Lecture #10: Anisotropic plasticity Crashworthiness Basics ... · D. Mohr 2/15/2016 Lecture #10...

2/15/2016 1 1Lecture #10 – Fall 2015 1D. Mohr

151-0735: Dynamic behavior of materials and structures

by Dirk Mohr

ETH Zurich, Department of Mechanical and Process Engineering,

Chair of Computational Modeling of Materials in Manufacturing

Lecture #10:

• Anisotropic plasticity• Crashworthiness

• Basics of shell elements

© 2015



ANISOTROPIC PLASTICITY



Hill’48 yield function

Hill (1948) proposed an anisotropic quadratic yield function. For plane stressconditions, the Hill’48 function reads



Experimental Characterization of flow anisotropy




In addition to the axial strain, the width strain is measured in uniaxial tensionexperiments using digital image correlation.

http://www.veryst.com/what-we-offer/materials-testing-modeling/Testing Library/uniaxial-tension-ut-testing

http://www.veryst.com/what-we-offer/materials-testing-modeling/Testing Library/uniaxial-tension-ut-testing



Hill’48 yield function

For steel sheets, it is recommended to combine the isotropic von Mises yieldfunction with an anisotropic Hill’48 flow potential. The Hill parameters can thenbe conveniently determined from the Lankford ratio measurements.



Crashworthiness



Axial crushing of square tubes

Results from Kohar et al. (Thin-walled structures, 2015)

P



SEA of a structure

The Specific Energy Absorption (S.E.A.) is often used to evaluate thecrashworthiness of thin-walled structures. It is defined as the ratio of the WorkW performed (“absorbed energy”) during the crushing/crashing of a structureand the total mass m of the structure. The SEA of structures is typically given inJ/g.

m

WSEA

PduW

P



Folding Mode

Consider a hollow square box structure of with B and wall-thickness t subject toan axial force P. In our simplified analysis, it is assumed that the structuredeforms in a so-called symmetric axial folding mode.



Kinematics

Each of the four initially flat constituent plates, needs to form three plastic hingelines in order to create a fold. To ensure kinematic compatibility (i.e. noformation of opening cracks along the plate intersections), each plate needs tobe stretched along the circumferential direction.

Plastic hinge lines



Plastic hinge lines

For an ideal plastic material, the plastic work performed during the folding of aplate of width B/2 reads

24 0

BMEb

2/B

with M0 denoting the fully plasticbending moment. For completing anentire fold, we have =p/2 andhence

BMEb p0



Fully Plastic Bending Moment

The fully plastic bending moment (per unit width) for a plate of thickness t is

2

004

1tM

assuming an ideal plastic material behavior and a uniaxial stress state in thesheet material (strong simplifications).

0

00

0

elastic Partially-plastic fully-plastic

t

http://www.codecogs.com/users/23287/Plastic-Theory-0002.png

http://www.codecogs.com/users/23287/Plastic-Theory-0002.png



Corner stretching

The amount of circumferential stretching required depends on the plastic foldingwavelength 2H.

2/B



Corner stretching

For a plastic folding wave length of 2H, we have a maximum hypothetical corneropening of

H

H

2/B

b

Hb 2

b

3D-viewtop view (after completing fold)

H

2/B



Corner stretching

The plastic work required to stretch a plate of initial dimension of 2H x B/2 to theunfolded geometry shown above (right) would be

2

0

2/

0

02

1)(4 HtdtE

H

m

b

top view (after completing fold)

H

2/B

H

H

2/B

Unfolded side view

2/H

The above is only an estimate of the plastic work knowing that the real kinematics are more complicated.



Work balance

A square column is composed of eight plates of width B/2. The balance ofexternal work performed by the force P and the internal work then reads:

BttHPd

H

2

0

2

0

2

042

18

p

while the mean crushing force

H

m PdH

P

2

02

1

is

H

tBHtPm p 20



Folding wave length

The folding wave length is an unknown in our problem. It is common practice toassume that the folding wave length adjusts itself such as to minimize the meancrushing force (Alexander’s postulate),

][minarg HPH m

The minimization problems reads

which yields

2

tBH

p

0220

H

tBt

H

Pm p

According to this simple model, the folding wave length of homogenous columnsis independent of the material properties. It is monotonically related to the wallthickness and profile width.



Specific Energy Absorption

The corresponding mean crushing force then reads

tBtPm p 22 0

The mass of the column section of height 2H is

And hence we have a specific energy absorption of

B

tSEA p

2

2

0

tBHHtBm 8)2(4

This result suggest that the structural efficiency for absorbing energy is amonotonic function of the t/B ratio. In other words, “thick-walled” box columnsare the best energy absorbers, provided that they deform in a symmetric axialfolding mode.



SEA of a material

The specific energy absorption can also be evaluated at the material level. Thematerial SEA can be defined as the integral of the stress-strain curve (workdensity) as normalized by the mass density. The units are therefore the same asthose of the structural SEA, i.e. J/g.

pdm

WSEA

1

Vm

pdVW

p



SEA of a ideal plastic material

2

1 0 pdSEA

p

0

0.5



Structural efficiency

B

tSEA p

2

2

0

Material SEA

t=1mm

B=60mm

Example:

232.0 0SEA

The folding mode is not very efficient as far as the mass specific absorption ofenergy is concerned. This can be seen when comparing the structural SEA withthe material SEA:



Metallic Honeycombs

Honeycombs man-made low-density materials that feature a 2D hexagonalmicrostructure. The porosity of metallic honeycombs made from aluminum foilis typically greater than 95%.

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiGx566gKbJAhWH1xQKHSD_BsQQjRwIBw&url=http://rsta.royalsocietypublishing.org/content/364/1838/31&psig=AFQjCNGYhoc212FVbBa_ukLAEBS2L4kXug&ust=1448349478381861

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiGx566gKbJAhWH1xQKHSD_BsQQjRwIBw&url=http://rsta.royalsocietypublishing.org/content/364/1838/31&psig=AFQjCNGYhoc212FVbBa_ukLAEBS2L4kXug&ust=1448349478381861



Axial crushing of honeycombs

undeformed

“Crushed material”

Under uniaxial compression, honeycombs specimens exhibit two distinctphases: (1) crushed/folded material, and (2) undeformed material. These twophases are separated by a crushing front which travels through the specimen.

Crushing front

Crushed/folded



Axial crushing of honeycombs

Constant width (no plastic Poisson’s effect)

The stress-strain response exhibits (1) an initial peak, followed by (2) a plateauregime, followed by (3) a densification regime.



Detailed FE modelingThe axial folding of honeycombs can be described through a detailed FE modelof the characteristic unit cell of their periodic microstructure.



Detailed FE modeling

b c d e f g h i j ka

A plot of the plastic work density shows that most of the energy is absorbednear the intersection lines of neighboring cell walls. However, large proportionsof the folded microstructure contribute only little (blue areas) to the overallenergy absorption.



Detailed FE modeling

A longitudinal cut elucidates the progressive folding mechanism.



Comment on the SEA of honeycombs

As shown for thin-walled tubes, structures form folds to decrease the amount ofwork required to accommodate an applied axial deformation. Many portions ofthe structures do not contribute to the energy absorption. Consequently, thematerial SEA of thin-walled honeycombs is significantly lower than that of non-porous homogeneous materials!

Honeycombs are nonetheless advertised as “excellent materials for energyabsorption purposes”. This statement can be justified by the fact that many thedeceleration may not exceed a given threshold value.

critmaF

Due to the constant stress (plateau regime) and zero-Poisson effect, it is easy todesign protective structures with honeycombs that deform at a constant force(and hence constant deceleration).



FE Analysis with Shell Elements



Basic Shell Elements

Solid element Shell element

The main feature of shell elements is that all quantities are expressed withrespect to the shell mid-surface.

Dynamore (2013)

It is typically assumed that cross-sections remain straight (Bernoulli hypothesis).In addition to displacement Degrees-Of-Freedom (DOF), the rotation of thecross-section is included as an additional DOF.




There exist numerous shell element formulation and their review is far beyondthe scope of this class. Here, we will just comment on a few features of“standard” shell elements which are based on the Mindlin-Reissner kinematicassumptions (e.g. element S4R in Abaqus or element type 2 in LS-DYNA).

The main features are:

• In the global coordinate system, the shell element features 6 nodal DOF• The stresses are computed using the plane stress formulation of the

constitutive model• Thickness is updated using the Poisson behavior

Dynamore (2013)




The virtual work (see FEA class) is computed using numerical integration at theelement level. The constitutive law (and hence the stresses) are only evaluatedat the location of the integration points. The number of thickness integrationpoints usually needs to be specified by the user.

Dynamore (2013)



Reading Materials for Lecture #10

• Kocks, Argon and Ashby (1975), Thermodynamics and kinetics of slips

• M.A. Meyers, Dynamic behavior of Materials

• C. Roth and D. Mohr (2015), “Ductile fracture experiments with locally proportional loading histories”, Int. J. Plasticity, http://www.sciencedirect.com/science/article/pii/S0749641915001412

Lecture #10: Anisotropic plasticity Crashworthiness Basics ... · D. Mohr 2/15/2016 Lecture #10...

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