Insights into Small Strain Elasto-Plasticity Variational Viewpoint

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Explore the variational viewpoint on old and new results in small strain elasto-plasticity, discussing deformation, frame indifference, linear elasticity, crystalline plasticity, stress equilibration, thermodynamics, Von Mises flow rule, existence theory, and variational evolution in a nutshell.


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  1. Old and new results in small strain elasto-plasticity: The variational viewpoint Gilles A. Francfort WPI, February 6, 2017

  2. A pastiche of small strain kinematics 1 Deformation and frame indifference: a vector x (x) the deformation The deformation gradient F= is a good measure of the geometric changes of the body free energy density W(F) Frame indifference says that the value of the energy does not depend on the observer: W(F)=W(RF), R elt. of SO(3) T W(F) is really W(F F) If writing (the displacement field) and if (small deformations) then (the linearized strain) So, for small deformations W is a function of E(u) that might as well be quadratic Linear elasticity:

  3. A pastiche of small strain kinematics 2 Crystalline Plasticity F=EP (in textbooks F=F F ) E e p P + W function of E because of dislocations P is an internal variable In small deformation setting, P=I+2p, with tr p=0, E=I+2e, so E(u)=e+p So kinematics of small strain plasticity are p is an internal variable with a quadratic energy

  4. A pastiche of thermodynamics The stress is and, in the absence of inertia, it will equilibrate the external loads No temperature change thermo. says dissipation must be nonneg. deviatoric part of stress Now it seems plausible that macroscopically stresses cannot be too large wo. constantly triggering more plastic strains Then, to enforce dissipation rule, classical to impose normality rule or flow rule : K

  5. Summing up quasi-static setting A the Hooke s law has good coercivity prop. + appropriate b.c. s and i.c. s Special case: Von Mises Flow rule becomes For simplicity, from now on no forcesand pure Dirichlet b.c. s

  6. Existence theory bounded measures Structure of BD: Lebesgue part jump part Cantor part Suquet 81 using approximation, Dal Maso De Simone Mora 2006 using energetic evolutions, F. Giacomini 2012 for Lip. boundary. Why BD and why the boundary relaxation? Coming shortly

  7. Variational Evolution in a nutshell Define the dissipation and the total energy (u(t),e(t),p(t)) is abs. cont. and it satisfies, at each t, Global minimality: for all admissible triplets Energy conservation:

  8. Why minimality and why BD in a nutshell Imagine all fcts. are smooth =0 To minimize discretize in time and minimize, at time t , i+1 Minimizing seq. (un, en, pn) will only be bounded in W1,1x L2xL1 u un weak-lim of is a measure 0 b.c. Energy conservation =0 (equilibrium) =0 (flow rule)

  9. Recovering the original system Easy, except for flow rule. What is bd. meas.fct. measure issues of duality since the 80 s: Kohn, Temam, especially tricky for mixed b.c. s (F. Giacomini 2012) In particular, expect flow rule on boundary (F. Giacomini 2012) Indeed, since on boundary: with and since we expect a boundary flow rule Missing equation in the engineering or mechanics literature .

  10. Many questions The heterogeneous case: only solved for essentially finitely many phases Solombrino 2009, 2014, F. Giacomini 2012 Is one free to model the interface as an arbitrary plastic layer: NO F. Giacomini 2015 Is plasticity stable under homogenization? Periodic, NO F. Giacomini 2014 Can there really be a Cantor part to plasticity? Demyanov 2009 YES in 1d F. Giacomini Marigo YES in 3d. Question related to uniqueness Uniqueness? of stress and elastic strain classical, of plastic strain? see later Regularity of the stress? Essentially only known in Von Mises case Bensoussan Frehse 93, Demyanov 2009 Dynamic evolutions: more uniqueness; short time regularity using hyperbolic methods Babadjian Mifsud 2017, ..

  11. Uniqueness in the Von Mises case The easiest example the bi-axial test isotropic mat. initial state: b.c. s are all traction except: Homogeneous solution: elastic if t below tcrit , plastic else Can there be others?

  12. Uniqueness in the Von Mises case 2 Write that E(u)(t) must be a symmetrized gradient: This gives a 3x3 hyperbolic system in It has 0 determinant iff Conclusion: Uniqueness in that case. Case Spatial wave equation: Solution Hence Eu(t), hence u(t) with the b.c. s Assume d<l, then there exists an infinite number of sols.

  13. Departing from pure elasto-plasticity A whole range of constitutive laws that incorporate hardening (usually a nice thing) or softening (always a bad thing) Best models for metal plasticity: Nonlinear kinematic hardening: Armstrong Frederick type models give rise to non-associative plasticity deemed non-variationalizable Still something can be done variationally Departing from small strain: the great mystery that has exhausted prior generations .

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