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Key Concepts: Neutral Inclusions

11/11/2020

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PhD student Eleanor Russell discusses an important concept in the development of robust advanced materials

1    Inclusions: The good, the bad and the ugly
In material science, we refer to a material domain as a matrix and an inclusion as an object inside the matrix with different material properties.
Inclusions of all different shapes and sizes can be used to intentionally alter the effective properties of a composite material. For example, reinforcing brittle concrete with steel rods to increase its strength or, on a smaller scale, embedding highly conductive fibres throughout a less conductive matrix to improve heat dissipation. In both cases, the material properties of the matrix and inclusions complement each other and result in a more practical composite material.

​On the other hand, sometimes engineers are forced to incorporate inclusions into their designs; adding windows along the side of commercial aircraft for instance. In this case, the material properties can be affected in undesirable (and occasionally catastrophic) ways.

Picture
Picture
Fragment of the first production Comet at the Science Museum in London. Image credit: Krelnik, CC BY-SA 3.0 , via Wikimedia Commons
​A famous example is the world's first jetliner, the Comet, in 1952. The pressurised cabins were designed with large, squared-off windows causing pressure to repeatedly build-up at the corners. At high altitudes the corners of a window could be subjected to pressure up to three times higher than the rest of the cabin. Unfortunately, in several cases, this resulted in fatal structural damage. To avoid this problem, aircraft are now designed with the rounded, porthole-style windows we see today, however, neutral inclusions could offer an alternative approach.
2    Neutral Inclusions
​Neutral inclusions were first introduced by Mansfield in 1953 for holes in plates [1]. The idea is to provide inclusions with an additional coating to remove any undesired perturbations from the physical fields in the matrix. A perturbation is any deviation of a field from its state when the inclusion is not present. 
For example, we could surround the squared-off windows of the Comet with an additional coating to relieve the pressure from its corners. The required properties of the coating will depend on both the matrix (the cabin wall) and the inclusion (the window). If the necessary properties are satisfied the combination of the inclusion and its coating is referred to as a neutral inclusion and the fields in the matrix become unperturbed.

​All we need now is to determine the required properties within the coating. To show how we approach this problem we consider perturbations within the temperature field of a matrix.
Picture
2.1    Neutral Inclusions: thermal conductivity example
​In the following example we consider a steady-state, two-dimensional problem. The steady-state assumption removes any time-dependence from the solution. As a result, the only material property we need to consider is the thermal conductivity​, denoted \(k\). The thermal conductivity of a material measures its ability to conduct heat through diffusion.
​
We are interested in the temperature field, \(T(\mathbf{x})\), which must satisfy the heat equation. For this example, we assume the thermal conductivity is isotropic (independent of direction) and homogeneous (independent of position) and is therefore a scalar. As a result, together with the steady-state assumption, the heat equation reduces to Laplace's equation, given by:
$$\begin{equation}
\nabla ^2 T(\mathbf{x}) = 0,
\label{laplace}
\end{equation}$$
​​where \(\nabla ^2\) is the Laplace operator.​
​Consider the linear temperature field in Figure 1, where the black lines represent lines of constant temperature called isotherms. This temperature field is linear with respect to \(x\) and therefore has the form \(T(x)=\alpha x+\beta\) for some constants \(\alpha\) and \(\beta\).

​Let the matrix in Figure 1 have conductivity \(k_m\). Notice in Figure 2 that when a circular inclusion with conductivity \(k_i\), where \(k_i \neq k_m\), is embedded in the matrix, the field becomes perturbed and is no longer linear.
Picture
Figure 1: Linear temperature field with respect to x-coordinate
Picture
Figure 2: Inclusion leading to perturbations
By adding a coating with conductivity \(k_c\), as illustrated in Figure 3, we can remove the perturbations from the matrix and recover the linear temperature field
Picture
Figure 3: Configuration for neutral inclusion in polar coordinates, x = [r,θ].
​The temperature fields in the matrix, coating and inclusion are denoted by \(T_m(\mathbf{x}),\ T_c(\mathbf{x})\) and \(T_i(\mathbf{x})\) respectively and must all satisfy Laplace's equation. We solve (1) for all temperature fields to find:
$$ \begin{equation}
\begin{split}
&T_m(r,\theta) = \left(\alpha r + \dfrac{B_m}{r}\right) \sin \theta, \\
&T_c(r,\theta) = \left(A_c r + \dfrac{B_c}{r}\right) \sin \theta, \\
&T_i(r,\theta) = A_i r \sin \theta,
\label{temp fields}
\end{split}
\end{equation} $$​
where \(B_m,\ A_c,\ B_c\) and \(A_i\) are constants. ​​
​We wish to find a solution where \(B_m=0\) as this leads to a linear temperature field throughout the matrix given by \(T_m(r,\theta)=\alpha r \sin \theta = \alpha x\) (since \(x=r \sin \theta\)).
​
Let the radius of the inclusion and coating be given by \(r_i\) and \(r_c\) respectively. Solving the temperature fields in (2) with perfect contact conditions across each interface (continuity of heat flux and temperature) we find that \(B_m=0\) is satisfied when
\begin{equation}
D^2 = \dfrac{(k_m-k_c)(k_i+k_c)}{(k_i-k_c)(k_m+k_c)} \qquad \text{where} \qquad D= \dfrac{r_i}{r_c}
\label{k cond}
\end{equation}​
PictureFigure 4: Neutral Inclusion leading to unperturbed field in the matrix
Therefore, the two radii and the conductivities of all three components are dependent on each other. Figure 4 shows the resulting temperature fields when the condition in (3) is satisfied. We see that the temperature field in the matrix is unperturbed and once again linear with respect to \(x\) as required.

​
2.2    Neutral inclusions: extensions and limitations

​As we are not restricted to a single coating, we can extend the work in the previous section with as many additional coatings as we wish. For example, a bi-layer neutral inclusion where an inner coating insulates its core region, protecting the inclusion from external forces, ​and an outer coating counteracts any perturbations caused by the insulator. Since the inclusion may have any material properties, the bi-layer neutral inclusion acts as a cloak for the specific problem it is designed for.

In terms of limitations, firstly, an isotropic solution may not exist for a given field. For example, to find a neutral inclusion for a pressure field we must relax the isotropic assumption. Secondly, a neutral inclusion is not a perfect cloak as it is tailored for a specific configuration with fixed external forces. Therefore, if we change the nature of the external forces, the properties within each coating will no longer achieve the desired effects.

​A perfect cloak is one which, regardless of the external forces, leads to unperturbed fields in the matrix. Transformation theory is used to design perfect cloaks and other interesting concepts. We will discuss these in a future blog post.
[1] E. H. Mansfield, Neutral holes in plane sheet - reinforced holes which are elastically equivalent to the uncut sheet, The Quarterly Journal of Mechanics and Applied Mathematics, 1953, 6(3), 370–378
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3-Minute Papers: Lightweight, Damage Resistant Materials

4/24/2020

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We have prepared and tested a series of polymer-filled syntactic foams.  Foams with a high proportion of polymer microspheres showed excellent recovery and damage resistance in compression tests.

The first syntactic foam was developed in 1955 by the Bakelite Company of New York, and hailed as “A plastic foam, which promises to cut the partial cost of boat and airplane construction as much as 50%” [1].  It offered strength, insulation and tuneable properties in a lightweight material.

Syntactic foams are materials made up of hollow microspheres (commonly of glass, ceramic or plastic) held in a polymer matrix.  The strength and buoyancy of syntactic foams has led to their widespread use in marine applications, their sound absorbing properties find uses in acoustic applications, and they have even been used in World Cup footballs due to their low density and elastic recovery [2,3]. 
Picture
Syntactic foam sphere used as a subsurface float in oceanographic mooring. Photograph by Z22, distrubuted under a Creative Commons Attribution-Share Alike 4.0 International license
Picture
Adidas Fevernova 2002 World Cup Football containing a syntactic foam layer. Photograph by Warrenski, distributed under a Creative Commons Attribution-Share Alike 2.0 Generic license
Glass vs plastic

The mechanical behaviour of materials can be defined in terms of stress (the force applied to the material per unit area) and strain (the deformation in the material in response to stress).

Viscoelastic materials combine the behaviours of viscous fluids and elastic solids when deformed. A plot of stress vs strain shows hysteresis, where the unloading (reverse) curve follows a different path to the loading (forward) curve, because some energy is lost to the system as heat:
Picture
Stress (σ) vs strain (ε) for an elastic (a) and a viscoelastic (b) material. The red shaded hysteresis loop shows the energy dissipated as heat in the viscoelastic case. Image credit: Gene Settoon This work has been released into the public domain by its author at English Wikipedia.
Under increasing compression, syntactic foams with glass or ceramic microspheres typically respond in three stages:
  1. Elastic behaviour.
  2. Crushing of the microspheres (a region of low stiffness).
  3. Densification: the cavities fill up with debris from the crushed microspheres.

Stages 2 and 3 correspond to catastrophic damage to the microspheres, so glass microspheres are inappropriate for applications where foams will be under high strain.

Where plastic microspheres are used, the response of the material to compression also has three stages, but with less definition in-between:
  1. A small region of elastic behaviour.
  2. Buckling of the microsphere walls (a low stiffness region).
  3. Densification: the microsphere walls begin to touch.

The mechanical properties of syntactic foams with glass microspheres are well documented, but less work exists on plastic microsphere syntactic foams.
Our study
​

In this study we manufactured and tested polyurethane (PU) syntactic foams containing two grades of polymer microsphere.  The foams contained a 2%, 10% or 40% volume of microspheres.
Picture
Scanning electron microscope images of syntactic foams with microsphere grades 551 and 920.
The syntactic foams were tested up to medium (25% and 50%) strain alongside unfilled polyurethane.  Samples were compressed and unloaded five times and the response measured.
The foams showed viscoelastic behaviour, with hysteresis in the stress-strain curves.  Samples with a low volume of microspheres showed similar behaviour to unfilled PU, but increased proportions of microspheres led to some different results.  The 10% and 40% samples exhibited stress softening, where a smaller force is needed to achieve the same deformation in successive loadings.  This may be due to the buckled microspheres not fully recovering between load cycles.

After 1 week, the samples were re-tested.  They showed the same behaviour and little or no change in thickness, indicating that they had fully recovered from the previous testing.
Picture
Schematic showing the pattern adopted for cyclic compression testing.
Testing up to high strain (70%) revealed interesting behaviour.  Unfilled PU and foams with low concentrations of microspheres were damaged, but samples with higher concentrations of microspheres showed damage resistance.   Foams with 10% microsphere content were damaged only in the last of the 5 load/unload cycles, and foams with 40% appeared intact after test.
Picture
Stress-strain curves for unfilled PU and syntactic foams tested up to 70% strain. Inset: Images of the foams after testing. Damage appears reduced in foams with increasing content of polymer microspheres.
​Microscopy reveals cracks in the polyurethane in the damaged samples.  The microspheres mitigate the damage by presenting a barrier to crack propagation.  This is the opposite result to the case with glass and ceramic microspheres, where foams with higher volume fractions are less resistant to damage, due to the brittle nature of the microspheres.
Picture
SEM images of unfilled PU and syntactic foams with 2%, 10% and 40% polymer microspheres, after testing to 70% strain. The damage is significantly reduced in syntactic foams containing higher proportions of microspheres.
These results indicate that polymer-filled syntactic foams containing higher volume fractions of microspheres can show good elastic recovery and significant damage resistance.  They may provide an excellent alternative to glass and ceramic containing syntactic foams for applications that require low density materials with high resistance to damage under strain.
 
Read the full article here: doi.org/10.1016/j.compositesb.2020.107764

[1] Plastic Foam Developed for Boats and Planes, The Science News-Letter, 1955, 67(14),  213
[2] N. Gupta, S.E. Zeltmann, V.C. Shunmugasamy, D. Pinisetty Applications of polymer matrix syntactic foams, JOM, 2014, 66(2), 245-254
[3] For explanation of the anatomy of the 2014 Adidas Brazuca World Cup football: www.livescience.com/46299-microscopic-analysis-brazuca-world-cup-ball.html
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    The Mathematics of Waves and Materials group are are a research group in the Department of Mathematics at the University of Manchester.  We work on the theoretical, numerical and experimental aspects of both materials and waves. See our research page for more information and details.

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