Kirchhoff's modulus G – shear modulus, formula, unit and meaning

Kirchhoff's modulus G, also called shear module and/ or shear modulus, is one of the basic constants describing elastic material properties. Determines the relationship between shear stress and shear strain in the linear-elastic range.
The Kirchhoff modulus is denoted by the symbol G and determines the resistance of a material to elastic change of shape caused by shearing.
The larger the value of G, the smaller the shear strain that occurs at the same shear stress. The Kirchhoff modulus should not be equated with shear strength – these are two different material properties.

What does Kirchhoff's modulus G mean?

If a material is subjected to shear stress, it may change shape. In the linear-elastic range, this deformation disappears when the load is removed.
Kirchhoff's modulus describes the relationship between shear stress τ a shear deformation γ.
This relationship can be written as:
τ = G × γ
hence:
G = τ / γ
where:

• G – Kirchhoff modulus, i.e. shear modulus,
• τ – shear stress,
• γ – shear deformation.

This relationship is valid in the linear-elastic range of material behavior.

Kirchhoff's modulus unit

The shear strain γ is a dimensionless quantity, therefore the Kirchhoff modulus has the same units as the stress.
G can be administered in:

• Pa – pascals,
• MPa – megapascals,
• GPa – gigapascals.

For metals, the shear modulus is most often given in GPa.
1 GPa = 1000 MPa
1 MPa = 1 N/mm²
For example:
G = 80 GPa = 80,000 MPa = 80,000 N/mm²

What does high G modulus physically mean?

A high Kirchhoff modulus means that the material offers great resistance to elastic change of shape when subjected to shear stresses.
For the same shear stress, a material with a larger G will undergo less shear deformation than a material with a smaller G.
So we can say that G is a measure of the shear stiffness of a material.

What is shear strain γ?

Shear strain describes the change in shape of a material caused by shear stresses.
This can be illustrated using a rectangular element whose lower surface remains stationary, while the upper surface is moved parallel to the lower surface. Initially, the rectangular element assumes a shape close to a parallelogram.
The shear deformation is denoted by the symbol:
γ
and is expressed as a dimensionless quantity corresponding to the change in angle expressed in radians.
For small deformations the following can be assumed:
γ ≈ Δx / h
where:

• Δx – relative surface displacement,
• h – the distance between them.

Example of calculating Kirchhoff's modulus

Let us assume that the material is subjected to a shear stress:
τ = 160 MPa
and the corresponding shear deformation in the linear-elastic range is:
γ = 0,002
Then:
G = τ / γ
G = 160 / 0,002 = 80,000 MPa
that is:
G = 80 GPa

Kirchhoff's modulus and Young's modulus E

G and E describe two different types of elastic response of the material.

Young's modulus describes the relationship between normal stress and longitudinal strain, while Kirchhoff's modulus describes the relationship between shear stress and shear strain.

Relationship between E, G and Poisson's ratio ν

For a homogeneous, isotropic linear-elastic material Young's modulus E, Kirchhoff modulus G and Poisson's ratio ν are related to each other by the relationship:
G = E / (2(1 + ν))
The relationship can also be written as:
E = 2G(1 + ν)
love:
ν = E / (2G) – 1
where:

• E – Young's modulus,
• G – Kirchhoff modulus,
• ν – Poisson's ratio.

This means that for a material that meets the assumptions of linear isotropic elasticity theory, knowledge of two of these parameters allows the calculation of the third one.

Example of calculating G from E and ν

For steel, the following can be roughly assumed:
E = 210 GPa
and:
ν = 0,30
Then:
G = 210 / (2 × (1 + 0,30))
G = 210 / 2,6
G ≈ 80,8 GPa
This value corresponds to the typical order of magnitude of the Kirchhoff modulus for steel.

Typical values ​​of Kirchhoff's modulus for metals

The G value depends on the material type. The values ​​below are indicative and should not replace material data intended for specific calculations.

The exact value depends on, among other things, the material composition, temperature, structure and direction of measurement.

Kirchhoff's modulus and shear strength

Shear modulus G is not the same as shear strength.
Kirchhoff's modulus describes the elastic deformation of a material under the influence of shear stresses.
Shear strength, on the other hand, determines the load level associated with material failure under specific shear conditions.

A material with a high G does not necessarily have a proportionally high shear strength.

Kirchhoff's modulus and torsion

The G module is particularly important when analyzing twisting shafts, rods and other elements.
When twisting a material, shear stresses and shear deformations occur.
For a circular rod operating in the linear-elastic range, the torsion angle can be described by the relationship:
φ = TL / (GJ)
where:

• φ – twist angle,
• T – twisting moment,
• L – element length,
• G – Kirchhoff modulus,
• J – polar moment of inertia of the cross-section.

The relationship shows that for the same geometry, length and torsional moment, an element made of a material with a higher G will undergo less twisting.

G modulus and torsional stiffness of the element

Just as Young's modulus E is not the same as the stiffness of the entire element, G should not be identified with the torsional stiffness of a specific part.
For a prismatic rod with appropriate geometry, the torsional behavior is influenced by, among other things:

• Kirchhoff's modulus G,
• cross-section geometry,
• element length.

In the case of a rod with a circular cross-section, the size is important GJ, called the torsional stiffness of the cross-section.
This means that two shafts made of the same material may have completely different resistance to elastic torsion due to differences in their dimensions.

Kirchhoff modulus and Rm, Re and Rp0,2

The G module does not determine the strength of a material.
Rm describes the maximum engineering stress achieved during tensile testswhile Re i IDR 0,2 are related to the transition of the material into the range of permanent plastic deformations.
G describes, however, elastic response of the material to shear.
Therefore, materials with similar Kirchhoff moduli may have very different values ​​of Rm and Rp0,2.

Kirchhoff's modulus and hardness

G is not a measure of hardness.
Hardness describes the resistance of a material to a specific type of local deformation, most often caused by the action of an indenter.
Kirchhoff's modulus, on the other hand, determines the elastic response of a material to shear stresses.
Heat treatment can significantly increase the hardness of steel without a corresponding increase in its G modulus.

The effect of heat treatment on the Kirchhoff modulus

Heat treatment can have a very strong effect on:

• Rm,
• Re i IDR 0,2,
• hardness,
• ductility,
• impact strength.

The effect of typical heat treatment on the Kirchhoff modulus is usually much smaller than its effect on strength properties.
This is due to the fact that elastic moduli are largely related to the nature of the bonds between atoms.

The effect of temperature on the Kirchhoff modulus

Kirchhoff's modulus depends on temperature.
For many metals, as the temperature increases the value of G decreasesThe material then becomes less stiff in response to shear stresses.
When calculating components operating at high temperatures, it is therefore necessary to use material data appropriate to the actual operating temperature.
This is important, among others, in the case of shafts, turbines, engine components, power installations and high-temperature devices.

Kirchhoff's modulus in anisotropic materials

Simple relationship:
G = E / (2(1 + ν))
concerns isotropic linear-elastic material.
In anisotropic materials, elastic properties are directionally dependent. It may be necessary to use different shear moduli for different planes and directions of the material.
This is important for, among other things:

• single crystals,
• materials with a strong texture,
• composites,
• layered materials.

In such cases, it is not possible to determine G from a single value of E and ν without checking the relevant material data.

How is Kirchhoff's modulus determined?

The Kirchhoff modulus can be determined by measuring the relationship between the shear stress and the corresponding elastic strain.
Depending on the material and method, the following may be used:

• twisting tests,
• methods for measuring shear strain,
• dynamic methods,
• methods using the velocity of propagation of elastic waves.

For an isotropic material, G can also be calculated from known values ​​of E and ν.

Kirchhoff's modulus in FEM calculations

In finite element analysis, the elastic properties of a material can be defined by an appropriate set of material constants.
For an isotropic linear-elastic material, it is often sufficient to specify two independent constants, for example:

• Young's modulus E,
• Poisson's ratio ν.

Based on these, the software can determine the remaining dependent elastic constants, including the Kirchhoff modulus G.

Why is Kirchhoff's modulus important?

G is particularly important wherever a structural element is subjected to shearing or twisting.
It is used, among others, to calculate:

• shafts and axles loaded with torsional moment,
• torsion springs, made of, among others, carbon spring steel i alloy spring steels,
• torque transmitting elements,
• shear loaded connections,
• shape deformations of the structure,
• torsional vibrations,
• material models used in FEM analyses.

Does a larger Kirchhoff modulus mean a better material?

No. A higher G value means a higher stiffness of the material in shear deformation, but it does not automatically mean higher strength, hardness, ductility or resistance to cracking.
The choice of material depends on the requirements of the specific structure. In one application, high torsional stiffness may be key, while in another, low weight and high yield strength may be key. corrosion resistance or the ability to absorb energy.

Kirchhoff's modulus G – the most important information

• G stands for Kirchhoff modulus, i.e. shear modulus.
• G is also called shear modulus.
• Describes the stiffness of a material under elastic shear.
• In the linear-elastic range the relationship is valid τ = G × γ.
• Kirchhoff's modulus is most often given in GPa.
• The larger G, the smaller the shear strain at the same shear stress.
• G is not the same as shear strength.
• G is not a measure of Rm, Re, Rp0,2 or hardness.
• For an isotropic linear-elastic material the relationship holds G = E / (2(1 + ν)).
• For steel G it is approximately approx. 80 GPa.
• The G-modulus is of particular importance in shear and torsion analysis.
• The value of G usually decreases with increasing temperature.
• The torsional stiffness of a particular element depends on both G and its geometry.
• In anisotropic materials, the shear modulus may depend on direction and plane.

Mechanical Properties - List

• Tensile strength Rm
• Yield strength Re, ReH and ReL
• Conventional yield strength Rp0,2
• Extension A and A5
• Z Narrowing
• Young's modulus E
• Poisson's ratio ν
• Kirchhoff's modulus G
• KV and KU impact strength – Charpy test
• Metal Hardness – HBW, HRC, HRB, HV and Knoop
• Fatigue strength and material fatigue
• Resistance to cracking K_IC
• Compressive strength
• Flexural strength
• Shear strength
• Creep of materials and mechanical properties at high temperatures
• Static tensile test of metals
• Stress-strain curve