Some elastic materials are anisotropic, meaning that their elastic behavior, specifically the relationship between applied stresses and unit strains, is different for different directions. A common form of anisotropy is that presented by orthotropic elastic materials in which the elastic behavior is characterized by a series of elastic constants associated with three mutually perpendicular directions. The best-known example of an orthotropic material is wood, which has different modulus of elasticity along the grain, tangentially to the growth rings and perpendicular to the growth rings.

The elastic behavior of an orthotropic material is characterized by nine independent constants: 3 longitudinal elasticity modules (N, E, E), 3 rigidity modules (G, G, G) and 3 Poisson's ratios (ν, ν, ν). In fact, for an orthotropic material, the relationship between the components of the stress tensor and the components of the strain tensor is given by:.

Where:.

As can be seen, the components that govern elongation and those that govern distortion are decoupled, which means that in general it is possible to produce elongations around a point without causing distortions and vice versa. The inverse equations that give strains as a function of stresses take a somewhat more complicated form:

Where:.

In fact, the previous matrix, which represents the stiffness tensor, is symmetric.

A particular case of orthotropic material is that of transversely isotropic materials in which there is a preferred or longitudinal direction and all sections perpendicular to it are mechanically equivalent. Thus, in any cross section in the different direction there will be isotropy and the number of independent elastic constants necessary to characterize this material will be 5 and not 9, as in the case of a general orthotropic material. The five independent constants will be, in fact: 2 longitudinal elastic moduli (N, E), 1 rigidity modulus (G) and 2 Poisson's coefficients (ν (ν, ν). These constants are related to the other general constants of an orthotropic material through these relationships:

Specifying how stress and strain are to be measured, including directions, allows many types of elastic moduli to be defined. The three main ones are:

Two other elastic moduli are the first Lamé parameter, λ, and the wave modulus P"), M, as used in the modulus comparison table below the references. Homogeneous and isotropic (similar in all directions) (solid) materials have their elastic (linear) properties completely described by two elastic moduli, and any pair can be chosen. Given a pair of elastic moduli, all other elastic moduli Elastics can be calculated according to the formulas in the following table at the bottom of the page.

Inviscous fluids are special in the sense that they cannot withstand shear stresses, which means that the shear modulus is always zero. This also implies that the Young's modulus for this group is always zero.

In some texts, the modulus of elasticity is called elastic constant, while the inverse quantity is called elastic modulus.

Different types of elastic moduli corresponding to different types of deformation (examples):

Some elastic materials are anisotropic, meaning that their elastic behavior, specifically the relationship between applied stresses and unit strains, is different for different directions. A common form of anisotropy is that presented by orthotropic elastic materials in which the elastic behavior is characterized by a series of elastic constants associated with three mutually perpendicular directions. The best-known example of an orthotropic material is wood, which has different modulus of elasticity along the grain, tangentially to the growth rings and perpendicular to the growth rings.

The elastic behavior of an orthotropic material is characterized by nine independent constants: 3 longitudinal elasticity modules (N, E, E), 3 rigidity modules (G, G, G) and 3 Poisson's ratios (ν, ν, ν). In fact, for an orthotropic material, the relationship between the components of the stress tensor and the components of the strain tensor is given by:.

Where:.

As can be seen, the components that govern elongation and those that govern distortion are decoupled, which means that in general it is possible to produce elongations around a point without causing distortions and vice versa. The inverse equations that give strains as a function of stresses take a somewhat more complicated form:

Where:.

In fact, the previous matrix, which represents the stiffness tensor, is symmetric.

A particular case of orthotropic material is that of transversely isotropic materials in which there is a preferred or longitudinal direction and all sections perpendicular to it are mechanically equivalent. Thus, in any cross section in the different direction there will be isotropy and the number of independent elastic constants necessary to characterize this material will be 5 and not 9, as in the case of a general orthotropic material. The five independent constants will be, in fact: 2 longitudinal elastic moduli (N, E), 1 rigidity modulus (G) and 2 Poisson's coefficients (ν (ν, ν). These constants are related to the other general constants of an orthotropic material through these relationships:

Specifying how stress and strain are to be measured, including directions, allows many types of elastic moduli to be defined. The three main ones are:

Two other elastic moduli are the first Lamé parameter, λ, and the wave modulus P"), M, as used in the modulus comparison table below the references. Homogeneous and isotropic (similar in all directions) (solid) materials have their elastic (linear) properties completely described by two elastic moduli, and any pair can be chosen. Given a pair of elastic moduli, all other elastic moduli Elastics can be calculated according to the formulas in the following table at the bottom of the page.

Inviscous fluids are special in the sense that they cannot withstand shear stresses, which means that the shear modulus is always zero. This also implies that the Young's modulus for this group is always zero.

In some texts, the modulus of elasticity is called elastic constant, while the inverse quantity is called elastic modulus.

Different types of elastic moduli corresponding to different types of deformation (examples):