Advances in Engineering Plasticity and its Applications (AEPA 1996)

883 Pages·1996·75.478 MB·English

by  T. Abe and T. Tsuruta (Auth.)

ADVANCES IN
ENGINEERING
PLASTICITY
AND ITS
APPLICATIONS
Edited by
T.Abe
Faculty of Engineering, Okayama University, Okayama, Japan
and
T. Tsuta
Faculty of Engineering, Hiroshima University, Hiroshima, Japan
Proceedings of the Asia-Pacific Symposium on
Advances in Engineering Plasticity and its Applications—AEPA '96
21-24 August 1996
Hiroshima, Japan
1996
PERGAMON
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First Edition 1996
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INTERNATIONAL SCIENTIFIC COMMITTEE
T.Abe Okayama Univ., Japan
J. C. Choi Pusan National Univ., Korea
J. Fan Chongqing Univ., China
H. Huh Korea Adv. Inst. Sei. & Techn., Korea
K. Hwang Tsinghua Univ., China
T Inoue Kyoto Univ., Japan
H. Ishikawa Hokkaido Univ., Japan
D. W. Kim Seoul National Univ., Korea
K. S. Kim Brown Univ., USA
Z. Kuang Xi'an Jiaotong Univ., China
D. N. Lee Seoul National Univ., Korea
W. B. Lee The Hong Kong Polyt. Univ., Hong Kong
H. Lippmann Techn. Univ. of München, Germany
Y. W. Mai Univ. of Sydney, Australia
A. Makinouchi Inst, of Phys. & Chem. Research, Japan
D. McDowell Georgia Inst, of Techn., USA
R. McMeeking Univ. of California, USA
Z. Mroz IPPT, Polish Academy of Sciences, Poland
S. Murakami Nagoya Univ., Japan
K. W. Neale Shebrooke Univ., Canada
A. Needleman Brown Univ., USA
S. I. Oh Seoul National Univ.
N. Ohno Nagoya Univ., Japan
S. R. Reid UMIST, UK
M. B. Sayir Swiss federal Inst. of Techn., Switzerland
D. G. Suh Sung Kyun Kwan Univ., Korea
R. Sowerby MacMaster Univ., Canada
B. Storâkers Royal Inst. of Techn., Sweden
S. Stören Norwegian Inst. of Techn., Norway
S. Tanimura Univ. of Osaka Pref., Japan
M. Tokuda Mie Univ., Japan
Y. Tomita Kobe Univ., Japan
T. Tsuta Hiroshima Univ., Japan
V. Tvergaard Techn. Univ. of Denmark, Denmark
R. Wang Beijing Univ., China
T Wang Inst. of Mech., CAS, China
Z. Wang Harbin Inst. of Techn., China
G. Weng Rutgers Univ., USA
B. Xu Tsinghua Univ., China
G. Yang Taiyuan Univ. of Techn., China
W Yang Tsinghua Univ., China
N. Liang Inst. of Mech., CAS, China
V
ORGANIZING COMMITTEE
Chairman : T. Abe Okayama Univ.
Co-chairmen : T. Tsuta Hiroshima Univ.
Y. Tomita Kobe Univ.
M. Tokuda Mie Univ.
Asian coordinator : B. Xu Tsinghua Univ.
Pacific coordinator : W. B. Lee The Hong Kong Polyt. Univ.
Members : M. Gotoh Gifu Univ.
Y. lino Toyota Techn. Inst.
S. Imatani Kyoto Inst. of Techn.
K. Kaneko Science Univ. of Tokyo
M. Kawai Univ. of Tsukuba
S. Kawano Yamaguchi Univ.
H. Kitagawa Osaka Univ.
T. Kitamura Kyoto Univ.
H. Moritoki Akita Univ.
E. Nakamachi Osaka Inst. of Techn.
S. Nagaki Okayama Univ.
Y. Obataya Fukui Univ.
N. Ohno Nagoya Univ.
E Oka Gifu Univ.
K. Sato Science Univ. of Tokyo
Y Shibutani Kobe Univ.
S. Shima Kyoto Univ.
H. Takahashi Yamagata Univ.
T. Tamura Kyoto Univ.
E. Tanaka Nagoya Univ.
K. Tanaka Tokyo Metrop. Inst. of Techn,
S. Tanimura Univ. of Osaka Pref.
K. Uetani Kyoto Univ.
0. Watanabe Tsukuba Univ.
K. Yamaguchi Kyoto Inst. of Techn.
F. Yoshida Hiroshima Univ.
EXECUTIVE COMMITTEE
Chairman ' T. Tsuta Hiroshima Univ.
Members : T. Abe Okayama Univ.
Y Tomita Kobe Univ.
M. Tokuda Mie Univ.
N. Ohno Nagoya Univ.
F. Yoshida Hiroshima Univ.
M. Kawai Univ. of Tsukuba
VI
ORGANIZING COMMITTEE
Chairman : T. Abe Okayama Univ.
Co-chairmen : T. Tsuta Hiroshima Univ.
Y. Tomita Kobe Univ.
M. Tokuda Mie Univ.
Asian coordinator : B. Xu Tsinghua Univ.
Pacific coordinator : W. B. Lee The Hong Kong Polyt. Univ.
Members : M. Gotoh Gifu Univ.
Y. lino Toyota Techn. Inst.
S. Imatani Kyoto Inst. of Techn.
K. Kaneko Science Univ. of Tokyo
M. Kawai Univ. of Tsukuba
S. Kawano Yamaguchi Univ.
H. Kitagawa Osaka Univ.
T. Kitamura Kyoto Univ.
H. Moritoki Akita Univ.
E. Nakamachi Osaka Inst. of Techn.
S. Nagaki Okayama Univ.
Y. Obataya Fukui Univ.
N. Ohno Nagoya Univ.
E Oka Gifu Univ.
K. Sato Science Univ. of Tokyo
Y Shibutani Kobe Univ.
S. Shima Kyoto Univ.
H. Takahashi Yamagata Univ.
T. Tamura Kyoto Univ.
E. Tanaka Nagoya Univ.
K. Tanaka Tokyo Metrop. Inst. of Techn,
S. Tanimura Univ. of Osaka Pref.
K. Uetani Kyoto Univ.
0. Watanabe Tsukuba Univ.
K. Yamaguchi Kyoto Inst. of Techn.
F. Yoshida Hiroshima Univ.
EXECUTIVE COMMITTEE
Chairman ' T. Tsuta Hiroshima Univ.
Members : T. Abe Okayama Univ.
Y Tomita Kobe Univ.
M. Tokuda Mie Univ.
N. Ohno Nagoya Univ.
F. Yoshida Hiroshima Univ.
M. Kawai Univ. of Tsukuba
VI
PREFACE
The Asia-Pacific Symposium on Advances in Engineering Plasticity and Its
Applications (AEPA) was first organized by Professor W. B. Lee of Hong Kong
Polytechnic University in 1992. The Second Symposium was held by Professors XU
Bingye and YANG Wei of Tsinghua University, China in 1994. Due to the
overwhelming success of AEPA92 and AEPA94, and the sincere hope of all
participants to continue these Symposia in a series, AEPA96 has been planned
to be held at Hiroshima University in Higashi-Hiroshima City, Japan, on August
21-24, 1996.
The aim of the symposium is to provide a forum for discussion on the state-of-art
developments in plasticity, as approached from different scales. The close
interaction of the theories from macroplasticity, mesoplasticity and
microplasticity is emphasized, together with their applications in various
engineering disciplines such as solid mechanics, metal forming, structural
analysis, geo-mechanics and micromechanics. Case studies showing applications
of plasticity in inter-disciplinary or nonconventional areas are also included.
The Symposium is hosted and organized by Hiroshima University. The Symposium is
coorganized by The Japan Society of Mechanical Engineers, The Society of
Materials Science, Japan , and The Japan Society for Technology of Plasticity.
The Hong Kong Polytechnic University / and Tsinghua University cooperate in this
Symposium, together with The Iron and Steel Institute of Japan and The Japan
Society for Precision Engineering. The Symposium is supported by Japan Society
for the Promotion of Science, Hiroshima Prefecture, Electric Technology Research
Foundation of Chugoku, Congress of Hiroshima Central Technopolis Construction
and Promotion, Amada Foundation for Metal Work Technology and Higashi Hiroshima
City. We are sincerely thankful for these Organizations, without whose kind
cooperation we could not have this Symposium.
We also would like to express our deep appreciation to all the participants in the
Symposium. We would like to thank the members of the Executive Committee, the
Organizing Committee and the International Scientific Committee, general
lecturers, keynote speakers and session chairmen for their kind contribution to
the Symposium. We hope that the exchange of scientific or engineering ideas will
contribute to the mutual understanding and the development of Asia-Pacific area
as well as that of the world.
Takeji Abe and Toshio Tsuta
Chairmen of AEPA96
August 1996
xix
DYNAMIC ANALYSIS OF SHEAR FLOW OF GRANULAR MATERIALS
A J M SPENCER
Department of Theoretical Mechanics
University of Nottingham
Nottingham NG7 2RD England
ABSTRACT
This paper considers simple shearing flow of a cohesionless granular material in which the stress is
governed by the Coulomb-Mohr yield condition, with angle of internal friction of φ. It is supposed
that the flow conforms to the double-shearing theory, in which deformation takes place by
simultaneous shears on the two surfaces on which the critical shear stress is mobilised. Previous
studies of quasi-static shear flow are extended to include weight and inertia of the material. This
problem is of potential interest in earthquake engineering. It is shown that the horizontal velocity
component and the orientation of the principal stress axes are determined by a pair of hyperbolic first-
order partial differential equations. A qualitative description is given of the motion of a horizontal
layer of granular material on a rough rigid base that goes through a cycle of horizontal acceleration.
KEYWORDS
Granular material, shear flow, Coulomb-Mohr conditions, double-shearing theory, hyperbolic partial
differential equations, earthquake engineering.
INTRODUCTION
The problem considered is simple shearing flow of a cohesionless granular material. The stress is
assumed to be governed by the Coulomb-Mohr yield condition, with angle of internal friction φ, and
it is supposed that the flow conforms to the 'double-shearing' theory, in which the deformation takes
place by simultaneous shears on the two surfaces on which the critical shear stress is mobilised. The
relevant theory for quasi-static motions was described in Spencer (1986), where it was found that the
theory allows two modes of simple shearing flow; a steady-state mode, in which the orientation of the
principal stress axes is fixed relative to the shear plane, and an unsteady mode in which the principal
stress axes rotate during flow. It was shown that in many cases the steady-state mode is unstable, and
that the unsteady mode allows a strain-soflening behaviour which may be associated with the
commonly observed formation of shear bands in shear flows of dry granular materials.
3
4 Spencer
In this paper this analysis is extended to include the effect o finertia. This is of interest in relation to
earthquake engineering, since it has been observed that under certain conditions large shearing
deformations may occur in a sand layer overlying relatively rigid rock which undergoes horizontal
acceleration. These effects have also been demonstrated experimentally on the laboratory scale
(Richards et al, 1990).
FORMULATION
All quantities are referred to a system of plane rectangular cartesian coordinates (x,y), in which thej
axis is vertical. We consider a layer of ideal granular material of uniform thickness /?, lying on a rigid
base .y = 0. The in-plane stress components are denoted a ,a and a . The upper surface y-h
xx yy xy
supports a uniform vertical load w per unit area but is free from tangential force. Thus
ayy = -w> σ*ν = 0> y = h- 0)
The stress components can be expressed as
σ =-ρ + ςοο82ψ, a = -p-qcos2y/, a = qsin2y/, (2)
χχ yy xy
where
Ρ = -1^ + σ^, g = {i(a -a )2 + a2 }\ q>0, (3)
χχ xx yy xy
2σ
tan2^ = 1*—, (4)
so that ψ is the angle the largest (algebraically) principal stress direction makes with the x-axis. The
Coulomb-Mohr condition has the form
q<psinq, (5)
where φ is the angle of internal friction, assumed constant, and the equality in (5) applies whenever
the material is deforming. The material undergoes a shearing deformation in the x direction on the
planes y = constant, so that the vertical components of displacement, velocity and acceleration are
zero, and the horizontal components of these quantities are, respectively,
u = u(y,t), v(y,t) = âu/ât, f(y,t)=âv/ât. (6)
The stress is taken to be independent of x, and gravity to be the only body force, and then the
equations of motion reduce to
—ZL = pg, —- = pf, (7)
ôy ây
where p is the density. It follows from (1) and (7) that
<Jyy = -W+P^y-h)· (8)
For the velocity field we adopt the 'double-shearing' theory described, for example, in Spencer (1982)
which expresses the assumption that deformation occurs by simultaneous shears on the two surfaces
on which the critical shear stress required by the Coulomb-Mohr condition for flow is mobilised. For
deformation of the form (6), this theory requires (Spencer, 1986) that
dv â ψ
(cos2 ψ+sin φ)— + 2sin φ—^ = 0, (9)
ây ât
Proceedings ofAEPA '96 5
which implies that, in general, the principal stress axes rotate during shear. It is noted, however, that
(9) has a singular solution
cos2^=-sin^, (10)
or
Ψ=±ψ, where ψ = \π+\φ . (11)
5 $
This corresponds to the case in which the critical shear stress is mobilised on planes y = constant and
deformation is by shear on these planes only. However it was shown in Spencer (1986) that this
solution is unstable in the sense that (9) predicts that any perturbation from the steady-state values
ψ=±ψ will result in the principal axes rotating away from the direction determined by ±y/. A
5 s
further requirement is that the plastic work-rate must be non-negative, which means
âv
σ ,^->0, (12)
χ
ôy
and hence, from (9)
sin2v/ ^<Q, (13)
cos2y+sin^ at
de Josselin de Jong (1971) has proposed an additional condition which includes, but is stronger than,
the inequality (12). It is convenient to introduce the notations
l + tan|d , .
ΛΛ
r=tan^, r, = tan^, = - *--. (14)
Then, when the equality holds in (5), there follow
-2g(l+ r2r2) -2g(r2 + r2)
5
σχχ ~—; ; ■> σνν~—; ; ■> χ*-*)
(if-Dir2+ 1) yy (r;-l)(r2+l)
- - l qX (16)
" r2 + r
and (9) can be written as
\0v , ·, ,, δτ
(^)f (r;-l)f = 0, (17)
+
dy at
and the inequality (13) as
0 (18)
-TM^ ·
τ) - τ2 Ôt
It follows, as shown in Spencer (1986), that for positive plastic work rate, the following possibilities
exist:
(a) âv/ây>0 and 0<ψ<±π, with
ây//ât<0 for 0<ψ<ψ,
5
ây//ât>0 for ψ <ψ<\π.
3
(b) âv/ây<0 and -\π< ψ<0, with
ây//ât>0 for -ψ <ψ<0,
Λ
ây//ât<0 for -\π<ψ<ψ.
$
6 Spencer
If the shear strain âu/ây is denoted by χ, then (17) can be written as
('■•-^«••-»i-·
dt
which can be integrated to give
(19)
r)
where y (y) and r0>) denote values of γ and r respectively at t - 0. Alternatively (19) can be
0 0
inverted to give τ in terms of / as
τ (τ + τ )&χρ{-(χ-χ )οοΐφ}-(τ -τ )
ί 0 0 5 0
(20)
Ts (^ + ^o)exp{-(r-/o)cot^}+(r,-ro)
CHARACTERISTICS THEORY
From (18), (15) and (16) there follows
^ = T(T2-\){-w pg(y-h)}/(T2 + r2). (21)
y +
By substituting (16) and (21) into (7) it follows, after rearrangement, that
2
p( > + y^ + ( i-Ti)( i-l){ - (y-h)}^ = pg(T>-\)T(T2 + Tl). (22)
T T T T wpg
dt ay
It is noted that when T=±T, (22) reduces to
S
<?v- _= +ie (rs2-l) = ±gtan^, (23)
dt 2TS
so that the body undergoes a constant uniform acceleration in this singular case.
In the general case (17) and (22) are a pair of first-order partial differential equations for v and r.
They can be shown by standard methods to form a hyperbolic system, with characteristics in the (y,t)
plane given by
*y- Ûzl.\?L- -k)\ , (24)
dt ± τ] + τ2 [gρ{y
and characteristic relations along these curves
(r; + r2)</v±(r,2--l)< g(y-h)\ dr = g(r2 -\)xdt, (25)
where the ± signs in (25) correspond to the ± curves (24). From (24), when T=±T both families of
S
characteristics degenerate to lines y = constant.