LaméParametern

LaméParametern, known in English as Lamé parameters, are two fundamental constants used in linear elasticity to characterize isotropic homogeneous materials. They are usually denoted by the Greek letters λ (lambda) and μ (mu), the latter also being called the shear modulus or second Lamé parameter. These constants appear in Hooke’s law for isotropic materials, relating stress τ to strain ε through the tensor equation σ = 2με + λ tr(ε)I, where tr(ε) is the trace of the strain tensor and I is the identity tensor.

The Lamé parameters provide a convenient way to express material stiffness and are directly linked to the more familiar Young’s modulus E and Poisson’s ratio ν. The relationships are E = μ(3λ + 2μ)/(λ + μ) and ν = λ/(2(λ + μ)). Conversely, λ and μ can be derived from E and ν by λ = Eν/[(1 + ν)(1 – 2ν)] and μ = E/[2(1 + ν)]. These formulas reveal that for most engineering materials, μ is directly related to shear behavior while λ governs volumetric response.

Lamé parameters are used extensively in finite element analysis, seismic wave propagation, and mechanical design where stress and strain distributions must be predicted accurately. They also appear in the wave equations describing compressional (P‑wave) and shear (S‑wave) velocities in seismology, where vp = sqrt((λ + 2μ)/ρ) and vs = sqrt(μ/ρ), with ρ the density. Because they encapsulate both stiffness and compressibility, LaméParametern provide a compact and versatile description of elastic behavior across disciplines.