4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 1
Mao-Hong YU，Yue-Ming LI (speaker)
Xi’an Jiaotong University, Xi’an, China
Unified Strength Theory for Materials and Structures
Spring Workshop on Nonlinear MechanicsApril 4-7, 2011
Xi’an, China
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 2
CONTENTS
1 Significance2 Unified Strength Theory3 Unified Constitutive Relation 4 Unified Characteristics line field
Theory5 Application
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 3
Strength:comes from various materials, structures (on Ground, in Space, under Ground and water).
强度：那里使用材料、建造结构，那里就
有强度问题，就需强度理论。
1 Significance> Background
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 4
1 Significance> Strength Theory
Strength theory: • study the material’s yield or failure law under complex
stress condition,• provide necessary calculation criterion and safety
criterion for structure design.
强度理论：• 研究材料在复杂应力状态下屈服或破坏规律，
• 为工程结构设计提供必须的计算准则和安全判据。
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 5
yield surface under complex stress state
What is the meaning of outer convex limit loci ?最大外凸极限线的意义是什么？
yield point under simple stress state
π-plane
?
1 Significance> Problem
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 6
Three principal shear stresses Three principal normal stresses
1 313 2
σ στ −= 1 3
13 2σ σσ +
=
2 323 2
σ στ −=
1 212 2
σ στ −= 1 2
12 2σ σσ +
=
2 323 2
σ σσ +=
Stress State at One Point
2 Unified Strength Theory
obviously τ13 = τ12+τ23
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 7
1 313 2
σ στ −=
、
2 Unified Strength Theory
1 313 2
σ σσ +=
Single-Shear Stress Concept
principal shear stress（τ13,τ12,τ23）
stress state （σ1, σ2, σ3）
13 Cτ =Tresca
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 8
13 12
13 2
12 23
33 12 2
CC
τ ττ τ
τ ττ τ
= ≥ ≤
++ =
when when
、
since τ13 = τ12+τ23
2 Unified Strength Theory
1 313 2
σ στ −= 1 3
13 2σ σσ +
=
2 323 2
σ στ −=
1 212 2
σ στ −= 1 2
12 2σ σσ +
=
2 323 2
σ σσ +=
Twin-Shear Stress Concept
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 9
21 , , ( / )1 1
tt cC σαβ σ σ
αα
α−
= = =+ +
tσcσβ
1 2 3( )2
F σ σα σ σ= − + = t1 3
2 1σ ασσ
α+
≤+
when
1 2 31' ( )2
F σ σ σ σα= + − = t1 3
2 1σ ασσ
α+
≥+
when
where 、C can be determined with material properties,
'13 12
' '13 23
13 12
13 23
( )( )
F C when F FF C when F F
β σ σβ σ
ττ τ στ = + + = ≥
= = ≤++ +
+
Convert to pincipale stress state
2 Unified Strength Theory
For geo-material:
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 10
2 Unified Strength TheoryMechanical Model of the Unified Strength Theory
continuum, space filled
13 23τ τ，
13 12τ τ，
13 12 23τ τ τ，，
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 11
Bounds and region
yield loci for metal material limit loci for geo-material
2 Unified Strength Theory
What is for the region ?
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'13 12 13 12
' '13 23 13 23
( )( )
F C F FF
b bb b C F F
τ τ β σ στ τ β σ σ
= + + + = ≥
= + + + = ≤
whenwhen
0
0
1 (1 ), , 11 1 ( )
tt
t c
bC b σ ταβ σα α σ τ σ
− += = = −
+ + −
1 31 2 3 2( )
1 1tbb
F when σ ασασ σ σ σ σα
+= − + = ≤
+ +
1 31 2 3 2
1 ( ) 1 1t whenbF
bσ ασσ σ ασ σ σ
α+′ = + − = ≥
+ +
A parameter b is introduced, and the unified strength theory is modeled as:
where C、β、b could be determined by tension σt、compressionσc and pure shear τ0 as well as:
Convert to pincipale stress space:
2 Unified Strength Theory
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Limit Loci of the Unified Strength Theory at Deviatoric Plane
2 Unified Strength Theory
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( )( ) 2323121223132313
'
2323121212131213
βστβστσσβττ
βστβστσσβττ
+≤+=+++=
+≥+=+++=
ifCbbf
ifCbbf
C=13τb = 0
b= 0.5
b = 1
C=8τ
23122323'
23121213
ττττ
ττττ
≤=+=
≥=+=
ifCf
ifCf
Tresca
Mises approximate
Twin-shear criterion
b = 0
b= 0.5
b = 1
C=+ 1313 σβτ
C=+ 88 σβτ
( )( ) 2323121212132323
'
2323121212131213
βστβστσσβττ
βστβστσσβττ
+≤+=+++=
+≥+=+++=
ifCf
ifCf
Mohr-Coulomb
Druker-Prager approximate
Twin-shear strength theory
0≠β Geomaterial
= 0 β Metal material
β reflects pure shear property. reflects SD effect , b
2 Unified Strength Theory
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 15
Three yield criteriaUnified strength theory
( )( ) 2323121223132313
'
2323121212131213
βστβστσσβττ
βστβστσσβττ
+≤+=+++=
+≥+=+++=
ifCbbf
ifCbbf
π-plane
Twin-shear
2 Unified Strength Theory
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 16
Single-shear concept
Three-shear concept
Twin-shear concept
Mohr-Coulomb
Drucker-Prager
Twin shear Strength Theory
1962, 1982 1985-1990
Tresca 1864
Mises 1904
Unified Strength Theory
1990-
2 Unified Strength TheoryClassic Strength Theory Progress
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 17
single–shear 1900 twin–shear 1985 unified strengththeory 1991
unified yieldcriterion 1991
2 Unified Strength Theory
Classic Strength Theory Progress
Two things were done.
= 0 β
Unified on π-plane，not meridian plane
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Others b<0; b>1 meaning ?
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 19
Fb
b t= −+
+ =σα
σ σ σ1 2 31( ) , α
ασσσ
++
≤1
312
Fb
b t' ( ) ,=+
+ − =1
1 1 2 3σ σ ασ σαασσ
σ++
≥1
312when
when
2 Unified Strength Theory> Simplicity and linearity
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 20
2 Unified Strength Theory> Symmetry
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 21
Unified Strength Theory0 1b≤ ≤
Single-shearfailure criterion
Mohr 1900
Twin-shear failure criterion
YU 1985
New failure criteria
0<b<1
Single-shear Yield criterion
Twin-shearYield criterion
New yield criteria
0<b<1
0b = 1b =
1961
1985
1991
0 1b< <
Tresca1864
2 Unified Strength Theory>Unification of Linear Failure Criteria
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 22
Shear stress ratio
( ) ( ) ( )''' 321
213
232
221
σσσσσσσσσ
++−+−+−
Shear strain ( ) ( ) ( )213
232
221 εεεεεε −+−+−
31
21
13
12
σσσσ
τ
τµτ −
−==
shear stress ratio and shear strain is defined
Twin-shear stress state parameter (Yu 1998)
Experiment results of sands by Yoshimine.M.(1996)
2 Unified Strength Theory>The choice of b depending on the experiments
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 23
It is convenient for analytical solution andnumerical calculation.
A series of results can be obtained by using theunified strength theory, including:Tresca solution,
Mohr-Coulomb solution,
Twin-shear solution,
and a lot of new results.
2 Unified Strength Theory> Convenient for using
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 24
0 20 40 60 80 100 120 1400
100200300400500600700800900
10001100
δ(mm)
P(N)
Unified Strength Theory(b=1.0) Unified Strength Theory(b=0.5) Mohr-Coulomb(b=0.0)
Different b for different theory
Bearing capacity for weightless soil under a plane strain strip load using unified strength theory (b=0, b=0.5 and b=1).
2 Unified Strength Theory> Convenient for using
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 25
consolidation soil fine sand
2 Unified Strength Theory > experiments
comparing with experimental results
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 26
Comparison of cracked angle with various fracture criteria
167.0=µ
0 30 60 900
30
60
90
maximum tension stress criterionmin-J2 criterion
unified fracture criterion(¦ Á=1,b=1)
¦ Â(¡ã)
-¦ È(¡ã)
minimum strain energy density criterion
unified fracture criterion(¦ Á=1,b=0)
unified fracture criterion(¦ Á=0.1,b=0)unified fracture criterion(¦ Á=0.1,b=1)
test point[10]
comparing with experimental results
2 Unified Strength Theory > experiments
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 27
200 400 600 800 1000 12000.0
0.5
1.0
1.5
2.0
2.5
3.0
V0 (m/s)
z max (m
)
b=0.0 b=0.15 b=0.5 b=1.0 test data
Comparison of penetration deep with test data
Normal penetration
2 Unified Strength Theory > experiments
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 28
under plastic stress conditions under plastic strain conditions
1- Large aggregates 2- Small aggregates 3- Water bubble 4- Air bubble 5- Mortar
Three meso-concrete models with different aggregate gradation
Failure loci of three meso-concrete models
2 Unified Strength Theory > experiments
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 29
0 2 4 6 8 100.100.120.140.160.180.200.220.240.260.280.30
(Jiang-shen,1986) (本文解£¬b=1) 试验点(ÎÄÏ× [11] )
d/h
p/[π
h(d+
h)f' c]
m=20
strength theory calculating value comparing with test results for concrete plate punching
2 Unified Strength Theory > experiments
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 30
3 Unified Constitutive Relation
3.1 Elasto-Plastic Constitutive Relations
pdε
Treatment of coner singularity for twin-shear theory
pdεpdε
pdε
Treatment of coner singularity for single-shear theory
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 31
3 Unified Constitutive Relation
pdε
pdε
pdε
pdε
pdε
Full view of platic strain increments
pdε
Coner singularity treatment of unified strength theory
Unified Strength Theory
pdεpdε
pdε
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3.2 Elasto-plastic FEM Analysis
tIJ
bbJF σαθαθα
=−++−
++= )1(3
sin1
)1(cos3
2)2
1( 12/12/1
012/12/1
' )1(3
sin)1
(cos3
)12( t
IJb
bJbbF σαθαθα =−+
++++
+−
=
32
31
233cos
31
J
J−=θ
The unified strength theory can be expressed by stressinvariant and stress angle.
The expression can be rewritten as:
3 Unified Constitutive Relation
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 33
∂∂
∂∂
+
∂∂
∂∂
−=
σσ
σσFDFH
DFFDDD
e
T
e
T
e
eep
][
][][][][
'
σθ
θσσσσα
∂∂⋅
∂∂
+∂
∂⋅
∂∂
+∂∂⋅
∂∂
=∂∂
=FJ
JFIFFT
2/12
2/12
1 )()(
},,,,,{ xyyxyzzyx τττσσσσ =where
∂∂⋅−
∂∂⋅
−=
∂∂
σσθσθ 2/1
222
332/3
2
)(3)(1
3sin23 J
JJJ
J
332211 αααα CCC ++=
σα
∂∂
= 11
IT
σα
∂∂
=2/1
22
)(JT
σα
∂∂
= 33
JTwhere
3 Unified Constitutive Relation
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 34
)1(31
11 α−=
∂∂
=IFC
+−
++⋅++−
++=
∂∂⋅+
∂∂
=
θαθαθθαθα
θθ
cos1
)1(sin3
2/2)(1-ctg3sin1
)1(cos3
2)2
1(
)(3ctg
)( 2/12
2/12
2
bb
bb
FJJ
FC
+−
++−⋅−=∂∂⋅−= θαθα
θθθcos
1)1(sin
32)
21(
3sin23
)(3sin23
22/3
23 b
bJ
FJ
C
)1(31
1
''1 α−=
∂∂
=IFC
+
++++
⋅++
++++−
=
∂∂⋅
∂+
∂∂
=
θαθαθθαθα
θθ
cos)1
(3
sin)b1b-2(-ctg3sin)
1(
3cos)
12(
)(3ctg
)(
'
2/12
2/12
''2
bb
bb
bb
FJJ
FC
+
++++−
−⋅−=∂∂⋅−= θαθα
θθθcos)
1(
3sin)
12(
3sin23
)(3sin23
2
'
2/32
'3 b
bbb
JF
JC
)(21 '
2202 CCC +=0 '
1 1 11 ( )2
C C C= + )(21 '
3303 CCC +=
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 35
299.83
205.81
0.002.00 4.00
limit load 252.38 when b=1
143.84
200.28
0.002.00 4.00
limit load 171.77 when b=0
4 Unified Characteristics line field Theory
B
c
d
a
bc1
d1
q=rD
qu
a1
I
II
III
II
III45o+
Rough footing of foundation
B
Pp
C
Pp
Ca1 a
b
qu
wedge-type model
W
4.1 Unified Slip Field Theory for Plane Strain Problem
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 36
Unified solution of bearing capacity for Rough footing (基底完全粗糙)
Single-shear solution Twin-shear solution
4 Unified Characteristics line field Theory
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 37
0
1000
2000
3000
4000
5000
b=0.0 b=0.2 b=0.5 b=0.8 b=1.0
Unified solution of bearing capacity for Smooth footing (基底完全光滑)
Twin-shear solutionSingle-shear solution
4 Unified Characteristics line field Theory
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 38
4.2 Characteristics Line Field for Plane Stress Problem
-----------------------
4 Unified Characteristics line field Theory
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 39
0.0000000000.2000000030.4000000060.6000000090.8000000121.00000001513.70
13.75
13.80
13.85
13.90
13.95
14.00
14.05
14.10
14.15
14.20
1.00.4 0.6 0.80.2
13.8
13.9
14.0
14.1
0.013.7
b
q/c
0
4.3 Unified Characteristics Theory for Spatial Axisymmetric Plastic Problem
Yu M-H，et al.
Unified characteristics line theoryof spatial axisymmetric plasticproblem. Science in China (SeriesE), 2001 44(2): 207-215.
4 Unified Characteristics line field Theory
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0.0 0.2 0.4 0.6 0.8 1.0
1.4
1.6
1.2
1.0
2.2
2.0
1.8
b
Characteristic line
Finite element
Comparison of the FEM with Characteristics field theory
Single-shear solutionTwin-shear solution
Unified solution：serial results of bearing capacity for circle foundation
q◌ ਼/2c
4 Unified Characteristics line field Theory
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Plastic Analysis of Plate
5. Applications> Analytical Solution
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0.0 0.2 0.4 0.6 0.8 1.0
1.6
1.8
2.0
2.2
2.4
2.6
α=0.6α=0.7α=0.8α=0.9
b
p eD/(tσ t
)
α=1.0
A series of solutions obtained by using the unified strength theory
One solution obtained by using the Mohr-Coulomb strength theory
Limit pressures for thick cylinder with the parameter b of the unified strength theory
5. Applications> Analytical Solution
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Relation of limit pressure to parameter b
0.0 0.2 0.4 0.6 0.8 1.02.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
α=0.6
α=0.7
α=0.8
α=0.9
b
p eD
/(tσ t
)
α=1.0
Relation of wall thickness to parameter b
0.0 0.2 0.4 0.6 0.8 1.00.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
α=0.6
α=0.7
α=0.8
α=0.9
b
t [σ]
/(pD)
α=1.0
stress state
te Dt
bbbp σα
4221
−++
= 2 21 4[ ]
b b pDtbα
σ+ −
≥+
5. Applications> Analytical Solution
Thin-wall Pressure Vessel
a series solutions
Single shear
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Internal forces in a circular plate element
Plastic limit load of plate(b from 0 to 1)
5. Applications> Analytical Solution
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Internal moment fields with loading radius d=a
psprr MMmMMm /,/ θ==
Velocity field with loading radius d=a
00 / www =
5. Applications> Analytical Solution
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 46
Rotating discs and rotating cylinders
Relation of the strength theory parameter bto plastic limit angular velocity
0
123
rbb y
ρσ
ω++
=
pω
Relation of the angular velocity to the radius of the plastic zone
812 (3 )
ye
σω
ν ρ=
+
5. Applications> Analytical Solution
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 47
Plastic zone for mode I crack in plane stress
21
2sin
11
11
2cos
π21
+++
+−=
θαα
αθσ b
bbKrt
bθθ ≥
21 )
2sin
111(
2cos
π21
+−
+=θθ
σ bbKr
tbθθ ≤ α
αθ+
=2
arcsin2b
when
when
Unified solution for the plastic zone at crack tip:
1=α
Plastic zone for mode I crackin plane strain
5. Applications> Fracture
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Plastic zone of crack tip forMode II (plane stress)
1=α
Plastic zone of Crack tip forMode II (plane strain)
1=α
5. Applications> Fracture
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 49
The sample model: macro and meso fiber direction
Tresca von Mises twin-sheartransverse direction
Stress–strain curves
Plastic zones
5. Applications > Multi-scale computing
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Plastic zone when b=0 Plastic zone when b=1
5. Applications> Numerical Solution
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 51
1α =
Unified Yield Criterion
5. Applications> Numerical Solution
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 52
Plastic zone of weightless soil under the same load by using the unified strength theory
UST: b=0.5 UST: b=1UST: b=0 (Mohr-Coulomb)
5. Applications> Numerical Solution
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Underground cave and finite element mesh
Principal stress trace around underground cave
5. Applications> Numerical Solution
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The maximum principal stress σ1 around the cave
The principal stress σ2around the cave
5. Applications> Numerical Solution
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Plastic zone of rock excavation (A large power station at Yellow River) b=1 (Twin-shear theory)
Plastic zone of rock excavation (A large power station at Yellow River) b=0 (Mohr-Coulomb theory)
5. Applications> Numerical Solution
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 56
5. Applications ( Ship lock of the Yangtze Three Georges )
根据Mohr-Coulomb准则计算的塑性区
根据双剪强度理论计算的塑性区
Plastic zone of excavation b=0 (Mohr-Coulomb theory)
Plastic zone of excavation b=1 (Twin-shear theory)
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 57
UST b=0 (Mohr-Coulomb)
Deformation Analysis of foundation of Big Goose Pagoda
UST b=1 (Twin-shear)
5. Applications> Numerical Solution
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Applications
5. Applications> Numerical Solution
Xi’an City Wall
(a) Twin-shear (b) Single shear
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Shear strain spread with several yield criteria for a slope
5. Applications> Numerical Solution
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结构面
Safety factors with b
20
0
( 3 )l
l
J p k dlw
dl
α
τ
− −= ∫
∫
5. Applications> Numerical Solution
Safety factor cures for different b
Dynamic safety factor
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 61
Strength Theoriesfor Structures
Strength Theoriesfor Materials
Plastic analysis
of structures
Plane stress
problems
Plane strain
problems
Axial-symmetricproblems
NumericalAnalysis
of structures
Material Mechanics
Rock-Soil Mechanics
ConcreteMechanics
Metal PlasticMechanics
Summary
Unified Strength Theory
Single-shearfailutre criterion
1900Twin-shear
failure criterion1985
New failure criteria0<b<1
Single-shear yield criterion Twin-shear
Yield criterion
New yield criteria
0<b<11α =
1α < 1bα = =1, 0bα = =
1, 1bα < =1, 0bα < =
0 1b≤ ≤
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“The UTS (统一强度理论) is advantageous over other failurecriteria because it encompasses all other established criteria asspecial cases. Or, such criteria are merely the linear approximationsof the UST. Moreover, the parameters of the UST are easily obtainedby experiments. ”
Xuesong Zhang, Hong Guan, Yew-Chaye Loo (Dean, School ofEngineering, Griffith University Gold Coast Campus, Australia)“UST failure criterion for punching shear analysis of reinforcement
concrete slab-column connections”.: Computational Mechanics – New Frontiers for New Millennium,Valliappan S. and Khalili N. eds. Elsevier Science Ltd, 2001, 299-304.
澳大利亚Griffith大学教授
“统一强度理论超越其他各种准则之处在于它包含了所有其他已经建立的准则并把他们作为特例，或者是统一强度理论的线性逼近。此外，统一强度理论的参数可以由实验较容易得到。”
Evaluation
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“The beauty of the twin-shear-unified strength theory is her feasibilityin difining the convex shape of the surface. Setting the value of thecontrollable convex parameter b to 0 or 1 yields the lower and upper limitof the convex shape function. For any arbitrary value of b, the shapefunction can be written in the following form:
……”Fan S. C. and Qiang, H.F. , “Normal high-velocity impacton concrete slabs-a simulation using the meshless SPH procedures”. Computational Mechanics – New Frontiers for New Millennium,Valliappan S. and Khalili N. eds. Elsevier Science Ltd, 2001, 1457-1462.
“双剪统一强度理论的美在于她在确定外凸面形状时的可行性。取可控制的外凸参数b为0或1，可得出外凸形状函数的下限和上限，形状函数可写为：
……”
新加坡南洋理工大学教授
Evaluation
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• Mao-Hong Yu, “Unified Strength Theory and Its Applications”, Berlin: Springer, 2004
• Mao-Hong Yu, et al. “Generalized Plasticity”, Berlin: Springer, 2006
• Mao-Hong Yu, et al. “Structural Plasticity”, ZJU-Springer, 2009
More details can be found in themonograph published by Springer in Berlin
References
4/19/2011 State Key Lab. for Strength and Vibration Xi'an Jiaotong University 65
Thank you
Spring Workshop on Nonlinear MechanicsApril 4-7, 2011
Xi’an, China
