Strain and stress tensors

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 92-93

Section 1.3.6.3. Strain and stress tensors

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.6.3. Strain and stress tensors

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The displacement vector from the reference position of a particle to its new position has as components $[u_{i} = x_{i} - a_{i}. \eqno(1.3.6.5) ]$

The term strain refers to a change in the relative positions of the material points in a body. Let a final configuration be described in terms of the reference configuration by setting t equal to a constant in (1.3.6.1). Then t no longer appears as a variable and (1.3.6.1) can be written $[x_{i} = x_{i} (a_{1}, a_{2}, a_{3}), ]$ where the $[a_{i}]$ are the independent variables. It follows that $[\hbox{d}x_{j} = {\partial x_{j} \over \partial a_{i}} \hbox{d} a_{i} = \left({\partial u_{j} \over \partial a_{i}} + \delta_{ij}\right) \hbox{d}a_{i}. \eqno(1.3.6.6) ]$

Let now the particle initially at ( $[a_{1}]$ , $[a_{2}]$ , $[a_{3}]$ ) move to ( $[x_{1}]$ , $[x_{2}]$ , $[x_{3}]$ ). The square of the initial distance to a neighbouring particle whose initial coordinates were $[a_{j} + \hbox{d}a_{j}]$ is $[\hbox{d}s^{2} = \hbox{d}a_{j} \hbox{d}a_{j}. ]$ The square of the final distance to the same neighbouring particle is $[\hbox{d}s^{2} = \hbox{d}x_{j} \hbox{d}x_{j}.]$

In a material description, the strain components $[S_{ik} ]$ are defined by the following equations: $[\hbox{d}x_{j} \hbox{d}x_{j} - \hbox{d}a_{j} \hbox{d}a_{j} = 2 S_{ik}\hbox{d}a_{i} \hbox{d}a_{k}. \eqno(1.3.6.7) ]$ Substituting (1.3.6.6) into (1.3.6.7), it follows that $[\left({\partial u_{j} \over \partial a_{i}} + \delta_{ji}\right) \left({\partial u_{j} \over \partial a_{k}} + \delta_{jk}\right) \hbox{d}a_{i} \hbox{d}a_{k} - \hbox{d}a_{j} \hbox{d}a_{j} = 2 S_{ik}\hbox{d}a_{i} \hbox{d}a_{k}. ]$ Hence $[S_{ik} = \textstyle{1\over 2} \left(\displaystyle{\partial u_{k} \over \partial a_{i}} + {\partial u_{i} \over \partial a_{k}} + {\partial u_{j} \over \partial a_{i}} {\partial u_{j} \over \partial a_{k}}\right). ]$

If the products and squares of the displacement derivatives are neglected, the strain components reduce to the usual form of `infinitesimal elasticity' [see equation (1.3.1.8)]: $[S_{ik} = \textstyle{1 \over 2} \left(\displaystyle{\partial u_{i} \over \partial a_{k}} + {\partial u_{k} \over \partial a_{i}}\right). ]$

It is often useful to introduce the Jacobian matrix associated with the transformation (a, x). The components of this matrix are $[J = \pmatrix{\alpha_{11} &\alpha_{12} &\alpha_{13}\cr \alpha_{21} &\alpha_{22} &\alpha_{23}\cr \alpha_{31} &\alpha_{32} &\alpha_{33}\cr}, ]$ where $[\alpha_{ik} = {\partial x_{i} \over \partial a_{k}} = {\partial u_{j} \over \partial a_{k}} + \delta_{jk}. ]$

From the definition of matrix J, one has $[\hbox{d}{\bf x} = J\hbox{d}{\bf a}]$ and $[\hbox{d}x^{2} - \hbox{d}a^{2} = \left(\hbox{d}{\bf x}\right)^{T} \hbox{d}{\bf x} - \left(\hbox{d}{\bf a}\right)^{T} \hbox{d}{\bf a} = \left(\hbox{d}{\bf a}\right)^{T} \left(J^{T} J - \delta\right)\hbox{d}{\bf a}, ]$ where $[\left(\hbox{d}{\bf a}\right)^{T}]$ , $[\left(\hbox{d}{\bf x}\right)^{T}]$ and $[J^{T}]$ are the transpose matrices of da, dx and J, respectively, and δ is the Kronecker matrix.

The Lagrangian strain matrix S may then be written symbolically: $[S = \textstyle{1\over 2} \left(J^{T}J - \delta\right). \eqno(1.3.6.8) ]$

When finite strains are concerned, we have to distinguish three states of the medium: the natural state, the initial state and the final or present state: The natural state is a state free of stress. The initial state is deduced from the natural state by a homogeneous strain. The final state is deduced from the initial state by an arbitrary strain.

Concerning the stress tensor, as pointed out by Thurston (1964), the stress-deformation relation is complicated in nonlinear elasticity because `the strain is often referred to a natural unstressed state, whereas the stress $[T_{ij}]$ is defined per unit area of the deformed body'. For this reason, the differential of work done by the stress is not equal to the stress components times the differentials of the corresponding strain components. So, following Truesdell & Toupin (1960), we shall introduce a thermodynamic tension tensor $[t_{ij}]$ defined as the first derivative of the energy with respect to strain. If the internal energy U per unit mass is considered, the thermodynamic tension refers to an isentropic process. Then $[t_{ij}^{\sigma} = \rho_{0} \left({\partial U \over \partial S_{ij}}\right)_{\sigma}, ]$ where σ is the entropy and $[\rho_{0}]$ the volumic mass in the initial state.

If the Helmholtz free energy F is considered, the thermodynamic tension refers to an isothermal process. Then $[t_{ij}^{\Theta} = \rho_{0} \left({\partial U \over \partial S_{ij}}\right)_{\Theta}, ]$ where Θ is the temperature. It will be shown in Section 1.3.7.2 that $[T_{ij} = (1/J)\alpha_{ik}\alpha_{jl}t_{kl}. ]$

References

Thurston, R. N. (1964). Wave propagation in fluids and normal solids. Physical acoustics, Vol. 1A, edited by W. P. Mason, pp. 1–109. New York: Academic Press.Google Scholar

Truesdell, C. & Toupin, R. (1960). The classical field theories. Handbuch der Physik, Vol. III/1, edited by S. Flügge. Berlin, Göttingen, Heidelberg: Springer-Verlag.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 92-93