particle size is much smaller than the minimum gap (SI Appendix).Now, (6) predicts  ∝  0.22 , consistent with the observed collapse of  (ℎ 0 ) data taken at different speeds when we plot  (ℎ 0 )/ 5 √ , Fig. 5B.The prediction of  ∝ ℎ −0.55 does not capture the transient, early-stage response, but shows moderate agreement at intermediate ℎ 0 , Fig. 5B (dashed), with a prefactor of 2.4 consistent with a scaling analysis.The observed divergence in  as ℎ 0 → 0 is weaker than for  = 1, matching the predicted trend.However, it is also weaker than predicted for  = 0.4.Better agreement between theory and experiment here may require more careful modelling of shear-thinning fluids under squeeze flow conditions 22 .

√
collapsing the data (cf.Fig. 4C).This is consistent with the almost-constant viscosity of this fluid at low shear rates:  decreases from 3 to 1 Pa s as  increases from 10 −1 to 10 2 s −1 .A different behaviour is seen when  ≥ 50 mm min −1 , Fig. 5C (light lines): / √  no longer collapses the data, and the ℎ 0 dependence becomes stronger, although the smallℎ 0 limit could not be accessed in these high  experiments due to load cell limits.The shear rate at the onset of this change can be estimated by using (5) for  eff with  = 1 Pa s, so that  = 6 eff /ℎ 2 0 ∼ 160 s −1 at  = 50 mm min −1 and ℎ 0 = 0.6 mm.This is consistent with the shear rate at which we observe shear thickening in our fluid, Fig. 5A (filled squares), once again supporting the validity of our analysis in terms of an effective flat plate of width  eff , and an effective viscosity set by the edge shear rate,  eff = (   ).

Energy scaling
The speeds at which we have performed our experiments to validate our scaling analysis are far too low for realistic impact protection at  ≳ 1 m s −1 .Nevertheless, our analysis, now substantially validated by experiments, allows some predictions for higher speeds via energy scaling.
Under such conditions, our fumed silica suspensions may become brittle 23 , rendering manufacturing challenging, and post-impact 'self healing' may not be possible.A fluid with more complex rheology, e.g., one that thins only at the high  of impact, may be more suitable.This reduces stresses at slow deformation, facilitating manufacturing, self-healing, and, perhaps, even enabling fully flexible laminates.Such rheology could be achieved using suspensions that thin after thickening, due to asperity compression 24 or a brush-like coating 25 , or a polymer solution with a low-shear plateau 26 .

CONCLUSIONS
Inspired by the use of shear-thickening fluids in body armours, we have established a general scaling framework for analysing the impact response of solid-fluid laminates, which captures interactions through an effective rigid plate squeeze flow with width  eff , which scales only weakly with all parameters, (5).Insight can, therefore, be gained by thinking in terms of a simple rigid plate squeeze flow.Strikingly, we conclude that, not thickening, but shear thinning with  ∝  −0.6 optimises protection, Fig. 5D.This arises from reducing the  (ℎ 0 ) divergence, with a low  eff at small ℎ 0 (high ), while still absorbing the impact energy with a high  eff at large ℎ 0 (smaller ).These scaling predictions were substantially verified in controlled-velocity impact tests where we measured  (ℎ 0 ) and imaged the pressure distribution using photoelasticity.Together, these results establish the effective rigid plate squeeze flow approximation as a useful tool for analysing fluid-solid interactions in composites incorporating non-Newtonian fluids.
Further work including flow perpendicular to  and  27 or curvature 22 , as well as normal stress differences 28 , straindependence 29 and extensional viscosities 30 , could allow predictive design of optimised fluids for realistic impact velocities.More generally, our scaling approach may also apply to non-Newtonian fluid-solid interaction problems arising from rubbing skin ointments 31 or eating chocolate 32 .
Viscosities are shown relative to Newtonian glycerol (Fig. 5A grey triangles,  = 1.24Pa s).Hydrophilic silica initially weakly shear thins, before reaching a critical rate,   ≃ 100 s −1 , where further stress does not increase the rate (discontinuous shear thickening 36 ).This is consistent with previous results 37 , with the onset of thickening occurring when the stabilising force, attributed to the absorption of PEG onto the silica surface, is overcome and the particles enter frictional contact 38 .Compared to monodisperse spheres, DST occurs at a low volume fraction, ≈ 11%, which may be attributed to the fractallike nature of the particles with additional rolling constraints 39,40 .
Hydrophobic silane surface modification creates a strongly shearthinning material 41 , Fig. 5A (orange circles), similar to removing adsorbed surfactants 42 .At low  slip is observed 43 , above this shear thinning with  ≈ 0.4 (dashed line,  = 38 Pa s 0.4 ) occurs up to sample fracture.Around  = 100 s −1 ,  for all fluids are comparable, at the range of  for low-velocity impact testing.The three fluids, with comparable absolute  but different  dependence, allow isolation of the role of fluid rheology.
Our quasi-2D controlled-velocity impact apparatus is based on a universal testing machine (Lloyd Instruments LS5, AMETEK).The force-displacement response (20 or 100 N load cell, 1 kHz sampling) is measured with  = 0.5 mm min −1 to 200 mm min −1 .Combined with a dark-field circular polariscope (FL200, G.U.N.T. Gerätebau GmbH) and a photo-elastic base, qualitative pressure measurements can be made.For force-displacement measurements, the initial gap, ℎ  , and zero displacement, Δ = 0, were set with no fluid.After loading the fluid, the laminate was allowed to come to equilibrium,  = 0 and Δ = 0.The impactor was then moved down 0.8 mm at a fixed speed, , recording  () and Δ() from which  (Δ) was reconstructed.The gap, ℎ 0 = ℎ  − Δ + /.
To infer the fluid pressure, we used a polariscope to probe stress in the base, giving finer spatial resolution than transducer arrays 45,46 .Stress-induced intensity patterns in the PDMS,  (, , ), were recorded using a camera (Nikon Z6, 3840 × 2160 30 Hz, 8-bit grey-scale).Instead of precisely quantifying the stress 47 , we sought to establish the extent of any high-pressure region.A narrow region at the top of the base layer is isolated in recording, 700 × 10 px 2 , Fig. 4a (red outline).The change in intensity from the quiescent state at the start of recorded movies, Δ (, , ), is averaged vertically, Δ (, ), and smoothed on short length scales using a Savitzky-Golay filter.The intensity is normalised to saturation (ISO 1200 and shutter speed 1/125).

FIG. 1 .
FIG. 1. Non-Newtonian fluid-solid interaction.(A) Diagram of point impact on simplified laminate geometry.(B) Schematic of full coupling between fluid rheology, fluid flow and glass deformation.(C) Diagram of simplified effective plate.(D) Schematic of simplified closure with single effective plate width variable.