Energy Elasticity

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Energy elasticity

Structural causes of energy elasticity

The structural cause of energy-elastic behaviour is the change in the average atomic distances and bond angles when mechanical stresses are applied. The mechanical work required to do this is stored in the form of potential energy (increase in internal energy) and is recovered completely and immediately when the stress is removed (1st law of thermodynamics) [1]. Due to its structural causes, energy-elastic behaviour is limited to the range of small deformations. Here, a linear relationship between stress and strain is observed, which is described by HOOKE's law.

HOOKE's law for energy elastic behaviour

For the simple case of uniaxial tensile stress, Eq. (1) applies:

$\sigma \,=\,E\cdot \varepsilon$.

(1)

The proportionality constant between stress and strain is referred to as the modulus of elasticity E. It is related to the binding forces in the material. Alternatively, the compliance C can also be determined (Eq. 2):

$\varepsilon \,=\,C\cdot \sigma$.

(2)

In addition to the change in length, a test specimen under tensile stress also undergoes a reduction in cross-section if it is in a plane stress state due to its geometry. The magnitude of this cross-sectional change is described by the transverse contraction coefficient (Poisson's ratio) ν. It expresses the ratio of the strain in the transverse direction (ε_y, ε_z) and longitudinal direction (ε_x). For uniaxial stress, Eq. (3) applies:

$v\,=\,-{\frac {\varepsilon _{y}}{\varepsilon _{x}}}\,=\,-{\frac {\varepsilon _{z}}{\varepsilon _{x}}}$.

(3)

In the case of shear stress, Hooke's law applies as follows Eq. (4), where G denotes the shear modulus, τ the corresponding shear stress and γ the shear.

$\tau \,=\,G\cdot \gamma$

(4)

Relationships between elastic constants

With the Poisson's ratio ν, which indicates the ratio between transverse strain &epsilon_q and longitudinal strain &epsilon_l as an absolute value according to Eq. (5)

$\nu \,=\,{\frac {\varepsilon _{q}}{\varepsilon _{l}}}$

(5)

the relationship between the modulus of elasticity and the shear modulus for small elastic deformations is obtained as

$E\,=\,2\cdot \left(1+\nu \right)\cdot G$

(6)

In the case of incompressibility, as with rubber, the upper limit of ν = 0.5, whereby a Poisson's ratio of around 0.3 is recorded for most plastics due to volume effects occurring under tensile stress [2]. Assuming multiaxial compression on all sides (hydrostatic stress), the compression modulus can be calculated as a further elastic constant according to Eq. (7):

$K\,=\,{\frac {E}{3\left(1-2\mu \right)}}$

(7)

The equations given here apply only to ideal elastic behaviour with deformation that is very small in relation to the geometric dimensions of the test specimens used.

See also

• HOOKE´s law
• Linear-viscoelastic behaviour
• Elastic modulus
• Deformation

References

[1]	Lüpke, Th.: Material Behavior and Constitutive Equations. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich, (2022) 3rd Edition, pp. 75–77 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	Wehrstedt, A.: Neues auf dem Gebiet der Werkstoffprüfung. In: Frenz, H., Wehrstedt, A. (Eds.): Kennwertermittlung für die Praxis. Wiley VCH (2003) pp. 1–12, (ISBN 3-527-30674-9; see AMK-Library under M 10)