Nonlinear elasticity

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. 91-94

Section 1.3.6. Nonlinear elasticity

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. Nonlinear elasticity

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1.3.6.1. Introduction

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In a solid body, the relation between the stress tensor T and the strain tensor S is usually described by Hooke's law, which postulates linear relations between the components of T and S (Section 1.3.3.1). Such relations can be summarized by (see equation 1.3.3.2) $[T_{ij} = c_{ijkl} S_{kl}, ]$ where the $[c_{ijkl}]$ 's are the elastic stiffnesses.

For a solid under finite strain conditions, Hooke's law, valid for infinitesimal deformations, does not hold, and the fundamental definitions for stress and strain must be revisited.

1.3.6.2. Lagrangian and Eulerian description

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Finite elastic strains may be treated from two different viewpoints using either the Lagrangian (material) or the Eulerian (spatial) descriptions.

Let us consider a fixed rectangular Cartesian coordinate system with axes $[x_{i}]$ ( [i = 1, 2, 3] ). Any particular position vector r of components ( $[x_{1}]$ , $[x_{2}]$ , $[x_{3}]$ ) denotes a point in space. A point that always moves with the material is called a particle or material point. Let every particle be identified by its coordinates at some reference time $[t_{0}]$ . These reference coordinates, referred to the same Cartesian system, will be denoted by ( $[a_{1}]$ , $[a_{2}]$ , $[a_{3}]$ ) and the corresponding position vector a. A particular vector a can serve as a name for the particle located at that position at the reference time $[t_{0} ]$ .

The vectors r and a both specify a position in a fixed Cartesian frame of reference. At any time, we associate each r with an a by the rule that r is the present position vector of the particle initially at a. This connection between r and a is written symbolically as $[{\bf r} = {\bf r}(t, a) \quad \hbox{ or } \quad x_{i} = x_{i} (t, a_{1}, a_{2}, a_{3}), \eqno(1.3.6.1) ]$ where $[{\bf a} = {\bf r}(t_{0}, {\bf a}) \quad \hbox{ or } \quad a_{i} = x_{i} (t_{0}, a_{1}, a_{2}, a_{3}). \eqno(1.3.6.2) ]$

The coordinates $[a_{i}]$ that identify the particles are called material coordinates. A description that, like (1.3.6.1), uses (t, $[a_{1}]$ , $[a_{2}]$ , $[a_{3}]$ ) as independent variables is called a material or Lagrangian description.

The converse of (1.3.6.1) and (1.3.6.2) may be written $[{\bf a} = {\bf a} (t, {\bf r}) \quad \hbox{ or } \quad a_{i} = a_{i} (t, x_{1}, x_{2}, x_{3}), \eqno(1.3.6.3) ]$ where $[{\bf r} = {\bf a}(t_{0}, {\bf r}) \quad \hbox{ or } \quad x_{i} = a_{i} (t_{0}, x_{1}, x_{2}, x_{3}). \eqno(1.3.6.4) ]$

A spatial description or Eulerian description uses the independent variables (t, $[x_{1}]$ , $[x_{2}]$ , $[x_{3}]$ ), the $[x_{i}]$ being called spatial coordinates.

Now, for the sake of simplicity, we shall work with the Lagrangian formulation exclusively. For more details see, for instance, Thurston (1964) and Wallace (1970, 1972).

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 $[T_{ij} = (1/J)\alpha_{ik}\alpha_{jl}t_{kl}. ]$

1.3.6.4. Second-order and higher-order elastic stiffnesses

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Following Brugger (1964), the strain-energy density, or strain energy per unit volume Φ, is assumed to be a polynomial in the strain: $[\Phi = \Phi_{0} + c_{ij}S_{ij} + {1\over 2!}c_{ijkl}S_{ij}S_{kl} + {1\over 3!}c_{ijklmn}S_{ij}S_{kl}S_{mn}, \eqno(1.3.6.9) ]$ where $[\Phi = \rho_{0} U(X,S_{ij})]$ , $[\Phi_{0} = \rho_{0} U(X,0)]$ , X denotes the configuration of the initial state and the $[S_{ij}]$ 's are the Lagrangian finite strain-tensor components.

If the initial energy and the deformation of the body are both zero, the first two terms in (1.3.6.9) are zero. Note that $[c_{ij}]$ is a stress and not an intrinsic characteristic of the material. In this expression, the elastic stiffnesses $[c_{ijkl}]$ and $[c_{ijklmn}]$ are the second- and third-order stiffnesses , respectively. Since the strain tensor is symmetric, pairs of subscripts can be interchanged [see equation (1.3.3.4)]: $[\eqalign{c_{ijkl} &= c_{jikl} = c_{ijlk} = c_{jilk},\cr c_{ijklmn} &= c_{jiklmn} = c_{ijlkmn} = c_{jilkmn} = c_{ijklnm} \cr&= c_{jiklnm} = c_{ijlknm} = c_{jilknm}.\cr} ]$

More accurately, the isentropic and the isothermal elastic stiffnesses are defined as the nth partial derivatives of the internal energy and the Helmholtz free energy, respectively. For example, the third-order isentropic and isothermal stiffnesses are, respectively, $[\eqalign{c_{ijkl mn}^{\sigma} &= \rho_{0} {\partial^{3} U \over \partial S_{ij}\partial S_{kl}\partial S_{mn}},\cr c_{ijkl mn}^{\Theta} &= \rho_{0} {\partial^{3} F \over \partial S_{ij}\partial S_{kl}\partial S_{mn}},\cr} ]$ where the internal energy, U, is a function of X, $[S_{ij}]$ and σ, and the Helmholtz free energy, F, is a function of X, $[S_{ij}]$ and Θ.

From these definitions, it follows that the Brugger stiffness coefficients depend on the initial state. When no additional information is given, the initial state is the natural state.

The third-order stiffnesses form a sixth-rank tensor containing $[3^{6} = 729]$ components, of which 56 are independent for a triclinic crystal and 3 for isotropic materials (the independent components of a sixth-rank tensor can be obtained for any point group using the accompanying software to this volume). The three independent constants for isotropic materials are often taken as $[c_{123}]$ , $[c_{144}]$ and $[c_{456}]$ and denoted respectively by $[\nu_{1} ]$ , $[\nu_{2}]$ , $[\nu_{3}]$ , the `third-order Lamé constants '.

The `third-order Murnaghan constants' (Murnaghan, 1951), denoted by [l, m, n] , are given in terms of the Brugger constants by the relations $[l = \textstyle{1\over 2} c_{112} ; \ m = c_{155} ; \ n = 4c_{456}. ]$

Similarly, the fourth-order stiffnesses form an eighth-rank tensor containing $[3^{8} = 6561]$ components, 126 of which are independent for a triclinic crystal and 11 for isotropic materials (the independent components of a sixth-rank tensor can be obtained for any point group using the accompanying software to this volume).

For a solid under finite strain conditions, the definition of the elastic compliance tensor has to be reconsidered. In linear elasticity , the second-order elastic compliances $[s_{ijkl}]$ were defined through the relations (1.3.3.2): $[S_{ij} = s_{ijkl} T_{kl} \quad \hbox{or} \quad s_{ijkl} = {\partial S_{ij} \over \partial T_{kl}}, ]$ while, in nonlinear elasticity, one has $[s_{ijkl} = {\partial S_{ij} \over \partial t_{kl}}, ]$ where $[t_{kl} = \rho_0 {\partial U \over \partial S_{kl}}. ]$

1.3.6.5. Expansion of elastic constants for small initial stress

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In most experiments, the initial stress is small compared with the second-order elastic constants (for example, 1 GPa hydrostatic pressure compared with the usual value c_ijkl = 100 GPa). Consequently, the deformation between the initial (stressed) state and the natural (unstressed) state is small compared with 1. For this reason, it is convenient to expand the elastic constants in the initial state as a power series in the strain about the natural state. To avoid confusion, we introduce new notations: $[\bar{X}]$ now represents the coordinates in the natural or unstressed state; X represents the coordinates in the initial or homogeneously strained state; $[u_{i} = x_{i} - X_{i} ]$ are the components of displacement. All letters with superscript bar refer to the natural state; for example, $[\bar{S}_{ij}]$ denotes the Lagrangian strain in the natural state; $[\bar{U} = U(\bar{X}, \bar{S}_{ij}) ]$ .

Now, in order to relate the properties at X to those at $[\bar{X}]$ , we need to specify the strain from $[\bar{X}]$ to X. Let $[a_{ij} = {\partial X_{i} \over\partial \bar{X}_{j}} = \bar{\alpha}_{ij}. ]$ Consequently, $[{\partial \bar{S}_{ij} \over \partial S_{mn}} = a_{mi}a_{nj}. ]$ The second-order elastic constants at X can be expressed in terms of the second- and third-order elastic constants at $[\bar{X}]$ : $[c_{ijkl} = \rho_{0} {\partial^{2} U \over \partial S_{ij}\partial S_{kl}} = \rho_{0} {\partial^{2}\bar{U} \over \partial \bar{S}_{mn}\partial \bar{S}_{pq}} a_{im}a_{jn}a_{kp}a_{l q} ]$ or $[c_{ijkl} = {\rho_{0} \over \bar{\rho}_0} \left(\bar{c}_{mnpq} + \bar{c}_{mnpqrs}S_{rs} + {1\over 2!} \bar{c}_{mnpqrstu}S_{rs}S_{tu} + \dots\right) a_{im}a_{jn}a_{kp}a_{l q}. ]$ This expression holds for both isentropic and isothermal elastic constants.

1.3.6.6. Elastic strain-energy density

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The elastic strain-energy density has appeared in the literature in various forms. Most of the authors use the Murnaghan constants as long as isotropic solids are concerned. However, most of the literature uses Brugger's thermodynamic definition when anisotropic media are under consideration (Brugger, 1964).

The elastic strain-energy density for an isotropic medium, including third-order terms but omitting terms independent of strain, may be expressed in terms of three strain invariants, since an isotropic material is invariant with respect to rotation: $[\Phi = {\lambda + 2\mu \over 2} (I_1)^{2} - 2\mu I_{2} + {l + 2m \over 3} (I_{1})^{3} - 2 m I_{1} I_{2} + nI_{3}, ]$ where λ and μ are the second-order Lamé constants , [l, m, n] are the third-order Murnaghan constants, and $[I_{1}]$ , $[I_{2}]$ , $[I_{3}]$ are the three invariants of the Lagrangian strain matrix. These invariants may be written in terms of the strain components as $[\eqalignno{I_{1} &= S_{11} + S_{22} + S_{33} &\cr I_{2} &= \left| \matrix{S_{11} &S_{12}\cr S_{21} &S_{22}\cr}\right| + \left| \matrix{S_{22} &S_{23}\cr S_{32} &S_{33}\cr}\right| + \left| \matrix{S_{33} &S_{31}\cr S_{13} &S_{11}\cr}\right| &\cr I_{3} &= \left| \matrix{S_{11} &S_{12} &S_{13}\cr S_{21} &S_{22} &S_{23}\cr S_{31} &S_{32} &S_{33}\cr}\right|. &\cr} ]$ The elastic strain-energy density for an anisotropic medium (for example a medium belonging to the most symmetrical groups of cubic crystals) is (Green, 1973) $[\eqalignno{\Phi &= \textstyle{1\over 2} c_{11} \left[(S_{11})^{2} + (S_{22})^{2} + (S_{33})^{2}\right] + c_{12} \left[S_{11}S_{22} + S_{22}S_{33} + S_{33}S_{11}\right] &\cr &\quad + c_{44}[(S_{12})^{2} + (S_{21})^{2} + (S_{23})^{2} +(S_{32})^{2} + (S_{31})^{2} + (S_{13})^{2}] &\cr &\quad + c_{111} \left[(S_{11})^{3} + (S_{22})^{3} + (S_{33})^{3}\right] &\cr &\quad + c_{112} \left[(S_{11})^{2}(S_{22} + S_{33}) + (S_{22})^{2}(S_{33} + S_{11})\right.\cr&\quad + \left.(S_{33})^{2}(S_{11} + S_{22})\right] &\cr &\quad + \textstyle{1\over 2} c_{144} \left\{S_{11}[(S_{23})^{2} + (S_{32})^{2}] + S_{22}[(S_{31})^{2} + (S_{13})^{2}] \right.\cr&\quad + \left. S_{33}[(S_{12})^{2} + (S_{21})^{2}]\right\} &\cr &\quad + \textstyle{1\over 2}c_{166} \left\{\left[(S_{12})^{2} + (S_{21})^{2}\right](S_{11} + S_{22})\right. \cr&\quad+ \left[(S_{23})^{2}+(S_{32})^{2}\right](S_{22} + S_{33}) &\cr &\quad + \left.\left[(S_{13})^{2} + (S_{31})^{2}\right] (S_{11} + S_{33})\right\} &\cr &\quad + c_{123}S_{11}S_{22}S_{33} + c_{456} \left[S_{12}S_{23}S_{31} + S_{21}S_{32}S_{13}\right]. &\cr} ]$

References

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 91-94