Stress and Strain in Plastics Machinery's Handbook, 31st Edition
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Fig. 8. Setup for Testing Shear Strength of Plastics in ASTM Test D732. Specimen may be Molded or Cut from Sheet
Shear modulus G is a constant otherwise called the modulus of rigidity. For materials that behave according to Hooke’s law, it is directly comparable to the modulus of elasticity used in direct stress calculations. The constant is derived from (9) For isotropic materials in their Hooke’s-law range at low deformations, it may be assumed that the modulus in compression is equal to E . Then a simple theoretical relation ship links the tensile and shear moduli with Poisson’s ratio: (10) ASTM D1043 provides a method for determining the shear modulus G of plastic speci mens at various temperatures by a torsion test. Material suppliers can furnish values of G for their compounds. Relating Material Constants: Although only two material constants are required to characterize a material that is linearly elastic, homogeneous, and isotropic, three such constants have been introduced here. These three constants are tensile modulus E , Poisson’s ratio ν , and shear modulus G , and they are related by Equation (10) or rewritten in the following form, based on elasticity principles: (11) This relationship holds for most metals and is generally applicable to injection-moldable thermoplastics. It must be remembered, however, that most plastics, particularly fiber- reinforced and liquid crystalline materials, are inherently either nonlinear, anisotropic, or both. Bulk Modulus B: When a sample of material is subjected to high pressure from all sides, its volume diminishes with a corresponding increase in density. Bulk modulus B is defined as the increase in pressure divided by the fractional decrease in specific volume: (12) Since V 1 ρ = , it follows that (13) The dimensions of B are pressure per unit of fractional volume contraction. In English units B is in Mpsi/(in 3 /in 3 ), or just Mpsi; the SI units for B are GPa/(cm 3 /cm 3 ) or just GPa. The last equation indicates that what happens to volume (and specific volume) also happens in the same degree to density, but with the opposite sign, i.e., as the volume increases, the density decreases and vice versa. G Shear Strain Shear Stress = γ τ = G E 2 1 ν = + ^ h G E 2 1 v = + ^ h ln d V B V ∆ P dP " ∆ ^ h ⁄ V = B 1 P / ρ ρ ∆ " ∆ ^ h = ln dP d ρ
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