Chapter 5. Applicability of honeycomb as a sacrificial composite to resist low-
5.4 Numerical simulations
Shear energy s b bs
b s
E K E
K K
Membrane energy 1 4
m 4 m o
E K w
5.3.2 Static equivalent contact force model
The energy balance concept is used to predict the equivalent static contact force and indentation in the sandwich panel. The kinetic energy of the impactor is balanced by strain energy induced due to bending-shear, and membrane effect and work done by the contact force (Eq.(5.7)).
2 1
1
2M Vo EbsEiEm (5.7)
The work done by the contact force P is given as follows
om
i o
E Pd
(5.8)The contact force is nonlinearly dependent on the global displacement (Abrate 2005) as given below.
Substituting contributing energy terms in Eq. (5.7) yields
2 2 4
1
0
1 1 1
2 2 4
om
o bs o m o
M V K w K w Pd
(5.9)Here, the global deflection (wo) and indentation ( ) are unknowns in the energy balance equation. Combining Eq.(5.6) and Eq. (5.9), gives a nonlinear relationship between global deflection and indentation.
2 2
3 2
(7505 4250 2791 )
16 1
3 17640
o
f o bs o m o
f
D q K w K w
h
(5.10)
Global deflection (wo) and indentation( ) are evaluated by solving coupled nonlinear equations using numerical methods like Newton-Rapson (Eqs. (5.9) and (5.10)). Subsequently, Eq. (5.6) is used to quantify the contact force due to impact on the honeycomb structure.
characterisation
conducting low-velocity impact experiments on honeycomb sandwich specimen. Once validated, the FE simulation results are treated as virtual experiment data and used for calibrating the probabilistic model. The reason to resort to FE results as virtual experiments is the practical limitation to perform repetitive experiments with all possible combination of input variables. Gardoni et al. (2002); Sharma et al. (2014), adapted a similar approach to propose a probabilistic demand and capacity modeling for impact problems. A Similar approach has been implemented in the current study to formulate the probabilistic model. Once the probabilistic model is developed, it circumvents any further dependency of conducting expensive FE simulations and predict the contact force using an explicit equation. Numerical analysis is conducted using commercial finite element software Abaqus 6.8. The mean geometrical and material parameters considered for the representative FE model are shown in Table 5.2 (a) and (b), respectively.
Table 5.2 a) Mean geometrical properties of representative honeycomb sandwich structure
Description Geometrical property Symbol Value
Facesheet
Thickness of facesheet (mm)
hf 1
Radius of facesheet (mm)
Rf 62.5
Honeycomb core
Height of core (mm)
hc 27
Cell wall thickness (μ) t 70
Cell size (mm) c 6.5
Impactor
Diameter of impactor (mm) 20
Lumped mass (kg)
M 2
Drop height (m) h 0.5
Table 5.2 b) Mean material properties of representative honeycomb sandwich structure
The impactor is modeled as a solid deformable hemisphere and meshed using four- node linear tetrahedral continuum elements. These elements are stiff and are easily adaptable to the curved geometry. To simulate the experimental conditions, a nonstructural lumped mass of two kg is added to the impactor. The impactor is made of steel and is stiffer when compared to thin walled honeycomb core. For low impact energies, the impactor is assumed to undergo small deformations in the linear elastic range and hence is defined as an elastic material. A predefined velocity is imparted to the indentor resting on the deformable facesheet. The facesheet is modelled as solid
Material property Facesheet Honeycomb
core Impactor
Young’s modulus (GPa) 70 70 200
Density (kg/m3) 2700 2700 7850
Poisson’s ratio 0.3 0.3 0.3
Yield strength of core (MPa) 250 250 -
deformable body and is discretized using eight node hexahedral continuum elements (C3D8R). During the impact, the facesheet experiences high stress concentration within the contact region. Hence, a fine mesh is adapted in the contact region to accurately capture the response. To resemble the boundary conditions of the experiment, the bottom facesheet is restrained in all the three directions and the top facesheet is clamped along the periphery. Mesh optimization study is conducted to achieve convergence and reduce the rigid body modes of the deformation. Thin facesheet undergoes excessive bending and membrane stretching which results in induces enormous stress in the contact region. Therefore, the facesheet is divided into five elemental layers along the thickness to incorporate shear deformation and control excessive distortion at the contact area.
In the current study, FE simulation results are used to generate data for probabilistic model. The credibility of the probabilistic model depends on the numerical results and therefore a suitable material model is to be adopted for achieving accurate results. Honeycomb core shows similar behavior to elastic perfectly plastic material from the quasi static compression test. However, this represents the global behavior of specimen at the structural level and may differ from the actual stress-strain behavior of the material. Foo et al. (2008) extensively investigated the response of honeycomb sandwich structures for different material models like elastic perfectly plastic, bilinear, Ramberg-Osgood. Though nonlinear strain hardening models like Ramberg-Osgood provide enhanced accuracy, the computational demand is very high. Consideration of material non-linearity into model, significantly increases the simulation runtime. In addition to the time taken for each simulation, it also increases the number of input variables and design points, which makes it very cumbersome for probabilistic study.
FE simulations are conducted for different material models to determine the influence on contact force and indentation. The elastic perfectly plastic assumption is also in accordance with existing literature on honeycomb sandwich specimen for low-velocity impacts (Foo et al. 2008; Jen and Chang 2008; Sun et al. 2017; Zhou and Stronge 2006). Therefore, in the current study, the honeycomb sandwich specimen is assumed to behave as an elastic-perfectly plastic material.
The circularly shaped honeycomb core is modelled using four-node shell element with reduced integration (S4R) because of their capability to capture large deformations. To capture core-crushing behavior, the honeycomb core is finely meshed with total number of 2,00,000 elements. The finer mesh controls excessive distortion of elements within the contact region. Surface-to-surface contact pair is applied between the colliding surfaces. In addition, general contact algorithm is also employed to account for any possible interactions between the cell walls during the crushing of honeycomb core. The compatibility of displacement between facesheet and honeycomb core is ensured through surface based tie constraint (Foo et al. 2008). The facesheet acting as master surface governs the displacement of slave nodes on core surface and prevents delamination during deformation.
characterisation
As the probabilistic model is calibrated using numerical results it is essential to validate these FE results with benchmark experiments. Figure 5.5 shows the validation of the numerical and experimental contact force at an impact energy of 13.28 J for the specimen having dimensions as shown in section 5.2. The overall dynamic response is in good agreement with each other. This behavior is in accordance with the previous investigation on low-velocity impact on honeycomb structures (Foo et al. 2008). Figure 5.6 shows the comparison of the experimental and numerical deformation pattern of the honeycomb composite at an impact energy of 13.28 J. Numerical results are compared with experimental observations for impact energies mentioned in section 5.2.
It is observed from Figure 5.7 that the analytical predictions deviate from both experimental and numerical results significantly. The divergence in the results predicted by deterministic model makes it a compulsion for development of the probabilistic model which could correct the biased results.
Figure 5.5 Force-time history of experimental and FE simulation for an impact energy of 13.28J
Figure 5.6 Deformation contour of honeycomb sandwich specimen subjected to impact energy of 13.28J (a) Experiment specimen, and (b) FE model
a) b)
Figure 5.7 Analytical, numerical, and experimental comparison of contact force for different drop height