CH.8. PLASTICITY Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Previous Notions Lecture 2 Lecture 3 Principal Stress Space Lecture 4 Normal and Shear Octahedral Stresses Stress Invariants Lecture 5 Effective Stress Rheological Friction Models Elastic Element Frictional Element Elastic-Frictional Model 2 Overview (cont’d) Rheological Friction Models (cont’d) Frictional Model with Hardening Elastic-Frictional Model with Hardening Phenomenological Behaviour Notion of Plastic Strain Notion of Hardening Lecture 6 Bauschinger Effect Elastoplastic Behaviour 1D Incremental Theory of Plasticity Additive Decomposition of Strain Lecture 7 Hardening Variable Yield Stress, Yield Function and Space of Admissible Stresses Lecture 8 Lecture 9 Constitutive Equation Lecture 9 Elastoplastic Tangent Modulus Lecture 10 Uniaxial Stress-Strain Curve 3 Overview (cont’d) 3D Incremental Theory of Plasticity Additive Decomposition of Strain Hardening Variable Lecture 11 Yield Function Loading - Unloading Conditions and Consistency Conditions Constitutive Equation Lecture 12 Elastoplastic Constitutive Tensor Yield Surfaces Lecture 13 Von Mises Criterion Tresca Criterion Mohr-Coulomb Criterion Lecture 14 Drucker-Prager Criterion 4 8.1 Introduction Ch.8. Plasticity 5 Introduction A material with plastic behavior is characterized by: A nonlinear stress-strain relationship. The existence of permanent (or plastic) strain during a loading/unloading cycle. Lack of unicity in the stress-strain relationship. Plasticity is seen in most materials, after an initial elastic state. 6 Previous Notions PRINCIPAL STRESSES Regardless of the state of stress, it is always possible to choose a special set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system. The three planes perpendicular to the principle axes are the principal planes. The normal stress components in the principal planes are the principal σ stresses. σ 33 x 31 σ x 3 32 3 ′ σ x3 13 σ σ1 00 23 σ σ x′ σ σ = σ 11 12 σ 21 σ 1 3 [] 002 22 σ1 00σ 3 σ 2 x1 x1 x2 x2 x2′ 7 Previous Notions PRINCIPAL STRESSES The Cauchy stress tensor is a symmetric 2nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers. Computing the eigenvalues λ and the corresponding eigenvectors v : σσ⋅=vλλ v[] −1 ⋅= v0 σ11 − λσ12 σ13 not det[]σσ−λ11= −= λ σ12 σ22 − λσ23 =0 σ σ σλ− INVARIANTS 13 23 33 σ σ 33 32 characteristic x 31 σ x λλ−I1 − II 23 λ −=0 3 32 3 equation x3′ σ13 σ 23 λσ≡ σ11 σ12 σ x1′ σ 11 21 σ 22 3 λσ≡ σ1 22 σ λσ33≡ 2 x1 x1 x2 x2 x2′ 8 Previous Notions STRESS INVARIANTS Principal stresses are invariants of the stress state. They are invariant w.r.t. rotation of the coordinate axes to which the stresses are referred. The principal stresses are combined to form the stress invariants I : I= Tr σ =σ =++ σσσ 1 () ii 123 REMARK 1 2 The I invariants are obtained II2=()σσ: −=−+1 (σσ12 σσ 13 + σσ 23) 2 from the characteristic equation I3 = det ()σ of the eigenvalue problem. These invariants are combined, in turn, to obtain the invariants J: JI11= = σ ii REMARK The J invariants can be 12 11 J2=() II 12 +=2:σσij ji =()σσ 2 22 expressed the unified form: 1 = σi ∈ 13 11Ji Tr() i {}1, 2, 3 J3=( I 1 +33 I 12 I + I 3) = Tr ()σσσ ⋅⋅ =σσij jk σ ki i 3 33 9 Previous Notions SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR Given the Cauchy stress tensor σ and its principal stresses, the following is defined: Mean stress 1 11 σ=Tr ()σ =σ =() σσσ ++ m 3 33ii 123 REMARK Mean pressure In a hydrostatic state of stress, the 1 p =−=−++σm () σσσ123 stress tensor is isotropic and, thus, 3 its components are the same in any Cartesian coordinate system. A spherical or hydrostatic As a consequence, any direction state of stress : σ 00 is a principal direction and the σσσ= = 123 σ ≡=00σσ1 stress state (traction vector) is the 00σ same in any plane. 10 Previous Notions SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR The Cauchy stress tensor σ can be split into: σ=σsph + σ′ The spherical stress tensor: Also named mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body 11 σσ:=σσ1 =Tr () 11 = sph m 33ii The stress deviatoric tensor: Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body σ′ =dev σ = σ−σ m 1 11 Previous Notions STRESS INVARIANTS OF THE STRESS DEVIATORIC TENSOR The stress invariants of the stress deviatoric tensor: I1′′= Tr ()σ = 0 1 II′=σσ ′′: − 2 212 () 2221 I3′=det ()σ ′ =σσσ11 ′′′22 33 +2 σσσ 12 ′′′ 23 13 −−− σσ 12 ′′ 33 σσ 23′′ 11 σσ 13 ′′ 22 =() σσσij ′′′ jk ki 3 These correspond exactly with the invariants J of the same stress deviator tensor: JI11′′= = 0 1 1 JI′′= 2 +==2:II′ ′()σσ ′′ 212 ()222 1 3 11 JI31′′= + 3II12′′+3I33′ = I ′ = Tr ()σσσ ′′′ ⋅⋅ =()σσij ′′′ jk σ ki 3 ( ) 33 12 Previous Notions EFFECTIVE STRESS The effective stress or equivalent uniaxial stress σ is the scalar: 33 σ =3J ' = σσ′′ = σσ´: ´ 2 22ij ij It is an invariant value which measures the “intensity” of a 3D stress state in a terms of an (equivalent) 1D tensile stress state. It should be “consistent”: when applied to a real 1D tensile stress, should return the intensity of this stress. 13 Example Calculate the value of the equivalent uniaxial stress for an uniaxial state of stress defined by: E, ν y σ 00 uσ σ σ ≡ x x 0 00 σ u σ u 0 00 x z 14 σ u 00 σ ≡ 0 00 Example - Solution 0 00 1 σ σ =σ )= u σ u Mean stress: m Tr( 00 33 3 σ m 00 σ σ ≡=0σ 00u 0 Spherical and deviatoric parts sph m 3 00σ of the stress tensor: m σ 00 u 3 2 σ 00 3 u σσum− 00 1 σ′ =−≡ σσ 0 −σσ 00 = − 0 sph m 3 u 00−σ m 1 00− σ 3 u 3 3 411 32 σσ= σσ′′ =2 () + + = σσσ= 2ij ij 2u 9 9 9 23 u u 15 8.2 Principal Stress Space Ch.8. Plasticity 16 Principal Stress Space The principal stress space or Haigh–Westergaard stress space is the space defined by a system of Cartesian axes where the three spatial axes represent the three principal stresses for a body subject to stress: σσσ123≥≥ 17 Octahedral plane Any of the planes perpendicular to the hydrostatic stress axis is a octahedral plane. 1 1 Its unit normal is n = 1 . 3 1 σσσ123≥≥ 18 Normal and Shear Octahedral Stresses Consider the principal stress space: The normal octahedral stress is defined as: 1/ 3 3σoct =OA = OP ⋅=n []σσσ123, , 1/ 3 = 1/ 3 3 =()σσσ ++ =3 σ 3 123 m I σσ= = 1 oct m 3 19 Normal and Shear Octahedral Stresses Consider the principal stress space: The shear or tangential octahedral stress is defined as: 3τ oct = AP Where the AP is calculated from: 2 2 22222 3τoct =AP = OP − OA =(σσσ123 ++) − 1 2 ' −()σσσ123 ++ =2J 2 3 Alternative forms of τ oct : 1/2 112 τ= σσσ222 ++− σσσ ++ oct 1 2 3() 123 3 3 1/2 12 1 222 2 τ= σσ − +− σσ +− σσ τ = []J′ oct ()()()1 2 23 13 oct 3 2 33 20 Normal and Shear Octahedral Stresses In a pure spherical stress state: 1 σ=σ11 →σ =3 σσ →σσ = = m 3 esf σ′ =−= σσesf 0 J2′ = 0 τ = A pure spherical stress state is oct 0 located on the hydrostatic stress axis. In a pure deviator stress state: ′ σ = σσ= σ m =Tr()σσ = Tr (′ ) = 0 oct 0 A pure deviator stress state is located on the octahedral plane containing the origin of the principal stress space 21 Stress Invariants Any point in space is unambiguously defined by the three invariants: The first stress invariant I 1 characterizes the distance from the origin to the octahedral plane containing the point. The second deviator stress invariant J 2′ characterizes the radius of the cylinder containing the point and with the hydrostatic stress axis as axis. The third deviator stress invariantJ 3′ characterizes the position of the point on the circle obtained from the intersection of the octahedral plane and the cylinder. It defines an angle θ () J 3′ . 22 Projection on the Octahedral Plane The projection of the principal stress space on the octahedral plane results in the division of the plane into six “sectors”: These are characterized by the different principal stress orders. Election of a criterion, FEASIBLE e.g.: σσσ123≥≥ WORK SPACE 23 8.4 Phenomenological Behaviour Ch.8. Plasticity 38 Notion of Plastic Strain PLASTIC STRAIN εε=ep + ε elastic limit: σ e LINEAR ELASTIC BEHAVIOUR σε= E e 39 Bauschinger Effect Also known as kinematic hardening. σ f σεe + K K σ e K −+σεe K K −σ e 41 Elastoplastic Behaviour Considering the phenomenological behaviour observed, elastoplastic materials are characterized by: Lack of unicity in the stress-strain relationship. The stress value depends on the actual strain and the previous loading history. A nonlinear stress-strain relationship. There may be certain phases in the deformation process with incremental linearity. The existence of permanent (or plastic) strain during a loading / unloading cycle.