Continuum Mechanics, Stress Tensors, and Material Response

Manual: General · Subject: Applied Physics

Analyze solids and fluids through tensorial descriptions of deformation, stress, and constitutive behavior.

Kinematics of Deformation

Displacement and strain

In small-deformation theory, the displacement field is $\mathbf{u}(\mathbf{x})$ and the strain tensor is $\varepsilon_{ij}=\tfrac{1}{2}(\partial_i u_j+\partial_j u_i)$ . This linearization is valid when gradients are small, enabling tractable modeling of engineering structures and soft matter.

Core tensor quantities

Stress tensor

\sigma_{ij}

Internal force per unit area

Strain tensor

\varepsilon_{ij}

Symmetric measure of deformation

Young's modulus

E

Uniaxial stiffness

Poisson ratio

\nu

Transverse contraction under axial loading

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Modeling caution

The linear strain approximation fails for large rotations, large strains, and many biological materials; use finite-strain formulations when appropriate.

Constitutive relations

For isotropic linear elasticity, Hooke's law in tensor form is $\sigma_{ij}=\lambda \varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}$ , where $\lambda$ and $\mu$ are Lamé parameters. This relation closes the balance laws and links internal stress to deformation.

Elastic vs. viscous response

Elastic solids

Stress depends on strain
Store recoverable energy
Oscillatory response with wave propagation

Viscous fluids

Stress depends on strain rate
Dissipate energy
Support diffusion-like momentum transport

What does the stress tensor represent physically?

Temperature anisotropy Internal force flux per unit area Mass density distribution Electrical potential

Write the small-strain tensor in terms of the displacement field.

Solving a linear elasticity problem

1

Step 1: Define geometry, boundary conditions, and material constants.
2

Step 2: Write equilibrium equations $\partial_j \sigma_{ij}+f_i=0$ .
3

Step 3: Substitute the constitutive law.
4

Step 4: Solve for displacement $\mathbf{u}$ , then compute strain and stress.

In isotropic linear elasticity, how many independent elastic constants are needed?

1 2 3 6

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