solid 2

Cauchy's first law of motion (direct form)

div σ + ρb = ρa = ρ(Dv/Dt)

Cauchy's first law of motion (indicial)

∂σ_ij/∂x_j + ρb_i = ρa_i

Local linear momentum balance, static/quasi-static

div σ + ρb = 0

What converts GLOBAL momentum balance into LOCAL form?

Cauchy's relation t=σn, the divergence theorem, and conservation of mass (Reynolds transport)

In the spatial (Eulerian) momentum balance, what is b?

Body force per unit MASS; ρb is body force per unit current volume

Which configuration does the spatial form live in?

The current (deformed) configuration; fields depend on current position x and time

How many equations vs unknowns in div σ + ρb = ρa?

3 scalar PDEs, 6 unknown stress components → needs a constitutive law to close

Balance of angular momentum (non-polar continuum) gives?

Symmetry of the Cauchy stress, σ = σᵀ

Angular momentum symmetry in index/permutation form

ε_ijk σ_jk = 0 ⇔ σ_ij = σ_ji

Define a non-polar (Boltzmann) continuum

No body couples and no couple stresses; angular momentum carried only by r×(linear momentum)

Why is angular momentum balance a constraint, not a new PDE?

After subtracting r×(linear momentum balance), only an algebraic remainder on σ survives → it constrains σ, adds no differential equation

When can Cauchy stress be NON-symmetric?

In polar/Cosserat continua with couple stresses (an antisymmetric part remains)

Independent components of a symmetric stress tensor

6 (since σ_ij = σ_ji collapses 9 → 6)

Which balance law makes σ symmetric?

Balance of angular momentum

Why are σ's eigenvalues guaranteed real?

σ is a real symmetric tensor → real eigenvalues (principal stresses)

What are the eigenvectors of σ called, and their property?

Principal directions; mutually orthogonal

Traction on a principal plane?

Purely normal — zero shear stress

Spectral decomposition of σ

σ = Σ_a σ_a (n_a ⊗ n_a), a = 1,2,3

Characteristic equation for principal stresses

det(σ − σI) = 0 → σ³ − I₁σ² + I₂σ − I₃ = 0

First invariant I₁

I₁ = tr σ = σ_ii

Second invariant I₂

I₂ = ½[(tr σ)² − tr(σ²)]

Third invariant I₃

I₃ = det σ

What does "invariant" mean physically?

Value is independent of the coordinate frame (unchanged by rotation)

Maximum shear stress formula

τ_max = ½(σ_max − σ_min)

Material (referential) form of linear momentum balance

Div P + ρ₀B = ρ₀A

What is P in the material form?

First Piola–Kirchhoff (nominal) stress

Relation between first P–K stress and Cauchy stress

P = J σ F⁻ᵀ, J = det F > 0

Define ρ₀, B, A in the material form

ρ₀ = reference density; B = body force per unit reference mass; A = material acceleration ∂²x/∂t²|_X

Why prefer the material form in solid mechanics?

Reference geometry is known and fixed; avoids tracking the unknown, moving current domain

Piola identity (links spatial and material forms)

Div P = J div σ

Cauchy traction (force per current area)

t(n) = σn, t_i = σ_ij n_j

Cauchy's lemma

t(n) = −t(−n) (Newton's third law for surfaces)

Cauchy's theorem

Traction depends LINEARLY on n; that linearity guarantees a stress tensor σ exists

Nominal (first P–K) traction

T(N) = P N, measured per unit REFERENCE area dA

Same-force relation between Cauchy and nominal traction

t da = T dA

Normal component of a traction vector

σ_n = t · n

Shear magnitude of a traction vector

τ = √(|t|² − σ_n²)

True (Cauchy) stress, uniaxial

σ = f/a (current area a)

Engineering / nominal stress, uniaxial

P = f/A₀ (original area A₀)

Engineering vs nominal stress — the distinction

Engineering stress is the SCALAR f/A₀; nominal stress is its TENSOR generalization (first P–K, P), reducing to f/A₀ in pure tension

Uniaxial incompressible relation between σ and P

σ = λP (λ = stretch)

In tension, true vs engineering stress — which is larger?

True stress > engineering stress (cross-section shrinks); eng curve droops after necking, true curve keeps rising

Is the first P–K stress P symmetric?

No — it's a two-point tensor, generally non-symmetric

Symmetric referential stress measure

Second Piola–Kirchhoff stress S = F⁻¹P = J F⁻¹ σ F⁻ᵀ (no direct traction meaning)

True stress from engineering stress (uniaxial, incompressible)

σ_true = σ_eng (1 + e), e = engineering strain

True (logarithmic) strain from engineering strain

ε_ln = ln(1 + e)

Local material momentum balance (indicial)

∂P_iJ/∂X_J + ρ₀B_i = ρ₀A_i

In ∂P_iJ/∂X_J, which index does the divergence act on?

The referential (uppercase) index J — divergence w.r.t. X

Why does P carry one lowercase and one uppercase index?

It's a two-point tensor: lowercase i = spatial (current), uppercase J = referential (reference)

Static local material form

∂P_iJ/∂X_J + ρ₀B_i = 0

Relations connecting material and spatial forms

ρ₀ = Jρ and B = b