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Clear out junk files and repair common Windows errorsFree Scan →Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Repair Windows errors before they cause bigger problemsFix Now →An elastic-body transformation maps each material point from a body’s reference configuration to its current, deformed configuration. Its displacement field can include translation and rotation as well as deformation; strain specifically measures changes in distances between points. The material’s constitutive law determines how it responds to that strain.
What does an elastic-body transformation map?
In continuum mechanics, a body is described by the positions of its material points before and after deformation. Let X denote a point’s position in the reference configuration and x its position in the current configuration. A deformation map sends each reference point to its current position:
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x = x(X)
The displacement of that material point is the difference between its current and reference positions:
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u(X) = x(X) − X
The deformation map describes where points go; the displacement field describes how far and in what direction they move. Neither displacement nor movement alone tells you how much the body has changed shape.
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How is displacement different from strain?
A displacement field can include rigid translation, rigid rotation, and deformation. Translation and rotation move points while preserving the distances between them. Strain captures changes in those distances, so a pure rigid-body movement does not constitute strain.
This distinction matters when describing an elastic body: the body may move substantially as a whole without changing shape, or it may deform while its overall position barely changes. Elasticity concerns the material response associated with deformation, not simply the fact that points have moved.
Which strain measure applies?
Finite deformation
For finite-deformation analysis, the Green–Lagrange strain tensor retains nonlinear terms associated with the displacement gradient. It is appropriate when those terms cannot be neglected.
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Small displacement gradients
When displacement gradients are small, linearized elasticity uses the infinitesimal strain tensor, which is the symmetric part of the displacement gradient. This is an approximation about local gradients, not necessarily about total displacement: a long, slender rod can move a large overall distance while its local displacement gradients remain small.
How does a material respond to strain?
Strain describes deformation; stress describes the forces induced in the body. A constitutive law specifies how a particular material relates its deformation to its response. The deformation map alone therefore does not determine stress without information about the material’s constitutive behavior.
A constitutive response should also be frame-indifferent: superimposing a rigid-body movement should not change the material response. In other words, the response should depend on the deformation, not on an arbitrary choice of overall position or orientation.
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What does “elastic transformation” mean in other contexts?
The phrase is context-dependent, and these uses should not be mistaken for the general continuum-mechanics definition.
Image registration
In image registration, an elastic transformation can mean a nonrigid displacement field used to align a moving image with a template. Elastic models can balance a similarity-driven external force against internal elastic forces. Fluid-based models are described as allowing more deformation than elastic models; this is an application-specific distinction, not a definition of an elastic body in mechanics. See the image-registration method paper.
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Experimental deformation measurement
In NASA Ames’ photogrammetry method, “elastic-body targets” are points on a test model whose deformation is measured after accounting for rigid-body movement. The described workflow tracks rigid-body and elastic-body targets, estimates the rigid-body transformation, applies it to the measurements, and then computes bending and elastic twist. It is an example of a measurement method, not a universal prescription. See NASA Ames’ description.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to read the phrase accurately
- If the subject is continuum mechanics, look for a mapping from a reference configuration to a current one, then distinguish displacement from strain.
- If the subject is image registration, determine what image displacement field or model is being described.
- If the subject is experimental measurement, check whether rigid-body motion has been separated from elastic deformation.
For a broader mechanics treatment, the University of Osnabrück thesis section on the strain tensor introduces the deformation mapping and strain concepts: “The strain tensor”.
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