Demo: Deformation of a Unit Square - EPS 164 Seismology Course Materials

Colab note: This notebook is designed to run on Google Colab. The first code cell installs dependencies.

# Install dependencies (for Google Colab or missing packages)
import sys

# Check if running in Colab
try:
    import google.colab
    IN_COLAB = True
    print("Running in Google Colab")
except:
    IN_COLAB = False
    print("Running in local environment")

# Install required packages if needed
required_packages = {
    'numpy': 'numpy',
    'matplotlib': 'matplotlib',
    'ipywidgets': 'ipywidgets'
}

missing_packages = []
for package, pip_name in required_packages.items():
    try:
        __import__(package)
        print(f"✓ {package} is already installed")
    except ImportError:
        missing_packages.append(pip_name)
        print(f"✗ {package} not found")

if missing_packages:
    print(f"\nInstalling missing packages: {', '.join(missing_packages)}")
    import subprocess
    subprocess.check_call([sys.executable, "-m", "pip", "install", "-q"] + missing_packages)
    print("✓ Installation complete!")
else:
    print("\n✓ All required packages are installed!")

Define the unit square.

def unit_square():
    return np.array([
        [0,0],
        [1,0],
        [1,1],
        [0,1],
        [0,0]
    ])

Define strain and rotation tensor.

def build_tensors(exx=0, ezz=0, exz=0, omega=0):
    # Symmetric strain tensor
    E = np.array([
        [exx, exz],
        [exz, ezz]
    ])
    # Antisymmetric rotation tensor
    W = np.array([
        [0, -omega],
        [omega, 0]
    ])
    I = np.eye(2)
    F = I + E + W
    return E, W, F

Apply Deformation

def deform(points, F):
    return points @ F.T

Plot the deformation.

def plot_deformation(exx=0, ezz=0, exz=0, omega=0):

    pts = unit_square()

    E, W, F = build_tensors(exx, ezz, exz, omega)

    pts_strain = deform(pts, np.eye(2) + E)
    pts_rot = deform(pts, np.eye(2) + W)
    pts_total = deform(pts, F)

    fig, ax = plt.subplots(figsize=(7,7))

    ax.plot(pts[:,0], pts[:,1],'w--',linewidth=2,label='Undeformed')

    ax.plot(pts_strain[:,0], pts_strain[:,1],linewidth=2,label='Strain')

    ax.plot(pts_rot[:,0], pts_rot[:,1],linewidth=2,label='Rotation')

    ax.plot(pts_total[:,0], pts_total[:,1],linewidth=3,label='Total')

    ax.set_aspect('equal')

    ax.grid(True,alpha=.3)

    ax.set_xlabel("x")
    ax.set_ylabel("z")

    ax.set_title("Strain and Rotation Effects on a Unit Square")

    ax.legend()

    plt.show()

interact(
    plot_deformation,

    exx=FloatSlider(min=-0.5,max=0.5,step=0.01,value=0),
    ezz=FloatSlider(min=-0.5,max=0.5,step=0.01,value=0),
    exz=FloatSlider(min=-0.5,max=0.5,step=0.01,value=0),

    omega=FloatSlider(min=-0.5,max=0.5,step=0.01,value=0)
);