Elastic cloth model#
Start with a piece of cloth laid flat without stretching it. Let \(A\) be its area in square metres, \(\mu\) its mass per square metre (kg/m²), and \(M\) its total mass in kilograms. For uniform cloth, meaning the same mass per area throughout, its mass is simply
This follows directly from what mass per square metre means: multiply it by the number of square metres. It is a definition-based calculation, not a separate material law.
For example, a piece 0.50 m long by 0.20 m wide has area \(A=0.10\) m². With \(\mu=0.22\) kg/m², meaning 0.22 kg for each square metre, its mass is \(M=0.22\times0.10=0.022\) kg. This mass per area matches the demonstration’s chosen cloth density; the patch dimensions are example geometry, not material constants.
Folding or bending this piece, making its surface turn rather than stay flat, does not change its mass. Nor does its mass change if it stretches: the same material is still present. The area of its shadow or its deformed surface must not be used to assign it a new mass.
A newton (N) measures force; a joule (J) measures energy. Kilograms per square metre means mass for each square metre of cloth.
Gravity gives this mass a downward force:
Bold symbols describe quantities with both a size and a direction. Here \(\mathbf{F}_g\) is the gravitational force and \(\mathbf{g}\) is gravitational acceleration, the change of velocity per second caused by gravity. Force \(\mathbf F_g\) is in newtons; choose \(\mathbf g=(0,-9.81,0)\) m/s² before evaluating the equation.
Here the chosen gravity matches the demonstration. Acceleration is in m/s²: velocity changes by that many metres per second each second. Our patch’s three acceleration components are along the boom, upward, and sideways: only the upward component is nonzero, and its negative sign means downward. Its weight magnitude is \(0.022\times9.81=0.21582\) N. Gravity is a real load even when the wind is zero.
The force formula is the weight relation explained in OpenStax: Mass and Weight. The chosen 9.81 m/s² is an approximate near-Earth value, not a universal gravitational acceleration.
Threads, extension, and shear#
A membrane is an ideal thin sheet that resists changes of length and changes of angle within its surface, but has no bending resistance. These two changes are called stretching and shearing; we explain each below. Real cloth has some bending resistance, which we model separately. This distinction explains how a sheet can be difficult to stretch but easy to fold.
Stiffness describes how much load, or applied force, is needed for a given deformation, meaning a change of shape or size. More stiffness means more load for the same deformation. In this chapter, elastic energy means energy stored in an ideal reversible deformation: it can be returned as the load is removed and the cloth returns toward its original shape.
Woven cloth is made by crossing two sets of threads over and under each other. These crossing thread families are called warp and weft. Call the warp direction 1 and the weft direction 2. They initially cross at right angles in the patch used here. Begin with the simpler case of equal extension stiffness along both thread directions. These directions belong to the cloth and rotate with it; they are not necessarily the world’s horizontal and vertical axes.
To understand extension, mark a short line along either thread direction. Let its original, unstretched length be \(\ell_0\) and its current length be \(\ell\), both in metres. The zero means the original or reference value, not a counter or a zero length. The reference state is the original unstretched cloth against which changes are measured. Strain, written \(\epsilon\), is the fractional change in length, with no units. For the definition using original length, called engineering strain,
This is the engineering-strain definition in MIT: The Concept of Strain, Section 2.1, equation (2.2a).
The length units cancel in this fraction: strain is dimensionless, meaning it has no physical unit. A 0.50 m warp line becoming 0.505 m long has \(\epsilon=0.01\), or 1% extension. Negative strain means shortening, not a negative length. This definition describes local material patches; strain may differ from patch to patch.
Equal stiffness does not mean equal extension. Pull a patch along one thread direction while holding its length fixed in the other: the pulled direction can lengthen while the held direction does not. Different loads or constraints can therefore produce different strains even when the threads have equal extension stiffness.
Shear is an angle change. Imagine a rectangular patch becoming a slanted parallelogram, with opposite sides still parallel, while its sides barely change length. Let \(\phi\) be the current angle between its thread directions, in radians. A right angle is \(\pi/2\) radians, or 90 degrees, where \(\pi\) is approximately 3.14159. Let \(\gamma\) name dimensionless engineering shear, choosing its positive sign to mean that the initially right angle closes. The symbol \(\approx\) means approximately equal. For small angle changes,
Radians are dimensionless. Closing the angle by 0.10 rad, approximately 5.73 degrees, corresponds to \(\gamma\approx0.10\). This angle interpretation is suitable for small changes, not an exact law for large deformations. Turning the whole patch without deforming it changes neither its lengths nor the angle between its threads, and must not create elastic strain energy.
The angle-change interpretation is developed in MIT: Shear as an Angle Change, Section 2.3. Those notes use a shear quantity equal to half the small angle change; our engineering shear \(\gamma\) uses the whole change, so the convention must be accounted for when comparing formulas.
Different parts of a sail can stretch or shear by different amounts. To find the whole sheet’s energy, we therefore need to add the local energy of all its small patches. This is where an area integral becomes useful.
Let \(S_0\) be the original, unstretched sheet over which we add those contributions. Let \(dA_0\) be an infinitesimal area: an arbitrarily tiny patch of that original sheet, in m². The symbol \(\int_{S_0}\) means to sum across it. The zero again labels the original state. For the earlier example patch, its whole original area is \(A_0=0.10\) m², the area previously called \(A\); \(dA_0\) is a tiny part of it, not another name for the whole area. In the same notation, \(\mu_0=0.22\) kg/m² is the mass per original area previously called \(\mu\).
The energy must account for both thread directions. Write \(\epsilon_1\) for the value of the same strain definition measured along warp, and \(\epsilon_2\) for its value along weft. They are two measurements, not two different definitions, and need not be equal.
Before writing the energy, choose three named stiffnesses for our equal-thread example:
Extension stiffness \(K=20{,}000\) N/m, the same along warp and weft: resistance to length change in either thread direction.
Coupling stiffness \(C=2{,}000\) N/m: coupling means extension in one thread direction affects the force needed in the other. It is not shear stiffness.
Membrane shear stiffness \(K_s=500\) N/m: resistance to changing the angle between threads. Here N/m means force per width of sheet, not force per area of a solid material.
These are material-dependent quantities. Their values here are an internally consistent teaching example, not universal constants, typical sailcloth values, or measured properties of the actual sail. Nor are they the stiffnesses of the demonstration’s numerical links, which join pairs of simulated cloth points.
Let \(E_m\) be elastic energy stored by stretching and shearing the cloth, measured in joules. The subscript names membrane energy, excluding the separate bending energy introduced later. Our chosen expression is
Where:
\(\epsilon_1,\epsilon_2\) are dimensionless strains along warp and weft.
\(\gamma\) is dimensionless engineering shear strain.
\(K,C,K_s\) are the three stiffnesses defined above (N/m).
\(E_m\) is the elastic energy stored by stretching and shearing the cloth (J). It does not include the separate bending energy introduced later.
This is a specialization of the thin-sheet extension law in Roylance: Laminated Composite Plates, equations (4) and (20)–(21), combined with the linear elastic work-to-energy principle in Alberta: Linear Elastic Strain Energy, equation (2). We choose equal directional extension stiffness and no direct extension-to-shear coupling; bending is treated separately. Neither source supplies our illustrative numerical stiffness values.
Why one half? If a pulling force increases in direct proportion to extension, the force rises from zero to its final value. Its average over that extension is half its final value, and work is average force times extension. The same principle gives the quadratic energy above, with the coupling term accounting for the interaction of the two extensions. We use an unloaded original state: no initial elastic pulling force is stored in this example sheet.
An illustrative material, with units attached#
The table collects our illustrative stiffnesses alongside the chosen density, gravity, and bending parameters. The extension stiffness is equal in the two thread directions; coupling and shear remain separate material properties.
Two additional quantities describe bending. Curvature is how much a surface turns per unit distance along it, measured in 1/m; flat cloth has zero curvature. We use \(\kappa\) for its current curvature and \(\kappa_0\) for its original curvature. Bending rigidity \(B\) is the stiffness resisting a change of curvature. Its energy calculation comes later, separately from the stretching and shearing energy.
Quantity |
Example value in SI units |
Meaning |
|---|---|---|
\(\mu_0\) |
0.22 kg/m² |
Mass per manufactured area; matches the demo |
\(\mathbf{g}\) |
(0, −9.81, 0) m/s² |
Chosen near-Earth gravity; matches the demo |
\(K\) |
20,000 N/m |
Extension stiffness in both thread directions |
\(C\) |
2,000 N/m |
Coupling between warp and weft extension |
\(K_s\) |
500 N/m |
Engineering-shear coefficient |
\(B\) |
\(2\times10^{-4}\) N m |
Illustrative bending rigidity, introduced below |
\(\kappa_0\) |
0 1/m |
Chosen flat reference curvature, not a stiffness |
The coupling stiffness \(C\) is not another shear stiffness. It expresses how extension in one thread direction affects the force in the other. A force per width is obtained from how quickly elastic energy per area increases as strain increases. Calculus calls this differentiating the energy with respect to strain.
Before writing the results, let \(N_1,N_2,N_{12}\) name these forces per width, in N/m:
\(N_1\): the pulling force along warp,
\(N_2\): the pulling force along weft,
\(N{12}\): the sliding force associated with shear between the two thread directions.
They are often called membrane force resultants because the forces through the sheet’s thickness have been combined into a force per width. They are not stress, meaning force per area, measured in pascals (Pa = N/m²). For our energy expression, the results are
These are derived from our displayed energy, not independently assumed formulas. For example, the rate of change of \(K\epsilon_1^2/2+C\epsilon_1\epsilon_2\) with respect to \(\epsilon_1\) is \(K\epsilon_1+C\epsilon_2\). Alberta: Linear Elastic Strain Energy explains the underlying energy derivative principle; here the energy is per sheet area rather than per volume.
With positive \(C\), stretching warp while holding weft at its original length requires tension, or pulling force, along weft too. If there is instead no pulling force along weft (\(N_2=0\)),
Thus 1% warp extension would accompany 0.1% weft contraction under that particular loading condition. Which edges are held or pulled, not the material stiffnesses alone, determines the two strains.
For positive elastic energy away from the zero-strain state, the equal-thread example requires the following conditions. The bars in \(|C|\) mean its magnitude, ignoring its sign:
Our values give \(20{,}000>2{,}000\) N/m and \(500>0\) N/m. The coupling term therefore cannot make the total elastic energy negative.
This can be checked directly, without importing another material law. The local elastic energy per original area, denoted \(w_m\) (J/m²), can be rewritten by expanding the squares:
Both extension coefficients in this rewritten expression are positive when \(K>|C|\), and a square cannot be negative. Positive \(K_s\) does the same for shear.
Worked example: extension and shear cost different amounts#
Take the uniform 0.10 m² reference patch above, stretch warp by 1%, hold weft at its original length, and apply no shear. These are example states, not new constants. The energy per reference area \(w_m\) is
Since N/m equals J/m², the units agree. The warp force per width is \(N_1=200\) N/m. Across the patch’s 0.20 m width this is 40 N of warp force. Holding the weft length also requires \(N_2=20\) N/m, because of the mixed coefficient. Releasing that constraint would change the strain and energy; the 40 N calculation specifically assumes zero weft extension.
Now return both lengths to their reference values and apply pure shear \(\gamma=0.10\). This separate state gives
The shear coefficient is much smaller than the extension coefficients, but this example’s shear strain is ten times its extension strain. The energies depend on both the stiffness and the square of the deformation.
Do not equate \(K\) with a numerical edge spring constant merely because both can be written in N/m. Here \(K\) relates force per width to dimensionless strain. A spring constant relates force to extension in metres. For this uniform rectangular patch with weft extension constrained, strain is extension divided by original length, and force is force per width times patch width. Substituting those two relations gives an effective warp spring constant of \(k_{\text{strip}}=K(0.20\,\text{m})/(0.50\,\text{m})=8{,}000\) N/m. Geometry and transverse constraints are essential to that conversion. The demo’s mesh links are not a calibrated discretization of this example.
Generalizing to unequal thread stiffness#
So far one extension stiffness \(K\) serves both thread directions. If the thread families or their weave respond differently, give the warp and weft squared-strain terms separate extension stiffnesses, each in N/m, rather than using the same \(K\) for both. The coupling and shear terms still apply. A membrane with two perpendicular material symmetry directions is called orthotropic: reflecting its loading about either direction leaves its elastic response correspondingly reflected, rather than introducing a new material behavior. The thread directions are those special directions here. This permits unequal stiffness along them but does not require it.
Equal extension coefficients along the threads do not mean that the material responds equally to loading at every angle. That stronger property is called isotropy. For the linear membrane expression used here, isotropy would also require \(K_s=(K-C)/2\), which would be 9,000 N/m for our extension and coupling values. We instead chose 500 N/m: our example is equally stiff in extension along the two thread directions but much easier to shear. Its shear stiffness remains an independent choice, not a consequence of making the two extension coefficients equal.
The isotropic relation follows from the membrane force and energy laws in MIT: Membrane Energy, Section 2.3: the shear coefficient is half the difference between the extension and coupling coefficients when engineering shear is used. Our independently chosen shear stiffness deliberately does not make that isotropic specialization.
This energy expression is a material law: a rule relating deformation to stored elastic energy and force. It is a simplified rule, not a claim that real cloth’s force always increases in direct proportion to strain. Actual threads may straighten before stretching; loading and unloading may follow different paths and lose energy. Thin cloth also forms wrinkles when pushed along its surface instead of carrying large sustained pushing forces. For large shape changes, a better law must still distinguish actual deformation from simply turning the patch without changing its lengths or thread angles.
Bending is a separate resistance#
Bending changes curvature without necessarily changing in-plane lengths. An ideal membrane has none of this resistance, but a cloth model can add a similarly simplified bending energy \(E_b\), measured in joules:
MIT: Bending Energy, Section 2.2.2, equations (45)–(46), derives the single-direction energy per area \(B\kappa^2/2\) for a flat reference. We rename its bending rigidity and express curvature relative to the chosen stress-free value \(\kappa_0\). This scalar, single-curvature example is not the full two-direction plate or shell energy; the source derives that more general energy separately in equations (47)–(53).
Here \(B\) is bending rigidity (N m), \(\kappa\) is curvature (1/m), and \(\kappa_0\) is reference curvature. Use the illustrative \(B=2\times10^{-4}\) N m and flat reference \(\kappa_0=0\) 1/m from the table. A current cylindrical bend of radius 0.50 m has curvature magnitude \(\kappa=2\) 1/m. For our uniform patch, this example state stores
Bending rigidity times curvature squared times area has units N m × 1/m² × m² = N m = J. This small example bending energy explains why the chosen sheet can bend more easily than it stretches. A more complete thin-sheet model accounts for bending in different directions and twisting rather than using just one curvature value. Both energies have units of joules, but they penalize different deformations.
The constants in the table characterize our chosen material and environment; \(\kappa_0\) specifies its reference condition. Position (m), time (s), local strains \(\epsilon_1,\epsilon_2,\gamma\) (dimensionless), and current curvature \(\kappa\) (1/m) are evolving variables, not fixed constants. The original sheet and patch lengths specify its chosen size and shape. In particular, the example strains and bend above are not imposed everywhere on a sail.
Attachments and fullness#
To describe which points are attached, label a material point by two coordinates \(\mathbf{a}\), two numbers locating it on the original sheet. Its current position at time \(t\) is \(\mathbf{r}(\mathbf{a},t)\), in metres. The labels identify the same piece of material as it moves; they do not prescribe its current position. The luff is the edge beside the mast; the foot is the bottom edge along the boom. In this model both are fixed in position. The other two edges, the leech away from the mast and the head at the top, are free. Let \(\mathbf{r}_{\text{rig}}\) be the specified position of an attachment on the fixed mast or boom, in metres. At the attachments, position is prescribed for all time:
An ideal fixed attachment supplies a support reaction, the force needed to hold the attached cloth in place. These prescribed positions are boundary conditions: instructions for what happens at the sheet’s edges. A free edge has no attachment force or attachment bending moment, the turning effect of a support force, although air can still load the cloth there. A real attachment may also fix the direction the cloth leaves the support. Fixing position alone does not impose that extra restriction on rotation.
Fullness means extra manufactured cloth that can form a curved bag without requiring the threads to stretch. It comes from the cut shape and seams, the joins between pieces of cloth, or more generally from the original lengths and angles built into the material. It must not be confused with elastic stretch under wind. Specifying a curved shape every frame would remove the very question we want to solve: which shape follows from the loads?
Sources and our choices#
Try the stretchy flag on the Moon in Chapter 2: pull its top-right corner, then release it under lunar gravity. It deliberately uses much softer cloth and a finite-extension limit added to the linear law above. Its model notes explain that nonlinear teaching choice; it is not a measurement of this sailcloth or a claim that real fabric cannot tear.
The linked university notes establish the mechanics principles, not measured properties of this sail. The three membrane stiffnesses and bending rigidity are illustrative choices. The mass per area and gravity match the demo’s inputs; that does not experimentally validate them. Our examples, positivity check, force formulas, and strip-stiffness conversion are calculated here from the stated definitions and energy law.
The source texts use more advanced notation than this chapter. The section and equation pointers above identify the relevant definitions or derivations; their letter names need not match ours.