Announcement

MATH 6760         Continuum Mechanics: Solids

Fall 2002    3 credit hours    M,W,F  10:45-11:35

What is continuous mechanics?
Continuous mechanics studies general princples that govern forcible changes of shapes and forms which we daily observe around. It provides a uniform mathematical framework for investigating specific areas of mechanics such as elasticity, plasticity, viscosity, liquids, etc.

The course will introduce methods and general principles of continuum mechanics and discuss various kinds of solids.

Particularly, we will discuss heterogeneous materials, stress localization and instabilities, expandable thermo-elastic and shape-memory materials, phase transition in solids, optimization, and self-remodeling of biomaterials.
Text: Lawrence E. Malvern. Introduction to the Mechanics of a Continuous Medium

Instructor: Professor Andrej Cherkaev
        email: cherk@math.utah.edu,
        phone 581-6822,
        homepage: http://www.math.utah.edu/~cherk

        This course is addressed to students in Applied Math, Physics, and Engineering

Why study continuous mechanics?
For those who value the beauty of mathematical theories, it is enough to cite Lagrange: "The admirers of the Analysis will be pleased to learn that Mechanics became one of its new branches" (Mécanique analytique)

For more practically inclined people, we recall the parable:
Five foreign travelers found an elephant in a dark barn. Each felt the elephant and described it to the others: 
    "The elephant is like a rope," said the first traveler, feeling the tail. 
    "The elephant is like a wall," said the second traveler, feeling the side. 
    "The elephant is like a blanket," said the third traveler, feeling its ear. 
    "The elephant is like a tree," said the fourth traveler, feeling its trunk. 
    "The elephant is like a spear," said the fifth traveler, feeling its tusk. 
In came the Elephant Keeper and opened the door of the barn, and everyone discerned the whole elephantness. 


Syllabus (preliminary)
  • Vectors and tensors: algebra and calculus
  • Stress, Strain, and Deformation
  • General Principles of Continuum Mechanics:
  • Conservation of mass
  • Momentum principles
  • Energy balance
  • Entropy and the Second Law of Thermodynamics
  • Gibbs' principle
  • Constitutive equations:
    • Nonlinear and classical elasticity
    • Expandable and thermoelastic materials
    • Ideal fluid
    • Viscoelasticity
    • Plasticity
  • Heretogeneous materials: Homogenization and Local phenomena
  • Adjustable materials:
  • "Smart" materials and actuators
  • Optimization
  • Bio-materials
  • Theories of Constitutive Equations
  • Fundamental Postulates of a Purely Mechanical Theory
  • Frame-indifference
  • Symmetries etc.
  • Waves in elastic materials

    • Howe work 1

    1. Express vector Z through vectors A, B, C, and constants a, b, c, if
        A.Z=a,  B . Z = b,  C . Z= c
    (Here, (.) is the scalar product

    2. Express all solutions R to the equation
    R. A =m
    through A and m.

    3. Prove that
        a) curl(curl A)= grad (div A)- div( grad A)
        b) div( A x B ) = (B . curl A )- (A . curl B)

    4. Let S  and   T  be the square matrices . Differentiate
         d /d S (  Tr ( S^{-1} T)

    First challenge project (due in 2 weeks, at September 28); will be followed by the oral presentation in class.

     
    Consider a (generally anisotropic)  structure made of a conducting material with the conductivity k and a perfect isolator. Denote by  c  the fraction of the conductor in the structure . It is known that
     
    1. The conductivity tensor K of conducting structures of an arbitrary geometry satisfies the inequality

     (*)    Tr[( K - k I) ^{-1} ]< c/(1-c)  (1/k) S

    where S is a nonnegative matrix with unit trace,  S >=0, Tr S =1.

    2. The bound (*) is exact: There exist structures that deliver equality in (*) for all permitted values of S. Tensor S depends on the structure and determines the degree of anisotropy (see for details Cherkaev, 2000).


    Suppose that this structure (composite) is sequentially submerged into several different homogeneous fields v_i,    i= 1..N.
    The dissipated energy density   w_i  of each field is
    w_i= v_i^T S v_i= Tr [S . (v_iv_i^T)]. Suppose also that you are free to choose the structure from the set (*)

    Find

    1. The minimal value of the sum of the energies w_1 + ... w_n.

    2. Optimal value of  structural matrix  S  and anisotropic conductivity  K


    HW 2. (due in 1 week at Sept 11).
    1. Consider a tensor E
    E=  def u = (1/2) [grad u + (grad u)^T]
    where  u is a differentialble vector
    Show that
    Ink (E) = curl ( curl E)^T=0
    Here Ink is a llinear second-rank differential operator (from inkompabilit"{a}t).

    Show that

    div (Ink G) =0 forall tensors G.
     
    2. Consider two 1-periodic vectors
    v_1= grad f1  and    v_2= grad f2, and the antisymmetric tensor  R:  R^T = -R
    Show that
    <v_1. R . v_2>=<v_1> . R . <v_2>
    where <  > is the average over a unit cube.
     

    Challenge 2  (due in two weeks, will be discussed in class)

    Consider a tensor  V   and assume that
    (*)     div V=0 .

    1. Find a quadratic function phi(V)=V:Q:V of elements of V  such that
    (i) phi (V) >= 0 if (*) holds
    (ii) There exist V such that phi(V) < 0

    2. Find all such quadratic forms.