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The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law: . Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
Elastic properties describe the reversible deformation (elastic response) of a material to an applied stress.They are a subset of the material properties that provide a quantitative description of the characteristics of a material, like its strength.
E i is the Young's modulus along axis i; G ij is the shear modulus in direction j on the plane whose normal is in direction i; ν ij is the Poisson ratio that corresponds to a contraction in direction j when an extension is applied in direction i. The Poisson ratio of an orthotropic material is different in each direction (x, y and z). However ...
Under such a condition, the above equation can obtain the direct-related stiffness for the degree of unconstrained freedom. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses. The elasticity tensor is a generalization that describes all possible stretch and shear parameters.
The bulk modulus (which is usually positive) can be formally defined by the equation K = − V d P d V , {\displaystyle K=-V{\frac {dP}{dV}},} where P {\displaystyle P} is pressure, V {\displaystyle V} is the initial volume of the substance, and d P / d V {\displaystyle dP/dV} denotes the derivative of pressure with respect to volume.
The governing formula for this mechanism is: Δ σ y = G b ρ {\displaystyle \Delta \sigma _{y}=Gb{\sqrt {\rho }}} where σ y {\displaystyle \sigma _{y}} is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector , and ρ {\displaystyle \rho } is the dislocation density.
In 1820, the French engineer A. Duleau derived analytically that the torsion constant of a beam is identical to the second moment of area normal to the section J zz, which has an exact analytic equation, by assuming that a plane section before twisting remains planar after twisting, and a diameter remains a straight line. Unfortunately, that ...
where is the shear modulus, which can be determined by experiments. From experiments it is known that for rubbery materials under moderate straining up to 30–70%, the Neo-Hookean model usually fits the material behaviour with sufficient accuracy.