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 t 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 = σn e2 And hence, the Cauchy stress tensor in the rotated configuration e reads: 1 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 The current position of a material point e2 initially located at the position X is described by the function e x = x[X,t] 1 e3 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.