Figure 1.
Geometry and components of the proposed uniaxial graded auxetic damper (UGAD).
Figure 1.
Geometry and components of the proposed uniaxial graded auxetic damper (UGAD).
Figure 2.
Boundary conditions and constraints of the bearing plate, piston and damper body.
Figure 2.
Boundary conditions and constraints of the bearing plate, piston and damper body.
Figure 3.
Mesh sensitivity study. Finding the most accurate-less expensive auxetic core model (different SM/L ratios), based on comparing plastic dissipation energy (PDE) and reaction force (RFd), for an auxetic core of L = 10 mm, t = 1 mm, S4R elements, AL3 aluminium, pulse load of 0.5 × 106 N in 0.002 s.
Figure 3.
Mesh sensitivity study. Finding the most accurate-less expensive auxetic core model (different SM/L ratios), based on comparing plastic dissipation energy (PDE) and reaction force (RFd), for an auxetic core of L = 10 mm, t = 1 mm, S4R elements, AL3 aluminium, pulse load of 0.5 × 106 N in 0.002 s.
Figure 4.
Stress-strain relationship for the three aluminium grades, at different strain rates (y-axis is the same for each subfigure), (a) Grade AL7075-T6 (AL1), (b) Grade AL6061-T6 (AL2), (c) Grade AL6063-T4 (AL3).
Figure 4.
Stress-strain relationship for the three aluminium grades, at different strain rates (y-axis is the same for each subfigure), (a) Grade AL7075-T6 (AL1), (b) Grade AL6061-T6 (AL2), (c) Grade AL6063-T4 (AL3).
Figure 5.
Ratio of compressed length to total length per time, for an auxetic core loaded in two different directions D1 and D2 ( = 0.75 mm, L = 5 mm, t/L = 0.15, = 60°, AL2 grade).
Figure 5.
Ratio of compressed length to total length per time, for an auxetic core loaded in two different directions D1 and D2 ( = 0.75 mm, L = 5 mm, t/L = 0.15, = 60°, AL2 grade).
Figure 6.
Plastic dissipation energy PDE with respect to time, for an auxetic core loaded in two different directions D1 and D2 ( = 0.75 mm, L = 5 mm, t/L = 0.15, = 60°, AL2 grade).
Figure 6.
Plastic dissipation energy PDE with respect to time, for an auxetic core loaded in two different directions D1 and D2 ( = 0.75 mm, L = 5 mm, t/L = 0.15, = 60°, AL2 grade).
Figure 7.
Ratio of the reaction force to the applied load (RFd/P) with respect to time, for an auxetic core loaded in two different directions D1 and D2, (t = 0.75 mm, L = 5 mm, t/L = 0.15,
= 60°, AL2 grade). It shows that direction D1 (with higher auxetic behaviour (
Table 4), gives less RFd/P.
Figure 7.
Ratio of the reaction force to the applied load (RFd/P) with respect to time, for an auxetic core loaded in two different directions D1 and D2, (t = 0.75 mm, L = 5 mm, t/L = 0.15,
= 60°, AL2 grade). It shows that direction D1 (with higher auxetic behaviour (
Table 4), gives less RFd/P.
Figure 8.
Ratio of PDE/Mass with respect to time, for three different cell dimensions A, B and C with = 60°, t/L = 0.2, subjected to the same loading conditions.
Figure 8.
Ratio of PDE/Mass with respect to time, for three different cell dimensions A, B and C with = 60°, t/L = 0.2, subjected to the same loading conditions.
Figure 9.
RFd/P–time history, for 3 different cell dimensions A, B and C with = 60°, t/L = 0.2, subjected to same loading conditions.
Figure 9.
RFd/P–time history, for 3 different cell dimensions A, B and C with = 60°, t/L = 0.2, subjected to same loading conditions.
Figure 10.
Peak value of RFd/P, for the 3 cell dimensions A, B and C.
Figure 10.
Peak value of RFd/P, for the 3 cell dimensions A, B and C.
Figure 11.
Ratio of PDE/Mass with respect to time, for 3 different Aluminium grades AL1, AL2 and AL3, of an auxetic core with L = 10 mm, t = 2mm, t/L = 0.2.
Figure 11.
Ratio of PDE/Mass with respect to time, for 3 different Aluminium grades AL1, AL2 and AL3, of an auxetic core with L = 10 mm, t = 2mm, t/L = 0.2.
Figure 12.
RFd/P time history, for 3 different Aluminium grades AL1, AL2 and AL3, of an auxetic core with L = 10 mm, t = 2mm, t/L = 0.2.
Figure 12.
RFd/P time history, for 3 different Aluminium grades AL1, AL2 and AL3, of an auxetic core with L = 10 mm, t = 2mm, t/L = 0.2.
Figure 13.
Mass of auxetic cores with 3 different cell angles, and L = 10 mm, t = 2.6 mm, t/L = 0.26.
Figure 13.
Mass of auxetic cores with 3 different cell angles, and L = 10 mm, t = 2.6 mm, t/L = 0.26.
Figure 14.
PDE/Mass with respect to time, for 3 different cell angles, of an auxetic core with L = 10 mm, t = 2.6 mm, t/L = 0.26.
Figure 14.
PDE/Mass with respect to time, for 3 different cell angles, of an auxetic core with L = 10 mm, t = 2.6 mm, t/L = 0.26.
Figure 15.
RFd/P with respect to time, for 3 different cell angles, of an auxetic core with L = 10 mm, t = 2.6 mm, t/L = 0.26.
Figure 15.
RFd/P with respect to time, for 3 different cell angles, of an auxetic core with L = 10 mm, t = 2.6 mm, t/L = 0.26.
Figure 16.
Max. value of RFd/P for 3 different cell angles.
Figure 16.
Max. value of RFd/P for 3 different cell angles.
Figure 17.
Deformation patterns of three auxetic cores with different numbers of layers of the same geometrical properties and loading conditions, having the same loading direction D1, Grade AL3, Cell dimension B (L= 10 mm), t = 2.6 mm, t/L= 0.26, = 60°, (a) four layers, (b) eight layers, (c) twelve layers..
Figure 17.
Deformation patterns of three auxetic cores with different numbers of layers of the same geometrical properties and loading conditions, having the same loading direction D1, Grade AL3, Cell dimension B (L= 10 mm), t = 2.6 mm, t/L= 0.26, = 60°, (a) four layers, (b) eight layers, (c) twelve layers..
Figure 18.
PDE with respect to time, for auxetic cores of different number of layers, having the same geometrical properties and loading conditions, L = 10 mm, t = 2.6 mm, t/L = 0.26, cell angle = 60°, AL3.
Figure 18.
PDE with respect to time, for auxetic cores of different number of layers, having the same geometrical properties and loading conditions, L = 10 mm, t = 2.6 mm, t/L = 0.26, cell angle = 60°, AL3.
Figure 19.
PDE/Mass with respect to time, for auxetic cores of different number of layers, having the same geometrical properties and loading conditions, L = 10 mm, t = 2.6 mm, t/L = 0.26, cell angle = 60°, AL3.
Figure 19.
PDE/Mass with respect to time, for auxetic cores of different number of layers, having the same geometrical properties and loading conditions, L = 10 mm, t = 2.6 mm, t/L = 0.26, cell angle = 60°, AL3.
Figure 20.
RFd/P with respect to time, of 3 auxetic cores with different number of layers of the same geometrical properties and loading conditions, having the same loading direction D1, Grade AL3, Cell dimension B (L = 10 mm), t = 2.6 mm, t/L = 0.26, = 60°.
Figure 20.
RFd/P with respect to time, of 3 auxetic cores with different number of layers of the same geometrical properties and loading conditions, having the same loading direction D1, Grade AL3, Cell dimension B (L = 10 mm), t = 2.6 mm, t/L = 0.26, = 60°.
Figure 21.
Peak value of RFd/P of 3 auxetic cores with different number of layers; having the same geometrical properties and loading conditions.
Figure 21.
Peak value of RFd/P of 3 auxetic cores with different number of layers; having the same geometrical properties and loading conditions.
Figure 22.
Uniaxial graded auxetic damper (UGAD) cross-section with three auxetic cores for three different blast levels.
Figure 22.
Uniaxial graded auxetic damper (UGAD) cross-section with three auxetic cores for three different blast levels.
Figure 23.
Stress-strain curve of Aux. 1 under 20 m/s impact velocity, showing the four stages of crushing re-entrant auxetics.
Figure 23.
Stress-strain curve of Aux. 1 under 20 m/s impact velocity, showing the four stages of crushing re-entrant auxetics.
Figure 24.
Numerical stress-strain curve of Aux. 1 under different impact velocities, compared to the analytical “dynamic crushing strength”.
Figure 24.
Numerical stress-strain curve of Aux. 1 under different impact velocities, compared to the analytical “dynamic crushing strength”.
Figure 25.
Numerical stress-strain curve of the three auxetic cores together in the UGAD under different impact velocities, 1 m/s, 20 m/s and 40 m/s.
Figure 25.
Numerical stress-strain curve of the three auxetic cores together in the UGAD under different impact velocities, 1 m/s, 20 m/s and 40 m/s.
Figure 26.
A three dimensional (3D) printed prototype of the UGAD.
Figure 26.
A three dimensional (3D) printed prototype of the UGAD.
Table 1.
Fixed and variable geometrical parameters of the UGAD auxetic core.
Table 1.
Fixed and variable geometrical parameters of the UGAD auxetic core.
Fixed Parameters | | Variable Parameters |
---|
UGAD chamber internal space 210 × 210 × 430 mm | dimensions: | Cell dimensions L1, L2, L and H while L1 = 2 L |
Auxetic core extrusion depth = 200 mm | | Cell wall thickness t |
Auxetic core height = 200–210 mm | | Cell angle |
Cell wall aspect ratio = t/L = 0.10, 0.15, 0.20 | | Number of layers |
Table 2.
The three aluminium grades used for the auxetic core and their applications.
Table 2.
The three aluminium grades used for the auxetic core and their applications.
Symbol | AL Grade | Strength | Yield Point (MPa) | Applications |
---|
AL 1 | 7075-T6 | High | 546 | Aerospace and defence |
AL 2 | 6061-T6 | Medium | 324 | General Structural Applications |
AL 3 | 6063-T4 | Low | 90 | Door, windows, furniture |
Table 3.
Material parameters of the three aluminium grades used in the UGAD auxetic core.
Table 3.
Material parameters of the three aluminium grades used in the UGAD auxetic core.
| Description | Unit | AL7075−T6 [72] | AL6061−T6 [73] | AL6063−T4 [74] |
---|
E | Modulus of Elasticity | MPa | 71.7 × 103 | 69 × 103 | 68.9 × 103 |
ν | Poisson’s ratio | − | 0.33 | 0.33 | 0.33 |
ρ | Mass density | t/mm3 | 2.81 × 10−9 | 2.703 × 10−9 | 2.703 × 10−9 |
A | Yield Strength | MPa | 546 | 324 | 89.6 |
B | Ultimate Strength | MPa | 678 | 113 | 172 |
n | Work−hardening exponent | − | 0.71 | 0.42 | 0.42 |
| Reference Strain rate | s−1 | 1 × 10−4 | 1 × 10−4 | 1 × 10−4 |
C | Strain rate factor | − | 0.024 | 0.002 | 0.002 |
| Critical Damage | − | 0.3 | 0.3 | 0.3 |
| Damage threshold | − | 0 | 0 | 0 |
| Specific heat | mm2 k/s2 | 960 × 106 | 910 × 106 | 910 × 106 |
χ | Inelastic heat fraction | − | 0.9 | 0.9 | 0.9 |
| Melting Temperature | k | 750 | 925 | 616 |
| Room Temperature | k | 293 | 293.2 | 293.2 |
m | Thermal−softening exponent | − | 1.56 | 1.34 | 1.34 |
| − | − | −0.068 | −0.77 | −0.77 |
| − | − | 0.451 | 1.45 | 1.45 |
| − | − | −0.952 | 0.47 | 0.47 |
| − | − | −0.036 | 0.00314 | 0.00314 |
| − | − | 0.697 | 1.6 | 1.6 |
Table 4.
Loading directions D1 and D2 and their effect on the collapse mode and deformation of an auxetic core (t = 0.75 mm, L = 5 mm, t/L = 0.15, = 60°, AL2 grade).
Table 5.
Auxetic cores with three different cell dimensions and their properties.
Table 6.
Auxetic cores with three different cell angles and their properties.
Table 7.
The three auxetic cores with their geometric and mechanical properties.