The shear modulus of orthotropic thin sheets from three advanced high-strength steels (AHSS) is measured using the anticlastic-plate-bending (APB) experiment. In APB, a thin square plate is loaded by point forces at its four corners, paired in opposite directions. It thus assumes the shape of a hyperbolic paraboloid, at least initially. The principal stress directions coincide with the plate diagonals, and the principal stresses are equal and opposite. Hence, at 45 deg to these, a state of pure shear exists. A finite element (FE) study of APB is reported first, using both elastic and elastoplastic material models. This study confirms the theoretical predictions of the stress field that develops in APB. The numerical model is then treated as a virtual experiment. The input shear modulus is recovered through this procedure, thus validating this approach. A major conclusion from this numerical study is that the shear modulus for these three AHSS should be determined before the shear strain exceeds 2 × 10−4 (or 200 με). Subsequently, APB experiments are performed on the three AHSS (DP 980, DP 1180 and MS 1700). The responses recorded in these experiments confirm that over 3 × 10−4 strain (or 300 με) the response differs from the theoretically expected one, due to excessive deflections, yielding, changing contact conditions with the loading rollers and, in general, the breaking of symmetry. But under that limit, the responses recorded are linear, and can be used to determine the shear modulus.

References

References
1.
Deng
,
N.
, and
Korkolis
,
Y. P.
,
2018
, “
Elastic Anisotropy of Dual-Phase Steels With Varying Martensite Content
,”
Int. J. Solids Struct.
,
141–142
, pp. 264–278.
2.
ASTM
,
2015
, “
Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio by Impulse Excitation of Vibration
,” ASTM, West Conshohocken, PA, Standard No.
ASTM E1876-15
.https://www.astm.org/Standards/E1876.htm
3.
Wolfenden
,
A.
,
1990
, “
Dynamic Elastic Modulus Measurements in Materials
,” ASTM International, West Conshohocken, PA, Standard No.
STP-1045
.https://www.astm.org/DIGITAL_LIBRARY/STP/SOURCE_PAGES/STP1045.htm
4.
Martinček
,
G.
,
1965
, “
The Determination of Poisson's Ratio and the Dynamic Modulus of Elasticity From the Frequencies of Natural Vibration in Thick Circular Plates
,”
J. Sound Vib.
,
2
(
2
), pp.
116
127
.
5.
Spinner
,
S.
,
Reichard
,
T. W.
, and
Tefft
,
W. E.
,
1960
, “
A Comparison of Experimental and Theoretical Relations Between Young's Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars
,”
J. Res. Natl. Bur. Stand. Phys. Chem.
,
64A
(
2
), pp.
147
155
.https://nvlpubs.nist.gov/nistpubs/jres/64A/jresv64An2p147_A1b.pdf
6.
Iosipescu
,
N.
,
1967
, “
New Accurate Procedure for Single Shear Testing of Metals
,”
J. Mater.
,
2
, pp.
537
566
.
7.
Arcan
,
M.
,
1984
, “
The Iosipescu Shear Test as Applied to Composite Materials
,”
Exp. Mech.
,
24
(
1
), pp.
66
67
.
8.
Walrath
,
D. E.
, and
Adams
,
D. F.
,
1983
, “
The Losipescu Shear Test as Applied to Composite Materials
,”
Exp. Mech.
,
23
(
1
), pp.
105
110
.
9.
ASTM
,
1997
, “
Standard Test Methods for Shear Properties of Composite Materials by the V-Notched Beam Method
,” ASTM International, West Conshohocken, PA, Standard No.
ASTM D 5379
.https://www.astm.org/Standards/D5379
10.
Hung
,
S. C.
, and
Liechti
,
K. M.
,
1997
, “
An Evaluation of the Arcan Specimen for Determining the Shear Moduli of Fiber-Reinforced Composites
,”
Exp. Mech.
,
37
(
4
), pp.
460
468
.
11.
Arcan
,
M.
,
Hashin
,
Z.
, and
Voloshin
,
A.
,
1978
, “
A Method to Produce Uniform Plane-Stress States With Applications to Fiber-Reinforced Materials
,”
Exp. Mech.
,
18
(
4
), pp.
141
146
.
12.
Mohr
,
D.
, and
Doyoyo
,
M.
,
2003
, “
A New Method for the Biaxial Testing of Cellular Solids
,”
Exp. Mech.
,
43
(
2
), pp.
173
182
.
13.
Diel
,
S.
,
Huber
,
O.
,
Steinmann
,
P.
, and
Winter
,
W.
,
2014
, “
Design and Validation of a New Fixture for the Shear Testing of Cellular Solids
,”
Arch. Appl. Mech.
,
84
(
3
), pp.
309
321
.
14.
Zhang
,
Y.
,
Sun
,
F.
,
Wang
,
Y.
,
Chen
,
L.
, and
Pan
,
N.
,
2013
, “
Study on Intra/Inter-Ply Shear Deformation of Three Dimensional Woven Preforms for Composite Materials
,”
Mater. Des.
,
49
, pp.
151
159
.
15.
Liu
,
L.
,
Chen
,
J.
,
Li
,
X.
, and
Sherwood
,
J.
,
2005
, “
Two-Dimensional Macro-Mechanics Shear Models of Woven Fabrics
,”
Composites Part A
,
36
(
1
), pp.
105
114
.
16.
Nosrat-Nezami
,
F.
,
Gereke
,
T.
,
Eberdt
,
C.
, and
Cherif
,
C.
,
2014
, “
Characterisation of the Shear–Tension Coupling of Carbon-Fibre Fabric Under Controlled Membrane Tensions for Precise Simulative Predictions of Industrial Preforming Processes
,”
Composites Part A
,
67
, pp.
131
139
.
17.
Hosseini
,
A.
,
Kashani
,
M. H.
,
Sassani
,
F.
,
Milani
,
A. S.
, and
Ko
,
F. K.
,
2018
, “
Identifying the Distinct Shear Wrinkling Behavior of Woven Composite Preforms Under Bias Extension and Picture Frame Tests
,”
Compos. Struct.
,
185
, pp.
764
773
.
18.
Farley
,
G. L.
, and
Baker
,
D. J.
,
1983
, “
In-Plane Shear Test of Thin Panels
,”
Exp. Mech.
,
23
(
1
), pp.
81
88
.
19.
Chen
,
Q.
,
Boisse
,
P.
,
Park
,
C. H.
,
Saouab
,
A.
, and
Bréard
,
J.
,
2011
, “
Intra/Inter-Ply Shear Behaviors of Continuous Fiber Reinforced Thermoplastic Composites in Thermoforming Processes
,”
Compos. Struct.
,
93
(
7
), pp.
1692
1703
.
20.
Wittenberg
,
T. C.
,
van Baten
,
T. J.
, and
de Boer
,
A.
,
2001
, “
Design of Fiber Metal Laminate Shear Panels for Ultra-High Capacity Aircraft
,”
Aircr. Des.
,
4
(
2–3
), pp.
99
113
.
21.
Horrocks
,
D.
, and
Johnson
,
W.
,
1967
, “
On Anticlastic Curvature With Special Reference to Plastic Bending: A Literature Survey and Some Experimental Investigations
,”
Int. J. Mech. Sci.
,
9
(
12
), pp.
835
IN2
.
22.
Timoshenko
,
S. P.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
McGraw-Hill
, New York.
23.
Theocaris
,
P. S.
, and
Hazell
,
C. R.
,
1965
, “
Experimental Investigation of Subsequent Yield Surfaces Using the Moiré Method
,”
J. Mech. Phys. Solids
,
13
(
5
), pp.
281
294
.
24.
Farshad
,
M.
, and
Flüeler
,
P.
,
1998
, “
Investigation of Mode III Fracture Toughness Using an Anti-Clastic Plate Bending Method
,”
Eng. Fract. Mech.
,
60
(
5–6
), pp.
597
603
.
25.
Zamrik
,
S. Y.
, and
Davis
,
D. C.
,
1993
, “
STP-1191: A Simple Test Method and Apparatus for Biaxial Fatigue and Crack Growth Studies
,”
Advances in Multiaxial Fatigue
,
D.
McDowell
and
J. R.
Ellis
, eds.,
ASTM International
, West Conshohocken, PA.
26.
ASTM
,
2005
, “
Standard Test Methods for Shear Modulus of Wood-Based Structural Panels
,” ASTM, West Conshohocken, PA, Standard No. ASTM D 3044-94.
27.
ISO
,
1999
, “
Fibre-Reinforced Plastic Composites—Determination of the In-Plane Shear Modulus by the Plate Twist Method
,” International Organization for Standardization, Geneva, Switzerland, Standard No.
ISO 15310:1999
.https://www.iso.org/obp/ui/#iso:std:iso:15310:ed-1:v1:en
28.
Farshad
,
M.
,
Wildenberg
,
M. W.
, and
Flüeler
,
P.
,
1997
, “
Determination of Shear Modulus and Poisson's Ratio of Polymers and Foams by the Anticlastic Plate-Bending Method
,”
Mater. Struct.
,
30
(
6
), pp.
377
382
.
29.
Galuppi
,
L.
,
Massimiani
,
S.
, and
Royer-Carfagni
,
G.
,
2014
, “
Buckling Phenomena in Double Curved Cold-Bent Glass
,”
Int. J. Non Linear Mech.
,
64
, pp.
70
84
.
You do not currently have access to this content.