fang

Dynamic mechanical behavior of light-weight

lattice cellular materials

IUTAM Symposium: Mechanics of Liquid and Solid Foams May 8-13, 2011 ~ Austin, Texas

Authors: Daining Fang1,2, Liming Chen2, Xiaodong Cui2

Cooperator: Han Zhao3

1College of Engineering, Peking Univ2Department of Engineering Mechanics, Tsinghua Univ

3ENS-Cachan, France

Outline

1. Introduction

2. Fabrication of lattice sandwich material

3. Air blast experiment

4. Simulation analysis

5. Theoretical model

6. Conclusions

Cellular materials

Open

Closed

Bone Open Aluminum foamsponge

Polymer foamWood

Honeycomb Square gridTriangular grid

Octahedron lattice Triangular lattice 3D Kagome

Foam

Lattice material

2D

3D

Closed Aluminum foam

Cellular Materials

2D latticeHexagonal Square Mixed

Kagome Triangular SI-Square

3D lattice

Triangular Octahedron Kagome

Geometric Topology

Fabrication Method

Investment CastingCambridge University (2001)

Punch and folding formingVirginia University (2002)

Extrusion formingStanford University (2001)

hybrid toolingNASA (1999)

InterlockingTsinghua University (2004)

InterpenetrationTsinghua University(2007)

latticematerial

Light-weightHigh Strength

Energyabsorption

Heat resistance

Wave absorption

Excellent Performance

Attracting Interests

USS Cole, a guided missile destroyer was attacked by a boat full of explosive charges On 12 October 2000. The blast created a hole in the port side of the ship about 40 feet (12 m) in diameter, killing 17 crewmembers and injuring 39.

HSLA-80 Steel, Yielding strength: 550MPa

Conventional solid materials is incapable to resist the severe blast！

Impulsive Resistance

Energy absorption Blast experiment

Lattice sandwich structure is composed of front and back face sheets and lattice core

Xue et al (2004)

Previous Studies

Dynamic behavior of lattice materials

Experiment：The previous experimental investigation of lattice sandwich structure subject to air explosion was mainly focused on 2D lattice materials.

Simulation：The air effect was ignored in the simulation of TNT explosion.

Theory：The core was treated as equivalent continuum media, with the deformation mechanism of the microstructure ignored.

Existing problems

Experiment：Dharmasena et al (2008), Wadley et al (2008), Zhu et al (2008)

Simulation：Xue and Hutchinson (2003, 2004, 2005), Zhu et al (2009)

Theory：Fleck’s group (2004, 2005), Zhu et al (2008)

Motivation of our investigation

Design and fabricate the metal tetrahedral lattice sandwich structure

Perform the air blast experiment to investigate the deformation mechanism and the impulsive resistance of lattice sandwich structure

Establish a finite element model to simulate the deformation response

Propose an analytical model considering microstructure deformation mechanism to predict the dynamic response of lattice sandwich plates under impulsive loading

To study the deformation mechanism and the impulsive resistance ofthe tetrahedral lattice sandwich structures subject to air explosion

Aim

Specific research aspects

Outline

1. Introduction





6. Conclusions

Material (Face sheet：Al2024; Core：Al5052)

Geometry (Tetrahedral lattice and its relative density is 0.036)

Fabrication method: Punch and folding forming

hfhc

l

bh

l

Fabrication of Lattice Sandwich Material

Sheet perforation Node folding tetrahedral lattice

Welding (brazing) Lattice sandwich plate


Technical process

Outline

1. Introduction





6. Conclusions

Air Blast Experiment

Sketch of the frame and clamping device Sketch of the experimental set-up

Specimen

Charge

Frame

Pendulum

Laserdisplacement transducer

TNT charge mass：15g, 20g, 25g, 30g

Measurement：Deflection and impulse

Specimen subjected to air blast

Experiment Arrangement


1

2

1 ln22 x

xpI Me

T

Impulse:


Experimental results for different masses of charges

Deformation and failure modes of the front face

(b) (c)

(d) (e)

Tearing

Large deformation zone (a)

18.12 Ns 18.87 Ns

21.97 Ns 24.29 Ns concave-convex deformation


The front face sheet of the lattice sandwich plate exhibits a large global deformation and local concave-convex deformation

The front face sheet suffers tearing failure at an abundant high impulse level


Deformation and failure modes of the back face

Global deformation

18.12 Ns 24.29 Ns

28.63 Ns When the applied impulse has less intensity, only global deflection was observed.

Local punctate convex zone appears when the applied impulse is intense enough.

A crack was observed at the centre of back face sheet under an impulse of 28.63 Ns.


Deformation and failure modes of the lattice samdwich plate

Shear failure was observed in the transition region due to the incompatible deformation of the front and back face sheets.

No delamination failure between the front sheet and the core occurred in our experiment.

Deflection of the back face sheet

Tetrahedral lattice sandwich plate possesses better impulsive resistance than that of hexagonal honeycomb sandwich plate.


2o

of c

wwh h

Yf f

IIAM

Maximum non-dimensional deflection

Non-dimensional impulse

0.20 0.25 0.30 0.35 0.40 0.450.5

1.0

1.5

2.0

2.5

fY fI AM

wo/(

h f+2h c)

Triangular pyramid Hexagonal honeycombTetrahedral latticeHexagonal honecomb(Zhu et al., 2008)

Outline

1. Introduction





6. Conclusions

Simulation analysis

Xue et al (2003, 2004, 2005) Square honeycomb, pyramidal latticeABAQUS: Applied impact load directly

Dharmasena et al (2008) Square honeycombABAQUS: Applied impact load directly

Zhu et al (2009)Hexagonal honeycombLS-DYNA: Ignored air effect

This study aims to simulate the explosion induced wave propagation process in the air and the resulting deformation response of lattice sandwich structures.

Existing numerical simulations

Zhu et al (2009)

TNT

Air

Face sheet

Lattice coreyz

x

Software: ANSYS/LS-DYNA

Geometric model Boundary conditions Air: non-reflecting boundary conditionFour sides of the structure: fully clamped Interaction between the structure, air and TNT: fluid-structure couplingInteraction between the air and TNT: share the same nodesInteraction between the face and core: share the same nodes

Part Elements number Elments type Algorithm

TNT 1100 solid164 Euler

Air 155410 solid164 Euler

Lattice core 51736 solid164 Lagrange

Face sheet 58800 solid164 Lagrange

Simulation analysis

Face sheet and lattice core (bi-linear elasto-plastic constitutive relation)

TNT (high explosion burn and Jones Wilkins Lee（JWL）equation)

Air (equations of state (EOS) -Linear polynomial equation)

1 2

1 2

1 1R V R V Ep A e B eRV R V V

2 3 20 1 2 3 4 5 6p C C V C V C V C C V C V E

Material model

Simulation analysis

Material property, JWL and EOS input data

1.0E+02.50E-60.00.40.40.00.00.0-1.0E-6

V0E0C6C5C4C3C2C1C0

*EOS_LINEAR_POLYNOMIAL

1.2929E-3

ρ

*MAT_NULL

MediumAir

1.07.0E-20.300.954.153.23E-23.71

V0E0ωR2R1BA

*EOS_JWL

0.190.671.63

pDρ

*MAT_HIGH_EXPLOSIVE_BURN

ChargeTNT

7.0E-32.65E-30.330.702.68

EtσyνEρ

*MAT_PLASTIC_KINEMATIC

coreAl5052

7.37E-37.58E-40.330.722.68

EtσyνEρ

*MAT_PLASTIC_KINEMATICFacesheetAl2024

LS-DYNA material type, material property , JWL and EOS input data (unit = cm, g, s)PartMaterial

Simulation analysis

Peak value of pressure Duration

0 3.7MPap

0.15mst

压力（

102 M

Pa）

时间（µs）

3

2

1

压力

（M

Pa）

时间（µs）

4

Time

Pres

sure

TNT explosion in the air

Front elevation view

Top view

Simulation analysis

t=5μs t=25μs

t=50μs t=70μs

Formation and propagation of shock wave in the air

Expansion of explosive starts at the detonation pointA series of compress wave form in the front of the air and a shock wave is created with a

strong discontinuity Shock wave reaches the surface of structure at t=70μs

Simulation analysis

Fluid structure interaction

t=75μs t=90μs

t=210μst=130μs

A dent deformation is first formed at the central area of sandwich front face, and then the deformation extends both outwards and downwards with the transfer of impulse

After a time period of approximately 210μs, the contact force between explosive and target structure almost reduces to 0.

Simulation analysis

（a）t=100μs （b）t=200μs

（c）t=400μs （d）t=2000μs

Deformation response of sandwich structure due to inertia

The plate continues to deform under its own inertia. The deformation zone gradually extends to the external clamped boundaries from the central region.

Simulation analysis

Deformation Comparison

Front face sheet

Lattice core

Back face sheet

From the simulation results, we also can see the local concave-convex deformation of front face, the local punctate convex of back face, and the plastic buckling of lattice core.

The simulation results are in good agreements with the experimental results.

Outline

1. Introduction



4. Simulation of air blast test


6. Conclusions

In the previous studies, the deformation mechanism of the microstructure was ignored, and the core was treated as equivalent continuum media..

In the previous studies, only 2D lattices have been considered in the theoretical models.

However, 3D lattices exhibit a different deformation mode at high deformation rates, and the microstructure deformation mechanism must be included.

Theoretical model

Fleck’s group (2004, 2005), Zhu et al (2008)…Existing theoretical models

Theoretical model

The core is considered as continuum in their model, which is abandoned in this study, and we developed a mechanism-based stress-strain relation for the core compression.

Stage I:Fluid-Structure

interaction

Stage II:Core compression

Stage III:Global deflection

Face sheetLattice core

ch

fh

ch

w

Continuum media

Fleck and Deshpande (2004) divides the response of the sandwich structure into three sequential stages. Based on this three-stage framework, they proposed an analytical model to predict the dynamic response of clamped sandwich beam.

Overpressure equation (Sadovsky equation):

The momentum per unit area of the front face-sheet (Kambouchev, 2006)：

The overall transmitted momentum

3

0 2 3

1.07 0.1, 10.076 0.255 0.65 , 1 15

Z Zp

Z Z Z Z

11

0 0

ss

ssa R R

R sR

I C fp t

16 20 24 28 3210

15

20

25

I (N

s)

Mc (g)

Experiment Prediction

0 04 d d

L L

aI I x y

Stage I: The Initial Fluid-structure Interaction

1 3

cZ R M

Unit cell of pyramid latticeLee et al (2006)

Step 1: Axial compression of struts before buckling

Stage II: The Core Compression

Step 2: Plastic buckling of the struts before the contact of buckled struts with face sheet

Step 3: Densification of lattice core

Step 1: Axial compression of struts

4 , 3 292 , 3 2 0.23

dc c c

cd d

c c

E E

E

00 3 2 , where 1 6 30d d Y Y

c E v c

Assumptions: The core material is treated as an ideally elastic-plastic material, and the plastic buckling of the strut is assumed to occur when the strain reaches 20% or 30%, according to the experimental measurements of Lee et al.(2006). The equivalent stress-strain relation of the tetrahedral lattice core before plastic buckling can be derived as

Elastic deformation region

Plastic deformation region

49 cE EThe equivalent modulus —— micromechanics analyses of unit cell

The equivalent yielding strain —— based on the numerical simulation of Vaughn et al (2005)

Plastic hinge analysis

Step 2: Plastic buckling of struts

(a) Unit cell of the triangular lattice; (b) Collapse mechanism of the struts in a unit cell.

2

1

2

1

Assumption: the middle plastic hinge occurs at the location with 1/4 length of the strut from the joint at the front face sheet.

The rotation of struts around the plastic hinges occurs in plane ABC or the plane parallel to it, due to the rectangular cross section of the struts and the deformation compatibility.

9 plastic hinges in each unit cell

2 2 22

3

8 2 1632 3

d dc c

l h l hbh Pl

21 2

21 2

1arcsin 16 20 852 171

1arcsin 32 40 3652 459

c

c

Simple geometric analysis

2

16P

P

PM MN

The bending moment of the cell strut, 2 4d

PM bh

2 3dPM bh

for strut-1for strut-2 and strut-3

dPN bh

The relation of equivalent compression stress with the strain increment

2 211 16arcsin 1.5 1.87 1.23 6arcsin 1.22 1.53 0.8

3c c

cc

24 ( )dP Pl bh

1 1 2 2 1 22 4M M P

The work done by the force is equal to the plastic energy dissipation23 2cP l

2

1

2

1

Step 2: Plastic buckling of struts

Step 3: Densification of lattice core

The deformation mechanisms are very complicated in this step. For simplicity, the nonlinear deformation induced by contact of the struts with face sheet is ignored, thus the overall stress of lattice core

II DD IIc c c c

c c cD II D IIc c c c

1D Dc c c the critical overall stress of densificationD

c

The overall strain of the lattice core is in the range of II Dc c c

——is the overall densification strain of the core, which is near 100%.Dc

where

—— the strain at the end of step 2, indicating the initial contact of the plastichinge with the plate

IIc

Compression model of the lattice core

0.0 0.2 0.4 0.6 0.80

5

10

15

20

25

30

nom

inal

stre

ss (M

Pa)

nominal strain

Gas gun experiment Simulation Prediction

Stress–strain curves(The experimental result and numerical simulation are derived by Lee et al, 2006)

Good agreement is obtained with the prediction compared to theexperiment result and numerical simulation

Comparisons of the experiment, simulation and theoretical predictions for pyramid lattice core

Plate bending and stretching

The exact maximum normal stress yield locus of sandwich plate (Qiu et al, 2004)

0 0

1M NM N

0Yf f f c fM h h h W

The plastic bending moment of the sandwich plate

The plastic membrane force

0 2 Yf fN h

The contributions of the lattice core to the bending moment andmembrane force are ignored due to the low relative density and plastic hinge.

Approximate circumscribing and inscribing square yield locus are adopted to simplify the analyses

Comparison of Experiment and Analytical Predictions

The predictions using inscribing yield locus is much closer with the experimental results at lower transmitted impulses, while at higher transmitted impulses using circumscribing locus gives better predictions.

The maximum back face deflection of the sandwich plate under impulsive loading

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400.0

0.5

1.0

1.5

2.0

Wo

fY fI AM

Experiment Prediction by inscribing locus Prediction by circumscribing locus

2

1 221

21 12 32 2

oo

f c f c

W IWh h h h

2

1 221

41 12 32 2

oo

f c f c

W IWh h h h

for circumscribing square yield locus

for inscribing square yield locus

Outline

1. Introduction





6. Conclusions

Conclusions

Experiment: The tetrahedral lattice sandwich structures are designed and fabricated through perforated metal sheet forming and welding technology. The explosion experiments provide insight into the deformation and failure mechanisms of the sandwich structures.

Simulation: A finite element model with consideration of air effect is established to simulate the explosion induced wave propagation process in the air and the resulting deformation response of lattice sandwich structures.

Model: An analytical model considering the microstructure deformation is developed to predict the deformation response of clamped tetrahedral lattice sandwich plates subject to air shock loading. The microstructure deformation of the lattice core is well captured, and the analytical results agree well with the testing results.

Thank you!

Enstock and Smith（2007） Nurick and Martin (1989) Zhu et al（2008）

Dharmasena et al（2008） McShane et al（2006）

Set-up of Blast Experiment


Nonuniform strut width Strut fracture

Mould modification Orientation pin

Two-stage folding

Perforation Folding Welding

Welding between two face sheets

Entire brazing in protective gas

Technical difficulties


Four-cable ballistic pendulum system

Specimen

Charge

Frame

Pendulum

Laserdisplacement transducer

fang

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