1. Linear Algebra Review

(1) Systems of Linear Equations

A linear equation in general form:

[ a_1 x_1 + a_2 x_2 + \dots + a_n x_n = b ]

For a system of linear equations:

[ A X = B ]

where:

  • ( A ) is the coefficient matrix,
  • ( X ) is the vector of unknowns,
  • ( B ) is the vector of constants.

(2) Matrices and Operations

  • Matrix Addition & Subtraction: ( A \pm B = [a_{ij} \pm b_{ij}] )
  • Scalar Multiplication: ( k A = [k a_{ij}] )
  • Matrix Multiplication: ( (AB){ij} = \sum{k=1}^{p} a_{ik} b_{kj} )

(3) Properties of Matrices

  • Associative Law: ( A(BC) = (AB)C )
  • Distributive Law: ( A(B+C) = AB + AC )
  • Transpose Properties:
    • ( (A^T)^T = A )
    • ( (A+B)^T = A^T + B^T )
    • ( (AB)^T = B^T A^T )

(4) Determinants and Inverses

  • Determinant:

[ \det(A) = \sum_{i=1}^{n} a_{i1} C_{i1} ]

where ( C_{ij} ) is the cofactor:

[ C_{ij} = (-1)^{i+j} M_{ij} ]

  • Inverse of a Matrix: If ( A ) is invertible:

    [ A^{-1} = \frac{\text{adj}(A)}{\det(A)} ]

    where adj(A) is the adjugate matrix.

(5) Eigenvalues and Eigenvectors

For a square matrix ( A ), an eigenvalue equation is:

[ A K = \lambda K ]

Rearranging:

[ (A - \lambda I) K = 0 ]

For nontrivial solutions:

[ \det(A - \lambda I) = 0 ]

which gives the eigenvalues ( \lambda ), and solving for ( K ) gives the eigenvectors.

2. Basics of Stress and Strain

(1) Definition of Stress

Stress at a point is defined as:

[ \sigma = \lim_{\Delta A \to 0} \frac{\Delta F}{\Delta A} ]

(2) Stress Components

For an infinitesimal volume:

[ \mathbf{\sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz}
\sigma_{yx} & \sigma_{yy} & \sigma_{yz}
\sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} ]

where:

  • ( \sigma_{xx}, \sigma_{yy}, \sigma_{zz} ) are normal stresses.
  • ( \sigma_{xy}, \sigma_{xz}, \sigma_{yz} ) are shear stresses.

From moment equilibrium:

[ \sigma_{xy} = \sigma_{yx}, \quad \sigma_{xz} = \sigma_{zx}, \quad \sigma_{yz} = \sigma_{zy} ]

(3) Stress Transformation

For a rotated coordinate system:

[ \sigma_P = \sigma N ]

where ( N ) is the unit normal vector.

3. Principal Stresses and Invariants

(1) Principal Stresses

Principal stresses occur on planes with no shear stress.

Solving:

[ \det(\sigma - \lambda I) = 0 ]

yields principal stresses ( \sigma_1, \sigma_2, \sigma_3 ).

(2) Stress Invariants

Stress tensor properties do not change with coordinate system rotations:

[ I_1 = \sigma_{xx} + \sigma_{yy} + \sigma_{zz} ]

[ I_2 = \sigma_{xx} \sigma_{yy} + \sigma_{yy} \sigma_{zz} + \sigma_{zz} \sigma_{xx} - \sigma_{xy}^2 - \sigma_{yz}^2 - \sigma_{zx}^2 ]

[ I_3 = \det(\sigma) ]

4. First Law of Thermodynamics in Solid Mechanics

(1) Energy Balance

[ \delta W + \delta H = \delta U + \delta K ]

where:

  • ( W ) = work done by external forces.
  • ( H ) = heat transfer.
  • ( U ) = internal energy.
  • ( K ) = kinetic energy.

For static equilibrium and adiabatic conditions:

[ \delta W = \delta U ]

(2) Work Done by Stresses

Using the divergence theorem:

[ \delta W = \int_V \sigma_{ij} \delta \varepsilon_{ij} \, dV ]

Internal energy per unit volume:

[ \delta U_0 = \frac{1}{2} (\sigma_{xx} \delta \varepsilon_{xx} + \sigma_{yy} \delta \varepsilon_{yy} + \sigma_{zz} \delta \varepsilon_{zz} + 2\sigma_{xy} \delta \varepsilon_{xy} + 2\sigma_{yz} \delta \varepsilon_{yz} + 2\sigma_{zx} \delta \varepsilon_{zx}) ]

5. Hooke’s Law for Isotropic Elasticity

For an isotropic material (same properties in all directions), Hooke’s law is:

[ \varepsilon_{xx} = \frac{1}{E} (\sigma_{xx} - \nu (\sigma_{yy} + \sigma_{zz})) ]

[ \varepsilon_{yy} = \frac{1}{E} (\sigma_{yy} - \nu (\sigma_{xx} + \sigma_{zz})) ]

[ \varepsilon_{zz} = \frac{1}{E} (\sigma_{zz} - \nu (\sigma_{xx} + \sigma_{yy})) ]

where:

  • ( E ) = Young’s modulus.
  • ( \nu ) = Poisson’s ratio.

For shear stress:

[ \gamma_{xy} = \frac{\tau_{xy}}{G}, \quad \gamma_{xz} = \frac{\tau_{xz}}{G}, \quad \gamma_{yz} = \frac{\tau_{yz}}{G} ]

where ( G ) is the shear modulus:

[ G = \frac{E}{2(1+\nu)} ]

6. Von Mises Stress

To determine failure under multiaxial loading, we use the von Mises stress:

[ \sigma_e = \sqrt{\frac{1}{2} \left( (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right)} ]

For stress components:

[ \sigma_e = \sqrt{\frac{1}{2} \left( (\sigma_{xx} - \sigma_{yy})^2 + (\sigma_{yy} - \sigma_{zz})^2 + (\sigma_{zz} - \sigma_{xx})^2 + 6(\sigma_{xy}^2 + \sigma_{yz}^2 + \sigma_{zx}^2) \right)} ]

7. Summary

✅ Key Topics

  • Linear Algebra Review: Matrices, determinants, eigenvalues.
  • Stress and Strain Basics: Stress tensors, transformation.
  • Principal Stresses and Invariants: Coordinate-independent properties.
  • First Law of Thermodynamics: Work-energy relation in solids.
  • Hooke’s Law for Isotropic Materials: Linear elasticity.
  • Von Mises Stress: Failure criterion for materials.