Yes, Poisson's ratio can indeed be linked to the concept of mass conservation. Mass conservation can be used to evaluate it. Mathematically, if a material is stretched along the x-axis, causing a strain 𝜖_𝑥, the resulting strains in the perpendicular y and z directions are given by: 𝜖_𝑦=𝜖_𝑧=−𝜈𝜖_𝑥 ν is Poisson's ratio, which is typically a positive number for most materials. Mass conservation implies that during the deformation of a material, the mass of the material remains constant. For a given volume element of the material, if we assume the material is incompressible, then its density remains constant, which implies that any increase in one dimension must be exactly compensated by a decrease in other dimensions to maintain the same volume. Consider a small cubic element of the material with an initial volume 𝑉0=𝐿𝑥 𝐿𝑦 𝐿𝑧, where 𝐿𝑥, 𝐿𝑦, and 𝐿𝑧 are the initial lengths along the 𝑥, 𝑦, and 𝑧,axes, respectively. After deformation, the new dimensions become 𝐿𝑥′=𝐿𝑥(1+𝜖_𝑥), 𝐿𝑦′=𝐿𝑦(1+𝜖_𝑦), and 𝐿𝑧′=𝐿𝑧(1+𝜖_𝑧) The new volume 𝑉′ after deformation is: 𝑉′=𝐿𝑥′𝐿𝑦′𝐿𝑧′=𝐿𝑥(1+𝜖_𝑥)⋅𝐿𝑦(1+𝜖_𝑦)⋅𝐿𝑧(1+𝜖_𝑧) For small strains, the volume can be approximated as: 𝑉′≈𝑉0(1+𝜖_𝑥+𝜖_𝑦+𝜖_𝑧) For mass conservation in the case of an incompressible material, the volume must remain constant, so 𝑉′=𝑉0 This implies: 𝜖_𝑥+𝜖_𝑦+𝜖_𝑧=0 Substituting the relationship between longitudinal and transverse strains (using Poisson's ratio): 𝜖_𝑥−𝜈𝜖_𝑥−𝜈𝜖_𝑥=0 This implies that for incompressible materials, Poisson's ratio 𝜈=1/2
w(x,y) is the displacement of the generic point (x,y) of the plate in the direction z perpendicular to the plate. In the classical theory of thin plates, the derivatives of w are related to stresses, strains, bending moments, twisting moments, shear forces, etc.
Tricky equations 😅 but you did the job well
Wonder if the parameter nu in the constitutive relation can be linked to the conservation of mass
Yes, Poisson's ratio can indeed be linked to the concept of mass conservation. Mass conservation can be used to evaluate it.
Mathematically, if a material is stretched along the x-axis, causing a strain 𝜖_𝑥, the resulting strains in the perpendicular y and z directions are given by:
𝜖_𝑦=𝜖_𝑧=−𝜈𝜖_𝑥
ν is Poisson's ratio, which is typically a positive number for most materials.
Mass conservation implies that during the deformation of a material, the mass of the material remains constant. For a given volume element of the material, if we assume the material is incompressible, then its density remains constant, which implies that any increase in one dimension must be exactly compensated by a decrease in other dimensions to maintain the same volume.
Consider a small cubic element of the material with an initial volume 𝑉0=𝐿𝑥 𝐿𝑦 𝐿𝑧, where
𝐿𝑥, 𝐿𝑦, and 𝐿𝑧 are the initial lengths along the
𝑥, 𝑦, and 𝑧,axes, respectively. After deformation, the new dimensions become
𝐿𝑥′=𝐿𝑥(1+𝜖_𝑥),
𝐿𝑦′=𝐿𝑦(1+𝜖_𝑦), and
𝐿𝑧′=𝐿𝑧(1+𝜖_𝑧)
The new volume 𝑉′ after deformation is:
𝑉′=𝐿𝑥′𝐿𝑦′𝐿𝑧′=𝐿𝑥(1+𝜖_𝑥)⋅𝐿𝑦(1+𝜖_𝑦)⋅𝐿𝑧(1+𝜖_𝑧)
For small strains, the volume can be approximated as:
𝑉′≈𝑉0(1+𝜖_𝑥+𝜖_𝑦+𝜖_𝑧)
For mass conservation in the case of an incompressible material, the volume must remain constant, so
𝑉′=𝑉0
This implies:
𝜖_𝑥+𝜖_𝑦+𝜖_𝑧=0
Substituting the relationship between longitudinal and transverse strains (using Poisson's ratio):
𝜖_𝑥−𝜈𝜖_𝑥−𝜈𝜖_𝑥=0
This implies that for incompressible materials, Poisson's ratio 𝜈=1/2
@@math.physics thank you for the comprehensive explanation
Ah so the equation including W is for bending?
w(x,y) is the displacement of the generic point (x,y) of the plate in the direction z perpendicular to the plate. In the classical theory of thin plates, the derivatives of w are related to stresses, strains, bending moments, twisting moments, shear forces, etc.