Vicinity Transformation

Central to the computation of stresses and strains is the Jacobian matrix which relates the initial and deformed configuration:

d x i = ∂ x i ∂ X j d X j = D j x i d X j = F i j d X j
D j = ∂ ∂ X j

The transformation is fully described by the elements of the Jacobian matrix F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGgbaaaa@39A2@ :

F i j ≡ D j x i

So that Equation 1 can be written in matrix notation:

d x = F d X

The Jacobian, or determinant of the Jacobian matrix, measures the relation between the initial volume d Ω and the volume in the initial configuration containing the same points:

d Ω = | F | d Ω 0

Physically, the value of the Jacobian cannot take the zero value without cancelling the volume of a set of material points. So the Jacobian must not become negative whatever the final configuration. This property insures the existence and uniqueness of the inverse transformation:

d X = F − 1 d x

At a regular point whereby definition of the field u ( X ) is differentiable, the vicinity transformation is defined by:

F i j = D j x i = D j ( X i + u i ( X , t ) ) = δ i j + D j u i

or in matrix form:

F = 1 + A

So, the Jacobian matrix F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaaaa@36E7@ can be obtained from the matrix of gradients of displacements:

A ≡ D j u i