Mechanics of deformable solids Definition

Deformation

Builds on Continuum mechanics Jacobian matrix Determinant

Definition

Let $B \subseteq \R^3$ be a nonempty open set, the reference configuration of a continuum body. A deformation of $B$ is a map $\varphi \colon B \to \R^3$ such that:

$\varphi$ is continuously differentiable;
$\varphi$ is injective;
$\det \nabla \varphi(X) > 0$ for every $X \in B$ , where $\nabla \varphi$ is the Jacobian matrix of $\varphi$ .

The displacement of the deformation is the map $u \colon B \to \R^3$ , $u(X) = \varphi(X) - X$ . The set $\varphi(B)$ is the deformed or current configuration.

Remarks

The uniform stretch $\varphi(X) = (2X_1, X_2, X_3)$ is a deformation: it is smooth, injective, and $\det \nabla \varphi = 2 > 0$ ; its displacement is $u(X) = (X_1, 0, 0)$ . The reflection $\varphi(X) = (-X_1, X_2, X_3)$ is smooth and injective but fails clause 3, since $\det \nabla \varphi = -1$ : it would turn the body inside out. Clause 3 together with the inverse function theorem makes $\varphi$ a diffeomorphism of $B$ onto the open set $\varphi(B)$ . A deformation is called rigid when $\varphi(X) = c + QX$ with $Q$ a rotation matrix and $c \in \R^3$ .

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Tag	`174T9R5XE8`
URI	`m2://174T9R5XE8`
Kind	definition
Writing	W0 stub
MSC	74A05
Defines	Deformation
Uses	Continuum mechanics, Jacobian matrix, Determinant
Changed	3 November 2026

Deformation

Definition

Remarks

Reading path

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