Elastic modulus

The modulus of elasticity (also: tensile modulus, coefficient of elasticity, modulus of elongation, modulus of elasticity or Young's modulus, named after the English physician and physicist Thomas Young) is a material parameter of the material technology, the relationship between stress and strain in the deformation of a solid body with linear describes elastic behavior.

The modulus of elasticity is abbreviated as E- module or as symbols with E and has the unit of a mechanical stress. The plural of " modulus of elasticity " is " the moduli of elasticity ".

The magnitude of the Young's modulus is greater, the more resistance to a material is its elastic deformation. A component made of a material having high elastic modulus ( e.g. steel ) is thus stiffer than a member of the same construction ( the same geometric dimensions ), which (for example rubber) is made of a material having a low modulus of elasticity.

The modulus of elasticity (or in other notation, the spring constant) is the proportionality constant in Hooke's law. For crystalline materials, however, the elastic modulus is directional and must be described by the elasticity tensor whose components are shown in simplified form by the elastic constants. The elastic constants are material constants, and can vary within a real solid. The elasticity tensor is described in the article elasticity theory in more detail.

  • 3.1 " reference modulus to other material constants? "
  • 3.2 "Power reduction through better material? "
  • 3.3 " E- module ≠ stiffness "
  • 3.4 " σ = E ⋅ ε "

Definition

The modulus of elasticity is defined as the slope of the graph in the stress-strain diagram for uniaxial loading within the linear elastic range. This linear range is precisely referred to as Hooke.

This refers to ( = force / area ) the mechanical stress ( normal stress, not shear stress ) and elongation. The elongation is the ratio of the change in length to the original length. The unit of modulus of elasticity is the voltage:

The modulus of elasticity is called a material constant, because with him and the Poisson's ratios of the elasticity law is set up. The modulus of elasticity is not constant with respect to all physical quantities. It depends on various environmental conditions such as temperature, humidity or the deformation rate.

Application

In an ideal linear elastic constitutive law (proportionality in the stress-strain diagram ), the spring constant c of a straight bar from its cross-sectional area A, its length and modulus of elasticity E. results

With the expressions for the voltage and for the elongation of the Hooke's law to obtain the uniaxial stress condition of the above equation

And from this the Young's modulus

Typical numerical values

Notes for unit conversion:

  • ( Newtons per square millimeter, a is a mega Pascals)
  • ( a kilo Newtons per square millimeter is a Gigapascal )

In two-dimensional components is expected to rivers instead of voltages. Therefore, one uses a thick -related modulus here, which corresponds to a stiffness. This quantity has the unit.

Relationships elastic constants

In addition to the elastic modulus, the shear modulus, also known as shear or G module is used which is measured in torsion and 0.33, depending on Poisson's ratio - is up to 0.5 times the modulus of elasticity. For stiff materials, the elastic modulus is usually measured in soft ( gels, polymer melts) the shear modulus, as the modulus of elasticity in such systems usually not possible to measure well because the specimen deforms under its own weight, the so-called sagging.

It is valid for a linear elastic, isotropic material, the following relationship between the shear modulus G, the bulk modulus K and the Poisson's ratio ν:

Common misconceptions

" Reference modulus to other material constants? "

Frequently, the modulus of elasticity with different material characteristics is associated. However, this is not easy:

  • The modulus of elasticity has no strict relation to the hardness of the material
  • The modulus of elasticity has no strict relation to the yield point of the material
  • The modulus of elasticity has no strict relation to the tensile strength of the material

A simple structural steel has ( almost) the same modulus of elasticity as a high-alloy high-strength stainless steel, ie both deform almost identical at any given load. However, the "better" material charged significantly higher ( and of course even more deformed) than the "simple".

But there is a general trend:

  • The modulus of elasticity of a metal increases with its melting temperature.

Tungsten has a higher modulus of elasticity than iron, a copper, an aluminum, than lead.

In addition:

  • The modulus of elasticity of body-centered cubic metals ( with comparable melting temperature ) higher than the face-centered cubic of.

The reason for the relationships is that both the elastic modulus and the melting temperature of the metals of the force - distance curve of the atoms dependent.

"Power reduction through better material? "

Stresses in the material depend only on the load ( applied forces ) and the geometry from ( force per area ), not on the material selected. In cases " static overdetermination " (eg, continuous beams, disabled thermal expansion movements of floating body in waves or tides ) the forces and stresses are dependent on the stiffness of the structural system. In such cases, materials can lower modulus cause component stresses are reduced.

"E - module ≠ stiffness "

The stiffness of a component depends on the material used and the processing, but also on the geometry of the component. When uniaxial strained tension rod the axial stiffness is the product of Young's modulus and cross-sectional area divided by the length, the bending beam, the bending stiffness of the modulus of elasticity, the area moment of inertia and the length of the bar depends on. With ropes, the stiffness is very much dependent on the type of braiding except the material.

For complex geometries, we see no simple expression for the formulate " stiffness ". Using the finite element method, we can simulate this by means of individual elements and solve this with an established global stiffness matrix.

" Σ = E ⋅ ε "

The relationship is only valid for the uniaxial train. In general, 2D or 3D stress state Hooke's law must be applied in its general form - here several voltages come in any stretching term, and several strains in each voltage term, eg. Here, the Poisson's ratio.

A determination of the elongation, for example by means of strain gauges or speckle interferometry is thus no determination of the stresses in the component.

250873
de