General Theory of Finite Deformation

1. General Theory of Finite Deformation Kejie Zhao Instructor: Prof. Zhigang Suo May.21.2009 Harvard School of Engineering and Applied Sciences

2. Beyond linear theory Ingredients in linear theory • Deformation geometry • Force balance • Material model Beyond linear theory

3. In continuum mechanics, we model the body by a field of particles, and update the positions of the particles by using an equation of motion Framework of finite deformation

4. Kinematics of deformation • Name a material particle by the coordinate of the place occupied by the material particle when the body is in a reference state: particleX

5. Kinematics… • Field of deformation A central aim of continuum mechanics is to evolve the field of deformation by developing an equation of motion

6. Kinematics… • Displacement • Velocity • Acceleration

7. Kinematics… • Deformation gradient • F(X,t) maps the vector between • two nearby material particles in • reference state, dX, to the vector • between the same two material • particles in the current state, dx

8. Kinematics… • Polar decomposition: any linear operator can be written as a product Rotation vector Stretch vector C: Green deformation tensor

9. Conservation laws

10. Conservation of mass • Define the nominal density of mass • A material particle does not gain or lose mass, so that the nominal density of mass is time independent during deformation

11. Conservation of linear momentum • It requires that the rate of change of the linear momentum, in any part of a body, should equal to the force acting on the part Linear momentum: Rate of change:

12. Forces • Nominal density of body force • Nominal traction • Conservation of linear momentum

13. Linear momentum… • Conservation of linear momentum Inertial force

14. Linear momentum… • Stress-traction relation As the volume of the tetrahedron decreases, the ratio of area over volume becomes large, and the surface traction prevail over the body force

15. Linear momentum… • Divergence theorem • Conservation of linear momentum in differential form

16. Conservation of angular momentum • For any part of a body at any time, the momentum acting on the part equals to the rate of change in the angular momentum The conservation of angular momentum requires that the product be a symmetric tensor.

17. Conservation of energy • Displacement of a particle • Work • Recall The nominal stress is work-conjugate to the deformation gradient

18. Conservation of energy… • Heat • Nominal density of internal energy Conservation of energy requires the work done by the forces upon the part and the heat transferred into the part equal to the change in the internal energy

19. Conservation of energy… • q-IK relation • Divergence theorem • Conservation of energy in differential form Work done by external forces Energy received from reservoirs Energy due to net conduction

20. Conservation of energy… • When the body undergoes rigid-body rotation • Free energy is unchanged “…knowing the law of conservation of energy and the formulae for calculating the energy, we can understand other laws. In other words many other laws are not independent, but are simply secret ways of talking about the conservation of energy. The simplest is the law of the level” ---Richard Feynman

21. Product of entropy • To apply the fundamental postulate, we need to construct an isolated system, and identify the internal variables. • The body • A field of reservoirs in volume thermal contact • A field of reservoirs in surface thermal contact • All the mechanical forces The mechanical forces do not contribute to the entropy

22. Entropy… • Nominal density of entropy • To construct thermodynamics model of the material, we assume the system has two independent variables: the nominal density of energy u, and the deformation gradient F

23. Entropy… • An isolated system produces entropy by varying the internal variables • A list of internal variables: • Three types of constraints • Deformation kinematics • Conservation laws • Materials model

24. Entropy… • Conservation of energy • Differential form of • Deformation gradient • Entropy production of the composite

25. Entropy… • The variation of the independent internal variables are: • The inequality consists of contributions to the entropy product due to three distinct processes: the deformation of the body, the heat conduction in the body, and the heat transfer between the body and reservoirs Material model?

26. Material model… • The model of isothermal deformation of elastic body is specified by the following equations • Thermodynamic equilibrium: isothermal deformation of an elastic body

27. Material model… • Inverting the nominal density of entropy • In differential form: • Nominal density of Helmholtz free energy • As a material model, we assume the free energy is a function of temperature and deformation gradient

28. Equation of motion • Given free-energy function W(F), the field equations • Boundary conditions • Initial conditions

29. Summary • Kinematics of deformation • Conservation laws • Conservation of mass • Conservation of linear momentum • Conservation of angular momentum • Conservation of energy • Product of entropy • Material model Equation of motion

30. Research interest • Coupled diffusion and creep deformation of Li-ion battery electrode • Stress level induced by lithium ion insertion

31. Research interest… • For insertion processes, the deformation of the host material may be assumed to be linear with the volume of ions inserted • Assume lithium ion is much more mobile than the host particles Coupled partial differential equations of concentration and stress field With material law, appropriate boundary conditions, it’s solvable!!!

32. Thanks!