General conditions of equilibrium of a solid

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, p. 76

Section 1.3.2.1. General conditions of equilibrium of a solid

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

1.3.2.1. General conditions of equilibrium of a solid

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Let us consider a solid C, in movement or not, with a mass distribution defined by a specific mass ρ at each point. There are two types of force that are manifested in the interior of this solid.

(i) Body forces (or mass forces), which one can write in the form $[{\bf F} \hbox{ d}m = {\bf F} \rho \hbox{ d}\tau, ]$ where dτ is a volume element and dm a mass element. Gravity forces or inertial forces are examples of body forces. One can also envisage body torques (or volume couples), which can arise, for example, from magnetic or electric actions but which will be seen to be neglected in practice.

(ii) Surface forces or stresses. Let us imagine a cut in the solid along a surface element dσ of normal n (Fig. 1.3.2.1). The two lips of the cut that were in equilibrium are now subjected to equal and opposite forces, R and $[{\bf R}' = -{\bf R}]$ , which will tend to separate or draw together these two lips. One admits that, when the area element dσ tends towards zero, the ratio $[{\bf R}/\hbox{d}\sigma]$ tends towards a finite limit, $[{\bf T}_{n}]$ , which is called stress. It is a force per unit area of surface, homogeneous to a pressure. It will be considered as positive if it is oriented towards the same side of the surface-area element dσ as the normal n and negative in the other case. The choice of the orientation of n is arbitrary. The pressure in a liquid is defined in a similar way but its magnitude is independent of the orientation of n and its direction is always parallel to n. On the other hand, in a solid the constraint $[{\bf T}_{n}]$ applied to a surface element is not necessarily normal to the latter and the magnitude and the orientation with respect to the normal change when the orientation of n changes. A stress is said to be homogeneous if the force per unit area acting on a surface element of given orientation and given shape is independent of the position of the element in the body. Other stresses are inhomogeneous. Pressure is represented by a scalar, and stress by a rank-two tensor, which will be defined in Section 1.3.2.2.

Figure 1.3.2.1 | top | pdf |

Definition of stress: it is the limit of R dσ when the surface element dσ tends towards zero. R and R′ are the forces to which the two lips of the small surface element cut within the medium are subjected.

Now consider a volume V within the solid C and the surface S which surrounds it (Fig. 1.3.2.2). Among the influences that are exterior to V, we distinguish those that are external to the solid C and those that are internal. The first are translated by the body forces, eventually by volume couples. The second are translated by the local contact forces of the part external to V on the internal part; they are represented by a surface density of forces, i.e. by the stresses $[{\bf T}_{n}]$ that depend only on the point Q of the surface S where they are applied and on the orientation of the normal n of this surface at this point. If two surfaces S and S′ are tangents at the same point Q, the same stress acts at the point of contact between them. The equilibrium of the volume V requires:

(i) For the resultant of the applied forces and the inertial forces: $[\textstyle\int \int\limits_{S}\displaystyle {\bf T}_{n} \hbox{ d}\sigma + \textstyle\int \int\int\limits_{V}\displaystyle {\bf F}\rho \hbox{ d}\tau = {\hbox{d}\over \hbox{d}t} \left\{\textstyle\int \int \int\limits_{V}\displaystyle {\bf v} \hbox{ d}\tau \right\}. \eqno(1.3.2.1) ]$

Figure 1.3.2.2 | top | pdf |

Stress, $[{\bf T}_{n}]$ , applied to the surface of an internal volume.

(ii) For the resultant moment: $[\textstyle\int \int\limits_{S}\displaystyle{\bf OQ} \wedge {\bf T}_{n} \hbox{ d}\sigma + \textstyle\int \int\int\limits_{V}\displaystyle {\bf OP} \wedge {\bf F} \rho \hbox{ d}\tau = {\hbox{ d}\over \hbox{ d}t}\left\{\textstyle\int \int \int\limits_{V}\displaystyle {\bf OP} \wedge {\bf v}\hbox{ d}\tau \right\}, \eqno(1.3.2.2) ]$ where Q is a point on the surface S, P a point in the volume V and v the velocity of the volume element dτ.

The equilibrium of the solid C requires that:

(i) there are no stresses applied on its surface and
(ii) the above conditions are satisfied for any volume V within the solid C.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 76