ABSTRACT
We propose the concept of periodic cellular materials with programmable effective properties and present initial results from a computational study of a prototypical material that exhibits this behavior. Nonlinear FEA shows that programmed geometric imperfections at the cell level can be used to modify the effective compressive storage modulus of shape memory polymer (SMP) based periodic cellular materials after they have been manufactured. The ability of SMPs to freeze a temporary deformation for an extended period of time and the low modulus of these materials in the rubbery regime allow us to freeze controlled and reversible imperfections at the cell level following the typical temporary shape programming process for SMPs. Small geometric imperfections (2% global strain) are observed to produce variations of up to 40% in the effective initial compressive storage modulus in the prototypical material.
KEYWORDS: cellular materials, smart materials, shape memory polymers, morphological imperfections
INTRODUCTION
Cellular materials
These materials comprise an assembly of cells with solid edges or faces, packed together to fill space [1]. They combine properties inherited from the base material with those imparted by the cellular (micro-) structure to yield unique combinations of properties that are not seen in solid materials. Examples of unique property combinations reported in cellular materials include high stiffness per unit mass, high compressive failure strains at nearly constant stress, low thermal conductivity along with high mechanical strength at elevated temperatures, high surface area per unit volume along with high mechanical strength, low density, etc. These properties are driving growing use of cellular material forms in industries ranging from medicine (lightweight and biocompatible scaffolds) to transportation (padding for occupant and pedestrian protection) to chemical processing and waste management (catalysts structures) to aerospace and construction (lightweight sandwich panels) [1,2].
Cellular materials with programmable effective properties
Once a cellular material is manufactured, it’s properties are fixed. This work is motivated by the question: can we create cellular materials ‘whose effective properties can be adjusted after they have been manufactured? If this can be done without complete reprocessing of the materials, the range of application of cellular materials can be increased significantly. Facile adjustment of effective properties will allow customized instances of a cellular material to be derived cost-effectively from a common mass-produced cellular primitive. A reduction in cost driven by economies of scale and the ability to customize the effective properties of a material for a particular application has the potential to change how and where these materials are used. The objective of this paper is to study the feasibility of modifying the effective mechanical properties of cellular materials after they have been manufactured by imposing controlled morphological imperfections at the cell level.
Morphological imperfections in cellular materials
Unavoidable morphological imperfections cause the properties of both natural and commercial cellular materials to deviate from those of their regular idealizations. As an example, these imperfections are responsible for an order of magnitude reduction in the deviatoric strength and greater reductions in the hydrostatic strength of commercial metal foams [3, 4], Morphological imperfections are broadly classified into periodic, that are repeated in every cell, and random ones [5], Examples of periodic morphological imperfections include the rounding of corners and under or over expansion in honeycombs [6], curved or wavy cell walls [7] and cell walls with non-uniform thicknesses [5], The first mentioned imperfection arises due to permitted variations in the manufacturing process used to create Al honeycombs. Pressure variations across adjacent cells during solidification of a foam or the pressure exerted by a blowing agent used to foam a melt give rise to curved cell walls or edges [7], Local buckling caused by premature handling of foams during the solidification process is known to cause waviness in the cell walls [7], Random morphological defects include missing or torn cell walls, misaligned cell walls, missing cells and non-uniform cell sizes. These, too, are caused by insufficient control over the manufacturing process or by errors in material handling. The dramatic changes in effective properties caused by morphological imperfections are attributed to a change in the deformation mode from a stretching-dominated one to a bending- dominated one. However, not all morphological imperfections result in reductions of effective properties: a 10% under expansion of honeycombs has been known to increase the Young’s modulus and plastic collapse stress by approximately 50% and 10% respectively [6]; a 10% compression of a commercial Al foam to cause rupture of some cell walls is shown to increase the sound absorption of the foam across the entire audible spectrum [2], Despite this, most of the past work in this area studies morphological imperfections in cellular materials with a view to mitigate their deleterious effects. We treat morphological imperfections as material design variables whose magnitude and spatial distribution may be chosen to modify the properties of a cellular material in a desired manner.
Shape memory polymers
Shape memory polymers (SMP) undergo a large change (e.g. a factor of 10 – 100) in their storage modulus over a relatively narrow temperature range (e.g. 20-50°C) when they are heated above a characteristic switching temperature. This is the consequence of thermally induced rupture of the low energy cross-links (known as molecular switches) between the polymer chains [8], On further heating, this soft or rubbery state persists for a finite temperature range (15-30°C) before the material decomposes or is irretrievably damaged. Elastically recoverable strains in excess of 50% may be imposed on the material in the rubbery state as the relative motion between polymer chains is restrained only by a small number of high energy crosslinks (known as netpoints) in this regime. If the deformed material is cooled to a temperature below its switching temperature while maintaining the imposed deformation, the deformation is ‘frozen’ in the material due to the re-formation of the low energy cross-links. In the absence of external loads the frozen deformation has been known to persist for extended periods of time (e.g. 6 months under zero external stress in a polyurethane foam, [9]). If the material with a frozen-in deformation is reheated to above the switching temperature in the absence of external loads, the frozen deformation is lost when the low energy cross-links rupture due to thermal activation. As these cross-links cleave, the strain energy stored in the material during its deformation in the rubbery state drives the recovery of its original (i.e. un-deformed) configuration. In addition to their ability to store a temporary deformation, SMPs also possess the following attributes which make them uniquely suited for the study of programmable cellular materials: (a) the switching temperatures for many commercially important SMPs are low (30-100°C) and can be readily achieved in thermal chambers of standard universal test machines, (b) a sample can be reused for the study of different imperfections, and (c) they have a low storage modulus (1-10 MPa) in the rubbery state which makes the imposition of controlled geometric imperfections easier than in metallic cellular materials with a high storage modulus.
Outline
The rest of this paper is organized as follows. The selection of a cell topology, imperfection type and the resulting design space are discussed in Section 2. Section 3 covers the process of programming imperfections at the cell level. A description of the finite element (FE) analysis procedure is given in Section 4. The results of these simulations are presented and interpreted in this section as well. The paper concludes with a summary and limitations of the reported work, and outlines the scope for further work.
THE DESIGN SPACE
Unit cell
After its relative density, the properties of a cellular material are influenced most by the topology and shape of the unit cells; size is less important [1]. Cellular materials deform by bending or stretching of the cell walls, or by a combination of these deformation modes. Stretching deformation requires significantly higher energy than bending for cells with thin walls, therefore low relative density foams loaded in a manner that emphasizes stretching are stiffer and stronger than foams of the same relative density that are loaded in a manner that emphasizes bending [10], Deshpande et al. [10] use Maxwell’s criterion for static determinacy of a pin jointed frame to establish topological criteria for stretch dominated behavior in 2- and 3-D foams. The resulting topologies are rigid in the Maxwell sense. They attain the Hashin-Shtrikman upper bounds for shear and bulk moduli of 2-phase isotropic composites, thereby maximizing the stiffness to weight ratio. This work also mentions minimally or partially stretch- dominated topologies that cannot attain the Hashin-Shtrikman bounds and hence would be unsuitable for optimally lightweight and stiff cellular materials. However, such topologies offer the potential of allowing the effective properties of the cellular material to be modified by changing the relative contribution of bending and stretching to the deformation experienced by the material in normal use.
Fig. 1 Cellular material (a) is composed by stacking the unit cell (b) in two dimensions
Our choice of unit cell trades optimal stiffness (or strength) to weight performance for the ability to vary its performance after manufacture. Figure i shows a partially stretch dominated, closed unit cell described in [10] that is used in this work. The diagonal brace makes the unit cell partially rigid in the Maxwell sense and imparts anisotropy to the effective properties of the resulting material. The relative density for this unit cell can be expressed in terms of the parameters shown in Fig. 1(b) as
for low values of
We use eq. 1 for the relative density in this work as we only consider cellular materials with relative densities in the [0.05, 0.2] range. For a given aspect ratio (^i), the relative density is linear in %2 as is usual for closed cell foams [1]. The initial storage modulus for this material as obtained from beam and truss theory is given by
Geometric imperfections
Chen et al. [5] note that of the common morphological imperfections in metal honeycombs, fractured cell walls produce the largest reduction in yield strength followed in order by missing cells, wavy cell walls, cell wall misalignment, dispersion in cell sizes and non-uniformity in cell wall thicknesses. Of these, only waviness and misalignment of cell walls are reversible. Waviness or cell wall curvature can be induced at the cell level by loading the material in compression to produce Y-periodic (i.e. in every cell) buckling, while cell wall misalignment can be produced by loading the material in shear.
In this work we focus on reversible geometric imperfections corresponding to the first Y-periodic buckling mode. This approach is feasible when the critical buckling mode in the cellular material is local (i.e. the characteristic length of the resulting displacement field is comparable with the characteristic dimension of a unit cell) instead of global (i.e. characteristic length is much larger than a unit cell). Infinite planar rectangular grids with low relative density and slender edges have a critical buckling mode that is local [11]. In general, this holds for other cell topologies as well. If the lowest buckling mode is local but is separated from the next higher buckling mode by a small margin, these modes can interact to produce a deformation field that is not Y-periodic [12], Therefore, the above choice of a geometric imperfection works only when the chosen material exhibits a Y-periodic critical buckling mode and there is sufficient separation between the critical and next higher buckling modes. Other approaches for imposing controlled geometric imperfections may be explored when the above requirements are not met; these will be left for future study.
Design space
The aspect ratio fa) and thickness ratio fa) are two dimensionless parameters that describe the shape of an idealized cell i.e. a cell without a geometric imperfection. We restrict all members of the cell to have the same thickness (/). Low thickness ratios describe cells with a low relative density (see Eq. 1) and a low storage initial modulus (see Eq. 2). These have a lower critical buckling load than cells with a higher thickness ratio. Given the linear relationship between the thickness ratio and the relative density (see Eq. 1) and the primacy of relative density as a determinant of properties in cellular materials, we retain the relative density instead of the thickness ratio in all results.
The shape of the chosen geometric imperfection is given by the first local buckling mode. The magnitude of this imperfection is characterized by the post-buckling global strain (eg p). This dimensionless parameter is chosen to characterize the shape of a cell with a geometrical imperfection as it arises naturally out of the imperfection programming process described in the next section.
The design space is defined by the triplet {pr, n-[, egp}. The relative density is restricted to low relative density cellular materials (pr e [0.05,0.2]) to ensure that the critical buckling mode is local rather than global as discussed earlier. The magnitude of the imperfection is limited to small values (egiP 6 [0,10]%) so that the critical buckling stress is greater than the normal operating stress. This permits the material with the imperfection to have a finite linear elastic regime under normal operating conditions. The aspect ratio is restricted to n1 e [0.5,1.5] because cells aspect ratios outside this range are more difficult to manufacture.
IMPERFECTION PROGRAMMING
The standard thermo-mechanical process used for freezing temporary deformations in SMPs is used to impose geometrical imperfections at the cell level in this work. This is illustrated schematically in Fig. 2. We use a polyurethane based thermoplastic SMP of the polyester polyol series from Mitsubishi Heavy Industries in this work [13], It has a switching temperature of 55°C corresponding to a glass transition in the material; its storage modulus below and above the switching temperature is 907 MPa (T 6 [20,40]°C) and 27.7 MPa (T 6 [70,100]°C) respectively. The parts of the process where the material is at 70°C can be identified by the red line, while the parts where the material is at 25 °C are plotted using a blue line in fig. 2.
Fig. 2 Process for programming a geometrical imperfection in an SMP-based cellular material
The imperfection programming process starts at the point 1 where the material is under zero stress and is in its rubbery state at a uniform temperature of 70°C.. It is then deformed in a universal test machine under displacement control to 2 where a compressive strain of approximately 4% is imposed on it. As the storage modulus of the material is quite low in the rubbery regime, it undergoes elastic buckling at approximately 2% strain. The cross-head of the universal test machine is then held fixed to hold the strain constant while the material is cooled uniformly to 25° C. The increase in modulus corresponding to the transition from the rubbery to the glassy regime causes the stress in the sample to increase sharply at point 3.
Thermal contraction and stress relaxation in the viscoelastic material counteract this stress increase leading to a moderate jump in stress from point 2 to 3. The material is then unloaded at a uniform temperature of 25°C to 4. Some elastic strain recovery occurs during the unloading resulting in a material with a Y-periodic geometric imperfection corresponding to a global strain of approximately 3.75% at 4. The material with imperfect cell geometry is now ready for evaluation of its effective properties at 25°C. A different magnitude of the imperfection can be imposed on the material subsequently by heating it to 70°C under zero stress (point 5) and repeating the programming process described above.
COMPUTATIONAL STUDY
Finite element analysis
Boundary conditions
The computational effort of simulating the mechanical response of a specimen of a periodic cellular material can be reduced significantly by exploiting the periodicity of the material. Significant insight into the behavior of a large specimen comprising multiple unit cells can be obtained with a modest computational effort by simulating a representative volume element (RVE) of the material instead of the whole specimen [6], Analysis of a suitable RVE is especially effective for determining the elastic properties such as the initial compressive modulus, the critical buckling stress and the propagation stress. We choose an RVE (see Fig. 3) and a set of periodic boundary conditions that satisfies the conditions established in [6], Periodic boundary conditions represent the boundary conditions that would apply to an RVE, which is sufficiently distant from the boundaries, if it were to be isolated from the rest of the material. They impose the requirements of periodicity and continuity of the displacement and traction fields across adjacent RVEs. This ensures that the stress and strain fields obtained from analyzing an RVE can be arrayed and ‘stitched’ together without gaps or overlaps to form the stress and strain fields in the overall material. We impose periodic boundary conditions using a linear multi-point constraint following the Dummy Node Method [14],
Elements and discretization
A fine mesh of approximately 2000 elements is generated for the RVE. Four noded plane strain elements (CPE4 elements in Abaqus/Standard) with bi-linear interpolation of the displacement field are used for all simulations because the out-of-plane dimension of the simulated materials is much greater than the in-plane thickness of the RVE segments. A fine mesh is used to capture the onset of local yielding in the material.
The geometrically nonlinear FE analysis is terminated at the onset of plastic yielding in the RVE or on reaching a global strain of 10%. Large deflection simulations using a hyperelastic material response showed that this model was capable of reproducing the complete three-phase response commonly seen in cellular materials [6].
However, the simulations reported here were limited to a linear elastic material response and small local strains. No unstable behavior (e.g. snap through) or contact between segments was encountered in these simulations. A family of stress-strain responses for a specific cell design ({pr=0.2, K\= 0.625}) corresponding to different levels of geometric imperfection are shown in Fig. 3. As with most low density polymeric foams, these material designs also exhibit an initial linear elastic response that ends in elastic buckling.
Fig. 3 Normalized compressive stress-strain responses for a cellular material designwith different levels of geometrical imperfections 
Results
The initial (zero external strain) compressive storage modulus of the cellular material (£*) normalized by that of the solid material (Es) is plotted in Fig. 4a for different relative densities and programmed imperfection magnitudes. The analytical result for the ideal (or perfect) cell geometry from Eq. (2) is overlaid for comparison. The moduli of both: the perfect and imperfect geometry RVEs increase approximately linearly with the relative density. There is a significant drop in the storage modulus corresponding to the lowest simulated imperfection level (egp =2%) for all relative densities. The reduction in the storage modulus decreases for higher levels of the imperfection. Figure 4b shows the variation of the normalized initial compressive modulus with the magnitude of the programmed geometric imperfection for different values of the aspect ratio. The initial axial stiffness of slender beam columns with a small initial imperfection corresponding to the first buckling mode is known to vary inversely as the amplitude of the imperfection [15], As the global strain (eg p) used for programming the imperfection is a linear function of the amplitude of the imperfection for small values of the global strain, the results shown in Fig. 4b are consistent with the above observation.
Fig. 4 Variation of the normalized, initial, compressive storage modulus with respect to (a) the relative density for different levels of programmed geometric imperfections at ^=0.625 and (b) the magnitude of programmed geometric imperfections for different aspect ratios at pr=0.2
The curvature of cell walls resulting from programming of the imperfection results in a change in the relative density of the material. As the properties of a cellular material are strongly correlated with its relative density, the properties of the imperfect geometry specimens should be expressed in terms of the changed relative density [7], However, the change in density due to cell wall curvature is small for low values of eg p. We use the relative density of the perfect geometry cell to simplify graphical representation of the variation in properties that can be produced by imposition of various geometric imperfections.
CONCLUSION
Nonlinear FE simulations show that programmed geometric imperfections at the cell level can be used to modify the effective storage modulus of an SMP-based cellular material. Y-periodic buckling is used to program an initial curvature in the walls of a unit cell in a prototypical material. A curvature corresponding to 2% global strain in this material resulted in a 40% reduction in the effective compressive storage modulus. These preliminary results open the door to the development of programmable cellular materials using smart materials for reversible changes in effective mechanical properties.
Programmed morphological imperfections, like unintentional imperfections, cause a reduction in the effective storage moduli of cellular materials. This disadvantage in performance can be offset by a) the flexibility to modify the effective properties of the material after it has manufactured or b) a lower cost due to the ability to derive materials with several different effective properties from a common mass produced cellular primitive. Future effort will focus on the study of other properties: mechanical (e.g. critical buckling stress, propagation stress, storage moduli at non-zero strains, fundamental frequency), thermal (conductivity), electrical (conductivity), acoustic etc. The programmed geometric imperfections were assumed to persist indefinitely at temperatures below the switching temperature in this work. Further work will address the evolution of the programmed imperfection within the context of a viscoelastic material model. Interactions between programmed and randomly distributed unintentional morphological imperfections will also be considered in future analyses.
