13.1 Kinematics of continuous media

Kinematics of continuous media is all about how materials move and deform. It's the foundation for understanding how stuff bends, stretches, and flows. This topic covers the math tools we use to describe these motions.

We'll look at key concepts like the deformation gradient, strain measures, and velocity fields. These ideas help us analyze everything from tiny vibrations in metals to huge ocean currents. It's pretty cool stuff!

Deformation and Strain

Deformation Gradient Tensor and Strain Measures

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• Deformation gradient tensor $\mathbf{F}$ maps material points from reference to current configuration
• Components of $\mathbf{F}$ given by $F_{ij} = \frac{\partial x_i}{\partial X_j}$, where $x_i$ are current coordinates and $X_j$ are reference coordinates
• Strain tensor quantifies local deformation of a material body
• Green-Lagrange strain tensor $\mathbf{E}$ defined as $\mathbf{E} = \frac{1}{2}(\mathbf{F}^T\mathbf{F} - \mathbf{I})$
• Cauchy strain tensor $\boldsymbol{\varepsilon}$ used for small deformations, given by $\boldsymbol{\varepsilon} = \frac{1}{2}(\nabla\mathbf{u} + \nabla\mathbf{u}^T)$
• Strain measures essential for analyzing material behavior under loading conditions (structural analysis, material testing)

Finite and Infinitesimal Strain Theories

• Finite strain theory accounts for large deformations and rotations
• Applies to materials undergoing significant shape changes (rubber, soft tissues)
• Utilizes nonlinear strain measures like Green-Lagrange strain tensor
• Infinitesimal strain theory assumes small deformations and rotations
• Valid for materials with small displacements relative to their dimensions (metals under normal loading)
• Uses linearized strain measures like Cauchy strain tensor
• Choice between finite and infinitesimal strain theories depends on expected deformation magnitude and material properties

Motion Description

Displacement Field and Coordinate Systems

• Displacement field $\mathbf{u}(\mathbf{X},t)$ describes motion of material points from reference to current configuration
• Defined as $\mathbf{u}(\mathbf{X},t) = \mathbf{x}(\mathbf{X},t) - \mathbf{X}$, where $\mathbf{x}$ is current position and $\mathbf{X}$ is reference position
• Eulerian description focuses on spatial points in the current configuration
• Observes physical quantities at fixed points in space (fluid mechanics applications)
• Lagrangian description follows material points from reference to current configuration
• Tracks individual particles throughout their motion (solid mechanics applications)
• Material coordinates $\mathbf{X}$ identify specific material points in the reference configuration
• Spatial coordinates $\mathbf{x}$ represent positions in the current, deformed configuration
• Relationship between material and spatial coordinates given by $\mathbf{x} = \chi(\mathbf{X},t)$, where $\chi$ is the motion function

Velocity Gradient Tensor

• Velocity gradient tensor $\mathbf{L}$ describes spatial variation of velocity field
• Defined as $L_{ij} = \frac{\partial v_i}{\partial x_j}$, where $v_i$ are velocity components and $x_j$ are spatial coordinates
• Can be decomposed into symmetric and antisymmetric parts: $\mathbf{L} = \mathbf{D} + \mathbf{W}$
• $\mathbf{D}$ represents the rate of deformation tensor (symmetric part)
• $\mathbf{W}$ represents the spin tensor (antisymmetric part)
• Velocity gradient tensor used to analyze fluid flow patterns and material deformation rates
• Applications include studying vorticity in fluid dynamics and strain rates in solid mechanics

Rotation

Rotation Tensor and Its Applications

• Rotation tensor $\mathbf{R}$ describes rigid body rotation of material elements
• Obtained from polar decomposition of deformation gradient: $\mathbf{F} = \mathbf{R}\mathbf{U}$
• $\mathbf{R}$ orthogonal tensor ($\mathbf{R}^T\mathbf{R} = \mathbf{I}$) and $\mathbf{U}$ right stretch tensor
• Rotation tensor preserves angles and distances between material points
• Used to separate rigid body rotation from pure deformation in continuum mechanics
• Applications include analyzing large deformations in nonlinear elasticity and plasticity
• Important for describing material behavior in finite strain theory and computational mechanics
• Rotation tensor can be parameterized using various methods (Euler angles, quaternions, Rodriguez formula)