Elastic strain

See also: Elasticity

Introduction

The elastic strain is a reversible deformation : the medium turns over in its initial state when the requests are removed.

The elastic strain is an important field of the Mécanique continuous mediums (MMC) and Thermodynamique (compression of gases).

Linear elasticity relates to the small deformations proportional S to the request. In this range, lengthening is proportional to the force in the case of a stretching, and the Angle is proportional to the couple in the case of a Torsion.

With the greatest deformations, elasticity becomes nonlinear for certain materials. For others, the fracture or creep intervenes. See low.

Let us note that the concept of elasticity is not specific to mechanics of the continuous mediums and in physics, generally, one speaks about elastic interaction when it there with the conservation of a size. For example, in a elastic shock , the objects exchange to them kinetic energy, but there is no loss by friction or residual deformation (in fact, the deformation is elastic). In the case of the interaction of an atom with a electromagnetic Radiation, one speaks about elastic scattering when the Rayonnement preserves same the Wavelength (Diffusion Rayleigh).

Case of gases

A Gaz consists of Molécule S which flies and are entrechoquent. They are also knocked with the walls of the container containing gas, which creates the Pression. The kinetic energy average of a molecule is proportional to the absolute Température (in Kelvin):

$E\_c\; =\; \backslash \; frac\; \{3\}\; \{2\}\; \backslash \; cdot\; K\; \backslash \; cdot\; T$

$k$ being the Boltzmann constant.

The pressure of gas on the walls thus depends on the number of shocks a second and of the force of each shock, this force depending on the kinetic energy. If one decreases the volume of the envelope by maintaining the temperature constant (isothermal compression), one thus increases the frequency of the shocks the pressure. Contrary, if the envelope is increased, one decreases the frequency of the shocks, and thus the pressure is decreased. This is found in the laws of reaction of gases, for example in the law of the Perfect gas S, the pressure is inversely proportional to the Volume:

$P\; \backslash \; propto\; \backslash \; frac\; \{1\}\; \{V\}$

(the constant of proportionality is worth $nRT$ where $N$ is the quantity of gas and $R$ is the constant of perfect gases). If one takes a cylinder of constant section $S$ and length $L$ variable by the action of a piston, one has

$F\; =\; P\; \backslash \; cdot\; S\; =\; \backslash \; frac\; \{has\}\; \{L\}$

maybe for small variations (Development limited of the function $\backslash \; frac\; \{1\}\; \{L\}$ around $l\_0$):

$\backslash \; Delta\; F\; \backslash \; simeq\; -\; \backslash \; frac\; \{has\}\; \{l\_0^2\}\; \backslash \; cdot\; (l-l\_0)$

who is a law linear (or rather closely connected) in $l$.

There is well an elastic behavior for isothermal gases subjected to weak variations of volume.

See the articles kinetic Theory of the gases and kinetic Pressure.

Elastic strain of the solids

Example of the springs

The simplest case of elastic strain is that of the Ressort S.

Three example of springs: arises with not-jointed whorls solicited according to its axis (fig. of left), arises with blade solicited in inflection (in the center), arises with blade solicited in torsion (on the right)

On the drawings, we did not represent the reaction of the support to which the spring is hung. But it should well be seen that the deformation results from the application of two opposite forces ; if there is only one force, pursuant to the basic principle of dynamics, the force accelerates the spring without causing deformation, one is reduced to the Mécanique of the point.

When the Lois of deformation are linear, the proportionality factor is called stiffness spring and is noted K :

• F = K ₁ · Δ L for a traction and compression;
• F = K ₂ · θ for an inflection;
• C = K ₃ · θ for a torsion.

It is noticed that the coefficients K ₁, K ₂ and K ₃ are not homogeneous (they do not have same the Dimension). The angle θ must be expressed in Radian S.

In the case of a spring with not-jointed whorls, the energy of elastic strain W is the work force

$W\; =\; \backslash \; int\_0^\; \{\backslash \; Delta\; L\}\; F\; \backslash \; cdot\; dl$

It is thus the surface of the triangle delimited by the line in the graph (Δ L , F ), that is to say

W = 1/2 K ₁ Δ L ² = 1/2 · F · Δ L

graphic Illustration of the elastic deformation energy in the case of a spring with not-jointed whorls

---- Note : on the first figure, we used a graph showing the deformation according to the force, for example ( F , Δ L ). On the second figure, we reversed center and represented the force according to the deformation (Δ L , F ). If the first representation seems more intuitive to us (one represents the force as the cause of the deformation), both are equivalent. It is in fact the second, (Δ L , F ), which is used, tensile tests being done with increasing imposed deformation (see the explanation in the article the article mechanical Essais). ----

Elastic limit

The elastic strain intervenes for the weak requests. If the requests are increased, the mode of deformation is changed:

• Rupture (damage) for materials known as “fragile”:
• Plastic deformation (irreversible and nonlinear) then rupture for materials known as “ductile S”;
• possibly Creep for ductile materials if the speed of deformation is slow and the high temperature.

The constraint delimiting the elastic range of the other fields is called Limit elastic ( yield strength in English).

Constraint and deformation

One uses two models of elastic strain: the traction and compression and the shearing . The inflection can be modelled according to the cases like a traction and compression or like a shearing, torsion is modelled like a shearing (see low).

Uniaxial traction and compression

Let us take the case of the traction or compression of a cylindrical or parallelepipedic part according to its axis. The traction and compression corresponds to forces being exerted perpendicular to the sections of these parts; it is known as uniaxial because the sides of the part are not constrained, all the forces are on the same axis.

Lengthening in the axis

By taking parts of various dimensions, one notices that for a force given:

• lengthening Δ L is proportional to the initial length L ₀ of the cylinder;

this is conceived well: if two end to end identical springs are put, the first spring completely transmits the force to the second spring, both thus lengthen same quantity; thus so with a spring there is a lengthening Δ L ₁, with two springs total lengthening is 2Δ L ₁;

• lengthening Δ L is inversely proportional to the section of the cylinder;

one conceives also easily whom if one puts two identical springs in parallel, each spring will exert half of the force of traction, final lengthening will be thus Δ₁/2; if the section of the part is doubled, it is as if one put two parts side-by-side. If one wants to characterize material by disregarding form of the part and his dimensions, one thus defines:

• the relative lengthening or deformation ( strain in English), noted ε

$\backslash \; varepsilon\; =\; \backslash \; frac\; \{\backslash \; Delta\; L\}\; \{l\_0\}\; =\; \backslash \; frac\; \{L\; -\; l\_0\}\; \{l\_0\}$

ε is without dimension, one sometimes expresses it in % (100×Δ L / L ₀)

• the forced ( stress in English), noted σ

$\backslash \; sigma\; =\; \backslash \; frac\; \{F\}\; \{S\}$

σ is homogeneous with a Pression; because of the concerned enormous values, one generally expresses it in Méga - Pascal (MPa).

The elastic law is written then:

σ = E ε

it is the Loi of Hooke ; E is the Modulus Young ( Young' S modulus in English), who is a characteristic of material. E is also homogeneous with a pressure, because of the very high values which it takes, it is generally expressed in Giga - Pascal (GPa).

It is seen easily that density of deformation energy W , i.e. the elastic energy divided by the volume of the part, is worth:

W = 1/2 · σ · ε = 1/2 · E ε ²

Widening

When one exerts a traction or a compression, one notes that the width of the part also varies, contrary to lengthening. The relative variation of dimension is proportional to relative lengthening ε, the proportionality factor is called the Poisson's ratio or report/ratio of Poisson ( Poisson' S ratio in English) in homage to the French mathematician Siméon Denis Poisson. It is noted ν and is without unit:

• for a cylinder:

$\backslash \; frac\; \{\backslash \; Delta\; R\}\; \{r\_0\}\; =\; -\; \backslash \; naked\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; Delta\; L\}\; \{l\_0\}\; =\; -\; \backslash \; naked\; \backslash \; cdot\; \backslash \; varepsilon$

• for a right-angled parallelepiped:

$\backslash \; frac\; \{\backslash \; Delta\; has\}\; \{a\_0\}\; =\; -\; \backslash \; naked\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; Delta\; L\}\; \{l\_0\}\; =\; -\; \backslash \; naked\; \backslash \; cdot\; \backslash \; varepsilon$

$\backslash \; frac\; \{\backslash \; Delta\; B\}\; \{b\_0\}\; =\; -\; \backslash \; naked\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; Delta\; L\}\; \{l\_0\}\; =\; -\; \backslash \; naked\; \backslash \; cdot\; \backslash \; varepsilon$

Let us consider the volume of the part. For a cylindrical part, there is

V = L × π R ²

for small variations, there is thus

Δ V / V ₀ = Δ L / L ₀ + 2·Δ R / R ₀

(development limited to the first order), that is to say

Δ V / V ₀ = (1 - 2ν) · ε

In the same way for a parallelepipedic part, there is

V = L × has × B

Δ V / V ₀ = Δ L / L ₀ + Δ has / has ₀ + Δ B / B ₀

thus of the same

Δ V / V ₀ = (1 - 2ν) · ε

it is thus seen that:

if ν > 0,5 volume decreases in traction and increases in compression (exceptional case);

if ν < 0,5 volume increases in traction and decreases in compression (the most general behavior).

For a steel, ν is worth approximately 0,3, one is thus in the second case

So now one maintains the width constant - for example a compression is carried out but the part is in a ultra-rigid sheath and cannot extend -, then, the deformation is not uniaxial, the sheath exerts a pressure (a constraint) on the sides of the part. It is then necessary to use another elastic coefficient, noted C ₁₁, different of E :

σ = C ₁₁ · ε

Shearing

If a right-angled parallelepiped is considered, shearing is a variation of the angle, which is not right any more. That corresponds to forces being exerted the face parallel to.

One defines in the same way the constraint as being the force divided by the surface on which she is exerted; this constraint is called scission (always expressed in MPa) and is noted τ.

The deformation is the variation with the right angle γ, called shearing , expressed in radian.

There is always a linear law:

τ = G · γ

where G is the modulus of rigidity or module of Coulomb , generally expressed in GPa. In the case of an isotropic medium, the modulus of rigidity is related on the Young modulus and the Poisson's ratio by the following relation: $G\; =\; \backslash \; frac\; \{E\}\; \{2\; \backslash ;\; (1\; +\; \backslash \; naked)\}$

---- Note : in the Tensor article of the deformations , the definite angle γ is worth half of the definite angle γ here. ----

Isostatic pressing

An isostatic pressing is the exercise of a isotropic Pression, i.e. which with the same value in all the directions. If one indicates by V the volume of the object, the relative variation of volume is proportional to the variation of the pressure P :

$\backslash \; Delta\; P\; =\; -\; K\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; Delta\; V\}\; \{V\_0\}$

where K is the module of compressibility or modulus of elasticity ( bulk modulus in English). It is noticed that K is the reverse of the isothermal coefficient of Compressibilité χ _T defined in thermodynamics by:

$\backslash \; frac\; \{1\}\; \{K\}\; =\; \backslash \; chi\_T\; =\; -\; \backslash \; frac\; \{1\}\; \{V\}\; \backslash \; cdot\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; P\}\; \backslash \; right)\; \_T$

K is also homogeneous with a pressure and is generally expressed in a giga-Pascal (GPa). One a:

In the case of an isotropic medium, the module of compressibility K , the Young modulus E and it modulus of rigidity G are bound by the following relation: $\backslash \; frac\; \{1\}\; \{E\}\; =\; \backslash \; frac\; \{1\}\; \{9\; \backslash ;\; K\}\; +\; \backslash \; frac\; \{1\}\; \{3\; \backslash ;\; G\}$

Case of the great deformations

The definition which one took of ε depends on the followed way. Let us consider a final deformation of ε₁ + ε₂. If one makes the deformation in a stage, the final length is

L = L ₀ (1 + ε₁ + ε₂)

So on the other hand one deforms initially ε₁, one has a first length

L = L ₀ (1 + ε₁)

who becomes the initial length for the following stage, therefore when a deformation ε₂ is added, one obtains

L = L ₀ (1 + ε₁) (1 + ε₂)

By developing this last formula, one sees that both are equivalent if

ε₁ · ε₂ < < 1

maybe, in a synthetic way, if

ε ² < < 1

it is the assumption of the small deformations .

For the great deformations, one can use another definition of ε:

$\backslash \; varepsilon\; =\; \backslash \; ln\; \backslash \; left\; (\backslash \; frac\; \{L\}\; \{l\_0\}\; \backslash \; right)$

it is seen that if L and L ₀ are close, the limited development of this formula gives again the definition of ε small deformations

Why the laws are linear?

In a general way, any law can locally (i.e. for small variations) be replaced by a Développement limited first order, or “linear approximation”, provided that the tangent of the law is not horizontal around the point considered. The elastic laws are thus linear approximations of the real behavior, more complex.

More precisely, the explanation of the linearity is in the form of the interatomic Potentiel W ( R ), where R is the distance between two atoms.

With a Temperature of 0 K, the distance between two atoms is R ₀. If one moves away a little this value, energy W increases; one can locally approach the law of W by a parabola (it is acted in fact of a Développement limited to the second order), one can thus write:

W ( R ) = W ₀ + K · ( R - R ₀) ².

The force being the derivative of the potential energy, it is seen that the atoms are subjected to a force of recall (which tends to make return to R ₀) which is worth:

F = 2 K · ( R - R ₀)

who is well a linear law.

Complex deformations

We saw up to now simple examples of deformation: uniaxial traction, shearing, isostatic pressing, on a right-angled parallelepiped. The real applications correspond to more complex parts and requests, requiring to describe the strain and the stresses by matrices, Tenseur S, to see the articles:

• Tensor of the deformations
• Tensor of the constraints

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