Hyperelasticity

A material is said to be hyperelastic (Green elastic) if the work needed to deform it can be expressed as a change in a potential. The potential is then a function of the deformation. Another way of expressing this is that there shall be a function f(E) such that the deformation work can be expressed as:

ω = φ . = ∂ φ ∂ E ⋅ E . = ∂ φ ∂ E i j E .

φ is thus an expression for the work needed to deform the volume. Since we are considering an elastic material, the energy stored in the volume when deforming it, can be reversed to cause work to be done on the surroundings. Thus φ represents a potential energy often called the elastic energy or strain energy.

Since the internal and external work done on the volume is equal, w can also be expressed as:

ω = T ⋅ G = T i j v i , j

This equality is used to create a relationship between the stress tensor and strain energy φ. We start with the relationship between deformation and deformation rate:

E i , j       = 1 2 ( u i , j     +     u j , i   )

u . i , j       =       ∂ ∂ t     ∂ u i ∂ x j       =       ∂ ∂ x j ∂ u i ∂ t       =       v i , j

⇒           E .     =   1 2 (   v i , j     +   v j , i   )

Since the strain tensor is symmetric:

∂ φ ∂ E i j         =   ∂ φ ∂ E j i

ω     =     ∂ φ ∂ E i j E . i j     = 1 2   ∂ φ ∂ E i j v i , j       +     1 2   ∂ φ ∂ E j i v j , i       = ∂ φ ∂ E i j v i , j

The above equations can be combined:

ω     =       T i j       v i , j     =     ∂ φ ∂ E i j     v i , j         ⇒       T i j       =     ∂ φ ∂ E i j    

The above result shows that hyperelasticity implies elasticity, but not the other way round.

The number of elasticities can be reduced for a hyperelastic material by combining the strain-tension relationship for an elastic material and the equation above:

T K     =     L K L ⋅ E L     = ∂ φ ∂ E K

Since:

∂ 2 φ ∂ E K ∂ E L     =     ∂ 2 φ ∂ E L ∂ E K

It then follows that:

L K L     =     L L K

The matrix L is thus symmetrical.The number of independent elasticities is thus reduced from 36 to 21 for the hyperelastic material. The material can be described with 3 non-orthogonal basis vectors with unequal length (a triclinic system).