UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Lagrangian and Eulerian Forms of Finite Plasticity Permalink https://escholarship.org/uc/item/55v4x159 Author de Vera, Giorgio Tantuan Publication Date 2016 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Lagrangian and Eulerian Forms of Finite Plasticity By Giorgio Tantuan De Vera A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering – Mechanical Engineering in the Graduate Division of the University of California, Berkeley Committee in charge: Professor James Casey, Co-chair Professor George C. Johnson, Co-chair Professor Oliver M. O’Reilly Professor Filippos Filippou Fall 2016 Lagrangian and Eulerian Forms of Finite Plasticity Copyright 2016 by Giorgio Tantuan De Vera Abstract Lagrangian and Eulerian Forms of Finite Plasticity by Giorgio Tantuan De Vera Doctor of Philosophy in Mechanical Engineering University of California, Berkeley Professor James Casey, Co-chair Over the past half century, much work has been published on the theory of elastic-plastic materials undergoing large deformations. However, there is still disagreement about certain basic issues. In this dissertation, the strain-based Lagrangian theory developed by Green, Naghdi and co-workers is re-assessed. Its basic structure is found to be satisfactory. For the purpose of applications, the theory is recast in Eulerian form. Additionally, a novel three-factor multiplicative decomposition of the deformation gradient is employed to define a unique intermediate configuration. The resulting theory of finite plasticity contains an elastic strain tensor measured from the intermediate stress-free configuration. The consti- tutive equations involve the objective stress rate of the rotated Cauchy stress, which can be expressed in terms of the rate of deformation tensor. In their general forms, the Lagrangian theory can be converted into the Eulerian theory and vice versa. Because the Green-Naghdi theory has a strain measure that represents the difference between total strain and plastic strain, rather than representing elastic strain, it does not lend itself to a physically realistic linearization for the case of small elastic deformations accompanied by large plastic defor- mations. The proposed theory is well suited to describe this case as it can be linearized about the intermediate configuration while allowing the plastic deformations to be large. Differences between the linearized Green-Naghdi theory and the new theory are illustrated for uniaxial tension and compression tests. 1 For Jimmy. Thank you for saving my life. i Contents 1 Introduction 1 1.1 The Classical Theory of Plasticity: Infinitesimal Deformation . ...... 1 1.2 Finite Plasticity: Theories and Issues from the ’60s . ... 2 2 Deformable Continua 8 2.1 Mathematical Preliminaries . 8 2.2 Kinematics and Field Equations . 10 2.3 InvarianceRequirements . .. .. 13 2.4 ElasticSolids ................................... 14 3 The Green-Naghdi Theory of Elastic-Plastic Materials 16 3.1 The elastic regions in strain space and stress space . ..... 17 3.2 Loadingcriteria.................................. 19 3.3 Strain hardening response, flow rules and hardening rules . ...... 20 3.4 An equivalent set of kinematical measures . .. 22 3.5 A class of elastic-perfectly plastic materials . ... 24 4 Multiplicative Decompositions of the Deformation Gradient 25 4.1 Lee’sdecomposition................................ 25 4.2 A three-factor decomposition of F ........................ 30 4.3 Analternatethree-factordecomposition . .... 32 5 Fields and Functions on the Intermediate Configuration κ∗ 36 5.1 Yield functions and loading criteria . 36 5.2 The rotated Cauchy stress T anditsrate.................... 43 5.3 The second Piola-Kirchhoff stress tensor S∗ anditsrate............ 46 5.4 Application of the von Mises criterion to the expression of stress rates. 49 5.5 The yield function in E∗ space.......................... 52 5.6 The loading index in other spaces . 55 6 Stress Measures, Their Rates, and Constitutive Equations 57 6.1 The symmetric Piola-Kirchhoff stress tensor S ................. 57 6.2 The rotated Cauchy stress tensor T ....................... 58 ii 6.3 An alternate rotated Cauchy stress T∗ as a function of E∗ .......... 63 6.4 The Piola-Kirchhoff stress S∗ as a function of E∗ ................ 66 6.5 General objective rates of the rotated Cauchy stress . ......... 67 6.6 Strain hardening criteria and objective rates . .... 71 6.7 The back stress tensor α ............................. 71 7 Special Constitutive Equations 73 7.1 The Piola-Kirchhoff stress tensor S and a special class of materials . 73 7.2 The rotated Cauchy stress T ........................... 76 7.3 The Piola-Kirchhoff stress tensor S∗ ....................... 77 7.4 Linearization and its physical validity . 78 7.4.1 Rotated stress T ............................. 79 7.4.2 Piola-Kirchhoff stress tensor S ...................... 80 7.5 Comparisonofstressmeasures. .. 81 7.5.1 Uniaxial tension and compression using the new Eulerian theory . .. 83 7.5.2 Uniaxial tension and compression using the Green-Naghdi theory .. 97 8 Summary and Conclusions 107 Bibliography 109 iii List of Figures 1.1 Multiplicative decomposition of the deformation gradient F .......... 4 1.2 Isotropichardening ................................ 5 1.3 Kinematichardening ............................... 5 2.1 Deformablecontinuum .............................. 12 3.1 Yield surface in strain space . 18 3.2 Yieldsurfaceinstressspace . .. .. 19 4.1 Mapping of a neighborhood N (X) ....................... 27 4.2 Lee’s decomposition: F = FeFp ......................... 27 4.3 Invariance under superposed rigid body motions . .... 29 4.4 A unique decomposition of F byBammannandJohnson . 31 4.5 Another unique decomposition of F ....................... 33 5.1 Yield surface in E space ............................. 53 5.2 Yield surface in E∗ space............................. 54 7.1 Stress T11 vs Strain E∗11 intension ....................... 89 7.2 Stress S11 vs E11 ................................. 90 7.3 Comparisonofstrainmeasures. 91 7.4 Comparison of stresses in tension for small strains . ..... 92 7.5 Comparison of stresses in tension for large strains . ..... 93 7.6 Comparison of stresses in tension for small plastic strain . ..... 94 7.7 Comparison of stresses in tension for large plastic strain . ...... 95 7.8 Comparison of strains in tension . 96 7.9 Stress S11 vs E11 ................................. 100 7.10 Stress T11 vs Strain E∗11 intension ....................... 101 7.11 Stress T11 vs E∗11 incompression ........................ 102 7.12 Comparisonofstrainmeasures. 103 7.13 Comparison of stresses in tension for small strains . ...... 104 7.14 Comparison of stresses in tension for large strains . ...... 105 7.15 Comparison of strainss in tension . 106 8.1 The decomposition F = R∗U∗Up......................... 107 iv List of Symbols A, B general notation for second-order tensors , A general notation for fourth-order tensors A A a,b, B a,b fourth-order tensors that relates objective rates of stress a acceleration vector B a continuous body in space b body force per unit mass C Cauchy-Green tensor D rate of deformation E Lagrangian strain E3 3D Euclidean space E1, E2, E3 basis vectors in the reference configuration Ee elastic strain Ep plastic strain E∗ strain tensor in the intermediate configuration F deformation gradient H displacement gradient I second-order identity tensor I , J fourth-order identity tensors J Jacobian K fourth-order tensor relating stress and strain L velocity gradient L fourth-order elasticity tensor N (X) neighborhood of X P first Piola-Kirchhoff tensor Q rotation tensor R rotation tensor S second Piola-Kirchhoff tensor T Cauchy stress tensor T, T∗ rotated Cauchy stress tensors U right stretch tensor v U plastic parameters Ep, αR, and κ V left stretch tensor v velocity vector W vorticity tensor X material point of a body X, x position vectors of a material point X fourth-order tensor involving the derivative of Up with respect to Ep Z a, b general vector quantities e Eulerian strain tensor e1, e2, e3 basis vectors in the current configuration f yield function in stress space fˆ loading index in stress space g yield function in strain space gˆ loading index in strain space k bulk modulus of elasticity o(ǫ) order of epsilon α back stress or shift tensor β constitutive function for the rate of back stress Γ push-forward of the normal to the yield surface in strain space γ deviatoric part of a strain tensor δij Kronecker delta ǫ an infinitesimal or small quantity κ, κ∗, κ0 configurations of the body B κ strain hardening measure λ constitutive function forκ ˙ λ, λp, λ∗ stretches µ shear modulus of elasticity π loading index constant π transformation of a kinetical tensor ρ density ρ constitutive function for the rate of plastic strain Σ representation of a kinetical tensor σ a rate-type tensor involving plastic parameters τ deviatoric part of a stress tensor Φ strain hardening index χ mapping between material points ψ plastic flow rule constant Ω rate of rotation tensor vi Chapter 1 Introduction 1.1 The Classical Theory of Plasticity: Infinitesimal Deformation The origins of classical plasticity can be traced back to the experiments of Henri Tresca, first published in 1868. He established that when metals reach a critical value of shear stress, they begin to flow like a fluid. Thus, he introduced