CH.8. PLASTICITY Continuum Mechanics Course (MMC) - ETSECCPB - UPC - PowerPoint PPT Presentation

CH.8. PLASTICITY Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Overview  Introduction  Previous Notions  Principal Stress Space  Normal and Shear Octahedral Stresses  Stress Invariants  Effective Stress  Principal Stress Space  Normal and Shear Octahedral Stress  Stress Invariants  Projection on the Octahedral Plane  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  Bauschinger Effect  Elastoplastic Behaviour  1D Incremental Theory of Plasticity  Additive Decomposition of Strain  Hardening Variable  Yield Stress, Yield Function and Space of Admissible Stresses  Constitutive Equation  Elastoplastic Tangent Modulus  Uniaxial Stress-Strain Curve 3

Overview (cont’d)  3D Incremental Theory of Plasticity  Additive Decomposition of Strain  Hardening Variable  Yield Function  Loading - Unloading Conditions and Consistency Conditions  Constitutive Equation  Elastoplastic Constitutive Tensor  Yield Surfaces  Von Mises Criterion  Tresca Criterion  Mohr-Coulomb Criterion  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  stresses .  33  x x x  31 3 3  32   3   0 0 13 23 1 x               0 0 11 12 1 3  21 22  2    1   0 0  3 2 x x 1 1  1   2   3 x x x  2 2 2 7

Previous Notions  PRINCIPAL STRESSES  The Cauchy stress tensor is a symmetric 2 nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers .   Computing the eigenvalues and the corresponding eigenvectors : v            1 v v v 0      11 12 13 not                1 1 det = 0 12 22 23      INVARIANTS 13 23 33   33 characteristic         x x 3 2 x  I I I 0 31 equation 3 3  32 1 2 3  3 13 23 x          11 12 1 3 21 1 1 22      1   2   3 1 2 2     2 x x 3 3 1 1 x x x  2 2 2 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 REMARK 1 ii 1 2 3   1   The I invariants are obtained               2 I : I 2 1 1 2 1 3 2 3 2 from the characteristic equation   I   of the eigenvalue problem. det 3  These invariants are combined, in turn, to obtain the invariants J : REMARK    J I 1 1 ii The J invariants can be   1 1 1           2 J I 2 I : expressed the unified form: 2 1 2 ij ji 2 2 2   1      i J Tr i 1,2,3   1 1 1                i 3 i J I 3 I I 3 I Tr 3 1 1 2 3 ij jk ki 3 3 3 9

Previous Notions  SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR  Given the Cauchy stress tensor and its principal stresses, the following is defined:  Mean stress 1 1 1                Tr m ii 1 2 3 3 3 3 REMARK  Mean pressure In a hydrostatic state of stress, the 1             stress tensor is isotropic and, thus, p m 1 2 3 3 its components are the same in any Cartesian coordinate system.  A spherical or hydrostatic As a consequence, any direction state of stress :    0 0 is a principal direction and the             stress state (traction vector) is the 1 0 0   1 2 3 same in any plane.      0 0 10

Previous Notions  SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR        The Cauchy stress tensor can be split into: sph  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 1 1          1 1 1 : Tr sph m ii 3 3  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         1 dev m 11

Previous Notions  STRESS INVARIANTS OF THE STRESS DEVIATORIC TENSOR  The stress invariants of the stress deviatoric tensor :        I Tr 0 1   1        2 I : I 2 1 2   1                                           2 2 2 I det 2 3 11 22 33 12 23 13 12 33 23 11 13 22 ij jk ki 3  These correspond exactly with the invariants J of the same stress deviator tensor :     J I 0 1 1    1 1              2 J I 2 I I : 2 1 2 2 2 2     1 1 1                             3 J I 3 I I 3 I I Tr 3 1 1 2 3 3 ij jk ki 3 3 3 12

Previous Notions  EFFECTIVE STRESS   The effective stress or equivalent uniaxial stress is the scalar: 3 3           ' ´: ´ 3 J 2 ij ij 2 2  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, G y      0 0 u   x x      0 0 0  u u     0 0 0 x z 14

   0 0 u   Example - Solution    0 0 0      0 0 0     1 Mean stress:      u Tr ( u 0 0   m 3 3 3      0 0 m          u 0 0 0 0   Spherical and deviatoric parts   sph m 3        0 0  of the stress tensor: m   u 0 0     3   2  0 0   u 3        0 0 u m     1             0 0 0 0     sph m u 3         0 0 m 1     0 0    u  3 3 3 4 1 1 3 2                2 ( ) ij ij u u u 2 2 9 9 9 2 3 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:      1 2 3 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          1 2 3 18