deformation
any change in distance between two material points of a continuous medium, representing the change in shape or size of an object under the effect of a force.
1/47
| Term | Definition |
|---|---|
| deformation | any change in distance between two material points of a continuous medium, representing the change in shape or size of an object under the effect of a force. |
| how do deformations help study cells? | they directly reflect how cells mechanically interact with their environment |
| biological examples of cell deformation | - RBCs pass through a narrow capillary must undergo strong deformation - a cell moving through a dense matrix changes its shape to slip between surrounding fibers - the cell nucleus can compress or elongate in response to internal cytoskeletal forces |
| biological impacts of deformations | - excessive nuclear deformations can cause nuclear envelope rupture = genomic instability - cancer cells actively modulate their ability to deform = better tissue invasion skills |
| deformations of cell migration | When a cell migrates through a tissue, it elongates in the direction of motion, emits protrusions (lamellipodia, filopodia), and generates traction forces via its cytoskeleton. These forces cause internal deformations visible at the cytoplasmic and nuclear levels. |
| material point | occurs when a material deforms, each initial point of the material is moved to a new position -> then, the displacement vector is deformed |
| displacement vector depends on ___ | considered point; it defines a displacement field throughout the material equation : ⃗u = ⃗x − X⃗ |
| displacement field | a vector function defined over the initial domain of the material |
| how can the displacement field be tracked how experimentally? | through particle tracking, fluorescence microscopy imaging, or cell contour tracking |
| biological example of displacement field | traction force microscopy - a cell placed on a soft gel can exert tractions on it. mark the gel with fluorescent beads and track the displacements of the beads between the initial and deformed stations |
| mathematical modelling of traction force microscopy | A bead is initially at position X⃗ = (10 μm, 5 μm) in a gel. After traction by a cell, it is observed at ⃗x = (12 μm, 6.5 μm). The displacement vector is therefore 2 μm horizontally and 1.5 μm vertically. |
| rigid displacement vs deformation | relative change between the object's points |
| uniaxial elongation deformation | an object is stretched in one direction. |
| biological examples of uniaxial elongation | - a cell pulled between two micropipettes - an actin fiber under tension. |
| compression deformation | shortening in one direction. |
| biological examples of compression deformation | nucleus compressed in a microchannel |
| shear deformation | parallel sliding of layers |
| biological examples of shear compression | a cell subjected to fluid flow or on a soft substrate. |
| deformation in biology | — Epithelial cells elongate during morphogenetic processes. — The cytoskeleton undergoes internal shear during migration. — Living tissues can be locally compressed by surrounding forces. |
| small strain tensor | denoted ε. used to describe the local deformation of a material by expanding the displacement field around a given point |
| small, shape variation displacements can be approximated by ___ | first derivatives of displacement |
| symmetric tensor | measures the relative variation of distance between two neighboring points of the material. |
| elongations or contractions are measured with | diagonal components (εxx, εyy, εzz) along the axes |
| shear deformation is measured with | off-diagonal components (εxy), signifies changes in angles between initial axes |
| when a cell migrates on a soft substrate... | it can locally stretch it - εxx > 0 |
| a compressed cell nucleus experiences contraction in several directions | εii < 0 |
| X | initial position coordinates |
| x | after deformation coordinates |
| small strain tensor is symmetric, true or false? | true |
| small strain tensors represent... | the shape variation of the material |
| deformation gradient tensor | signified by F describes how a small material volume transforms |
| if F = I | there is no deformation |
| if F does not = I | there is a significant local deformation of the material |
| biological examples of large deformation tensor | In tumors or embryonic tissues, cells exert significant traction on their environment (e.g., collagen, extracellular matrix). Large deformations of these matrices modify : — the distribution of forces around cells ; — the local tissue structure ; — the mechanical response of the cell via mechanosensitive sensors. |
| linear framework is no longer valid when | displacements are large |
| deformation gradient tensor describes ___ | stretching, shear, and rotation of a small volume |
| deformation gradient tensor can be applied to | cell migration, growth, tissue remodeling |
| deformation invariants | certain quantities derived from the tensor that remain unchanged |
| when reference frame changes, ____ changes | tensor components |
| trace as a common invariant of small strain tensor | trace -> tr(ε) = εxx + εyy + εzz represents the relative volumetric deformation positive = expansion, negative = compression |
| determinant of F (large deformations) as a common invariant of small strain tensor | det(F) = local volume change ratio if det(F) > 1, the material has expanded if det(F) < 1, the material has contracted |
| biological example of strain tensor invariants | During an osmotic shock, a cell swells or shrinks depending on the concentration gradient. The volume change can be described by : — the trace of ε if deformations are small ; — or det(F) if deformations are large. |
| what allows quantification of cytoplasmic or nuclear swelling from images? | defining trace and the determinant of F in the small strain tensor |
| invariants are dependent or independent on the chosen reference frame? | independent |
| the trace of ε measures what? | the volume change (for small deformations) |
| the determinant of F gives what? | local volumetric enlargement factor (non linear case) |
| biological applications of invariants? | osmosis nuclear swelling growth |