Читать книгу The Rheology Handbook - Thomas Mezger - Страница 187
4.2.2Shear modulus
ОглавлениеWhen measuring ideal-elastic solids at a constant temperature, the ratio of the shear stress τ and corresponding deformation γ is a material constant if testing is performed within the reversible-elastic deformation range, the so-called linear-elastic range. This material specific value is referred to as the shear modulus G and reveals information about the rigidity of a material. Materials showing comparably stronger intermolecular or crystalline cohesive forces exhibit higher internal rigidity, and therefore, also a higher G-value. Sometimes, the shear modulus is also called modulus of elasticity in shear or rigidity modulus. Definition of the shear modulus:
Equation 4.4
G = τ / γ
Note: The plural of shear modulus is shear moduli.
The unit of the shear modulus is [Pa], (pascal), and 1 Pa = 1 N/m2
For rigid solids, the following units are also used:
1 kPa (kilopascal) = 1000 Pa = 103 Pa
1 MPa (megapascal) = 1000 kPa = 1,000,000 Pa = 106 Pa (= 1 N/mm2)
1 GPa (gigapascal) = 1000 MPa = 1,000,000,000 Pa = 109 Pa
A previously used unit was [dyne/cm2], then: 1 dyne/cm2 = 0.1 Pa. However, this is not an SI-unit. In Table 4.1 are listed values of shear moduli of various materials.
| Table 4.1: Values of G- and E-moduli , and of Poisson’s ratio μ, at the temperature T = +20 °C; own data and from [4.1] [4.2] [4.3] [4.4] [4.5] | |||
| Material | G modulus | μ | E modulus |
| very soft gel structures (examples: spray coatings, salad dressings) | 5 to 10 Pa | ||
| soft gel structures (example: brush coatings) | 10 to 50 Pa | ||
| viscoelastic gels (typical dispersions, lotions, creams, ointments, pastes of food, cosmetics, pharmaceuticals, medicals) | 50 to 5000 Pa (often100 to 500 Pa) | ||
| puddings (containing 5/7.5/10/15 % starch) | 0.1/0.5/1/5 kPa | ||
| adhesives before hardening– soft paste structure– strong structure, e. g. filled sealants | 0.1 to 10 kPa50 to 500 kPa | ||
| gummy bears, jelly babies | 10 to 500 kPa | ||
| spread cheese/soft cheese/semi-hard/hard/extra hard cheese | 1 kPas/10 kPas/0.1 MPa/0.5 MPa/1 MPa | ||
| butter (example): at T = +10/+23 °C | 2 MPa/50 kPa | ||
| soft natural gumsunfilled gumsfilled gumseraser gum (India rubber)technical elastomershard rubbers (e. g. car tires) | 0.03 to 0.3 MPa0.3 to 5 MPa3 to 20 MPa1 MPa0.3 to 30 MPa10 to 100 MPa | 0.490.40 to 0.450.35 to 0.400.40 to 0.450.35 to 0.40 | 0.1 to 1 MPa1 to 10 MPa10 to 50 MPa1 to 100 MPa30 to 300 MPa |
| PU coating, highly viscous/rigidone-pack PU adhesivetwo-pack PU reactive adhesive | 30 kPa/1.0 GPa1 to 10 MPa200 to 600 MPa | 100 kPa/2.5 GPa | |
| bitumen (example):at T = 0/-10/-30/-50 °C | 10/50/200/500 MPa | ||
| thermoplastic polymers,unfilled, uncrosslinked (usually) | 0.1 to 2 GPa | 0.30 to 0.35 | 1 to 4 GPa |
| PE-LDPE-HD | 70 to 200 MPa300 to 800 MPa | 0.480.38 | 200 to 600 MPa0.7 to 2 GPa |
| PPPP, filled | 0.2 to 0.5 GPa1 to 3 GPa | 0.350.25 | 0.5 to 1.3 GPa1.8 to 6.5 GPa |
| PVC-P (plasticized, flexible,Tg > +20 °C)PVC-U (unplasticized, rigid)PVC, filled | 0.5 to 5 MPa0.3 to 1 GPa1 to 3 GPa | 0.400.350.25 | 1.5 to 15 MPa1 to 3 GPa3 to 8 GPa |
| PEEK-CF(with 40/65 % carbon fibers) | up to80/155 GPa | ||
| pure resinsfilled and fiber-reinforced resins(dependent on the fiber orientation) | 1 to 2 GPa2 to 12 (24) GPa | 0.400.25 to 0.35 | 3 to 5 GPa5 to 30 (60) GPa |
| wood (axial)wood (radial) | 4 to 18 GPa0.3 to 0.6 GPa | ||
| Material | G modulus | μ | E modulus |
| ice (at T = -4 °C) | 3.7 GPa | 0.33 | 9.9 GPa |
| bone | 18 to 21 GPa | ||
| ceramics, porcelain | 15 to 35 GPa, 25 GPa | 0.20 | 40 to 80 GPa |
| marble stone | 28 GPa | 0.30 | 70 GPa |
| (window) glass | 30 GPa | 0.15 | 70 GPa |
| aluminum (Al 99.9 %) | 28 GPa | 0.34 | 72 GPa |
| gold (Au) | 28 GPa | 0.42 | 81 GPa |
| brass (Cu-Zn) | 36 GPa | 0.37 | 100 GPa |
| cast iron | 40 GPa | 0.25 | 100 GPa |
| bronze (Cu-Sn) | 43 GPa | 0.35 | 116 GPa |
| steel | 80 GPa | 0.28 | 210 GPa |
| diamonds | 1200 GPa |
For “Mr. and Ms. Cleverly“
Information on parameters obtained from tensile tests
Conversion of G- and E-values
Equation 4.5
E = 2 · G (1 + μ)
with the tensile modulus E [Pa], often called modulus of elasticity or Young’s modulus , and Poisson’s ratio μ with the unit [1], (my, pronounced: mu or mew). For a brief information on Thomas Young (1773 to 1829 [4.6]) and Siméon D. Poisson (1781 to 1840, [4.7]), see Chapter 14.2. Poisson’s ratio µ is the value of the ratio of the lateral (transversal) deformation to the corresponding axial deformation, resulting from uniformly distributed axial stress below the proportional limit of the material (according to ASTM D4092; by the way, in this standard instead of the sign μ the sign ν is used). The following holds (e. g. according to DIN 13316):
Equation 4.6
0 ≤ μ ≤ 0.5
The higher the value of Poisson’s ratio, the more ductile is a material; or: The lower the μ-value, the more brittle is its behavior when breaking. Cork, showing μ = 0, is a material with one of the two extreme values. Therefore here
Equation 4.7
E = 2 · G
On the other hand, for viscoelastic liquids occurs the other extreme value of µ = 0.50. In this case, there is no volume change when stressing or straining these kinds of materials. Close to that value are soft and very flexible rubbers showing μ = 0.49. The same value occurs when testing polymers exhibiting behavior of viscoelastic liquids at temperatures above the glass-transition temperature (T > Tg), because then, they are in a soft-elastic (or rubber-elastic) state. This applies also to other incompressible and isotropic materials. Hence, for these kinds of materials counts:
Equation 4.8
E = 3 · G
Note: Conversion of G- and E-modulus values
In general, calculation of G-values from E-values, and vice versa, is not recommended since there is evidence that suggests the Poisson’s ratio varies from material to material in the same material class and may vary from temperature to temperature for the same material (according to ASTM D1043). Therefore, these conversions must be regarded as rough estimates only.
Stress/strain diagrams (SSD) of tensile tests
Performing tensile tests, E-modulus values are determined in the linear-elastic range, i. e. in a range of very low strain values. In this range of a σ/ε diagram, the curve function shows a constant slope. The following applies here:
Equation 4.9
E = σ/ε
with the tensile stress σ [Pa], (pronounced: sigma), and the tensile strain or elongation ε in [%], (pronounced: epsilon). Further information on tensile tests can be found in Chapters 10.8.4.1 and 11.2.14; ISO 6721, ISO 6892; DIN 50125 and [4.8] [4.9] [4.10] [4.11] [4.12].
End of the Cleverly section
