151-0735: Dynamic behavior of materials and structures Lecture #5:
• Introduction to Continuum Mechanics • Three-dimensional Rate-independent Plasticity by Dirk Mohr
ETH Zurich, Department of Mechanical and Process Engineering, 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 stress tensor Suppose that a mechanically loaded body is hypothetically cut into two parts. The created hypothetical surfaces can be described by the unit normal vector field n=n[x] with the associated infinitesimal areas dA. t n dA dA n t
x e2
e1 The traction vectors t=t[x] describe the forces 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 Cauchy stress tensor
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 matrix 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 coordinate system: 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 Symmetry of the Cauchy stress tensor Unlike other tensors 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 Shear 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 p1p = 0
Non-trivial solutions can be found for p if
3 2 detσ 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 invariant: I1 = tr[σ]
1 2 2 2 2 2 2 2 second 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 1pi = 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 tensor product of two vectors e1 and e2 defines the linear map (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 Deformation Gradient (3D)
u
X x
The displacement 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 Velocity gradient
The time derivative of displacement gradient is 2x[X,t] 2u[X,t] v[X,t] F[X,t] = = = tX tX 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 spin tensor W.
D. Mohr2/15/2016 Lecture #5 – Fall 2015 18 18 18 151-0735: Dynamic behavior of materials and structures Polar decomposition The deformation gradient F (non-symmetric tensor) 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 strain tensor A frequently used deformation measure in finite strain theory 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
D. Mohr2/15/2016 Lecture #5 – Fall 2015 21 21 21 151-0735: Dynamic behavior of materials and structures
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 Fn1 F @ tn+1
DEFORMED @ tn Vn1 Fn Vn
INITIAL ROTATED ROTATED @ tn @ tn+1 Rn R
Rn1
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: tn1 dxn1 = (ΔF)dxn F Fn1 Fn1 = (ΔF)Fn Vn1 • Incremental rotation F V n n R t t Rn1 = (ΔR)Rn 0 n Rn • Incremental left stretch tensor Rn1 T Vn1 = 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 basis (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 εn1 (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 constitutive equation 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 1 223 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 yield 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 pressure-independent, i.e. it is insensitive to changes of the trace 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 yield surface 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 volume 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 Q1 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 εn1 = εn State variables at time t Calculate trial n fn1 0 p p p p Trial State n1 = n εn , n trial trial σn1 , fn1 Stress at time tn+1
trial σn1 = σn C : Δε fn1 0
Solve: fn1[ ] = 0 0 OUTPUT: State variables at time tn+1 p p Stress at time tn+1 εn1 = εn ε p p p σn1 = σn C : (Δε Δε p ) n1 = 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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