lecture #5: introduction to continuum mechanics three ... · d. mohr 2/15/2016 lecture #5 –fall...
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2/15/2016 1 1Lecture #5 – 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 #5:
• Introduction to Continuum Mechanics• Three-dimensional Rate-independent
Plasticity
© 2015
2/15/2016 2 2Lecture #5 – Fall 2015 2D. Mohr
151-0735: Dynamic behavior of materials and structures
Introduction toContinuum Mechanics
2/15/2016 3 3Lecture #5 – Fall 2015 3D. Mohr
151-0735: Dynamic behavior of materials and structures
Cauchy stress tensor
Suppose that a mechanically loaded body is hypothetically cut intotwo parts. The created hypothetical surfaces can be described bythe unit normal vector field n=n[x] with the associatedinfinitesimal areas dA.
n
tdA
t
n
dA
x
1e
2e
The traction vectors t=t[x] describe the forces per unit area thatwould need to act on the hypothetical surfaces ndA to ensureequilibrium.
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Cauchy stress tensor
n
tdA
x
1e
2e
The Cauchy stress tensor s=s[x] provides the traction vector tthat acts on the hypothetical surfaces ndA at a position x (in thecurrent configuration).
)( dAnσt =
From a mathematical point of view, the above equation definesthe linear mapping of vectors in R3. The operator s is thus called atensor.
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151-0735: Dynamic behavior of materials and structures
Cauchy stress tensor
For a given set of orthonormal coordinate vectors {e1, e2, e3}, wecan also define the stress components sij:
jjjj σee =s
jiij σee =s
je
jσe
ie
je
jσe
iejjs je
ie
ijst
traction vector t actingon unit surface definedby normal vector ej
2/15/2016 6 6Lecture #5 – Fall 2015 6D. Mohr
151-0735: Dynamic behavior of materials and structures
Cauchy stress tensor
For a given set of orthonormal coordinate vectors {e1, e2, e3}, it canalso be useful to write the stress tensor in matrix notation:
=
333231
232221
131211
}{
sss
sss
sss
σ
1e
2e
3e
11s
22s
33s 13s 31s
21s12s
32s
23s
ijs
acting on surface ej
coordinate system:along direction ei
Stress component
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151-0735: Dynamic behavior of materials and structures
Symmetry of the Cauchy stress tensor
Unlike other tensors used in mechanics, the Cauchy stress tensoris symmetric,
Tσσ =jiij ss =
which can be demonstrated by evaluating the local equilibrium. Inother words, there are only six independent Cauchy stress tensorcomponents. Vector notation is therefore also frequentlyemployed,
=
33
2322
131211
}{
s
ss
sss
σ
Sym.
=
23
13
12
33
22
11
s
s
s
s
s
s
σ
or
2/15/2016 8 8Lecture #5 – Fall 2015 8D. Mohr
151-0735: Dynamic behavior of materials and structures
Change of the stress tensor due to rotations
Rtt =~
1e
2e
1e
2e
Rnn =~
n
t
RσnRRσσnRRtt ()~()(~
==== T
Rσσ =~
nσnR ~~~) =T
TR
Let s denote the Cauchy stress tensor in the unrotated configurationwhich provides the traction vector t for a given normal vector n. Thetraction vector after rotating the stress configuration reads:
And hence, the Cauchy stress tensor in the rotated configurationreads:
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151-0735: Dynamic behavior of materials and structures
Principal stresses & directions
III pσp s=
11s1e
2e 21s1σet =
Iσp
IpIIp
principal direction
principal stress
We seek the directions p for which the traction vector acting onthe surface pdA has no shear components.
11s
Shear component
normal component
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151-0735: Dynamic behavior of materials and structures
1ppσp pp ss == 0p1σ = ps
Non-trivial solutions can be found for p if
0det = 1σ ps 032
2
1
3 = III ppp sss
(characteristic polynomial)
The characteristic polynomial is a cubic equation for the principalstresses. It is determined through the stress tensor invariants
][1 σtrI =first invariant:
):( 2
121
2 σσ= IIsecond invariant:
]det[3 σ=Ithird invariant:
with 2
23
2
13
2
13
2
33
2
22
2
11 222: ssssss =σσ
Principal stresses & directions
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151-0735: Dynamic behavior of materials and structures
Principal stresses & directions
IIIIII sss
Solving the characteristic polynomial yields three solutions whichare called principal stresses. After ordering, we have
maximum princ. stress
Intermediate princ. stress
minimum princ. stress
The corresponding orthogonal principal stress directions {pI, pII, pIII}are found after solving
0p1σ = iis
1=ipip for IIIIi ,..,=
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Spectral decomposition(of symmetric tensors)
With the help of the principal stresses and their directions, thestress tensor may also be rewritten as
which is called the spectral decomposition of the Cauchy stresstensor.
IIIIIIIIIIIIIIIIII ppppppσ = sss
Recall that the tensor product of two vectors e1 and e2 defines thelinear map
)()( 2121 aeeaee =
In matrix notation, we have
=
000
000
010
}{ 21 ee
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151-0735: Dynamic behavior of materials and structures
Stress tensor invariants
The value of the principal stresses remain unchanged underrotations. Only the principal directions will rotate:
IIIIIIIIIIIIIIIIII
TRpRpRpRpRpRpRσR = sss
This is can also be explained by the fact that the values of I1, I2 andI3 remain unchanged under rotations (that is why these are called“invariants”), e.g.
Hence the characteristic polynomial remains unchanged as well asits roots sI, sII and sIII. The principal stresses are therefore alsoinvariants of the stress tensor.
] [][1
TtrtrI RσRσ ==
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151-0735: Dynamic behavior of materials and structures
Description of Motion in 3D
A body is considered as a closed set of material points.
1e
2e
3e
body in its INITIAL CONFIGURATION
body in its CURRENT CONFIGURATION
X x
u
],[ tXxx =
The current position of a material pointinitially located at the position X is describedby the function
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151-0735: Dynamic behavior of materials and structures
Deformation Gradient (3D)
The displacement vector is then given by the difference in position
XXxXuu == ],[],[ tt
X x
u
The deformation gradient is defined as
X
Xu1
X
XuX
X
XxXF
=
=
=
],[]),[(],[],[
tttt
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151-0735: Dynamic behavior of materials and structures
Deformation Gradient (3D)
It follows from the definition of the deformation gradient that thechange in length and orientation of an infinitesimal vector dXattached to a material point can be described by the linear mapping
)(dXFdx =
X x
dXdx
The deformation gradient is thus also considered as a tensor.
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Velocity gradient
The time derivative of displacement gradient is
X
Xv
X
Xu
X
XxXF
=
=
=
],[],[],[],[
22 t
t
t
t
tt
It corresponds to the spatial gradient of the velocity field withrespect to the material point coordinate X in the initialconfiguration. The spatial gradient of the velocity field withrespect to the current position coordinate x is called velocitygradient:
x
vL
=:
We have the relationship
LFX
x
x
v
X
vF =
=
=
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151-0735: Dynamic behavior of materials and structures
Rate of deformation tensor
As any other non-symmetric second-order tensor, the velocitygradient can be decomposed into a symmetric and skew part:
WDL =
with
)(2
1][: Tsym LLLD ==
)(2
1][: Tskw LLLW ==
In mechanics, the symmetric part of the velocity gradient is typicallycalled rate of deformation tensor D, while the skew part is calledspin tensorW.
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151-0735: Dynamic behavior of materials and structures
Polar decomposition
The deformation gradient F (non-symmetric tensor) is oftendecomposed into a rotation tensor R and a symmetric stretchtensor.
VRRUF == with 1RRRR == )()( TT
TUU =
TVV =
UV
The tensor U is called right stretch tensor, while V is called leftstretch tensor
2/15/2016 20 20Lecture #5 – Fall 2015 20D. Mohr
151-0735: Dynamic behavior of materials and structures
Interpretation of stretch tensors
1. Stretching2. Rotation
VF
R
FR
U
1. Rotation2. Stretching
Left stretch tensor Right stretch tensor
VRF = RUF =
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151-0735: Dynamic behavior of materials and structures
Logarithmic strain tensor
A frequently used deformation measure in finite strain theory isthe so-called logarithmic strain tensor or Hencky strain tensor:
=
==3
1
)](ln[lni
iiiH uuUε
Its evaluation requires the spectral decomposition of the rightstretch tensor,
=
=3
1
)(i
iii uuU iii uUu =i.e.
The values i are called the principal stretches. The latter mayalso be computed using the left stretch tensor due to theidentity:
=
==3
1
)(i
iii
TRuRuRURV
2/15/2016 22 22Lecture #5 – Fall 2015 22D. Mohr
151-0735: Dynamic behavior of materials and structures
Three-dimensional Rate-independent Plasticity
2/15/2016 23 23Lecture #5 – Fall 2015 23D. Mohr
151-0735: Dynamic behavior of materials and structures
3D Kinematics: Incremental problem
nV
1nF
nR
nF
1nR
1nV
F
RINITIAL
DEFORMED @ tn
DEFORMED@ tn+1
ROTATED @ tnROTATED
@ tn+1
2/15/2016 24 24Lecture #5 – Fall 2015 24D. Mohr
151-0735: Dynamic behavior of materials and structures
3D kinematics: Incremental problem
• Incremental deformation gradient:
nn dxΔFdx )(1 =
nn FΔFF )(1 =
• Incremental rotation
nn RΔRR )(1 =
• Incremental left stretch tensor
) ( 1
T
nn ΔRVΔRΔVV =
With the above definitions in place, it can be shown that the incremental rotation can be obtained from the polar decomposition of the incremental deformation gradient:
) ( ΔRΔVΔF = 1ΔRΔR =T) )( (with and TΔVΔV =
1nFF
1nR
R
nR
nF nV1nV
1nt
1ntnt
nt
0t
2/15/2016 25 25Lecture #5 – Fall 2015 25D. Mohr
151-0735: Dynamic behavior of materials and structures
Strain rate and total strainThe rate of deformation tensor is work-conjugate to the Cauchy stress tensor and is thus frequently used to define the strain rate:
==
T
x
v
x
vDε
2
1:
To obtain a total strain measure, the strain rate is integrated on a fixed basis (e.g. initial configuration) and then rotated forward to the basis of the current time t:
][][][][][][0
tdtt T
t
TRRDRRε
=
In commercial finite element software, this integration is often approximated by
)ln()()(1 ΔVΔRεΔRε
T
nn
In the absence of rotations, the strain tensor obtained after integration is the same as the Hencky strain tensor.
2/15/2016 26 26Lecture #5 – Fall 2015 26D. Mohr
151-0735: Dynamic behavior of materials and structures
Additive strain rate decomposition
The strain rate is decomposed into an elastic and a plastic part,
pe εεε =
The corresponding algorithmic decomposition of the strain increment associated wit finite time increments t reads
pe εεΔVε == )ln(
The above decomposition is an approximation of the well-established multiplicative decomposition of the total deformation gradient,
peFFF =
The approximation (*) of (**) yields reasonable results in finite strain problems when the elastic strains are small compared to unity.
(*)
(**)
2/15/2016 27 27Lecture #5 – Fall 2015 27D. Mohr
151-0735: Dynamic behavior of materials and structures
Elastic constitutive equation
The linear elastic isotropic constitutive equation reads
eεCσ :=
with C denoting the fourth-order elastic stiffness tensor. For notational convenience, the above stress-strain relationship is rewritten in vector notation
=
e
e
e
e
e
e
E
23
13
12
33
22
11
23
13
12
33
22
11
21
021
0021
0001
0001
0001
)21)(1(
s
s
s
s
s
s
with the Young’s modulus E and the Poisson’s ratio n.
Sym.
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151-0735: Dynamic behavior of materials and structures
Equivalent stress definition
The yield function is often expressed in terms of an equivalentstress, i.e. a scalar measure of the magnitude of the Cauchy stresstensor. The most widely used scalar measure in engineering practiceis the von Mises equivalent stress:
SSσ :2
3][ == ss
with the deviatoric stress tensor
1σ
σσS3
][][
trdev ==
Note that the von Mises equivalent stress is a function of the deviatoricpart of the stress tensor only. It is thus pressure-independent, i.e. it is insensitive to changes of the trace of s.
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Equivalent stress definition
The von Mises equivalent stress is an isotropic function, i.e. it isinvariant to rotations of the Cauchy stress tensor:
] [][ TRσRσ ss = for any rotation
})()(){( 222
21
IIIIIIIIIIII sssssss =
R
As an alternative it may also be expressed as a function of the stresstensor invariants or the principal stresses, e.g.
23J=s SS :21
2 =Jwith
Von Mises plasticity models are therefore also often called J2-plasticity models.
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151-0735: Dynamic behavior of materials and structures
Yield function and surface
With the von Mises equivalent stress definition at hand, the yieldfunction is written as:
][][],[ pp kf s = σσ
Is
IIs
IIIs
The yield surface is
0],[ =pf σ
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151-0735: Dynamic behavior of materials and structures
Flow rule
In 3D, it has been demonstrated that the direction of plastic flow is aligned with the outward normal to the yield surface,
σε
=
fp with
s
s S
σσ 2
3=
=
f
σ
f
0=f
In other words, the ratios of the components of the plastic strain rate tensor are the same as the deviatoric stress ratios
p
kl
p
ij
p
kl
p
ij
S
S=
2/15/2016 32 32Lecture #5 – Fall 2015 32D. Mohr
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Flow rule
The proposed associated flow rule also implies that the plastic flow is incompressible (no volume change),
0][
2
3][ ==
s
Sε
trtr p
σ
f
0=f
The magnitude of the plastic strain rate tensor is controlled by the non-negative plastic multiplier . It is also called equivalent plastic strain rate.
0
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Isotropic strain hardening
The flow stress is expressed as a function of the equivalent plastic strain,
][ pkk =][
32
pk with
= dttp ][
It controls the size of the elastic domain (diameter of the von Mises cylinder in stress space).
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151-0735: Dynamic behavior of materials and structures
Isotropic hardening
The same parametric forms for are used in 3D as in 1D.
n
pS Ak )( 0 =
][ pkk =
]exp[10 pV Qkk =
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
SV kkk = )1(
Swift Voce
Qkkd
dk
p
0 ,0
Hardening saturation
p p p
k k k
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Loading/unloading conditions
=0f0 if
0=f0 if
0=f0 if
0fand
0=fand
The same loading and unloading conditions are used in 3D as in 1D:
2/15/2016 36 36Lecture #5 – Fall 2015 36D. Mohr
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Isotropic hardening plasticity (3D) - Summary
i. Constitutive equation for stress
)(: pεεCσ =
ii. Yield function][][],[ pp kf s = σσ
iii. Flow rule
iv. Loading/unloading conditions
=0f0 if
0=f0 if
0=f0 if
0fand
0=fand
v. Isotropic hardening law
][ pkk = with = dtp
σε
=
fp
2/15/2016 37 37Lecture #5 – Fall 2015 37D. Mohr
151-0735: Dynamic behavior of materials and structures
Return Mapping Algorithm (3D)
State variables at time tn+1p
n
p
n εε =1
OUTPUT:
ΔεCσσ :1 = nn
Stress at time tn+1
p
n
p
n =1
Applied total strain increment
Δε
Calculate Trial State
State variables at time tnp
n
p
n ,ε
0][1 = nf
01
trial
nf
Solve:
0
01
trial
nf
State variables at time tn+1
p
p
n
p
n εεε =1
=
p
n
p
n 1
Stress at time tn+1
trial
n
trial
n f 11 , σ
0=
OUTPUT:
)(:1 pnn ΔεΔεCσσ =
Simplified schematic assumes that all tensor variables at time tn have already been “pushed forward” to the basis at time tn+1.
2/15/2016 38 38Lecture #5 – Fall 2015 38D. Mohr
151-0735: Dynamic behavior of materials and structures
Reading Materials for Lecture #5
• M.E. Gurtin, E. Fried, L. Anand, “The Mechanics and Thermodynamics of Continua”, Cambridge University Press, 2010.
• Abaqus Theory Manual abaqus.ethz.ch:2080/v6.11/pdf_books/THEORY.pdf