A simple and practical representation of compatibility condition derived using a QR decomposition of the deformation gradient

Abstract

This paper examines a condition for the existence and uniqueness of a finite deformation field whenever a Gram–Schmidt (QR) factorization of the deformation gradient ${\mathbf {F}}$ is used. First, a compatibility condition is derived, provided that a right Cauchy–Green tensor ${\mathbf {C}} = {\mathbf {F}}^T {\mathbf {F}}$ is prescribed. It is well-known that under this condition a vanishing of the Riemann curvature tensor ${\mathbb {R}}$ ensures compatibility of a finite deformation field. We derive a restriction imposed on Laplace stretch $\varvec{{\mathcal {U}}}$, arising from a QR decomposition of the deformation gradient, through this compatibility condition. The derived condition on Laplace stretch is unambiguous, because a Cholesky factorization of the right Cauchy–Green tensor ensures the existence of a unique Laplace stretch. Although a vanishing of the Riemann curvature tensor provides a necessary and sufficient compatibility condition from a purely geometric point of view, this condition lacks a direct physical interpretation in a sense that one cannot identify the restrictions imposed by this condition on a quantity that can be readily measured from experiments. On the other hand, our compatibility condition restricts dependence of components of a Laplace stretch on certain spatial variables in a reference configuration. Unlike the symmetric right Cauchy–Green stretch tensor ${\mathbf {U}}$ obtained from a traditional polar decomposition of ${\mathbf {F}}$, the components of Laplace stretch can be measured from experiments. Thus, this newly derived compatibility condition provides a physical meaning to the somewhat abstract idea of the traditionally used compatibility condition, viz., a vanishing of the Riemann curvature tensor. Couplings between certain components of the Laplace stretch representing shear and elongation play a crucial role in deriving this condition. Finally, implications of this compatibility condition are discussed.

Appendices

Transpositions of tensors

1.1 Fourth-order tensor:

$$\begin{aligned} {\mathbb {A}}^T&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^T={\mathbb {A}}^{ijkl} e_j \otimes e_i \otimes e_l \otimes e_k ={\mathbb {A}}^{jilk} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^{ti}&=({\mathbb {A}}^{ijkl} e_i \otimes e_k \otimes e_j \otimes e_l )^{ti}={\mathbb {A}}^{ikjl} e_j \otimes e_i \otimes e_l \otimes e_k ={\mathbb {A}}^{ikjl} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^{to}&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^{to}={\mathbb {A}}^{ijkl} e_l \otimes e_j \otimes e_k \otimes e_i ={\mathbb {A}}^{ljki} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^t&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^t={\mathbb {A}}^{ijkl} e_l \otimes e_k \otimes e_j \otimes e_i ={\mathbb {A}}^{lkji} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^D&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^D={\mathbb {A}}^{ijkl} e_k \otimes e_l \otimes e_i \otimes e_j ={\mathbb {A}}^{klij} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^{dl}&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^{dl}={\mathbb {A}}^{ijkl} e_j \otimes e_i \otimes e_k \otimes e_l ={\mathbb {A}}^{jikl} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^{dr}&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^{dr}={\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_l \otimes e_k ={\mathbb {A}}^{ijlk} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ {\mathbb {A}}^d&=({\mathbb {A}}^{ijkl} e_i \otimes e_j \otimes e_k \otimes e_l )^d={\mathbb {A}}^{ijkl} e_j \otimes e_i \otimes e_l \otimes e_k ={\mathbb {A}}^{jilk} e_i \otimes e_j \otimes e_k \otimes e_l \nonumber \\ \end{aligned}$$

(A.1)

1.2 Third-order tensor:

$$\begin{aligned} \begin{aligned} \mathbf {\mathtt {A}}^T&=(\mathbf {\mathtt {A}}^{ijk} e_i \otimes e_j \otimes e_k )^T=\mathbf {\mathtt {A}}^{ijk} e_i \otimes e_k \otimes e_j =\mathbf {\mathtt {A}}^{ikj} e_i \otimes e_j \otimes e_k \\ \mathbf {\mathtt {A}}^t&=(\mathbf {\mathtt {A}}^{ijk} e_i \otimes e_j \otimes e_k )^t=\mathbf {\mathtt {A}}^{ijk} e_k \otimes e_j \otimes e_i =\mathbf {\mathtt {A}}^{kji} e_i \otimes e_j \otimes e_k \\ \mathbf {\mathtt {A}}^D&=(\mathbf {\mathtt {A}}^{ijk} e_i \otimes e_j \otimes e_k )^D=\mathbf {\mathtt {A}}^{ijk} e_j \otimes e_k \otimes e_i =\mathbf {\mathtt {A}}^{jki} e_i \otimes e_j \otimes e_k \\ \end{aligned} \end{aligned}$$

(A.2)

1.3 Second-order tensor:

$$\begin{aligned} {\mathbf {A}}^T=({\mathbf {A}}^{ij} e_i \otimes e_j )^T={\mathbf {A}}^{ij} e_j \otimes e_i ={\mathbf {A}}^{ji} e_i \otimes e_j \end{aligned}$$

(A.3)

Symmetries of ${\mathbb {R}}_m$

1.1 ${\mathbb {R}}_1$:

$$\begin{aligned} \begin{aligned} {\mathbb {R}}_1^{dr}&=-{\mathbb {R}}_1;\quad [{\mathbb {R}}_1^{dl}]^{dr}=-{\mathbb {R}}_1^{dl};\quad [[{\mathbb {R}}_1^D]^{dr}]=-{\mathbb {R}}_1^D;\\ [[{\mathbb {R}}_1^D]^{dl}]^{dr}&=-[{\mathbb {R}}_1^D]^{dl};\quad [{\mathbb {R}}_1^D]^{dr}=[{\mathbb {R}}_1^{dl}]^D \end{aligned} \end{aligned}$$

(B.4)

1.2 ${\mathbb {R}}_2$:

$$\begin{aligned} {\mathbb {R}}_2^{dr}=-{\mathbb {R}}_2;\quad {\mathbb {R}}_2^D={\mathbb {R}}_2;\quad {\mathbb {R}}_2^{ti}=[{\mathbb {R}}_2^{ti}]^D;\quad [{\mathbb {R}}_2^{ti}]^{dr}=[[{\mathbb {R}}_2^{ti}]^{dr}]^D \end{aligned}$$

(B.5)

1.3 ${\mathbb {R}}_3$:

$$\begin{aligned} \begin{aligned} {\mathbb {R}}_3^{dr}&={\mathbb {R}}_3;\quad [{\mathbb {R}}_3^{to}]^T=[{\mathbb {R}}_3^{dl}]^{ti};\quad [{\mathbb {R}}_3^{ti}]^T=[{\mathbb {R}}_3^{dl}]^{to}; \\ [[{\mathbb {R}}_3^{ti}]^T]^{dr}&=[{\mathbb {R}}_3^{ti}]^{dl};\quad [[{\mathbb {R}}_3^{to}]^{dl}]^{dr}=[{\mathbb {R}}_3^{to}]^T;\quad [[{\mathbb {R}}_3^{to}]^{dl}]^D=[{\mathbb {R}}_3^{ti}]^{dl};\quad [[{\mathbb {R}}_3^{ti}]^{dl}]^D=[{\mathbb {R}}_3^{to}]^{dl};\\ [[{\mathbb {R}}_3^{ti}]^{dr}]^D&=[{\mathbb {R}}_3^{to}]^{dr}\quad [[{\mathbb {R}}_3^{to}]^{dr}]^D=[{\mathbb {R}}_3^{ti}]^{dr};\quad [[{\mathbb {R}}_3^{to}]^T]^D=[{\mathbb {R}}_3^{dr}]^{ti};\quad [[{\mathbb {R}}_3^{ti}]^T]^D=[{\mathbb {R}}_3^{dr}]^{to}; \end{aligned} \end{aligned}$$

(B.6)

1.4 ${\mathbb {R}}_4$:

$$\begin{aligned} \begin{aligned} {\mathbb {R}}_4^t&={\mathbb {R}}_4;\quad [[{\mathbb {R}}_4^{to}]^T]^{dr}=[{\mathbb {R}}_4^{to}]^{dl};\quad [[[{\mathbb {R}}_4^{dr}]^{ti}]^{dr}]^D=[[{\mathbb {R}}_4^{dl}]^{ti}]^{dr};\\ [[{\mathbb {R}}_4^{to}]^{dl}]^D&=[{\mathbb {R}}_4^{to}]^{dl};\quad [[{\mathbb {R}}_4^{dr}]^{ti}]^D=[{\mathbb {R}}_4^{dl}]^{ti};\quad [[{\mathbb {R}}_4^{ti}]^{dr}]^D=[{\mathbb {R}}_4^{ti}]^{dr};\quad [{\mathbb {R}}_4^{ti}]^D=[{\mathbb {R}}_4^{to}]^T \end{aligned} \end{aligned}$$

(B.7)

System of equations

1.1 Equations arising from symmetries of $f_1({\mathbb {R}}_1)$

Because the diagonal blocks of $f_1({\mathbb {R}})$ are zero, we obtain:

$$\begin{aligned}&aW_{1,1}=0, \end{aligned}$$

(C.1.1)

$$\begin{aligned}&aW_{3,1}=0, \end{aligned}$$

(C.1.2)

$$\begin{aligned}&\beta W_{1,1}+\gamma W_{2,1}=\alpha W_{3,1}+b W_{4,1}, \end{aligned}$$

(C.1.3)

$$\begin{aligned}&\alpha W_{1,2}+b W_{2,2}=0, \end{aligned}$$

(C.1.4)

$$\begin{aligned}&\alpha W_{6,2}+b W_{7,2}=0, \end{aligned}$$

(C.1.5)

$$\begin{aligned}&\beta W_{1,2}=-aW_{6,2}, \end{aligned}$$

(C.1.6)

$$\begin{aligned}&\beta W_{6,3}+\gamma W_{7,3}+c W_{8,3}=0, \end{aligned}$$

(C.1.7)

$$\begin{aligned}&\beta W_{3,3}+\gamma W_{4,3}+c W_{5,3}=0, \end{aligned}$$

(C.1.8)

$$\begin{aligned}&\alpha W_{3,3}+bW_{4,3}=a W_{6,3}. \end{aligned}$$

(C.1.9)

From skew-symmetry of $f_1({\mathbb {R}}_1)$, we get

$$\begin{aligned}&aW_{1,2}=-\gamma W_{1,1}-b W_{2,1}, \end{aligned}$$

(C.1.10)

$$\begin{aligned}&a W_{3,2}=-\beta W_{1,1}-\alpha W_{2,1}, \end{aligned}$$

(C.1.11)

$$\begin{aligned}&\gamma W_{6,1}+b W_{7,1}=\alpha W_{1,2}+\alpha W_{2,2}-\gamma W_{3,2}-bW_{4,2}, \end{aligned}$$

(C.1.12)

$$\begin{aligned}&a W_{1,3}=a W_{6,1}-\gamma W_{1,3}-b W_{4,1}, \end{aligned}$$

(C.1.13)

$$\begin{aligned}&a W_{3,3}=-\beta W_{3,1}-\alpha W_{4,1}-cW_{5,1}, \end{aligned}$$

(C.1.14)

$$\begin{aligned}&\gamma W_{3,3}+b W_{4,3}=\beta W_{1,3}+\alpha W_{2,3}-\beta W_{6,1}-\alpha W_{7,1}-cW_{8,1}, \end{aligned}$$

(C.1.15)

$$\begin{aligned}&a W_{6,2}=\gamma W_{2,3}+b W_{2,3}+\gamma W_{3,2}+bW_{4,2}, \end{aligned}$$

(C.1.16)

$$\begin{aligned}&aW_{6,3}=-\beta W_{1,3}-\alpha W_{2,3}-\beta W_{3,2}-\alpha W_{4,2}-c W_{5,2}, \end{aligned}$$

(C.1.17)

$$\begin{aligned}&\gamma W_{6,3}+b W_{7,3}=-\beta W_{6,2}-\alpha W_{7,2}-c W_{8,2}. \end{aligned}$$

(C.1.18)

Because $a,\gamma ,\beta$ and $b,\alpha$ are the only coupled elements of $\varvec{{\mathcal {U}}}$, we conclude that

$$\begin{aligned} \begin{aligned} W_{1,1}&=W_{2,1}=W_{1,2}=W_{3,3}=W_{4,3}=W_{3,1}=W_{6,2}=W_{6,3}=W_{7,3}=W_{2,2}=W_{5,3}=W_{7,2}\\&=W_{8,3}=W_{4,1}=W_{5,1}=W_{8,2}=W_{3,2}+W_{6,1}=W_{4,2}+W_{7,1}=W_{1,3}-W_{6,1}=W_{8,1}\\&=W_{5,2}=W_{2,3}-W_{7,1}=0. \end{aligned}\qquad \end{aligned}$$

(C.1.19)

Thus, no term involving derivatives of $W_p$, $p=1,...,8$, appears in the 6 equations arising from equating off-diagonal elements of ${\mathbb {R}}$ to zero.

1.2 Equations arising from vanishing of Riemann curvature tensor

${\mathbb {R}}_{1313}=0$ leads to:

(C.2.1)

${\mathbb {R}}_{2323}=0$ yields:

(C.2.2)

where $\xi =\alpha \gamma -\beta b.$