Continuum mechanics

An elastic beam above a smooth obstacle cannot penetrate it, and a nonadhesive obstacle can only push. The displayed constraints express this when

R

is an upward distributed normal contact force; concentrated point forces must also be nonnegative. In the small-slope Euler-Bernoulli beam equation,

B w^{''''} - T w^{''} + p = R .

Integrating across a contact point with no applied couple gives jump conditions

[w] = [w^{'}] = [B w^{''}] = 0

and

[B w^{'''}] = R_{i}

, provided the tension is continuous. For a flat obstacle, no distributed load and zero tension, an open contact interval has

R = 0

; its entire normal contact force can therefore be concentrated at its endpoints.

Reduced elastic modulus ( $E^{*}$ )

The combined elastic response of two contacting isotropic bodies:

1/ E^{*} = (1 - ν_{1}^{2}) / E_{1} + (1 - ν_{2}^{2}) / E_{2}

. It includes compliance of both bodies.

Hertzian contact

The elastic contact of smooth nonadhesive bodies with locally quadratic surfaces. For a sphere on a flat surface, the load is proportional to the reduced elastic modulus, square root of sphere radius and indentation to the three-halves power.

Tabor parameter ( $μ$ )

A dimensionless measure comparing adhesive interaction range with elastic deformation in sphere contact. Small values favour the DMT contact approximation, while large values favour the JKR limit.

DMT contact model

An adhesive sphere-on-flat contact approximation combining Hertzian contact deformation with an attractive force outside the contact area. It is most appropriate for sufficiently stiff, small-radius contacts with a small Tabor parameter; a fitted reduced modulus is not automatically the sample Young's modulus.

Adhesion

Attraction between different contacting surfaces. Adhesion can be described by interfacial work per area, while a measured pull-off force additionally depends on geometry and the elastic response.

Stiction

Unwanted sticking that prevents intended relative motion. In surface micromachining, a receding liquid meniscus can pull a compliant released beam onto the substrate; adhesion can then keep it there. Avoiding a liquid-vapor drying interface addresses one route to stiction, but does not remove contact adhesion during use.

Pull-off force

The tensile force magnitude needed to detach contacting surfaces during retraction. A force-probe measurement reads it from the force minimum relative to the noncontact baseline; roughness, capillary forces and contact geometry affect its interpretation.

Normal contact force ( $N$ )

The component of a contact force perpendicular to the local contact plane.

Solid stress

solid stress is stress carried by solid constituents of a tissue or material. Compressive solid stress, interstitial fluid pressure and elastic stiffness are distinct quantities that can coexist in a tumour.

Continuum approximation

The continuum approximation describes a material using fields such as mass density, velocity field and stress, rather than following its individual molecules. Its averaging volumes must contain enough microscopic constituents to define bulk properties while remaining small relative to the distances over which those properties change appreciably. It supplies the model underlying much of fluid mechanics and elasticity.

Anisotropy

Anisotropy is directional dependence of a material response. A thermal conductivity tensor with unequal principal conductivities is an example. Layered arrangements can produce anisotropic effective conductivity even from isotropic phases.

Isotropy

Isotropy is invariance under rotations, so no spatial direction is preferred. For a three-dimensional second-rank conductivity tensor, invariance under every rotation forces

a = a_{s} I

. Isotropy of each constituent does not imply isotropy of an arranged composite microstructure. Statistical isotropy concerns the rotational invariance of its probability law.

Composite material

A composite material contains distinct constituent phases arranged in a microstructure. Its macroscopic response depends on both phase properties and geometry, not just volume fractions. Anisotropic effective conductivity can arise even when every constituent has isotropic thermal conductivity.

Carbon fibre

A carbon-based fibre reinforcement with strongly direction-dependent stiffness and strength.

Glass fibre

A slender glass reinforcement. Its mechanical role depends on orientation, aspect ratio and bonding to the matrix.

Long-fibre composite

A composite containing fibres long enough to transfer load effectively through the matrix. Continuous aligned fibres under axial loading approximately share strain with a well-bonded matrix.

Fibre volume fraction ( $V_{f}$ )

The fraction of composite volume occupied by reinforcing fibres; physically achievable values depend on fibre geometry, packing, matrix infiltration and voids.

Hybrid fibre composite

A composite containing two or more reinforcing fibre types. In the ideal longitudinal model its axial modulus is the volume-fraction-weighted sum of constituent moduli.

Longitudinal rule of mixtures ( $E_{L}$ )

For continuous, parallel, perfectly bonded fibres and uniform axial strain,

E_{L} = V_{f} E_{f} + (1 - V_{f}) E_{m}

. This idealization is the axial Voigt elastic approximation; voids, fibre waviness and poor load transfer reduce stiffness.

Carbon-fibre-reinforced polymer

Elastic laminate

An elastic laminate has planar perfectly bonded layers with a common normal. Its piecewise constant strains differ by

sym (a \otimes n)

, while its normal traction is common to every layer. For positive elastic stiffness tensors, these conditions and the mean strain uniquely determine the layer strains: the difference of two solutions has zero mean and zero total elastic work, and positivity forces it to vanish.

Effective elastic energy ( $W^{eff} (E) = ∣Ω ∣^{- 1} in f_{u = E x} \int_{Ω} W (sym \nabla u, x) d x$ )

The effective elastic energy is the minimum averaged strain energy density under prescribed affine boundary displacement. For differentiable stable elastic response without body force, varying the prescribed mean strain gives

D W^{eff} (E) = ⟨ σ ⟩

. The result follows by integration by parts, because the interior variation with zero boundary data does no work against an equilibrated stress tensor.

Phase polarization comparison bound ( $W^{eff} (E) \leq \frac{1}{2} E : C^{0} E + \overline{τ} : E - \frac{1}{2} \sum_{r, s} τ^{r} : A_{rs} τ^{s} - \sum_{r} c_{r} F_{r *} (τ^{r})$ )

Take a positive comparison elastic stiffness tensor

C^{0}

, excess energies

F_{r} (e) = W_{r} (e) - e : C^{0} e /2

, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where

A_{rs} = \int\int (χ_{r} - c_{r}) Γ (χ_{s} - c_{s})

. Stationarity in

τ^{r}

gives

c_{r} e^{r} + \sum_{s} A_{rs} τ^{s} = c_{r} E

, with

e^{r} = D F_{r *} (τ^{r})

. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.

Reuss elastic approximation ( $W_{R} = (⟨ W^{} ⟩)^{}$ )

The Reuss elastic approximation assumes uniform stress. The local strain is the derivative of the convex conjugate

W^{*} (S, x)

; its mean is

D ⟨ W^{*} ⟩ (S)

. Under invertible differentiable duality, this mean determines

S = D W_{R} (E)

. For positive linear elasticity,

C_{R} = ⟨ C^{- 1} ⟩^{- 1}

. Uniform stress is generally not accompanied by a compatible local displacement field, so this approximation is distinguished from an exact heterogeneous solution.

Voigt elastic approximation ( $W_{V} (E) = ⟨ W (E, x)⟩$ )

The Voigt elastic approximation assumes uniform strain. Differentiating its averaged strain energy density gives the averaged local stress. For linear elasticity the predicted elastic stiffness tensor is

C_{V} = ⟨ C ⟩

. The uniform strain is an admissible affine displacement trial field, so this gives an upper elastic energy bound for convex local energy.

Hill-Mandel energy identity ( $⟨ σ : ε ⟩ = ⟨ σ ⟩ : ⟨ ε ⟩$ )

For symmetric stress in static elastic equilibrium without body force, write

E = ⟨ ε ⟩

and

S = ⟨ σ ⟩

. The difference between microscopic and macroscopic work is

∣Ω ∣^{- 1} \int_{\partial Ω} (σ - S) n \cdot (u - E x)

. It vanishes for affine boundary displacement, uniform boundary traction, or matching periodic displacement fluctuations and opposite tractions on paired faces.

Volume fraction ( $p_{i} = ∣ Ω_{i} ∣/∣Ω∣$ )

The volume fraction of a constituent is its occupied volume divided by total sample volume. Fractions sum to one for a partition into phases. In a statistically homogeneous medium the phase fraction also equals the probability that a typical point lies in that phase, under the appropriate spatial or ensemble averaging assumptions.

Two-point phase correlation function ( $p_{ij} (x, x^{'}) = E [χ_{i} (x) χ_{j} (x^{'})]$ )

The indicator correlation gives the probability of specified phases at two points. Statistical homogeneity makes it depend on separation; isotropy makes it depend only on its length. In particular

p_{11} (0) = p_{1}

, so the connected correlation at zero separation is

p_{1} - p_{1}^{2} = p_{1} p_{2}

Effective conductivity ( $⟨ a \nabla u ⟩ = a^{*} ⟨ \nabla u ⟩$ )

Effective conductivity describes the averaged response of a microscopic conductivity field to a macroscopic temperature or electric-potential gradient. Here the positive conductivity flux is

a \nabla u

; physical heat flux has the opposite sign. For periodic media it is computed by cell correctors, and can be a tensor even for isotropic constituents.

Self-consistent conductivity approximation ( $C_{SC} = H (C_{SC})$ )

A self-consistent conductivity approximation sets the homogeneous comparison effective conductivity equal to the predicted response. For a uniformly oriented two-dimensional polycrystal with principal conductivities

a, b > 0

, the comparison formula gives

C_{SC}^{2} = ab

, so its positive solution is

ab

. This is an approximation defined by a fixed-point closure; exactness requires additional geometric or statistical hypotheses.

Hashin-Shtrikman conductivity variational principle

For a scalar reference

b

, choose a trial polarization

q

and solve

b Δ v + \nabla \cdot q = 0

with zero Dirichlet data. The trial functional is

F_{b} (q) = b ∣Ω∣∣ λ ∣^{2} + 2 λ \cdot \int_{Ω} q + \int_{Ω} q \cdot \nabla v - \int_{Ω} q \cdot (a - b I)^{- 1} q .

A reference above all phase conductivities gives

J (u) \leq F_{b} (q)

; one below all of them gives

J (u) \geq F_{b} (q)

. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.

Two-dimensional isotropic polycrystal conductivity ( $H (c) = \frac{2 ab + c ( a + b )}{a + b + 2 c}$ )

For uniformly distributed rotations of a two-dimensional conductivity tensor with positive principal values

a \leq b

, the Hashin-Shtrikman conductivity variational principle with isotropic comparison

c I

yields

H (c) I

. Averaging a rotated diagonal tensor replaces it by half its trace times the identity. Since

H^{'} (c) = (a - b)^{2} / (a + b + 2 c)^{2}

, the admissible comparison endpoints give

a (a + 3 b) / (3 a + b) \leq C_{eff} \leq b (3 a + b) / (a + 3 b)

within the statistically isotropic polycrystal model.

Hashin-Shtrikman bounds for conductivity ( $a_{1} + \frac{3 a _{1} ( a _{2} - a _{1} ) p _{2}}{3 a _{1} + p _{1} ( a _{2} - a _{1} )} \leq a^{*} \leq a_{2} + \frac{3 a _{2} ( a _{1} - a _{2} ) p _{1}}{3 a _{2} + p _{2} ( a _{1} - a _{2} )}$ )

These are the optimal volume-fraction bounds for three-dimensional statistically isotropic two-phase conductivity with

0 < a_{1} < a_{2}

. Constant phase polarization trial fields reduce the variational principle to a scalar quadratic optimization. The upper trial functional is minimized; the lower one is maximized.

Voigt bound ( $a^{*} ⪯ ⟨ a ⟩$ )

An affine trial temperature in the minimum-energy principle gives the upper tensor bound by the volume-averaged local conductivity. For an isotropic two-phase composite,

a^{*} \leq p_{1} a_{1} + p_{2} a_{2}

. The same uniform-field variational idea supplies the familiar corresponding elastic upper bound.

Laminate conductivity ( $a^{*} = a_{H} n \otimes n + a_{A} (I - n \otimes n)$ )

For isotropic phases with planar interfaces normal to

n

, normal conductivity is the weighted harmonic mean

a_{H} = (\sum p_{i} / a_{i})^{- 1}

and tangential conductivity is the weighted arithmetic mean

a_{A} = \sum p_{i} a_{i}

. Continuity of normal flux and tangential temperature gradient gives the two averages directly.

Conductivity homogenization

Conductivity homogenization replaces rapidly varying positive coefficients in a divergence-form elliptic equation by a macroscopic effective conductivity. Periodic media admit cell problems; statistically homogeneous media can instead be described by ensemble averages. The small-scale assumptions and boundary regime must be specified.

Locally periodic one-dimensional harmonic-mean coefficient ( $\overset{a}{^} (x) = (\int_{0}^{1} a (x, X)^{- 1} d X)^{- 1}$ )

For

(a (x, x / ϵ) y^{'})^{'} = f (x, x / ϵ)

with positive one-periodic fast coefficient, a periodic two-scale expansion has leading term independent of the fast variable. The first corrector satisfies

y_{1 X} = C (x) / a - y_{0}^{'}

. Its zero-mean derivative requires

C = \overset{a}{^} y_{0}^{'}

. Averaging the next equation gives

C^{'} = \int_{0}^{1} f d X

. Thus the leading equation has the local harmonic-mean transport coefficient and the arithmetic-mean source. Positivity is what makes the secular drift incompatible with periodicity.

Polarization field of a conductivity inclusion ( $P = (a - a_{0} I) \nabla u$ )

Relative to a uniform reference conductivity

a_{0}

, this polarization is the excess positive conductivity flux. The equation becomes

a_{0} Δ u + \nabla \cdot P = 0

. For a prescribed mean gradient

E

, averaging gives

a^{*} E = a_{0} E + ⟨ P ⟩

. This sign convention differs from the physical heat-flux sign.

Exciting field of a conductivity inclusion ( $E_{exc} = E + e_{regular} (y)$ )

The exciting field excludes an inclusion's own singular dipole field. Other inclusions and periodic images supply the regular correction at its centre

y

. The angular mean of a self-dipole gradient on a concentric sphere vanishes, while the mean-value property recovers the regular field at the centre. Consequently a spherical average can recover the exciting field, even though the total pointwise field is not constant on that sphere.

Spherical conductivity inclusion ( $β = 3 a_{0} (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

A sphere of conductivity

a_{1}

in a host

a_{0}

under a uniform incident gradient

E

has interior gradient

3 a_{0} E / (a_{1} + 2 a_{0})

. Outside a sphere of radius

R

, its potential correction is

- R^{3} (a_{1} - a_{0}) (E \cdot x) / [(a_{1} + 2 a_{0}) ∣ x ∣^{3}]

. Its polarization is

βE

inside and zero outside. Temperature and normal-flux continuity determine these coefficients.

Dilute spherical-inclusion conductivity ( $a^{*} = a_{0} I + pβ I + o (p)$ )

Well-separated shrinking spheres respond independently at leading order, so their polarization contributions add according to volume fraction. Clusters kept at fixed relative internal separation can have a different leading response; the dilute limit alone does not guarantee independent-sphere polarizability.

Maxwell approximation for conductivity ( $a^{*} = a_{0} (1 + 2 p k) / (1 - p k), k = (a_{1} - a_{0}) / (a_{1} + 2 a_{0})$ )

Matching the far dipole of a spherical composite with the sum of isolated spherical-inclusion dipoles gives this approximation. Its dilute expansion is

a_{0} + pβ + p^{2} β^{2} / (3 a_{0}) + O (p^{3})

. The rational expression is a useful closure, not the exact conductivity of every spherical-inclusion microstructure.

Dipole interaction correction to effective conductivity ( $a^{*} \approx a_{0} I + pβ I - β^{2} \sum_{i} Λ_{i}$ )

If the first exciting-field correction at inclusion

i

- β Λ_{i} E / p_{i}

, its revised polarization is

β χ_{i} (E - β Λ_{i} E / p_{i})

. Averaging weights it by

p_{i}

and gives the displayed second-reflection approximation. Further dipole reflections and finite-size multipoles are neglected.

Periodic conductivity cell problem ( $\nabla_{y} \cdot [a (y) (e_{j} + \nabla_{y} w_{j})] = 0$ )

The periodic zero-mean corrector

w_{j}

adjusts a unit imposed gradient so microscopic flux is divergence-free. For bounded uniformly elliptic coefficients, the weak problem has a unique solution modulo constants. The averaged flux is

a^{*} e_{j} = ⟨ a (e_{j} + \nabla w_{j})⟩

. For symmetric coefficients this also gives a symmetric positive-definite effective tensor.

Two-scale expansion for periodic conductivity ( $\nabla = \nabla_{x} + ϵ^{- 1} \nabla_{y}$ )

A periodic expansion with

y = x / ϵ

separates the macroscopic variable from the cell variable. Its leading cell equation forces

u_{0}

to be independent of

y

, the next equation gives

u_{1} = w_{j} (y) \partial_{x_{j}} u_{0}

, and averaging the order-one flux equation yields the homogenized divergence equation. Bulk correctors generally require boundary-layer adjustments at a physical Dirichlet boundary.

Boundary corrector in periodic homogenization ( $b^{ϵ} ∣_{\partial Ω} = - ϵ w_{j} (x / ϵ) \partial_{j} u_{0}$ )

A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.

Plasticity (physics)

Yield criterion ( $f (τ, q) \leq 0$ )

A yield criterion specifies the admissible elastic stress region, possibly depending on internal hardening variables

q

. Plastic loading occurs on its boundary. The Tresca yield criterion uses maximum principal-stress difference, while the Von Mises yield criterion uses a quadratic deviatoric-stress invariant. Thus a numerical stress difference must be compared to a yield stress using the convention of the selected criterion.

Plastic strain ( $e = e^{e} + e^{p}$ )

In small-strain elastoplasticity, total infinitesimal strain tensor splits into an elastic part and a generally irreversible plastic part. The elastic constitutive equation determines stress from

e^{e}

, while a yield condition and a plastic flow law determine the evolution of

e^{p}

. Removing external load need not remove plastic strain, so using Hooke's law on total strain after yielding is insufficient.

Plastic flow rule ( $\overset{e}{˙}^{p} = \dot{λ} \partial g / \partial τ$ )

A plastic flow rule prescribes the direction and rate of plastic straining once a yield condition is active. In a plastic-potential model its direction is the stress gradient of

g

, multiplied by a nonnegative plastic multiplier. An associated flow rule takes

g

equal to the yield function. The stress-strain split, loading conditions and hardening evolution are needed along with this equation to close an elastoplastic model.

Elastic unloading

Reduction of an applied load with purely elastic strain increments while accumulated plastic strain is fixed. The superposition principle applies to the increment, so the current stress equals the previous stress plus the linear elasticity response to the load change. The resulting stress must still satisfy the yield surface constraint; reverse yielding ends the purely elastic stage.

Reverse yielding

Renewed plastic deformation during unloading or reversed loading when the stress reaches the opposite part of the yield surface. For the spherical elastoplastic cavity with maximum pressure

P_{m} > 4 τ_{Y} /3

, elastic unloading first reaches the opposite Tresca yield criterion at the cavity wall after a pressure decrease

8 τ_{Y} /3

. Purely elastic return to zero pressure is possible when

P_{m} \leq 8 τ_{Y} /3

, with equality reaching the yield surface exactly at the endpoint.

Tresca yield criterion ( $f = \frac{1}{2} max_{i, j} ∣ σ_{i} - σ_{j} ∣ \leq τ_{Y}$ )

The Tresca yield criterion bounds the largest resolved shear stress, expressed through the principal stresses. In spherical symmetry, the two tangential principal stresses coincide, so

f = ∣ τ_{rr} - τ_{θθ} ∣/2

. Here

τ_{Y}

is a shear yield stress; the corresponding uniaxial yield strength is

2 τ_{Y}

Necking (engineering)

Localization of tensile deformation into a narrowing region of a specimen. The onset depends on the competition between declining cross-sectional area and work hardening.

Cold working

Plastic deformation below the temperature range for substantial concurrent recrystallization. It changes dislocation structure and texture; irradiation and later heating can modify the resulting microstructure.

Work hardening

An increase in resistance to further plastic deformation after deformation has changed the microstructure. Increasing dislocation density and interactions can impede subsequent dislocation glide.

Latent hardening

Increase in the resistance of one slip system caused by plastic shear on another. It can shift the onset of secondary slip away from a purely geometric equal-Schmid-factor prediction.

Forest hardening

Resistance to dislocation glide produced by intersections with other dislocations. Multiple active slip systems can form junctions and increase the density of obstacles. This is one mechanism of work hardening.

Yield point

A yield point marks onset of permanent deformation under the stated experimental criterion. A specified offset proof stress is an operational measure when a sharp yield transition is absent.

Proof stress

An offset proof stress is defined by the intersection of a loading curve with a line parallel to its initial linear elasticity response but shifted by a specified engineering strain. A 0.2 percent convention uses the line

σ = E (ε - 0.002)

, with Young's modulus

E

Von Mises yield criterion ( $σ^{'} : σ^{'} = 2 k^{2}$ )

The Von Mises yield criterion depends only on the second invariant of deviatoric stress. With shear yield stress

k

, its surface is

σ^{'} : σ^{'} = 2 k^{2}

; the uniaxial yield stress is

3 k

. An associated flow rule gives plastic strain rate proportional to

σ^{'}

, up to a positive normalization of the plastic multiplier.

Associated flow rule ( $D^{p} = \dot{λ} \partial f / \partial σ$ )

Plastic strain rate is normal to the yield surface in the work-conjugate stress variables. With symmetric stress variations, the pairing is

D : δ σ

; in two dimensions its off-diagonal contribution is

2 D_{12} δ σ_{12}

. This factor is essential when changing to reduced stress coordinates.

Perfect plasticity

A perfectly plastic material has a yield surface that does not change with accumulated plastic deformation. This removes hardening while retaining the distinction between elastic and plastic loading.

Spherical elastoplastic cavity ( $P_{c} = 4 τ_{Y} /3, s = a e^{(P - P_{c}) / (4 τ_{Y})}$ )

A spherical cavity of fixed reference radius

a

in an infinite solid with perfect plasticity, the Tresca yield criterion, and zero far-field stress first yields under internal pressure

P_{c}

. In the plastic region

a < r < s

, radial force balance and

τ_{θθ} - τ_{rr} = 2 τ_{Y}

give

τ_{rr} = - P + 4 τ_{Y} lo g (r / a)

. Matching to the outer linear elasticity solution, where

τ_{rr} (s) = - P_{c}

, gives the displayed plastic-front radius. These are small-strain reference-geometry formulae.

Antiplane perfect plasticity

In antiplane perfect plasticity, velocity

w = v_{3}

and shear stresses

a = σ_{13}, b = σ_{23}

depend only on

x_{1}, x_{2}

. Force balance is

a_{, 1} + b_{, 2} = 0

, the fixed yield surface is

f (a, b) = k

, and the associated flow rule is

\nabla w = 2 \dot{λ} (f_{a}, f_{b})

with

\dot{λ} \geq 0

. Differentiating the yield equation in each coordinate and combining it with equilibrium shows that

a, b, w

are constant along characteristic curves tangent to

(- f_{b}, f_{a})

. For example

- f_{b} a_{, 1} + f_{a} a_{, 2} = f_{b} b_{, 2} + f_{a} a_{, 2} = 0

. Nonzero regular yield gradient is needed for a characteristic direction.

Centred antiplane plastic fan

A centred fan consists of straight characteristic curves emanating from one point. Stress and antiplane velocity are constant on each ray and depend on the polar angle

ϕ

. The radial tangent condition is

(cos ϕ, sin ϕ) ∥ (- f_{b}, f_{a})

, so

f_{a} cos ϕ + f_{b} sin ϕ = 0

. A boundary ray carrying a traction-free crack-face stress

b = 0

has slope

tan ϕ_{b} = - f_{a} / f_{b}

, interpreted as vertical when

f_{b} = 0

. For the elliptic yield condition

a^{2} / A^{2} + b^{2} / B^{2} = 1

, a forward fan has

a = - A^{2} sin ϕ / H

b = B^{2} cos ϕ / H

H = (A^{2} sin^{2} ϕ + B^{2} cos^{2} ϕ)^{1/2}

. The boundary rays are vertical; the adjacent rear regions have constant stresses

(\mp A, 0)

. Velocity is a function of angle whose magnitude requires further loading data.

Slip-line field theory

Hencky stress relations ( $p - 2 k ϕ = constant on α, p + 2 k ϕ = constant on β$ )

In plane strain perfect plasticity with

σ_{11} = - p + k sin 2 ϕ

σ_{22} = - p - k sin 2 ϕ

and

σ_{12} = - k cos 2 ϕ

, equilibrium gives

\nabla p = 2 k (cos 2 ϕ sin 2 ϕ sin 2 ϕ - cos 2 ϕ) \nabla ϕ .

The slip line tangent directions

(cos ϕ, sin ϕ)

and

(- sin ϕ, cos ϕ)

are eigenvectors with eigenvalues

+ 1, - 1

. Taking directional derivatives proves the two Hencky stress relations. These sign choices must be changed consistently if the shear-stress convention changes.

Centred slip-line fan at a traction-free V-notch

For a symmetric reentrant notch opening towards negative

x_{1}

, let its semi-angle be

0 \leq γ \leq π /2

. On the upper free face, tensile tangential stress selects

p = - k

ϕ = π /4 - γ

. The adjacent centred fan has radial

β

lines and circular

α

lines:

ϕ = ϑ - π /2

p = - k + 2 k [ϑ - (3 π /4 - γ)]

, for

π /4 \leq ϑ \leq 3 π /4 - γ

. It joins a constant stress region containing the forward axis, where

ϕ = - π /4

and

p = - k (1 + π - 2 γ)

. Therefore

σ_{22} = k (2 + π - 2 γ)

in that fully plastic notch-tip region. This is a local stress field, not a claim that an arbitrary specimen is plastically yielded everywhere.

Geiringer velocity relations ( $d u - v d ϕ = 0 on α, d v + u d ϕ = 0 on β$ )

For an associated flow rule in the reduced plane-strain yield coordinates, the rate-of-strain tensor has zero extension along both slip line directions. Write velocity as

u e_{α} + v e_{β}

, with the local orthonormal axes turning through angle

ϕ

. Differentiating along an

α

curve gives longitudinal extension

d u - v d ϕ

; along a

β

curve it gives

d v + u d ϕ

. Setting these extensions to zero proves the relations for the corresponding anisotropic yield surface parameterization.

Stress characteristics for an anisotropic yield curve ( $d (σ - l) = 0 on α, d (σ + l) = 0 on β$ )

Parameterize a fixed yield surface by stress-space arc length

l

with

ξ_{, l} = - cos 2 ϕ

and

τ_{, l} = - sin 2 ϕ

. For mean normal stress

σ

, force balance gives

\nabla σ = M \nabla l

, where

M = (cos 2 ϕ sin 2 ϕ sin 2 ϕ - cos 2 ϕ)

. Its eigenvectors along angles

ϕ

and

ϕ + π /2

have eigenvalues

+ 1

and

- 1

, proving the displayed slip line invariants.

Slip line

A slip line is a characteristic curve of a yielded plane-strain plastic equilibrium system. The two orthogonal families carry stress and velocity compatibility relations. Their directions need not agree with principal stress directions.

Continuum thermodynamics

Reference-configuration jump balances

Let an interface move at signed speed

c

in a reference unit normal direction

n

, and set

[a] = a^{+} - a^{-}

with

n

pointing from minus to plus. For continuous reference density and no surface sources, distributional momentum conservation, conservation of energy and the Lagrangian entropy inequality give

ρ_{0} c [v_{i}] + n_{I} [P_{I i}] = 0, ρ_{0} c [u + ∣ v ∣^{2} /2] + n_{I} [P_{I i} v_{i} - q_{I}^{0}] = 0,

- ρ_{0} c [η] + n_{I} [q_{I}^{0} / θ] \geq 0.

(2)

For a level-set interface

ϕ = 0

ϕ_{t} = - c ∣\nabla ϕ ∣

, the singular part of

\partial_{t} a

- c [a] δ_{S}

and that of

Div b

n \cdot [b] δ_{S}

, proving these formulas. Using

V = - c

reverses the speed terms. Density jumps require jumps of the complete conserved densities.

Thermoelasticity

Thermoelasticity couples elasticity and temperature through a specific Helmholtz free energy

ψ (F, θ)

. Without constraints its reversible laws are

P_{I i} = ρ_{0} ψ_{, F_{i I}}

and

η = - ψ_{, θ}

. Mixed derivatives give the same coupling tensor in the thermal stress response and the strain dependence of specific entropy, with opposite signs.

Thermal stress ( $τ = C : (e - e^{th})$ )

Thermal stress is stress generated when thermal expansion or contraction is constrained or incompatible. A uniformly heated isotropic body whose entire boundary is fixed has

e = 0

and

τ = - K α_{v} (T - T_{0}) I

. For a nonuniform temperature field, interior displacement generally remains nonzero even with a fixed outer boundary; equilibrium and compatibility must be solved together. Cancelling the divergence of a thermal stress term with a body force is not the same as making that stress vanish.

Thermal strain ( $e^{th} = α_{v} (T - T_{0}) I /3$ )

Thermal strain is the strain that heating would produce without mechanical stress. In isotropic linear thermoelasticity it is a pure dilation: a volumetric expansion coefficient

α_{v}

corresponds to a linear expansion coefficient

α_{v} /3

. The constitutive law acts on the mechanical strain

e - e^{th}

. Spatially varying thermal strain need not be globally compatible with a stress-free displacement.

Uniform thermal stress in a fully constrained solid ( $τ = - K α (T - T_{0}) I$ )

For isotropic linear elasticity with volumetric expansion coefficient

α

, the stress-free thermal strain is

e^{th} = α (T - T_{0}) I /3

. Thus the thermal stress term is

- K α (T - T_{0}) I

, where

K = λ + 2 μ /3

is the bulk modulus. A uniformly heated body with its full boundary fixed has zero displacement and the displayed compressive stress. For spatially varying temperature and zero body force, the displacement instead solves

μ Δ u + (λ + μ) \nabla div u = K α \nabla T

with its boundary values; the uniform formula is not automatic. A body force per volume

+ K α \nabla T

cancels this thermal forcing in the equilibrium equation, but does not by itself cancel the stress or its boundary tractions.

Thermoelastic Maxwell relation for a stretched band ( $(\partial S / \partial x)_{T} = - (\partial f / \partial T)_{x}$ )

Let

f d x

be reversible work supplied to stretch a band. Its Helmholtz free energy obeys

d F = - S d T + f d x

. Equality of mixed derivatives gives the displayed relation. An entropy decrease on stretching therefore corresponds to increasing tensile force on heating at fixed extension. The sign follows the force/work convention; a restoring force exerted by the band has the opposite sign.

Thermoelasticity with a temperature-dependent constraint ( $ϕ (F) = h (θ)$ )

The admissible rates obey

ϕ_{, F} : \dot{F} = h^{'} (θ) \dot{θ}

. A reaction multiplier

q

in the reversible nominal stress tensor therefore has an accompanying specific entropy contribution:

P_{I i} = ρ_{0} ψ_{, F_{i I}} + q ϕ_{, F_{i I}}, η = - ψ_{, θ} + \frac{q}{ρ _{0}} h^{'} (θ) .

Their reaction contributions cancel in the Clausius-Duhem inequality on admissible rates. Omitting the entropy term would generally violate the constraint's power balance.

Linear thermoelasticity with prescribed thermal volume

Linearize

det F = h (θ)

F = I

θ = θ_{0}

h (θ_{0}) = 1

, and zero reaction. With displacement gradient

H

and temperature increment

ϑ

, one has

tr H = h^{'} (θ_{0}) ϑ

and

σ_{ji} = C_{ji l k} H_{k l} + β_{ji} ϑ + q δ_{ji}, ρ_{0} (η - η_{0}) = C_{e} ϑ - β_{ji} H_{ij} + q h^{'} (θ_{0}),

where

C_{ji l k} = ρ_{0} ψ_{, F_{ij} F_{k l}}

β_{ji} = ρ_{0} ψ_{, F_{ij} θ}

and

C_{e} = - ρ_{0} ψ_{, θθ}

at the reference state. The temperature factor converts this entropy coefficient to a volumetric heat capacity:

θ_{0} C_{e}

. Strictly constant volume is the case

h^{'} = 0

Clausius-Duhem inequality

Coleman-Noll procedure

Requiring the Clausius-Duhem inequality for independent reversible mechanical and thermal variations constrains a constitutive equation. For

U (F, η, ξ)

it gives

P_{I i} = ρ_{0} U_{, F_{i I}}

and

θ = U_{, η}

. Combining the Lagrangian internal energy balance with the Lagrangian entropy inequality leaves

f_{r} \dot{ξ}_{r} - q_{I} θ_{, I} / θ \geq 0

. The independence assumptions are part of the constitutive argument.

Lagrangian entropy inequality ( $ρ_{0} \overset{η}{˙} \geq ρ_{0} r / θ - Div (q / θ)$ )

The local entropy balance has nonnegative production, with heat supply

ρ_{0} r / θ

and thermodynamic entropy flux

q / θ

at positive temperature

θ

. Its integral version holds on every material reference subvolume.

Lagrangian internal energy balance ( $ρ_{0} \dot{U} = P_{I i} \dot{F}_{i I} + ρ_{0} r - q_{I, I}$ )

For the material-first nominal stress tensor, the displayed local conservation of energy law balances mechanical work, heat supply and outgoing heat flux. Integrating over a fixed reference material volume and using the divergence theorem gives its integral form. The full total-energy balance includes kinetic energy and external mechanical work; momentum conservation reduces it to this internal-energy balance.

Internal variable ( $ξ_{r}$ )

An internal variable records a material state beyond its instantaneous deformation gradient and temperature, such as a relaxing microstructural state. A constitutive equation specifies its evolution and its contribution to the free energy.

Dissipation potential ( $\dot{ξ}_{r} = \partial Ω/ \partial f_{r}$ )

A force-dependent potential specifies internal-variable rates by differentiation. Convexity with a minimum at zero force can ensure nonnegative force-rate dissipation. The name alone does not ensure the Clausius-Duhem inequality for an arbitrary potential.

Exponential memory from tensor relaxation ( $A (t) = e^{- t / τ} A (0) + α \int_{0}^{t} e^{- (t - s) / τ} ∥ Q (s) ∥^{n - 1} Q (s)^{T} d s$ )

The dissipation potential

Ω = α ∥ Q ∥^{n + 1} / (n + 1) - tr (A Q) / τ

, with the Frobenius norm, gives

\dot{A} + A / τ = α ∥ Q ∥^{n - 1} Q^{T}

. An integrating factor proves the displayed memory formula. For

n > 0

, its nonlinear driving term is defined continuously as zero at

Q = 0

. Mechanical dissipation is

tr (Q \dot{A}) = α ∥ Q ∥^{n + 1} - tr (A Q) / τ

, which must be nonnegative on thermodynamically admissible histories.

Thermodynamic force conjugate to an internal variable ( $f_{r} = - ρ_{0} \partial U / \partial ξ_{r}$ )

The derivative of internal energy at fixed deformation gradient and entropy, or of specific Helmholtz free energy at fixed temperature, defines this force. Its product with the internal-variable rate contributes mechanical dissipation to the Clausius-Duhem inequality.

Deformation (mechanics)

Deformation changes a body's shape or size. Strain quantifies relative changes in lengths and angles, whereas a rigid translation or rotation alone produces no strain. The continuum approximation describes these changes using displacement fields and their spatial variation.

Homogeneous deformation ( $y (X) = FX + b$ )

A homogeneous deformation has the same deformation gradient at every material point. On a connected reference domain it is an affine map

y (X) = FX + b

with constant matrix

F

and translation

b

. Orientation preservation requires

det F > 0

. Its stretches and local volume ratio are spatially constant. A homogeneous deformation describes the kinematics; a homogeneous hyperelastic material instead describes spatial independence of a constitutive law.

Finite strain theory

Elastic incompressibility ( $det A = 1$ )