151-0735: Dynamic behavior of materials and structures Lecture #5:

• Introduction to • Three-dimensional Rate-independent by Dirk Mohr

ETH Zurich, Department of Mechanical and Process , Chair of Computational Modeling of Materials in Manufacturing

© 2015

D. Mohr2/15/2016 Lecture #5 – Fall 2015 1 1 1 151-0735: Dynamic behavior of materials and structures

Introduction to Continuum Mechanics

D. Mohr2/15/2016 Lecture #5 – Fall 2015 2 2 2 151-0735: Dynamic behavior of materials and structures Cauchy Suppose that a mechanically loaded body is hypothetically cut into two parts. The created hypothetical surfaces can be described by the unit normal vector n=n[x] with the associated areas dA.  t n dA dA  n t

x e2

e1 The traction vectors t=t[x] describe the per unit area that would need to act on the hypothetical surfaces ndA to ensure equilibrium.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 3 3 3 151-0735: Dynamic behavior of materials and structures

n dA t

x e2

e1 The Cauchy stress tensor s=s[x] provides the traction vector t that acts on the hypothetical surfaces ndA at a position x (in the current configuration). t = σ(ndA)

From a mathematical point of view, the above equation defines the linear mapping of vectors in R3. The operator s is thus called a tensor.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 4 4 4 151-0735: Dynamic behavior of materials and structures Cauchy stress tensor

For a given set of orthonormal coordinate vectors {e1, e2, e3}, we can also define the stress components sij:

ei

σe j traction vector t acting on unit surface defined s ij = ei σe j

by normal vector ej e j t s ij

ei s jj σe j

e j s jj = e j σe j

D. Mohr2/15/2016 Lecture #5 – Fall 2015 5 5 5 151-0735: Dynamic behavior of materials and structures Cauchy stress tensor

For a given set of orthonormal coordinate vectors {e1, e2, e3}, it can also be useful to write the stress tensor in notation:

s 22 s11 s12 s13    s12 {σ} = s 21 s 22 s 23  s 32 s   21 s 31 s 32 s 33  s 23 s11 s s 31 Stress component s 33 13

s ij e2

along acting on : direction e surface e i j e1

e3

D. Mohr2/15/2016 Lecture #5 – Fall 2015 6 6 6 151-0735: Dynamic behavior of materials and structures of the Cauchy stress tensor Unlike other used in mechanics, the Cauchy stress tensor is symmetric, T s ij = s ji σ = σ

which can be demonstrated by evaluating the local equilibrium. In other words, there are only six independent Cauchy stress tensor components. Vector notation is therefore also frequently employed, s11    s 22 s11 s12 s13       s 33  {σ} =  s 22 s 23  or σ =    Sym.  s12   s 33  s   13  s 23  D. Mohr2/15/2016 Lecture #5 – Fall 2015 7 7 7 151-0735: Dynamic behavior of materials and structures Change of the stress tensor due to rotations

n~ = Rn ~ n t = Rt e2 t e1

Let s denote the Cauchy stress tensor in the unrotated configuration which provides the traction vector t for a given normal vector n. The traction vector after rotating the stress configuration reads: ~ T T t = Rt = R(σn) = Rσ(R n~) = (RσR )n~ = σ~n~ And hence, the Cauchy stress tensor in the rotated configuration reads: σ~ = RσRT

D. Mohr2/15/2016 Lecture #5 – Fall 2015 8 8 8 151-0735: Dynamic behavior of materials and structures Principal stresses & directions

σp component I p s II e 21 t = σe 2 1 pI

e1 s11 normal component

σpI = s I pI

principal principal stress direction

We seek the directions p for which the traction vector acting on the surface pdA has no shear components.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 9 9 9 151-0735: Dynamic behavior of materials and structures Principal stresses & directions

σp = s pp = s p1p σ s p1p = 0

Non-trivial solutions can be found for p if

3 2 detσ s p1 = 0 s p  I1s p  I2s p  I3 = 0 (characteristic polynomial) The characteristic polynomial is a cubic equation for the principal stresses. It is determined through the stress tensor invariants

first : I1 = tr[σ]

1 2 2 2 2 2 2 2 invariant: I2 = 2 (I1  σ : σ) with σ : σ = s11  s 22  s 33  2s13  2s13  2s 23

third invariant: I3 = det[σ]

D. Mohr2/15/2016 Lecture #5 – Fall 2015 10 10 10 151-0735: Dynamic behavior of materials and structures Principal stresses & directions

Solving the characteristic polynomial yields three solutions which are called principal stresses. After ordering, we have

Intermediate princ. stress

s I  s II  s III maximum minimum princ. stress princ. stress

The corresponding orthogonal principal stress directions {pI, pII, pIII} are found after solving

σ s i 1pi = 0 pi for i = I,.., III pi = 1

D. Mohr2/15/2016 Lecture #5 – Fall 2015 11 11 11 151-0735: Dynamic behavior of materials and structures Spectral decomposition (of symmetric tensors)

With the help of the principal stresses and their directions, the stress tensor may also be rewritten as

σ = s I pI  pI  s II pII  pII  s III pIII  pIII which is called the spectral decomposition of the Cauchy stress tensor.

Recall that the of two vectors e1 and e2 defines the (e1  e2 )a = e1(e2 a) 0 1 0 In matrix notation, we have   {e1  e2} = 0 0 0   0 0 0

D. Mohr2/15/2016 Lecture #5 – Fall 2015 12 12 12 151-0735: Dynamic behavior of materials and structures Stress tensor invariants

The value of the principal stresses remain unchanged under rotations. Only the principal directions will rotate:

T R σ R = s I RpI  RpI  s II RpII  RpII  s III RpIII  RpIII

This is can also be explained by the fact that the values of I1, I2 and I3 remain unchanged under rotations (that is why these are called “invariants”), e.g. T I1 = tr[σ] = tr[R σ R ]

Hence the characteristic polynomial remains unchanged as well as

its roots sI, sII and sIII. The principal stresses are therefore also invariants of the stress tensor.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 13 13 13 151-0735: Dynamic behavior of materials and structures Description of Motion in 3D

A body is considered as a closed set of material points. body in its CURRENT CONFIGURATION body in its INITIAL CONFIGURATION u

X x e2 The current position of a material point

e1 initially located at the position X is described

e3 by the function x = x[X,t]

D. Mohr2/15/2016 Lecture #5 – Fall 2015 14 14 14 151-0735: Dynamic behavior of materials and structures (3D)

u

X x

The vector is then given by the difference in position u = u[X,t] = x[X,t]  X

The deformation gradient is defined as x[X,t] (X  u[X,t]) u[X,t] F[X,t] = = = 1  X X X

D. Mohr2/15/2016 Lecture #5 – Fall 2015 15 15 15 151-0735: Dynamic behavior of materials and structures Deformation Gradient (3D) dx dX

X x

It follows from the definition of the deformation gradient that the change in length and orientation of an infinitesimal vector dX attached to a material point can be described by the linear mapping dx = F(dX) The deformation gradient is thus also considered as a tensor. D. Mohr2/15/2016 Lecture #5 – Fall 2015 16 16 16 151-0735: Dynamic behavior of materials and structures gradient

The time of displacement gradient is 2x[X,t] 2u[X,t] v[X,t] F[X,t] = = = tX tX X It corresponds to the spatial gradient of the velocity field with respect to the material point coordinate X in the initial configuration. The spatial gradient of the velocity field with respect to the current position coordinate x is called velocity gradient: v L := x We have the relationship v v x F = = = LF X x X

D. Mohr2/15/2016 Lecture #5 – Fall 2015 17 17 17 151-0735: Dynamic behavior of materials and structures Rate of deformation tensor

As any other non-symmetric second-order tensor, the velocity gradient can be decomposed into a symmetric and skew part: L = D  W with 1 D := sym[L] = (L  LT ) 2 1 W := skw[L] = (L  LT ) 2 In mechanics, the symmetric part of the velocity gradient is typically called rate of deformation tensor D, while the skew part is called W.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 18 18 18 151-0735: Dynamic behavior of materials and structures The deformation gradient F (non-) is often decomposed into a rotation tensor R and a symmetric stretch tensor. F = RU = VR with R(RT ) = (RT )R = 1 U = UT V = VT V  U

The tensor U is called right stretch tensor, while V is called left stretch tensor

D. Mohr2/15/2016 Lecture #5 – Fall 2015 19 19 19 151-0735: Dynamic behavior of materials and structures Interpretation of stretch tensors

Left stretch tensor Right stretch tensor

F = VR F = RU

F F V R

R U

1. Rotation 1. Stretching 2. Stretching 2. Rotation

D. Mohr2/15/2016 Lecture #5 – Fall 2015 20 20 20 151-0735: Dynamic behavior of materials and structures Logarithmic tensor A frequently used deformation measure in is the so-called logarithmic strain tensor or Hencky strain tensor: 3 εH = ln U = ln[i ](ui  ui ) i=1 Its evaluation requires the spectral decomposition of the right stretch tensor, 3 U = i (ui  ui ) i.e. Uui = iui i=1

The values i are called the principal stretches. The latter may also be computed using the left stretch tensor due to the

identity: 3 T V = RUR = i (Rui  Rui ) i=1

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Three-dimensional Rate-independent Plasticity

D. Mohr2/15/2016 Lecture #5 – Fall 2015 22 22 22 151-0735: Dynamic behavior of materials and structures 3D Kinematics: Incremental problem

DEFORMED Fn1 F @ tn+1

DEFORMED @ tn Vn1 Fn Vn

INITIAL ROTATED ROTATED @ tn @ tn+1 Rn R

Rn1

D. Mohr2/15/2016 Lecture #5 – Fall 2015 23 23 23 151-0735: Dynamic behavior of materials and structures 3D kinematics: Incremental problem • Incremental deformation gradient: tn1 dxn1 = (ΔF)dxn F Fn1 Fn1 = (ΔF)Fn Vn1 • Incremental rotation F V n n R t t Rn1 = (ΔR)Rn 0 n Rn • Incremental left stretch tensor Rn1 T Vn1 = ΔV (ΔR Vn ΔR ) 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: ΔF = ΔV (ΔR ) with (ΔR )(ΔR )T = 1 and ΔV = ΔVT

D. Mohr2/15/2016 Lecture #5 – Fall 2015 24 24 24 151-0735: Dynamic behavior of materials and structures Strain rate and total strain The rate of deformation tensor is work-conjugate to the Cauchy stress tensor and is thus frequently used to define the strain rate: T 1  v  v   ε = D :=        2  x  x   To obtain a total strain measure, the strain rate is integrated on a fixed (e.g. initial configuration) and then rotated forward to the basis of the current time t: t  T  T ε[t] = R[t] R [ ]D[ ]R[ ]d R [t]  0  In commercial finite element software, this integration is often approximated by T εn1  (ΔR)εn (ΔR)  ln(ΔV) In the absence of rotations, the strain tensor obtained after integration is the same as the Hencky strain tensor. D. Mohr2/15/2016 Lecture #5 – Fall 2015 25 25 25 151-0735: Dynamic behavior of materials and structures Additive strain rate decomposition The strain rate is decomposed into an elastic and a plastic part,

ε = ε e  ε p The corresponding algorithmic decomposition of the strain increment associated wit finite time increments t reads

ε = ln(ΔV) = εe  ε p (*) The above decomposition is an approximation of the well-established multiplicative decomposition of the total deformation gradient,

F = FeFp (**) The approximation (*) of (**) yields reasonable results in finite strain problems when the elastic strains are small compared to unity.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 26 26 26 151-0735: Dynamic behavior of materials and structures Elastic The linear elastic isotropic constitutive equation reads

σ = C : εe with C denoting the fourth-order elastic stiffness tensor. For notational convenience, the above stress-strain relationship is rewritten in vector notation

e s11  1   0 0 0 11     e  s  1  0 0 0    22    22  e s 33  E  1 0 0 0     =   33  s 1 2 0 0 e  12  (1)(1 2)  12     Sym.  e  s13 1 2 0 13     e  s 23   1 223  with the Young’s modulus E and the Poisson’s ratio n.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 27 27 27 151-0735: Dynamic behavior of materials and structures Equivalent stress definition The function is often expressed in terms of an equivalent stress, i.e. a scalar measure of the magnitude of the Cauchy stress tensor. The most widely used scalar measure in engineering practice is the von Mises equivalent stress:

3 s = s [σ] = S : S 2 with the deviatoric stress tensor tr[σ] S = dev[σ] = σ  1 3 Note that the von Mises equivalent stress is a function of the deviatoric part of the stress tensor only. It is thus -independent, i.e. it is insensitive to changes of the of s.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 28 28 28 151-0735: Dynamic behavior of materials and structures Equivalent stress definition The von Mises equivalent stress is an isotropic function, i.e. it is invariant to rotations of the Cauchy stress tensor:

s [σ] = s [R σ RT ] for any rotation R

As an alternative it may also be expressed as a function of the stress tensor invariants or the principal stresses, e.g.

1 s = 3J 2 with J 2 = 2 S : S

1 2 2 2 s = 2 {(s I s II )  (s I s III )  (s II s III ) }

Von Mises plasticity models are therefore also often called J2- plasticity models.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 29 29 29 151-0735: Dynamic behavior of materials and structures Yield function and surface With the von Mises equivalent stress definition at hand, the yield function is written as: s III

f [σ, p ] = s [σ] k[ p ] The is

f [σ, p ] = 0

s II

s I

D. Mohr2/15/2016 Lecture #5 – Fall 2015 30 30 30 151-0735: 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, f f s 3 S ε =  with = = p σ σ σ 2 s In other words, the ratios of the components of the plastic strain f σ rate tensor are the same as the f = 0 deviatoric stress ratios

p p ij Sij p = p kl Skl

D. Mohr2/15/2016 Lecture #5 – Fall 2015 31 31 31 151-0735: Dynamic behavior of materials and structures Flow rule

The proposed associated flow rule also implies that the plastic flow is incompressible (no change), 3 tr[S] tr[ε p ] =  = 0 f 2 s σ f = 0 The magnitude of the plastic strain rate tensor is controlled by the non-negative plastic multiplier    0 . It is also called equivalent plastic strain rate.

D. Mohr2/15/2016 Lecture #5 – Fall 2015 32 32 32 151-0735: Dynamic behavior of materials and structures Isotropic strain hardening

The flow stress is expressed as a function of the equivalent plastic strain, k = k[ ] p 2 3 k[ p ] with  [t] = dt p  It controls the size of the elastic domain (diameter of the von Mises cylinder in stress space).

D. Mohr2/15/2016 Lecture #5 – Fall 2015 33 33 33 151-0735: Dynamic behavior of materials and structures Isotropic hardening

The same parametric forms for k = k [  p ] are used in 3D as in 1D.

4.00E+02 4.00E+02 Hardening saturation 4.00E+02 k dk 3.50E+02 3.50E+02  0, k  k0  Q 3.50E+02 d p 3.00E+02 3.00E+02 3.00E+02

2.50E+02 2.50E+02 2.50E+02

2.00E+02 2.00E+02 2.00E+02

1.50E+02 1.50E+02 1.50E+02

1.00E+02 1.00E+02 1.00E+02 Swift Voce 5.00E+01 5.00E+01 5.00E+01

 p 0.00E+00 0.00E+00 0.00E+00

n kS = A( p  0 ) kV = k0  Q1 exp[ p ] k = (1)kV  kS

D. Mohr2/15/2016 Lecture #5 – Fall 2015 34 34 34 151-0735: Dynamic behavior of materials and structures Loading/unloading conditions

The same loading and unloading conditions are used in 3D as in 1D:

0 if f  0  =  0 if f = 0 and f = 0 0 if f = 0 and f  0

D. Mohr2/15/2016 Lecture #5 – Fall 2015 35 35 35 151-0735: Dynamic behavior of materials and structures Isotropic hardening plasticity (3D) - Summary i. Constitutive equation for stress

σ = C : (ε  ε p ) ii. Yield function

f [σ, p ] = s [σ]  k[ p ] f iii. Flow rule ε =  p σ iv. Loading/unloading conditions 0 if f  0  =  0 if f = 0 and f = 0 0 if f = 0 and f  0 v. Isotropic hardening law k = k[ ] with  = dt p p 

D. Mohr2/15/2016 Lecture #5 – Fall 2015 36 36 36 151-0735: Dynamic behavior of materials and structures Return Mapping Algorithm (3D) Applied total strain  = 0 increment Δε OUTPUT: State variables at time tn+1 p p εn1 = εn State variables at time t Calculate trial n fn1  0 p p p p Trial State n1 = n εn , n trial trial σn1 , fn1 Stress at time tn+1

trial σn1 = σn  C : Δε fn1  0

Solve: fn1[ ] = 0   0 OUTPUT: State variables at time tn+1 p p Stress at time tn+1 εn1 = εn  ε p p p σn1 = σn  C : (Δε  Δε p ) n1 = n  

Simplified schematic assumes that all tensor variables at time tn have already been “pushed forward” to the basis at time tn+1. D. Mohr2/15/2016 Lecture #5 – Fall 2015 37 37 37 151-0735: 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

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