Skip to main content

Isotropic vs anisotropic; Lame's constants

Short answer

Isotropic: two constants (E, ν), properties same in all directions. Anisotropic: up to 21 constants, properties direction-dependent. Lame's constants: λ=νE/[(1+ν)(1-2ν)], μ=E/[2(1+ν)]=G, giving σ_ij=λ e δ_ij+2μ ε_ij.

Differentiate between isotropic and anisotropic material body. Derive the expressions for Lame's constants for a linearly elastic isotropic material body.

Isotropic and anisotropic bodies

Isotropic bodyAnisotropic body
Elastic properties are the same in all directions at a pointProperties change with direction
Two independent constants, EE and ν\nuUp to 21 independent constants (general); 9 for orthotropic
Example: steel, concrete (usually), homogeneous soilTimber, rolled plates, layered rock and soil, fibre composites
Normal stress produces only normal strainNormal stress can also produce shear strain

Lame's constants

For a linearly elastic isotropic body, Hooke's law is εx=1E[(1+ν)σx−νΘ]\varepsilon_x=\dfrac{1}{E}[(1+\nu)\sigma_x-\nu\Theta] with Θ=σx+σy+σz\Theta=\sigma_x+\sigma_y+\sigma_z. Adding the normal equations gives e=εx+εy+εz=1−2νEΘe=\varepsilon_x+\varepsilon_y+\varepsilon_z=\dfrac{1-2\nu}{E}\Theta. Therefore

σx=E1+νεx+νE(1+ν)(1−2ν) e=2μ εx+λ e\sigma_x=\frac{E}{1+\nu}\varepsilon_x+\frac{\nu E}{(1+\nu)(1-2\nu)}\,e=2\mu\,\varepsilon_x+\lambda\,e

so that

λ=νE(1+ν)(1−2ν),μ=E2(1+ν)=G\lambda=\frac{\nu E}{(1+\nu)(1-2\nu)},\qquad \mu=\frac{E}{2(1+\nu)}=G

and τxy=μγxy\tau_{xy}=\mu\gamma_{xy}. These are Lame's constants; μ\mu is the shear modulus.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Computational Techniques in Civil Engineering chapter 3 questions →