Homework 3
- Your homework solution has to be handed in as a group solution via Moodle.
- Clearly state name and matriculation number of each student
1 Hooke’s Law
An oil well is being drilled. Each new section of the drill pipe supports its own weight and the weight of the pipe and drill bit beneath it.
Tasks
Task 1
Using Hooke’s law, calculate the stretch in a new 6 m long steel (Young’s Modulus \(E= 20 \times 10^{10} \text{Pa}\)) pipe that supports a \(100 \text{kg}\) drill bit and a \(3 \text{km}\) long pipe with a linear mass density of \(20 \text{kg} \text{m}^{-1}\). Treat the pipe as a solid cylinder with a \(5 \text{cm}\) diameter.
Task 2
Assuming a Poisson’s ratio \(\nu = \dfrac{1}{3}\), what is the corresponding cross-sectional thinning?
Task 3
Calculate the bulk modulus and the Lame constants for the steel pipe.
Task 4
Using the Voigt notation, calculate the Stiffness Matrix for the case of Linear Elasticity using the Lame’s constants.
2 Deformation Experiment
During a deformation experiment on a sample of an unknown material, the following deformation gradient was observed:
\[ F = \begin{bmatrix} 2.000 & 0.000 & 0.000 \\ 0.000 & 0.866 & 0.500 \\ 0.000 & -0.500 & 0.866 \end{bmatrix} \mathbf{e}_{i} \otimes \mathbf{e}_{j} \]
Tasks
Task 1
Decompose the deformation gradient and identify the dilation, rotation and stretch.
Task 2
Determine about which co-ordinate axis rotation takes place. By what angle is the sample rotated?
3 Material Testing - Rock
A deformation experiment is performed to determine the elastic constants of a rock sample. The rock sample is prepared so that it has the shape of a cube. This cube is compressed in a press parallel to the \(y\) and \(z\) axes with equal force. The two faces of the cube perpendicular to the \(x\) axis remain free. In the experiment, the normal stresses \(\sigma_{y} = \sigma_{z} = \sigma_{0}\) and the strains \(\varepsilon_{x}\), \(\varepsilon_{y}\) and \(\varepsilon_{z}\) are measured. For reasons of symmetry \(\varepsilon_{y} = \varepsilon_{z} = \varepsilon_{0}\) is valid. The following values are measured: \(\sigma_{0} = -40 \text{MPa}\), \(\varepsilon_{0} = -0.4 \times 10^{-3}\), \(\varepsilon_{x} = 0.2 \times 10^{-3}\),
Tasks
Task 1
Draw a sketch of the experiment. Write down the stress and strain tensor for this experiment. Does shear occur in the experiment?
Task 2
Using the constitutive relation for Isotropic Linear Elasticity, calculate the elastic constants \(\lambda\) and \(\mu\) for this rock sample.
Task 3
Calculate the Young’s Modulus and Poisson’s ratio for rock material.
Task 4
Using the Voigt Notation, compute the Compliance Matrix for the rock material.
4 Objectivity, Isotropy
The Cauchy Stress tensor \(\mathbf{\sigma}\) given in the deformed configuration can be transformed in the reference (undeformed) configuration with the Second-Piola Kirchoff Stress tensor \(\mathbf{S}\). The following relation holds
\[ \mathbf{S} = \text{det}(\mathbf{F}) \mathbf{F}^{-1} \mathbf{\sigma} \mathbf{F}^{-T} \] and is called the pull-back operation.
Consider the following material models or constitutive relations:
- \(\mathbf{S} = \alpha(\mathbf{C} - \mathbf{I})\)
- \(\mathbf{\sigma} = \alpha \mathbf{F}\)
- \(\mathbf{S} = \alpha \mathbf{I} + 2\beta [\text{tr}(\mathbf{B}) - 3]\mathbf{I}\)
where, \(\mathbf{C}\) and \(\mathbf{B}\) are the Right and Left Cauchy-Green tensors, \(\alpha, \beta\) are scalar parameters, and \(\mathbf{I}\) is the identity tensor.
Tasks
Task 1
Decide whether each of the material model is objective, and give reasons for the decision.
Task 2
Decide whether each of the material model is isotropic, and give reasons for the decision.
5 Asymtotics of a visoelastic equation
A very simple linear viscoelastic material model yields the following equation:
\[ \rho \frac{\partial^2}{\partial t^2} u = \alpha \frac{\partial^2}{\partial x^2} u + \beta \frac{\partial}{\partial t} \frac{\partial^2}{\partial x^2} u \quad, \tag{1}\]
with constants \(\rho, \alpha, \beta \in \mathbb{R}\) and unknown displacement field \(u = u(t, x) \in \mathbb{R}\). We want to study solution of this equation that obey the form
\[ u(t, x) = A \, exp(i (x-c \, t)) \tag{2}\]
with \(i^2 = -1\) and \(A, c \in \mathbb{R}\).
Recall Euler’s formula \[ exp(i \, \theta) = cos(\theta) + i \, sin(\theta) \]
A wave solution is damped if \(u \rightarrow 0\) for \(t \rightarrow \infty\).
Tasks
Task 1
The elastic limit is given for \(\alpha > 0\) and \(\beta = 0\). Show that Equation 1 admits solutions of type Equation 2 which are not dampened in time.
Task 2
The viscous limit is given for \(\alpha = 0\) and \(\beta > 0\). Show that Equation 1 admits stationary solutions of type Equation 2 which are dampened exponentially in time.
Task 3
How does the solution look like for the case \(\alpha > 0\) and \(\beta > 0\)? Differentiate the cases
- \(4 \, \rho \alpha > \beta^2\)
- \(4 \, \rho \alpha < \beta^2\)
6 Plane waves
The momentum balance for an isotropic, linear elastic body with \(\rho \in \mathbb{R}\) constant is given by
\[ \rho \ddot{\mathbf{u}} = \mu \Delta \mathbf{u} + (\mu + \lambda) \nabla ( \nabla \cdot \mathbf{u}) \tag{3}\]
where \(\mathbf{u}(\mathbf{X}, t)\) is the displacement field and \(\lambda, \nu \in \mathbb{R}\) are the Lam'e constants.
We consider plane wave solutions of the form
\[ \mathbf{u}(\mathbf{X}, t) = A \mathbf{n}_d \phi (k \mathbf{n}_p \cdot \mathbf{X} - \omega t) \tag{4}\]
where \(\phi\) is an arbirary scalar function describing the displacement profile, \(\mathbf{n}_d\) and \(\mathbf{n}_p\) are unit vectors corresponding to the direction of displacement and progapation. \(A, k\) and \(\omega\) are constants for the amplitude, wave number and angular frequency.
\(\ddot{u}\) denotes the acceleration of the displacement in Lagrangian coordinates.
Tasks
Task 1
Show that Equation 3 satisfies Equation 4 if and only the condition
\[ A \, k^2 \left( \mu \mathbf{n}_d + (\mu + \lambda) (\mathbf{n}_d \cdot \mathbf{n}_p)\mathbf{n}_p \right) = A\rho\omega^2 \mathbf{n}_d \]
is met.
Task 2
In the following, we want to show that for a linear elastic, isotropic solid with \(\lambda + \mu \neq 0\), plane waves only admit two types of solutions. The first wave type are longitudinal waves in direction \(\mathbf{n}_p\) where \(\mathbf{n}_p \, \parallel \, \mathbf{n}_d\). The second wave type are transverse waves where \(\mathbf{n}_p \, \bot \, \mathbf{n}_d\).
Assume \(\lambda + \mu \neq 0\). For any given \(\mathbf{n}_p \neq \mathbf{0}\) and \(A, k \neq 0\) show that the only independent solutions \((\mathbf{n}_d, \omega)\) are:
\[ \begin{aligned} & \mathbf{n}_d = \pm \mathbf{n}_p, \quad \omega^2 = k^2 (\lambda + 2 \mu) / \rho, \\ & \mathbf{n}_d \cdot \mathbf{n}_p = 0, \quad \omega^2 = k^2 \mu / \rho. \end{aligned} \]
You need to identify and solve an eigenvalue problem.
Task 3
What happens in the case \(\lambda + \mu = 0\).
7 Compatibility conditions
Assuming small deformations, the strain field \(\mathbf{D}\) is defined by
\[ \mathbf{D} = \frac{1}{2} \left( \nabla_X \mathbf{u} + \nabla_X \mathbf{u}^T \right) \tag{5}\]
Given a strain field \(\mathbf{D}\), we want to solve Equation 5 for \(\mathbf{u}\). Since \(\mathbf{D}\) is symmetric, it has six independent equations and \(\mathbf{u}\) has three unknons. Therefore \(\mathbf{D}\) needs to satisfy certain compatibility conditions in order to be solvable.
Tasks
Task 1
Assume \(\mathbf{u}\) is smooth and solves Equation 5. Show that \(\mathbf{D}\) must satify the following six compatibility equations:
\[ D_{nj, km} + D_{km, jn} - D_{kn, jm} - D_{mj, kn} = 0 \]
The subscripts before the comma denote the components. The subscript after the comma denote the derivatives.