Lame parameter mu = shear modulus derivation (rogue factor of 2)

In summary, the conversation is discussing the relationship between the shear modulus and Lame parameter in the context of linear, symmetric, isotropic stress-strain formulas. The individual is trying to derive this relationship but encounters an issue with their substitution of a shear strain, leading to the conclusion that μ = G. The conversation also mentions the importance of including a ½ in the equation to account for a tensor.
  • #1
Twigg
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Hello,

I am trying and failing to derive that the shear modulus ##G## is equal to the Lame parameter ##\mu##. I start with the linear, symmetric, isotropic stress-strain formula: $$\sigma = \lambda \mathrm{tr}(\epsilon) \mathrm{I} + 2\mu \epsilon$$ I then substitute a simple (symmetric) shear strain: $$\epsilon = \epsilon_{xy} (\hat{x} \otimes \hat{y} + \hat{y} \otimes \hat{x} )$$ But then I end up with $$\sigma_{xy} = 2\mu \epsilon_{xy}$$ or equivalently $$G = 2\mu$$ What did I goof up?

Thanks!
 
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1. What is the shear modulus and why is it important in materials science?

The shear modulus, also known as the modulus of rigidity, is a measure of a material's resistance to deformation under shear stress. It is an important parameter in materials science as it helps determine a material's ability to withstand forces and stresses, as well as its overall mechanical properties.

2. How is the shear modulus calculated?

The shear modulus is calculated by dividing the shear stress by the shear strain. This can be represented by the equation G = τ/γ, where G is the shear modulus, τ is the shear stress, and γ is the shear strain. It is typically measured in units of pascals (Pa) or gigapascals (GPa).

3. What is the significance of the "rogue factor of 2" in the shear modulus derivation?

The "rogue factor of 2" refers to a discrepancy that can arise in the calculation of the shear modulus when using different conventions for representing shear stress and strain. This factor is important to consider in order to ensure accurate and consistent results in materials testing and analysis.

4. How does the shear modulus affect a material's behavior under stress?

The shear modulus plays a key role in determining a material's stiffness and strength under shear stress. A higher shear modulus indicates a stiffer material that is less likely to deform under shear stress, while a lower shear modulus indicates a more flexible material that is more likely to deform under shear stress.

5. What factors can affect the shear modulus of a material?

The shear modulus of a material can be affected by various factors, including temperature, strain rate, and microstructural properties such as crystal structure and defects. It can also vary depending on the direction of the applied force, as some materials exhibit anisotropic behavior in which the shear modulus differs in different directions.

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