In the following, the component notation c_i = A_ijb_j with Einstein summation over repeated indices is preferred over the notation c = A⋅b, because the latter has to be translated to the former for every actual calculation, anyway.

1 Basic definitions

The elastic deformation can quantified by comparison of the deformed state to a reference state . We denote the coordinates in the reference frame by x_i⁽⁾ and use X_i⁽⁾ for the coordinates in the current state ¹ . Further, we introduce a mapping F that relates the coordinates of identical material points in A to the corresponding coordinates in B (see Fig. 1).

Figure 1: We distinguish two configurations: The reference state and the elastically deformed state . The mapping F relates both configurations to each other.

In order to simplify our notation, we introduce the shorthands ∂_i⁽⁾ := ∂∕∂x_i⁽⁾ and ∂_i⁽⁾ := ∂∕∂x_i⁽⁾ for the partial derivatives with respect to the coordinates in both frames. We define the deformation gradient tensor F by

(1)

and introduce the symbol J_F = |F| for its determinant. The deformation of the solid is quantified by the strain ω defined as ²

(2)

Nanson’s formula In order to relate the oriented area element dA_i⁽⁾ in frame to dA_i⁽⁾ in frame , we consider infinitesimal volume elements in both frames and and use dx_i⁽⁾ = F_ijdx_j⁽⁾:

(3)

and

(4)

From these two formulas we conclude

	JFdAi() = dA k()F ki
⇔	JF ildAi() = dA l()
⇔	JF ilni()dA() = n i()dA().	(5)

(6)

Helpful relations A few non-obvious identities can be found using Nanson’s formula and very simple geometric arguments. Consider an arbitrary closed surface δω of some volume ω in the reference frame .

(7)

Since we can choose the surface ∂ω arbitrarily, we find

(8)

In other words: in the sum ∂_k⁽⁾ and J_F^-1F_ki commute. Analogously, we can show that

(9)

Three different stress tensors The surface force per unit area on a vector area element dA⁽⁾ in the current frame is expressed by the Cauchy stress Σ_ij. Using relation (5) we can translate this to a force per unit area on a vector area element dA⁽⁾ in the reference frame .

(10)

where we have defined the first Piola-Kirchhoff stress σ. The inverse relation is

(11)

Note, that unlike the Cauchy stress Σ, the first Piola-Kirchhoff stress σ is not symmetric.

Both ΣdA⁽⁾ and σdA⁽⁾ are force vectors in the current frame . Using F, we can map it to the reference configuration and obtain

ijΣjknk()dA() =	=: silnl()dA()
=	=: siknk()dA()
=	sijnj()dA()	(12)

(13)

Note, that ∂_j⁽⁾Σ_ij on the one hand and ∂_j⁽⁾σ_ij and ∂_js_ij on the other hand are force densities with respect to different volumina and therefore differ by a factor of J_F. This can be easily seen by noting that

(14)

and

(15)

Relation of the Piola-Kirchhoff stresses to the elastic free energy In Hookean elasticity, where, the Cauchy stress is the derivative of the elastic free energy density per unit volume _el with respect to the linear strain ω_ij^(lin) = u_i,j + u_j,i:

(16)

In nonlinear elasticity, the second Piola-Kirchhoff stress takes the role of the Cauchy stress:

(17)

The first Piola-Kirchhoff stress is given by the derivative of ∂_el with respect to the deformation gradient tensor F_ij:

σij	= = = skl = Fik
	= Fikskj.	(18)

2 Elasticity and growth

When the considered material is growing, for example because it is a biological tissue consisting of proliferating cells, one has to seperate the deformation due to growth from elastic deformation. This is basically the distinction between growth of a spring (that might be caused by uniform thermal expansion) versus extension of the spring by external forces. To make this distinction quantitative, seperate the deformation gradient tensor F into a product of the two tensors G and A as proposed by Rodriguez et al. [RHM94] yielding

(19)

where the growth tensor G describes the deformation due to growth and A describes the elastic part of the deformation³ . We can then distinguish not only between the reference state and the current deformed state , but also define the virtual state , that describes the grown but otherwise undeformed material (see Fig. 2).

Figure 2: If we have growth and elasticity, we distinguish three configurations: The reference state , the virtual configuration of a stress-free grown state , and the actual grown and elastically deformed state . The mappings G, A, and F relate the three configurations to each other.

Analogously to the case without growth, we use the shorthands ∂_i⁽⁾ := ∂∕∂x_i⁽⁾, ∂_i⁽⁾ := ∂∕∂x_i⁽⁾, and ∂_i⁽⁾ := ∂∕∂x_i⁽⁾ for the partial derivatives with respect to the different coordinate frames and define J_F = |F|, J_G = |G|, and J_A = |A|. Elastic deformation is deformation from to , only. Therefore, we can for the moment forget about the frame and consider the reference state. We can then define the virtual Piola-Kirchhoff stress σ⁽⁾ that translates the force per unit area in the frame to a force per unit area in the frame as

(20)

Since the physical force is the same, no matter which frame we choose, the following two equalities have to hold:

(21)

Using the transformation of the spatial derivative⁴ ∂_j⁽⁾ = (G^-1)_kj∂_k⁽⁾ one can use Nanson’s formula (5) applied to , , and G to find

(22)

When we insert this result into eq. (21), we obtain

(23)

and have thus established the relation between the Piola-Kirchhoff stresses σ_ij and σ_ij⁽⁾.

References