Lecture #5: Introduction to Continuum Mechanics Three ... · D. Mohr 2/15/2016 Lecture #5 –Fall...

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



Introduction toContinuum Mechanics



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.




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.




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




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



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



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:



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



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





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 ,..,=



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



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 ==



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



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



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.



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 =

=

=



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.



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



Interpretation of stretch tensors

1. Stretching2. Rotation

VF

R

FR

U

1. Rotation2. Stretching

Left stretch tensor Right stretch tensor

VRF = RUF =



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



Three-dimensional Rate-independent Plasticity



3D Kinematics: Incremental problem

nV

1nF

nR

nF

1nR

1nV

F

RINITIAL

DEFORMED @ tn

DEFORMED@ tn+1

ROTATED @ tnROTATED

@ tn+1



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



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.



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.

(*)

(**)



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.



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.



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.



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



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=



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



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).



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



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:



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



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.



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

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