The dependence of the fracture toughness of two-dimensional (2D) elastoplastic lattices upon relative density and ductility of cell wall material is obtained for four topologies: the triangular lattice, kagome lattice, diamond lattice, and the hexagonal lattice. Crack-tip fields are explored, including the plastic zone size and crack opening displacement. The cell walls are treated as beams, with a material response given by the Ramberg–Osgood law. There is choice in the criterion for crack advance, and two extremes are considered: (i) the maximum local tensile strain (LTS) anywhere in the lattice attains the failure strain or (ii) the average tensile strain (ATS) across the cell wall attains the failure strain (which can be identified with the necking strain). The dependence of macroscopic fracture toughness upon failure strain, strain hardening exponent, and relative density is obtained for each lattice, and scaling laws are derived. The role of imperfections in degrading the fracture toughness is assessed by random movement of the nodes. The paper provides a strategy for obtaining lattices of high toughness at low density, thereby filling gaps in material property space.

References

References
1.
Davies
,
G. J.
, and
Zhen
,
S.
,
1983
, “
Metallic Foams: Their Production, Properties and Applications
,”
J. Mater. Sci.
,
18
(
7
), pp.
1899
1911
.10.1007/BF00554981
2.
Ashby
,
M. F.
,
Evans
,
A. G.
,
Fleck
,
N. A.
,
Gibson
,
L. J.
,
Hutchinson
,
J. W.
, and
Wadley
,
H. N. G.
,
2000
,
Metal Foams: A Design Guide
,
Butterworth Heinemann
,
Oxford, UK
.
3.
Quintana-Alonso
,
I.
, and
Fleck
,
N. A.
,
2009
, “
Fracture of Brittle Lattice Materials: A Review
,”
Major Accomplishments in Composite Materials and Sandwich Structures—An Anthology of ONR Sponsored Research
,
I. M.
Daniel
,
E. E.
Gdoutos
, and
Y. D. S.
Rajapakse
, eds.,
Springer
,
Dordrecht
, pp.
799
816
.
4.
Quintana Alonso
,
I.
, and
Fleck
,
N. A.
,
2007
, “
Damage Tolerance of an Elastic–Brittle Diamond-Celled Honeycomb
,”
Scr. Mater.
,
56
(
8
), pp.
693
696
.10.1016/j.scriptamat.2006.12.027
5.
Wadley
,
H. N. G.
,
2006
, “
Multifunctional Periodic Cellular Metals
,”
Philos. Trans. R. Soc. London, Ser. A
,
364
(
1838
), pp.
31
68
.10.1098/rsta.2005.1697
6.
Evans
,
A. G.
,
Hutchinson
,
J. W.
,
Fleck
,
N. A.
,
Ashby
,
M. F.
, and
Wadley
,
H. N. G.
,
2001
, “
The Topological Design of Multifunctional Cellular Metals
,”
Prog. Mater. Sci.
,
46
(
3
), pp.
309
327
.10.1016/S0079-6425(00)00016-5
7.
Wadley
,
H. N. G.
,
Fleck
,
N. A.
, and
Evans
,
A. G.
,
2003
, “
Fabrication and Structural Performance of Periodic Cellular Metal Sandwich Structures
,”
Compos. Sci. Technol.
,
63
(
16
), pp.
2331
2343
.10.1016/S0266-3538(03)00266-5
8.
Deshpande
,
V. S.
,
Ashby
,
M. F.
, and
Fleck
,
N. A.
,
2001
, “
Foam Topology: Bending Versus Stretching Dominated Architectures
,”
Acta Mater.
,
49
(
6
), pp.
1035
1040
.10.1016/S1359-6454(00)00379-7
9.
Fleck
,
N. A.
,
Deshpande
,
V. S.
, and
Ashby
,
M. F.
,
2010
, “
Micro-Architectured Materials: Past, Present and Future
,”
Proc. R. Soc. London, Ser. A
,
466
(
2121
), pp.
2495
2516
.10.1098/rspa.2010.0215
10.
Gibson
,
J. W.
, and
Ashby
,
M. F.
,
1997
,
Cellular Solids: Structures and Properties
,
Cambridge University Press
, Cambridge, UK.
11.
Romijn
,
N. E. R.
, and
Fleck
,
N. A.
,
2007
, “
The Fracture Toughness of Planar Lattices: Imperfection Sensitivity
,”
J. Mech. Phys. Solids
,
55
(
12
), pp.
2538
2564
.10.1016/j.jmps.2007.04.010
12.
Wang
,
A.-J.
, and
McDowell
,
D. L.
,
2004
, “
In-Plane Stiffness and Yield Strength of Periodic Metal Honeycombs
,”
ASME J. Eng. Mater. Technol.
,
126
(
2
), pp.
137
156
.10.1115/1.1646165
13.
Fleck
,
N. A.
, and
Qiu
,
X.
,
2007
, “
The Damage Tolerance of Elastic–Brittle, Two-Dimensional Isotropic Lattices
,”
J. Mech. Phys. Solids
,
55
(
3
), pp.
562
588
.10.1016/j.jmps.2006.08.004
14.
Choi
,
S.
, and
Sankar
,
B. V.
,
2005
, “
A Micromechanical Method to Predict the Fracture Toughness of Cellular Materials
,”
Int. J. Solids Struct.
,
42
(
5–6
), pp.
1797
1817
.10.1016/j.ijsolstr.2004.08.021
15.
Lipperman
,
F.
,
Ryvkin
,
M.
, and
Fuchs
,
M. B.
,
2007
, “
Fracture Toughness of Two-Dimensional Cellular Material With Periodic Microstructure
,”
Int. J. Fract.
,
146
(
4
), pp.
279
290
.10.1007/s10704-007-9171-5
16.
Huang
,
J. S.
, and
Gibson
,
L. J.
,
1991
, “
Fracture Toughness of Brittle Foams
,”
Acta Metall. Mater.
,
39
(
7
), pp.
1627
1636
.10.1016/0956-7151(91)90250-5
17.
Schmidt
,
I.
, and
Fleck
,
N. A.
,
2001
, “
Ductile Fracture of Two-Dimensional Cellular Structures
,”
Int. J. Fract.
,
111
(
4
), pp.
327
342
.10.1023/A:1012248030212
18.
Cui
,
X.
,
Xue
,
Z.
,
Pei
,
Y.
, and
Fang
,
D.
,
2011
, “
Preliminary Study on Ductile Fracture of Imperfect Lattice Materials
,”
Int. J. Solids Struct.
,
48
(
25–26
), pp.
3453
3461
.10.1016/j.ijsolstr.2011.08.013
19.
Symons
,
D. D.
, and
Fleck
,
N. A.
,
2008
, “
The Imperfection Sensitivity of Isotropic Two-Dimensional Elastic Lattices
,”
ASME J. Appl. Mech.
,
75
(
5
), p.
051011
.10.1115/1.2913044
20.
Sih
,
G. C.
,
Paris
,
P. C.
, and
Irwin
,
G. R.
,
1965
, “
On Cracks in Rectilinearly Anisotropic Bodies
,”
Int. J. Fract. Mech.
,
1
(
6
), pp.
189
203
.10.1007/BF00186854
21.
Williams
,
M. L.
,
1957
, “
On the Stress Distribution at the Base of a Stationary Crack
,”
ASME J. Appl. Mech.
,
24
(
1
), pp.
109
114
.
22.
Anderson
,
T. L.
,
1995
,
Fracture Mechanics: Fundamentals and Applications
,
CRC Press
, Boca Raton, FL.
23.
Li
,
F. Z.
, and
Pan
,
J.
,
1990
, “
Plane-Strain Crack-Tip Fields for Pressure-Sensitive Dilatant Materials
,”
ASME J. Appl. Mech.
,
57
(
1
), pp.
40
49
.10.1115/1.2888321
24.
De Kruijf
,
N. E.
,
Peerlings
,
R. H. J.
, and
Geers
,
M. G. D.
,
2009
, “
An Analysis of Sheet Necking Under Combined Stretching and Bending
,”
Int. J. Mater. Form.
,
2
(
Suppl. 1
), pp.
845
848
.10.1007/s12289-009-0543-4
25.
Onck
,
P. R.
,
Van Merkerk
,
R.
,
De Hosson
,
J. T. M.
, and
Schmidt
,
I.
,
2004
, “
Fracture of Metal Foams: In-Situ Testing and Numerical Modeling
,”
Adv. Eng. Mater.
,
6
(
6
), pp.
429
431
.10.1002/adem.200405156
26.
Mangipudi
,
K. R.
, and
Onck
,
P. R.
,
2011
, “
Multiscale Modelling of Damage and Failure in Two-Dimensional Metallic Foams
,”
J. Mech. Phys. Solids
,
59
(
7
), pp.
1437
1461
.10.1016/j.jmps.2011.02.008
27.
Mangipudi
,
K. R.
, and
Onck
,
P. R.
,
2011
, “
Notch Sensitivity of Ductile Metallic Foams: A Computational Study
,”
Acta Mater.
,
59
(
19
), pp.
7356
7367
.10.1016/j.actamat.2011.07.071
28.
Rice
,
J. R.
, and
Rosengren
,
G. F.
,
1968
, “
Plane Strain Deformation Near a Crack Tip in a Power-Law Hardening Material
,”
J. Mech. Phys. Solids
,
16
(
1
), pp.
1
12
.10.1016/0022-5096(68)90013-6
29.
Hutchinson
,
J. W.
,
1968
, “
Singular Behaviour at the End of a Tensile Crack in a Hardening Material
,”
J. Mech. Phys. Solids
,
16
(
1
), pp.
13
31
.10.1016/0022-5096(68)90014-8
30.
Ashby
,
M. F.
,
1989
, “
Overview No. 80: On the Engineering Properties of Materials
,”
Acta Metall.
,
37
(
5
), pp.
1273
1293
.10.1016/0001-6160(89)90158-2
31.
Deshpande
,
V. S.
, and
Fleck
,
N. A.
,
2001
, “
Collapse of Truss Core Sandwich Beams in 3-Point Bending
,”
Int. J. Solids Struct.
,
38
(
36–37
), pp.
6275
6305
.10.1016/S0020-7683(01)00103-2
32.
Hutchinson
,
R. G.
,
Wicks
,
N.
,
Evans
,
A. G.
,
Fleck
,
N. A.
, and
Hutchinson
,
J. W.
,
2003
, “
Kagome Plate Structures for Actuation
,”
Int. J. Solids Struct.
,
40
(
25
), pp.
6969
6980
.10.1016/S0020-7683(03)00348-2
33.
Olurin
,
O. B.
,
Fleck
,
N. A.
, and
Ashby
,
M. F.
,
2000
, “
Deformation and Fracture of Aluminium Foams
,”
Mater. Sci. Eng.: A
,
291
(
1–2
), pp.
136
146
.10.1016/S0921-5093(00)00954-0
34.
Chen
,
C.
,
Fleck
,
N. A.
, and
Lu
,
T. J.
,
2001
, “
The Mode I Crack Growth Resistance of Metallic Foams
,”
J. Mech. Phys. Solids
,
49
(
2
), pp.
231
259
.10.1016/S0022-5096(00)00039-9
You do not currently have access to this content.