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Mechanics of deformable solids Definition

Continuum mechanics

Builds on Diffeomorphism

Definition

Continuum mechanics is the branch of mechanics that studies bodies modelled as continua. A continuum body is given by the following data.

  1. A reference configuration: a nonempty open set B⊆ℝ3B \subseteq \R^3, whose points XX label the material points of the body.
  2. A motion: a map χ ⁣:B×I→ℝ3\chi \colon B \times I \to \R^3, with II a time interval, such that each χ(⋅,t)\chi(\cdot, t) is an orientation-preserving diffeomorphism of BB onto an open set Ωt\Omega_t.
  3. A mass density: a positive function ρ(x,t)\rho(x, t) on the current configuration Ωt\Omega_t, such that the mass of a part P⊆BP \subseteq B at time tt is ∫χ(P,t)ρ dx\int_{\chi(P, t)} \rho \, dx.

The motion, the density and the forces are related by balance laws, valid for every body, and by constitutive equations, which describe a particular material.

Remarks

A solid bar in B=(0,L)×(0,1)2B = (0, L) \times (0, 1)^2 stretched uniformly has the motion χ(X,t)=(λ(t)X1,X2,X3)\chi(X, t) = (\lambda(t) X_1, X_2, X_3) with λ(t)>0\lambda(t) > 0. It satisfies clause 2, since each χ(⋅,t)\chi(\cdot, t) is a diffeomorphism with Jacobian determinant λ(t)>0\lambda(t) > 0. Let BB contain points with X1X_1 of both signs. Then χ(X,t)=(X12,X2,X3)\chi(X, t) = (X_1^2, X_2, X_3) is not a motion, since it folds two material points onto one place and so is not injective. The point of view is macroscopic: the fields are assumed smooth, and the atomic structure is ignored.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Continuum_mechanics
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Changed3 November 2026

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