1.2 The Spatial State

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Because of the actions, the body B is mapped from its reference state into some other

state, a spatial state. Let the spatial state be described by rectangular coordinates {zk} and

curvilinear coordinates {xk}. These coordinates are called spatial coordinates. The same

definitions of the reference state apply to the spatial state, for instance,

ds2 = dxk dxlgkl (1.23)

where gkl is the metric tensor of the spatial state.Within the theory of continuum mechanics,

one tries to predict the spatial state from the reference state and the actions on it. Since

gk

= ∂x

∂xk

(1.24)

one can write

dx = dxkgk

= ∂xk

∂XK

dXKgk

= xk

,K dXKgk. (1.25)

This reveals that the spatial state can be predicted from the material state if xk

,K is known.

Defining the dyadic product of two vectors a and b, written as a ⊗ b such that

(a ⊗ b) · c = a(b · c) (1.26)

and

c · (a ⊗ b) = (c · a)b (1.27)

for an arbitrary vector c, and similar for two one-forms or a vector and a one-form, one

finds that

dx = xk

,Kgk(GK · dX)

= xk

,K (gk

⊗ GK) · dX. (1.28)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 5

Defining the deformation gradient F as

F = xk

,K (gk

⊗ GK) (1.29)

Equation (1.28) is transformed into

dx = F · dX. (1.30)

This shows that the deformation gradient is the Jacobian matrix of the motion from the

material into the spatial state.

The dyadic product of two vectors, of two one-forms and of a vector and a one-form is

called a contravariant tensor of rank two, a covariant tensor of rank two and a mixed-variant

tensor of rank two respectively. If

a = aKLGK ⊗ GL (1.31)

then one can also write (Equation (1.21))

a = aKLGKMGLNGM ⊗ GN = aMNGM ⊗ GN (1.32)

where aMN = aKLGKMGLN is obtained by raising the indices. Equation (1.31) is the

covariant expansion of a, Equation (1.32) is the contravariant expansion. To emphasize

this, the notation a_ will be used for the covariant expansion and a_ for the contravariant

one (this agrees with recent literature, see (Marsden and Hughes 1983), (Holzapfel 2000)).

Accordingly,

a_ = aKLGK ⊗ GL (1.33)

a_ = aKLGK ⊗ GL. (1.34)

F is called a mixed-variant two-point tensor since it is the dyadic product of basis vectors

belonging to different states (the material and the spatial state).

Notice that the dot on the right-hand side and on the left-hand side of Equation (1.26)

have a different meaning: the dot on the right-hand side denotes the contraction of two

vectors already encountered in Equation (1.8). The dot on the left-hand side symbolizes the

contraction of a tensor of rank two and a vector. Whereas the contraction of two vectors

is commutative, the contraction of a tensor of rank two and a vector is not

(a ⊗ b) · c = a(b · c) _= (c · a)b = c · (a ⊗ b). (1.35)

However,

(a ⊗ b) · c = a(b · c) = (c · b)a = c · (b ⊗ a) = c · (a ⊗ b)T (1.36)

where

(a ⊗ b)T := b ⊗ a (1.37)

is the transpose of a ⊗ b.

6 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

The length ds of a vector dx in the spatial state satisfies

ds2 = dx · dx

= xk

,K dXKgk

· xl

,L dXLgl

= xk

,Kxl

,L dXK dXLgk

· gl

= xk

,Kxl

,Lgkl dXK dXL. (1.38)

Defining the right Cauchy–Green tensor by

C := CKLGK ⊗ GL (1.39)

where

CKL = xk

,Kxl

,Lgkl (1.40)

one obtains

ds2 = CKL dXK dXL. (1.41)

Comparing Equation (1.23) and Equation (1.41), one notices that for the calculations of

ds2, the tensor C is the equivalent of g in the reference frame. One also says that C is the

pullback of g and, equivalently, g is the push-forward of C. Equation (1.41) also shows

that the Cauchy–Green tensor is positive definite. Furthermore, it satisfies

C = FT · F (1.42)

where FT is the transpose of F defined by

FT := xk

,K (GK ⊗ gk). (1.43)

Indeed, since (a ⊗ b) · (c ⊗ d) = (a ⊗ d)b · c, one finds

FT · F = xk

,Kxl

,L(GK ⊗ gk) · (gl

⊗ GL)

= xk

,Kxl

,LgklGK ⊗ GL. (1.44)

The stretch in a direction N = (dXK/dS)GK is defined by

λ(N) = ds

dS

=

_

CKL

dXK

dS

dXL

dS

= _CKLNKNL (1.45)

where NK = dXK/dS. Thus, λ(N) is the change of length of an infinitesimal vector in

direction N in the reference state.

If the mapping x(X) is one to one, it can be inverted to yield X(x). Since matter cannot

disappear, the Jacobian determinant

J := det(xk

,K ) = detF (1.46)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 7

cannot be zero and the mapping is one to one. Assuming the transformation to be continuous,

this means that J must be either everywhere positive or everywhere negative. Since

J = 1 for the identical transformation, it is everywhere positive.

dS2 can also be written as

dS2 = dXK dXLGKL = XK

,kXL

,l dxk dxlGKL

= (b

−1)kl dxk dxl (1.47)

where

(b

−1)kl := XK

,kXL

,lGKL. (1.48)

The tensor b (the inverse of b

−1) is called the left Cauchy–Green tensor and satisfies

bkl = xk

,Kxl

,LGKL (1.49)

or, equivalently,

b = F · FT. (1.50)

Consequently,

b

−1 = F

−T · F

−1 (1.51)

where

F

−1 = XK

,kGK ⊗ gk (1.52)

is the inverse of the deformation gradient and

F

−T = XK

,kgk ⊗ GK. (1.53)

Equation (1.47) shows that, with respect to dS2, b

−1 plays in the spatial state the role that

is assumed by G in the reference state, that is,

GKL dXK dXL = (b

−1)kl dxk dxl . (1.54)

Therefore, b

−1 is called the push-forward of G and equivalently G is called the pullback

of b

−1. Equation (1.41) and Equation (1.47) can also be written as

ds2 = dX · C · dX (1.55)

and

dS2 = dx · b

−1 · dx. (1.56)

Since J is the determinant of xk

,K , one also has

∂J

∂xk

,K

= cofactor(xk

,K ). (1.57)

8 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

The cofactor of xk

,K is defined as the determinant of the matrix one obtains after deleting

row k and column K in xk

,K (this is the so-called minor determinant of xk

,K ), multiplied by

(−1)k+K. Equation (1.57) is easily derived by recalling that the determinant of a matrix

can be obtained by taking the dot product of any row with the row of the corresponding

cofactors, for example, if the first row is used,

J = x1

,1cofactor (x1

,1) + x1

,2cofactor (x1

,2) + x1

,3cofactor (x3

,3). (1.58)

XK

,k is the inverse of xk

,K . Accordingly,

XK

,k

= 1

J

cofactor (xk

,K ). (1.59)

Indeed, the inverse of a matrix M satisfies (Greenberg 1978)

(M

−1)KL = 1

detM

cofactor (MLK). (1.60)

Comparing Equation (1.57) with Equation (1.59), one finds

∂J

∂xk

,K

= JXK

,k . (1.61)

This relationship will be needed for the time derivative of J .

So far, only length changes were considered. Since the determinant of a map describes

its volume change, one can write

dv = J dV (1.62)

where dv and dv are infinitesimal volume elements in the reference and spatial configuration

respectively. Denoting an infinitesimal surface element in the reference configuration by

the one-form dA orthogonal to the surface element and with size equal to the area of the

surface, and similarly for the spatial configuration, one obtains for Equation (1.62),

da · dx = J dA · dX (1.63)

or

da · F · dX = J dA · dX. (1.64)

Since this applies to an arbitrary vector dX, one finds

da = J dA · F

−1 (1.65)

or

da = JF

−T · dA. (1.66)

This is feasible since it expresses that for isochoric (volume-preserving, J = 1) motion,

the surface change is inversely proportional to the length change.

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 9

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