Vicinity Transformation

Central to the computation of stresses and strains is the Jacobian matrix which relates the initial and deformed configuration:(1)
d x i = x i X j d X j = D j x i d X j = F i j d X j
(2)
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@ :(3)
F i j D j x i
So that 式 1 can be written in matrix notation:(4)
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:(5)
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:(6)
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:(7)
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:(8)
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:(9)
A D j u i