14  Homework 1

Mathematical and Continuum Foundations

Due date: TBD — submitted via Moodle

14.1 Problem 1 — Tensor Operations

Let \(\mathbf{u}\), \(\mathbf{v}\) be vectors and \(\mathbf{S}\), \(\mathbf{T}\) second-order tensors in Cartesian coordinates.

  1. Compute \(\mathbf{u}\otimes\mathbf{v}\), \(\mathbf{S}\cdot\mathbf{v}\), and \(\mathbf{S}:\mathbf{T}\) using index notation.
  2. Compute the invariants \(I_1\), \(I_2\), \(I_3\) of \(\mathbf{S}\).
  3. Find the principal values and directions of \(\mathbf{S}\).
  4. Transform the components of \(\mathbf{S}\) to a coordinate system rotated \(45^\circ\) about the \(x_3\)-axis.

14.2 Problem 2 — Tensor Functions

Given a symmetric second-order tensor \(\mathbf{A}\) with eigendecomposition \(\mathbf{A} = \sum_\alpha \lambda_\alpha\,\mathbf{n}_\alpha\otimes\mathbf{n}_\alpha\):

  1. Write the eigenvalue problem using the three principal invariants.
  2. Write an expression for \(\ln\mathbf{A}\).
  3. Write an expression for \(\sqrt{\mathbf{A}}\).
  4. Derive \(\partial\mathbf{A}/\partial\mathbf{A}\) for symmetric \(\mathbf{A}\). Is the result symmetric?
  5. Repeat (d) for a non-symmetric \(\mathbf{A}\).

14.3 Problem 3 — Curvilinear Coordinates

Consider cylindrical coordinates \((\alpha^1, \alpha^2, \alpha^3) = (r, \phi, z)\) and Cartesian coordinates \((x^1, x^2, x^3)\) related by: \[ x^1 = r\cos\phi, \qquad x^2 = r\sin\phi, \qquad x^3 = z. \]

  1. Derive the covariant basis vectors \(\mathbf{g}_i\).
  2. Derive the contravariant basis vectors \(\mathbf{g}^i\).
  3. Verify \(\mathbf{g}_i\cdot\mathbf{g}^j = \delta_i^j\).

14.4 Problem 4 — Kinematics

Consider the deformation map \(\mathbf{x} = \mathbf{X} + k X_2\,\mathbf{e}_1\) (simple shear with constant \(k\)).

  1. Compute the deformation gradient \(\mathbf{F}\).
  2. Compute \(\mathbf{C} = \mathbf{F}^T\mathbf{F}\) and \(\mathbf{B} = \mathbf{F}\mathbf{F}^T\).
  3. Compute the Green-Lagrange strain \(\mathbf{E}\) and Euler-Almansi strain \(\mathbf{e}\).
  4. Evaluate and compare \(\mathbf{E}\) and \(\mathbf{e}\) for small (\(k=0.01\)) and large (\(k=1.0\)) shear at point \(\mathbf{X}=(1,1,1)^T\).

14.5 Problem 5 — Stress Measures

Using the deformation gradient from Problem 4 and a given Cauchy stress \(\boldsymbol{\sigma} = \sigma_0\,\mathbf{e}_1\otimes\mathbf{e}_1\):

  1. Compute the first Piola-Kirchhoff stress \(\mathbf{P} = J\boldsymbol{\sigma}\mathbf{F}^{-T}\).
  2. Compute the second Piola-Kirchhoff stress \(\mathbf{S} = J\mathbf{F}^{-1}\boldsymbol{\sigma}\mathbf{F}^{-T}\).
  3. Assess the objectivity of \(\boldsymbol{\sigma}\) and \(\mathbf{S}\).
  4. Derive the rate forms \(\dot{\boldsymbol{\sigma}}\), \(\dot{\mathbf{S}}\), \(\dot{\mathbf{P}}\) and analyse objectivity of each.