Biologically inspired crack delocalization in a high strain-rate environment

Abstract

Biological materials possess unique and desirable energy-absorbing mechanisms and structural characteristics worthy of consideration by engineers. For example, high levels of energy dissipation at low strain rates via triggering of crack delocalization combined with interfacial hardening by platelet interlocking are observed in brittle materials such as nacre, the iridescent material in seashells. Such behaviours find no analogy in current engineering materials. The potential to mimic such toughening mechanisms on different length scales now exists, but the question concerning their suitability under dynamic loading conditions and whether these mechanisms retain their energy-absorbing potential is unclear. This paper investigates the kinematic behaviour of an ‘engineered’ nacre-like structure within a high strain-rate environment. A finite-element (FE) model was developed which incorporates the pertinent biological design features. A parametric study was carried out focusing on (i) the use of an overlapping discontinuous tile arrangement for crack delocalization and (ii) application of tile waviness (interfacial hardening) for improved post-damage behaviour. With respect to the material properties, the model allows the permutation and combination of a variety of different material datasets. The advantage of such a discontinuous material shows notable improvements in sustaining high strain-rate deformation relative to an equivalent continuous morphology. In the case of the continuous material, the shockwaves propagating through the material lead to localized failure while complex shockwave patterns are observed in the discontinuous flat tile arrangement, arising from platelet interlocking. The influence of the matrix properties on impact performance is investigated by varying the dominant material parameters. The results indicate a deceleration of the impactor velocity, thus delaying back face nodal displacement. A final series of FE models considered the identification of an optimized configuration as a function of tile waviness and matrix properties. In the combined model, the optimized configuration was capable of stopping the ballistic threat, thus indicating the potential for bioinspired toughened synthetic systems to defeat high strain-rate threats.

1. Introduction

Biological materials have attracted the attention of researchers in both scientific and engineering disciplines owing to their unique structural and mechanical performance. An extensive review by Fratzl & Weinkamer [1] offers engineers an insight into the potential that these materials offer for the ongoing design of next-generation engineering structures. The adaption of material hierarchy [1,2], multi-functionality [3], morphing [4] and the potential for self-healing through embedded vascularity [5–7] are all active research areas within engineering, inspired by the behaviour and performance of biological materials.

In structural applications, the hierarchical nature of biological materials often yields superior mechanical performance from relatively mundane constituents (for example, bone, nacre and wood [1]). In particular, the composition and structure of nacre are of significant interest for crack delocalization as it comprises a brittle ceramic (aragonite, CaCO3, 95% vol.), arranged together with a small amount (5% vol.) of softer organic biopolymers [8]. This arrangement leads to fracture toughness values orders of magnitude greater than those of the base material aragonite, i.e. nacre is 3000× tougher than aragonite. Figure 1 illustrates the complex hierarchical structure of nacre. Many authors [9–15] have proposed various nanoscale-toughening mechanisms to account for this improved toughness via control of the interfaces between individual tiles: (i) biopolymer stretching; (ii) contact between aragonite asperities; and (iii) aragonite bridges which unlock/relock after shearing. As illustrated in figure 1f, individual tiles exhibit a unique tile waviness of ‘dovetail’-like features at the periphery. In a quasi-static tensile loading situation, the nacre tiles are stretched such that the tapered edges lock and their interfaces are subjected to normal compression [15], and hence resistance to further tile pull-out. The force equilibrium at the interfaces requires tensile tractions in the core of the tiles which could, through the appropriate selection of materials, yield a ‘double-knee’ in the stress–strain response [15]. This ‘interfacial hardening’ was recently captured by Espinosa et al. [10] in a scaled-up model using a synthetic material (acrylonitrile butadiene styrene). This feature can be considered as strain-stiffening and is not typically found in conventional engineering materials, especially in brittle systems such as engineering ceramics or fibre-reinforced polymer composites, which generally fail catastrophically at maximum strength.

Figure 1.

The multi-scale structure of nacre: (a) inside view of shell; (b) cross section of a red abalone shell; (c) schematic of brick wall-like microstructure; (d) optical micrograph showing tiling of tablets; (e) scanning electron microscope of fracture surface; (f) transmission electron microscopy showing tablet waviness (red abalone); (g) optical micrograph of nacre from fresh water mussel (Lampsilis cardium) and (h) topology of tablet surface using laser profilometry [9]. (Online version in colour.)

In nacre, the unique viscoplastic deformation of the organic interface and the crack delocalization of the layered microstructure of the inorganic aragonite leads to an increase in mechanical strength of the biocomposite above the baseline monolithic aragonite constituent. This is clearly shown in the stress–strain response in figure 2 and the extensive crack delocalization pattern that is generated by this type of structural geometry. Also highlighted is the importance of hydration for the tiles and the biopolymer adhesive; a feature not yet considered in any bioinspired variant.

Figure 2.

Effects of discontinuity found in nacre with (a) stress–strain curve of nacre in tension along the tiles, (b) transverse strain as function of longitudinal strain, (c) schematic of tile sliding and separation and (d) micrograph of a tensile specimen showing that all the potential sliding sites were activated [16]. (Online version in colour.)

An exact representation of natural materials, such as nacre, within finite-element analysis software packages is challenging and cumbersome owing to the structural and material complexity (i.e. multi-level hierarchy and varying mechanical and geometrical properties [17]) and the complicated kinematics that contribute at all length scales to the overall mechanical performance of the system. Consequently, the unification of all contributory effects into a constitutive material law, which leads to the prediction of mechanical behaviour of bioinspired structures via finite-element methods, is non-trivial and implies compromises when considering single mechanical effects.

Existing modelling approaches focus on the understanding of the exact kinematics and force distributions at the micro-scale [10,13,16,18,19]. For an engineering approach, the focus is to use novel concepts which ultimately lead to load and damage redistribution in a quasi-static or dynamic condition. This study investigates the response of a ‘damage-tolerant’ bioinspired system (evolved to counter a low strain-rate threat environment) to a high strain-rate environment, typical of a ballistic impact. The aim is to characterize the role of various microstructural features and investigate, via finite-element methods, the potential for a nacre-like discontinuous arrangement of engineering materials to counter a high-velocity impact threat. The features of nacre considered within the dynamic model are (i) tile interlocking, (ii) tile aspect ratio, and (iii) tile waviness. This paper describes the modelling methodology applied and the outputs from the parametric study of geometric and material properties with a recommendation for an optimal combination. Finally, it should be recognized that the work reported herein is inspired by the geometrical features found in nacre and the system's ability to ‘diffuse’ critical cracks and generate a form of ‘graceful degradation’—a very desirable characteristic for materials subject to a high strain-rate environment.

2. Finite-element model description

2.1. Modelling methodology

The finite-element (FE) model methodology employed in this study uses a discrete representation of the discontinuous material approach found in nacre. As the influence of multiple geometrical aspects on the impact performance is critical to this study, a programme was established to automate the finite-element mesh generation with respect to the tile size, aspect ratio, waviness and overall panel dimension. The connectivity between multiple tiles is established using cohesive elements (figure 3a), whereas a single tile within the arrangement is discretized by 16 solid elements (figure 3b).

Figure 3.

Schematic of (a) assembled panel and (b) single tile connectivity. (Online version in colour.)

In contrast to the biological counterpart, the tile geometry in this work is limited to square tiles rather than hexagonal (this will be considered in subsequent publications). Throughout this investigation, an aspect ratio of 20 × 20 × 1 mm was used for the tile geometry, which concurs with the average aspect ratio found in nacre [9]. While the connectivity within a single layer is accomplished by associating elements of neighbouring tiles via cohesive elements, the link between two layers using the same approach becomes cumbersome and could lead to high initial element deformations, especially when the layers are shifted relative to each other with a magnitude of half a tile length. Additionally, the resin interface between the layers is discretized with a higher number of elements to represent the occurrence of delaminations (defined as localized separation between tile and adhesive) with greater accuracy. To overcome the layer connectivity problem within the model, a tied connection is defined between the touching element surfaces of the tile and cohesive interface, respectively. Accordingly, the contact definition *CONTACT_TIED_SURFACE_TO_SURFACE available in LS-DYNA [20] is used in combination with segment set definitions extracted from the element faces. The use of the tied contact allows the establishment of connectivity between the two layers which is independent of the used mesh density or the tile shift.

2.2. Representation of tile waviness

At the mesoscale level, the staggered tile arrangement of nacre is the visually dominant feature of the structure. However, it has only recently been observed that the arrangement of the tiled layers is also subject to waviness [14]. This additional feature has a supplemental influence on dissipating the point of failure by enhancing the kinematics for tile interlocking. From a modelling perspective, the waviness is introduced to the FE model by shifting the nodal coordinates in the through-thickness direction. Therefore, a triangular wave function is used which is defined as follows:

2.1

Here t represents the coordinate and a is the period of the wave function. Furthermore, the parentheses symbol ⌊n⌋, used in equation (2.1), applies the floor function to its contents n. Although a sine function might be a more natural choice, the triangular wave function was chosen with respect to the mesh assembly and the need to avoid initial penetration of the nodes belonging to the interface regions. For the FE model, the triangular wave function is superimposed in the x- and y-directions and the amplitude of the wave function is governed by a scaling factor. In this study, the same scaling factor for the amplitudes in x- and y-direction is applied. Figure 4a,b illustrates a section through the generated mesh for the tiles and the resin interface, respectively.

Figure 4.

Section through the panel for visualization of (a) tile waviness, (b) distribution of cohesive elements, (c) tile waviness effect on element geometry and (d) schematic of individual tile indicating variable geometrical parameters. (Online version in colour.)

In order to achieve the assembly as shown in figure 4a,b, the tile shape is modified as displayed in figure 4c. Here, the tile thickness is 0.4 mm at the centre and increases gradually towards the outer edges until a thickness of 1 mm is reached. Figure 4d illustrates the sectional schematic of a profiled tile, where the minimal thickness is located in the centre of the tile and allows the angle of inclination to be measured by α.

2.3. Impact model assembly

The assembled finite element consists of an impactor, a support and the panel. For runtime efficiency, the model is designed as a quarter model with applied symmetry boundary conditions for the panel. For the simulations, an impactor with a radius of 4.5 mm and a mass of 8 g are used [21]. The impactor mesh (figure 5a) is generated in MSC. Patran [22] and discretized with 1410 nodes and 5590 elements. The mesh is exported to the LS-DYNA format and assigned to a rigid body material while the initial velocity in z-direction was 398 ms⁻¹ [21]. In a similar manner, the support (figure 5b) is also generated in MSC. Patran and exported to the LS-DYNA format. The part is discretized by 13 824 elements and 16 393 nodes and is modelled as a rigid body.

Figure 5.

Discretization of (a) the impactor, (b) the support and (c) the assembled quarter model. (Online version in colour.)

The separate parts are positioned and arranged as illustrated in figure 5c to represent a quarter model of the impact scenario. To ensure the physically correct behaviour of the model, three contact definitions of type *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE are defined. The contact definitions cover the combinations between (i) impactor/panel, (ii) support/panel, and (iii) self-contact for the panel. The self-contact covers the contact pairings within the panel, e.g. tiles/resin or tiles/tiles, but also addresses interactions between possible debris and the rest of the panel.

The panel is constructed as a discrete representation of nacre. A material model for a boron carbide (B₄C) ceramic is assigned to the solid elements that represent the tiles. B₄C was chosen, as this is often used for ballistic protection. For the resin behaviour, a user-defined cohesive material model is assigned to the elements located in the gaps between tiles and the interfaces [23]. Throughout this work, the overall dimensions of the panel are 40 × 40 × 7.45 mm. With the fixed tile geometry of 20 × 20 × 1 mm, this corresponds to an arrangement with seven layers of B₄C tiles. Thus, the panel is discretized with 45 802 nodes and 21 266 constant stress solid elements (ELFORM 1). In addition to the tiled panel, a single block of B₄C with the same overall dimensions was generated to demonstrate the effect of the discontinuous arrangement. In this case, the panel volume was meshed with 48 000 constant stress solid elements (ELFORM 1) corresponding to 52 111 nodes.

2.4. Material models

Three different material descriptions are used within the FE model (see table 3). The impactor and the support are modelled as rigid bodies; hence for both parts, the *MAT_RIGID material model is used. Parts associated with this material model do not undergo deformation. In case of the impactor, this assumption can be regarded as the worst-case scenario as any deformation of the impactor would be an additional form of energy dissipation. This rigid body material model requires the input of the density, elastic modulus and Poisson ratio. For the model, a density of 7186 kg m⁻³ is used while an elastic modulus of 210 GPa and a Poisson ratio of 0.3 are provided.

For a single ceramic plate, the impact causes compressive shock waves which propagate through the material. Once a free surface on the back face is reached, these waves are reflected back as tensile waves. Consequently, the ceramic material fractures if the magnitude of the reflected tensile wave exceeds the dynamic tensile strength [24]. Radial cracks are formed at the back of the structure owing to the tensile waves and then propagate forwards to the front of the plate. Simultaneously, a fracture cone forms in the impact zone and propagates through the structure towards the back. In the case of the tile arrangement used in this study, where multiple tiles are connected via a matrix, the pressure wave is partially transmitted through the resin region while the complementary part is reflected back into the ceramic plate. The proportion of transmitted pressure wave depends on the attenuation capability of the matrix and the thickness of the adhesive layer. In this study, the Johnson–Holmquist [25,26] constitutive model was used to describe the behaviour for ceramic materials under impact conditions. This model is capable of representing the material in terms of (i) intact and fractured strength, (ii) pressure/volume relation, and (iii) damage state [24]. The parameter set for B₄C [27] is detailed in table 4 below.

For quasi-static problems, LS-DYNA offers a large number of material models [28]. However, under impact conditions, a strong strain-rate dependency can be observed for polymer matrices [29,30]. Furthermore, the strain rate varies spatially and temporally under impact conditions. To obtain accurate behaviour, through a constitutive material model, the addition of viscous terms is required [28]. While such a material description requires an extensive number of parameters, the polymer response in the current work is captured using cohesive elements to address the delamination damage between the tiles. The connectivity of the tiles via the cohesive elements is established by solid elements with a finite thickness. Although LS-DYNA offers cohesive element formulations, a user-defined constitutive material model [23,31] with compressive strength enhancement and additional output options is used in this study. The cohesive law is defined via a bi-linear curve, which is defined by the parameter set listed in table 5 below.

3. Impact simulations

In the following sections, results are shown for simulations addressing: (i) the performance of a continuous versus a discontinuous material, (ii) the influence of resin properties, and (iii) the effect of tile waviness. Finally, the best-performing configuration of the resin property study and the tile waviness variation is combined and results are outlined.

3.1. Influence of material discontinuity

The impactor displacement and velocity profile evolution in time for a high-speed impact on a continuous and a discontinuous material are indicated in figure 6. The results indicate that for the discontinuous material, the configuration of flat tiles and base resin is capable of reducing the impactor velocity more effectively, with a constant impactor velocity of 65 ms⁻¹ attained after 10 µs. For the continuous monolithic block of B₄C, the time to reach a constant velocity is significantly longer and the impactor velocity is reduced to a much higher constant value of 297 ms⁻¹ after 30 µs. Figure 6 clearly indicates the success of the flat tile arrangement at absorbing the energy over the continuous material.

Figure 6.

Comparison of impactor displacement and velocity profiles. (Online version in colour.)

Although the impactor penetrates, the panel in both configurations the damage profiles differs greatly. In the case of the single block of B₄C, the impactor generated very localized damage while the flat tile assembly developed a significantly wider damage zone. Figure 7 shows the velocity resultant and plastic strain states for the continuous and discontinuous arrangement at the point in time where a constant impactor velocity is achieved. For the single block shown in figure 7a,c, the impactor passes through the back face of the panel, whereas for the discontinuous arrangement (figure 7b,d), the impactor has dispersed most of the initial velocity over a larger influence zone. However, it should be noted in the latter case that while the impactor has a residual velocity of 65 ms⁻¹, the tile debris travels with velocities up to 400 ms⁻¹.

Figure 7.

Velocity resultant plot of the impact simulation for (a) single monolithic block and (b) flat tile arrangement; contour plot of plastic strain for (c) single monolithic block and (b) flat tile arrangement.

3.2. Variation of resin properties

The influence of moisture content on the stress–strain properties of nacre has been reported in the study of Barthelat et al. [16] (figure 2a,b). The authors illustrate the importance of an active hydrated protein to suppress damage formation within the global structure. Clearly, the mechanical properties of the adhesive system play a critical role in the defence mechanism employed. Consideration of these properties will now be given to the engineering analogy.

To investigate the influence of the matrix properties on the residual impactor velocity, the fracture toughness values of the base resin are varied uniformly for the entire structure in the absence of any tile waviness. Table 1 lists the resin properties used in this comparative study, where the properties are changed by various orders of magnitude.

Table 1.

Parameter sets for resin property variation.

parameter	unit	base [32]	÷100	÷10	×10	×100	description
GIC	J m−2	200	2	20	2000	20 000	mode I energy release rate
GIIC	J m−2	1000	10	100	10 000	100 000	mode II energy release rate
velocity	ms−1	65	18	34	57	42	residual velocity

In table 1, the residual velocities of the impactor are compared. The lowest residual velocity of 18 ms⁻¹ is obtained from the matrix with the lowest fracture toughness. Velocities in the range of 34–57 ms⁻¹ are obtained for other variations, while still performing better than the baseline matrix. The velocity profiles for the different configurations are illustrated in figure 8. The configurations with factor ×10 and ×100 describe a similar profile as the base matrix up to 90 µs. After this point in time, the profiles diverge to exhibit a delayed onset but increase in velocity before arriving at the plateau level for a constant residual velocity in a much shorter time span compared with the other configurations. The residual velocity is also reduced when compared with the baseline material.

Figure 8.

Impactor velocity evolution for different resin properties. (Online version in colour.)

3.3. Influence of tile waviness amplitude

A unique feature of nacre concerns the incorporation of tile waviness as a damage-tolerance mechanism. The influence of the tile waviness on the impact performance of the engineering analogy was assessed by comparing the residual impactor velocity. In this study, the matrix properties and overall tile dimensions of 20 × 20 mm remain unchanged. The edges of the tiles are also fixed to a thickness of 1 mm. The amplitude of the tile waviness is varied while the period is fixed to a tile length. Figure 4d illustrates the sectional schematic of a profiled tile, where the minimal thickness is located in the centre of the tile and allows the angle of inclination to be measured by α.

In accordance with these definitions, the amplitude of the triangular wave function is changed to obtain the tile centre thicknesses listed in table 2. When using the tile length information, the angle of inclination α can be computed and is also listed for the corresponding tile centre thickness. Starting from a flat tile (t_1), the angle is gradually increased to 3.4°, which corresponds to a tile centre thickness of 0.4 mm.

Table 2.

Values for tile waviness amplitude.

parameter	unit	t_1	t_0.8	t_0.6	t_0.4	description
t	mm	1	0.8	0.6	0.4	tile centre thickness
α	°	0	1.1	2.3	3.4	waviness angle
velocity	ms−1	65	44	55	202	residual velocity

The results for the residual velocities obtained from the impact simulation with these profiled tiles are also included in table 2. The residual velocity of the flat tiles is 65 ms⁻¹. Improvements are observed for tiles which incorporate a slight inclination angle. The lowest residual velocity achieved is 44 ms⁻¹ as obtained from the tiles with a centre thickness of 0.8 mm. The worst performance, however, is obtained from the tiles with the thinnest centre thickness. Here, the reduction acts as a predetermined breaking point for the tile which shows the marginal benefit of a profiles tile. Figure 9 indicates the variation of the velocity profiles as obtained from each simulation. While the velocity gradient in the range of 3–6 µs is of similar magnitude for the tiles with a thickness of 1 and 0.8 mm, the curves extracted from the thinner tiles indicate less-compliant gradients.

Figure 9.

Impactor velocity evolution for different waviness amplitudes. (Online version in colour.)

3.4. Combination of best-performing configurations

To fully characterize the potential of an ‘engineered’ nacre withstanding a high strain-rate impact, the best-performing configurations from both the matrix property and waviness variation evaluations were combined. This methodology yielded a tile configuration with a tile centre thickness of 0.8 mm with the matrix properties set at ÷100 of the baseline material. In this simulation, it was observed that this geometrical and material configuration arrested the impactor and attained zero velocity. Figure 10a,b shows the velocity resultant and plastic strain map for the final state of the simulation. While the impactor is stopped, it should be noted that regions of the influence zone still have a significant velocity. Figure 10c–e illustrates the damage index throughout tile arrangement and shows the success of delocalizing a dynamic damage front.

Figure 10.

Presentation of results for a tile arrangement with waviness amplitude of 0.2 mm in combination with resin property set ÷100. (a) Velocity resultant map. (b) Plastic strain map. (c) Damage index distribution for B₄C tiles within arrangement and isolated with damage index equal to 1 in (d) top view and (e) side view using coordinate system as defined in (c).

The velocity evolution plots for the geometric and material variation studies are added for comparison purposes in figure 11. The results indicate a significant difference in the middle section of the profile between ÷100 and t_0.8 in terms of the velocity gradient and the point of inflexion of the plot. While the individual matrix or tile configurations are unable to stop the impactor, the combination of these is capable of reducing the speed of the impactor to a stationary state.

Figure 11.

Comparison of impactor velocity evolutions for best-case scenarios and combined properties. (Online version in colour.)

4. Discussion and concluding remarks

A discrete representation of a nacre-like structure, using current engineering materials, subjected to a high strain-rate threat is embodied in a complex FE model. In this study, a ceramic material model represents the individual tile constituents, whereas the matrix is discretized by a cohesive material law. The key results arising from these models are discussed below:

• — Influence of material discontinuity. The advantage of a discontinuous material shows notable improvements in sustaining impact relative to the continuous material. In the case of the continuous material, the shockwaves propagating through the material lead to localized failure while complex shockwave patterns are observed in the discontinuous flat tile arrangement. Furthermore, the time history recorded at the bottom node in the panel centre underneath the impactor shows that the panel back face shows instantaneous displacement for the continuous configuration while a time delay of 3.5 µs is observed for the discontinuous arrangement indicating that the shockwave propagation through the discontinuous material is inhibited. In the case of the discontinuous arrangement using profiled tiles, mechanisms found in nacre were observed in a similar fashion as reported in Espinosa et al. [10]. Albeit the presence of high rates and a complex loading condition, the model is capable of capturing the tile sliding after the matrix failure occurs leading to interlocking and ultimately tile failure.

• — Variation of matrix properties. The variation of binding matrix properties highlights the advantage of using matrix materials with lower energy fracture toughness. The velocity profile of the impactor is influenced in such a way that the slope in the middle section attains a steep gradient. Consequently, the corresponding deceleration profile suggests that a discontinuous arrangement, which leads to a small but intense deceleration peak, shows better results in arresting the impactor. The influence of the matrix properties suggests that it is able to steer the deceleration of the impactor velocity. From this study, the preferred velocity profile shows an initially low deceleration followed by a high deceleration. Furthermore, the shockwave propagation through the tile arrangement using lower fracture toughness matrix properties delays the back face node displacement from 3.5 (base, ×10 and ×100) to 4 µs (÷10 and ÷100). Hence, an additional advantage when using a low toughness matrix is to help reduce the reflected tensile shockwaves in the tile and attenuate the transmitted compressive shockwave to the adjacent tile.

• — Influence of tile geometry. The study to address the variation in tile waviness amplitude highlights the influence of the waviness on the later part of the impactor velocity profile. An increase in waviness amplitude leads to thinning of the tile in the centre. As can be seen for the configuration with a centre thickness of 0.4 mm, the impactor velocity is slowed to 202 ms⁻¹, whereas the configurations with less waviness amplitude achieve velocities of 65, 55 and 44 ms⁻¹ for centre thicknesses of 1, 0.6 and 0.8 mm, respectively. The inclination angle for the t_0.8 tile of 1.1° exhibits a suitable compromise to achieve tile interlocking and adequate centre thickness to prevent the premature formation of fracture.

• — Combination of the best-performing configurations. The model for the resin property study used flat tiles and the waviness study used the baseline matrix properties. Thus, the best-performing configuration for each of the above was combined such that the t_0.8 model from the waviness study was combined with the matrix properties from the ÷100 resin study. It was demonstrated that in this case, the combined model is capable of stopping the impactor within the same time frame.

For all the models, it was noted that the nodes of the tiles located on the back face can move with a velocity of up to the initial impactor velocity. Moreover, it should be noted that at this stage no validation for the model has been undertaken, hence this should be regarded as a feasibility study.

5. Future work

The current study can be enhanced further through the incorporation of a number of key functions. Firstly, the geometric parameters investigated should be expanded to include varying the aspect ratio of the tiles, and hence the size and thickness of the tiles, tile gap thickness and the percentage overlap of the tiles. Additional features found in nacre, such as hexagonal shaped tiles and in natural systems, such as functionally graded architecture, will also be considered in the subsequent FE model. The present study considered the effect of varying the amplitude of waviness, but the period could also be altered.

In reality, one of the key requirements of any high strain-rate protection system is to erode the projectile at the point of contact. However, the projectile in this study is modelled as a rigid body, and hence projectile deformation has been ignored. This should be considered as the worst-case scenario. The next step would be to alter the material definition in order to model the projectile as a deformable body, thereby allowing for projectile erosion/deformation to occur. In addition, it is also important to investigate oblique impacts on the crack delocalization and interfacial hardening of the discontinuous arrangement, as in reality, it is unlikely that any threat will approach the surface at an angle of exactly 90°.

Finally, for this work to be extended to fully simulate a high strain-rate event on personal protection systems, an absorber (backing plate) would need to be included within the FE model, which aims to absorb any remaining residual velocity while simultaneously containing debris propagation produced on impact. Materials such as Kevlar or Twaron could be considered and this would then allow for a comparison to be made with the existing commercially available armour systems.

Appendix. A

Table 3.

List of used parts and related LS-DYNA material models.

part description	LS-DYNA material mode
impactor	*MAT_RIGID
support	*MAT_RIGID
tiles	*MAT_JOHNSON_HOLMQUIST_CERAMIC
resin	*MAT_USER_DEFINED_MATERIAL_MODELS

Table 4.

Parameter set for boron carbide used in *MAT_JOHNSON_HOLMQUIST_CERAMIC material definition [27].

parameter	unit	value	description
RO	kg m−3	1140	density
G	GPa	197	shear modulus
A	—	0.927	intact normalized strength parameter
B	—	0.7	fractured normalized strength parameter
C	—	0.005	strength parameter (for strain rate dependence)
M	—	0.85	fractured strength parameter (pressure exponent)
N	—	0.67	intact strength parameter (pressure exponent)
EPSI	—	1.0	reference strain rate
T	GPa	0.26	tensile strength
SFMAX	GPa	0.2	maximum normalized fractured strength
HEL	GPa	19	Hugoniot elastic limit
PHEL	GPa	8.71	pressure component at the Hugoniot elastic limit
BETA	—	1.0	fraction of elastic energy loss converted to hydrostatic energy
D1	—	0.001	first parameter for plastic strain to fracture
D2	—	0.5	second parameter for plastic strain to fracture (exponent)
K1	GPa	233	first pressure coefficient (bulk modulus)
K2	GPa	−593	second pressure coefficient
K3	GPa	2800	third pressure coefficient
FS	—	1.5	failure criteria

Table 5.

Parameter set for Hexcel HexPly 8552 epoxy matrix [32] used in the cohesive element formulation via *MAT_USER_DEFINED_MATERIAL_MODELS [23,31].

parameter	unit	value	description
RO	kg m−3	1140	density
GIC	J m−2	200	mode I energy release rate
GIIC	J m−2	1000	mode II energy release rate
sImax	GPa	60	peak traction in normal direction
sIImax	GPa	90	peak traction in tangential direction
EI	N mm−3	467	initial elastic stiffness normal to the plane of the cohesive element in stress-separation diagram
EII	N mm−3	175	initial elastic stiffness in the plane of the cohesive element in stress-separation diagram
ENH	—	0.3	enhancement factor for compressive loading