^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France, and ^{b}Laboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence email: aauthier@wanadoo.fr
In this chapter, the strain and stress tensors are defined and their main properties are derived. The elastic tensors, elastic stiffnesses and elastic compliances are then introduced. Their variation with orientation, depending on the crystal class, is given in the case of Young's modulus. The next part is devoted to the propagation of waves in continuous media (linear dynamic elasticity and the Christoffel matrix); the relation between the velocity and the elastic constants is given for the cubic, hexagonal and tetragonal classes. The experimental determination of elastic constants and their pressure and temperature dependence are discussed in separate sections. The last two sections of the chapter concerns nonlinear elasticity (second and higherorder elastic constants) and nonlinear dynamical elasticity.
Keywords: Eulerian description; Hooke's law; Lagrangian description; Poisson's ratio; Voigt notation; Young's modulus; bulk modulus; compressibility; cubic dilatation; dynamic elasticity; elastic compliances; elastic constants; elastic stiffnesses; elastic strain energy; elastic waves; elasticity; elongations; energy density; harmonic generation; homogeneous deformation; polarization; pulseecho technique; pulsesuperposition method; resonance technique; shear; spontaneous strain; strain field; strain tensor; stress tensor.
Let us consider a medium that undergoes a deformation. This means that the various points of the medium are displaced with respect to one another. Geometrical transformations of the medium that reduce to a translation of the medium as a whole will therefore not be considered. We may then suppose that there is an invariant point, O, whose position one can always return to by a suitable translation. A point P, with position vector , is displaced to the neighbouring point P′ by the deformation defined by The displacement vector constitutes a vector field. It is not a uniform field, unless the deformation reduces to a translation of the whole body, which is incompatible with the hypothesis that the medium undergoes a deformation. Let Q be a point that is near P before the deformation (Fig. 1.3.1.1
After the deformation, Q is displaced to Q′ defined by
In a deformation, it is more interesting in general to analyse the local, or relative, deformation than the absolute displacement. The relative displacement is given by comparing the vectors and PQ. Thus, one has Let us set Replacing by its expansion up to the first term gives
If we assume the Einstein convention
If the components are constants, equations (1.3.1.3)
The fundamental property of the homogeneous deformation results from the fact that equations (1.3.1.4)
Some crystals present a twin microstructure
Let be the basis vectors before deformation. On account of the deformation, they are transformed into the three vectors The parallelepiped formed by these three vectors has a volume V′ given by where is the determinant associated with matrix B, V is the volume before deformation and represents a triple scalar product
The relative variation of the volume is It is what one calls the cubic dilatation
Let us project the displacement vector on the position vector OP (Fig. 1.3.1.2
Only the symmetric part of M occurs in the expression of the elongation:
The geometrical study of the elongation as a function of the direction of r is facilitated by introducing the quadric associated with M: where is a constant. This quadric is called the quadric of elongations, Q. S is a symmetric matrix with three real orthogonal eigenvectors
One can discuss the form of the quadric according to the sign of the eigenvalues
In order to follow the variations of the elongation with the orientation of the position vector, one associates with r a vector y, which is parallel to it and is defined by where k is a constant. It can be seen that, in accordance with (1.3.1.6)
Thus, the elongation is inversely proportional to the square of the radius vector of the quadric of elongations parallel to OP. In practice, it is necessary to look for the intersection p of the parallel to OP drawn from the centre O of the quadric of elongations (Fig. 1.3.1.3a
Equally, one can connect the displacement vector directly with the quadric Q. Using the bilinear form
One recognizes the components of the displacement vector u, which is therefore parallel to the normal to the quadric Q at the extremity of the radius vector Op parallel to r.
The directions of the principal axes of Q correspond to the extremal values of y, i.e. to the stationary values (maximal or minimal) of the elongation. These values are the principal elongations
If the deformation is a pure rotation Hence we have
The quadric Q is a cylinder of revolution having the axis of rotation as axis.
If the deformation is small but arbitrary, i.e. if the products of two or more components of can be neglected with respect to unity, one can describe the deformation locally as a homogeneous asymptotic deformation. As was shown in Section 1.3.1.2.4
In general, one is only interested in the pure deformation, i.e. in the form of the deformed object. Thus, one only wishes to know the quantities and the symmetric part of M. It is this symmetric part that is called the deformation tensor or the strain tensor. It is very convenient for applications to use the simplified notation due to Voigt:One may note that The Voigt strain matrix
Let us consider an orthonormal system of axes with centre P. We remove nothing from the generality of the following by limiting ourselves to a planar problem and assuming that point P′ to which P goes in the deformation lies in the plane (Fig. 1.3.1.4
As the coefficients are small, the lengths of P′Q′ and P′R′ are hardly different from PQ and PR, respectively, and the elongations
The components , , of the principal diagonal of the Voigt matrix
To summarize, if one considers a small cube before deformation, it becomes after deformation an arbitrary parallelepiped; the relative elongations
The cubic dilatation
Matrix M has only one coefficient, , and reduces to (Fig. 1.3.1.5a
This is a pure deformation
Matrix has one coefficient only, a shear (Fig. 1.3.1.5c
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.
Now consider a volume V within the solid C and the surface S which surrounds it (Fig. 1.3.2.2
The equilibrium of the solid C requires that:
Using the condition on the resultant of forces, it is possible to show that the components of the stress can be determined from the knowledge of the orientation of the normal n and of the components of a ranktwo tensor. Let P be a point situated inside volume V, , and three orthonormal axes, and consider a plane of arbitrary orientation that cuts the three axes at Q, R and S, respectively (Fig. 1.3.2.3
As long as the surface element dσ is finite, however small, it is possible to divide both terms of the equation by it. If one introduces the direction cosines, , the equation becomes When dσ tends to zero, the ratio tends towards zero at the same time and may be neglected. The relation then becomes This relation is called the Cauchy relation
Let us return to equation (1.3.2.1)
The equilibrium condition now becomes In order that this relation applies to any volume V, the expression under the integral must be equal to zero, or, if one includes explicitly the inertial forces, This is the condition of continuity or of conservation. It expresses how constraints propagate throughout the solid. This is how the cohesion of the solid is ensured. The resolution of any elastic problem requires solving this equation in terms of the particular boundary conditions of that problem.
Let us now consider the equilibrium condition (1.3.2.2)
In order that this relation applies to any volume V within the solid C, we must have or
Taking into account the continuity condition (1.3.2.5)
A volume couple can occur for instance in the case of a magnetic or an electric field
This result can be recovered by applying the relation (1.3.2.2)
For face 1, the constraint is if is the magnitude of the constraint at P. The force applied at face 1 is and its moment is
Similarly, the moments of the force on the other faces are
Noting further that the moments applied to the faces 1 and 1′ are of the same sense, and that those applied to faces 2 and 2′ are of the opposite sense, we can state that the resultant moment is where is the volume of the small parallelepiped. The resultant moment per unit volume, taking into account the couples in volume, is therefore It must equal zero and the relation given above is thus recovered.
We shall use frequently the notation due to Voigt (1910)
The Voigt matrix associated with the stress tensor
If the surface of the solid C is free from all exterior action and is in equilibrium, the stress field inside C is zero at the surface. If C is subjected from the outside to a distribution of stresses (apart from the volume forces mentioned earlier), the stress field inside the solid is such that at each point of the surface where the 's are the direction cosines of the normal to the surface at the point under consideration.
Consider a medium that is subjected to a stress field . It has sustained a deformation indicated by the deformation tensor S. During this deformation, the forces of contact have performed work and the medium has accumulated a certain elastic energy
In the deformation , the point P goes to the point P′, defined by A neighbouring point Q goes to Q′ such that (Fig. 1.3.1.1
Of sole importance is the relative displacement of Q with respect to P and the displacement that must be taken into account in calculating the forces applied at Q. The coordinates of the relative displacement are
We shall take as the position of Q the point of application of the forces at face 1, i.e. its centre with coordinates (Fig. 1.3.2.8
The work of the forces applied to the face 1′ is the same (, and change sign simultaneously). The works corresponding to the faces 1 and 1′ are thus and for the two stresses, respectively. One finds an analogous result for each of the other components of the stress tensor
Let us consider a metallic bar of length loaded in pure tension (Fig. 1.3.3.1
Young's modulus
Equations (1.3.3.2)
It is convenient to write the relations (1.3.3.2)
We have noted already that the matrix is the inverse of the matrix . These matrices can be written for cubic and isotropic materials as follows: where we have, for isotropic materials, We easily find that The coefficient is sometimes called the rigidity modulus
Expression (1.3.2.7)
Let us apply a hydrostatic pressure (Section 1.3.2.5.2
Under the action of a hydrostatic pressure, each vector assumes a different elongation
The coefficient of linear compressibility
If the applied stress reduces to a uniaxial stress
The elongation of a bar under the action of a uniaxial stress is characterized by and the diminution of the cross section is characterized by and . For a cubic material, the relative diminution of the diameter is One deduces from this that is necessarily of opposite sign to and one calls the ratio Poisson's ratio.
Putting this value into expression (1.3.3.12)
As the coefficient of compressibility
In practice, Poisson's ratio is always close to 0.3. It is a dimensionless number. The quantity represents the departure from isotropy of the material and is the anisotropy factor
It is interesting to calculate Young's modulus in any direction. For this it is sufficient to change the axes of the tensor . If A is the matrix associated with the change of axes, leading to the direction changing to the direction , then Young's modulus in this new direction is with The matrix coefficients are the direction cosines of with respect to the axes , and . In spherical coordinates, they are given by (Fig. 1.3.3.3
The representation surface
The isotropy relation between elastic compliances
These relations can equally well be written in the symmetrical form
If one introduces the Lamé constants
Two coefficients suffice to define the elastic properties of an isotropic material, and , and , μ and λ, μ and ν, etc. Table 1.3.3.3
We saw in Section 1.3.2.3
If we use the relations of elasticity, equation (1.3.3.2)
The elastic properties of materials have been considered in the preceding section in the static state and the elastic constants
The purpose of the next sections is to establish relations between the wavelength – or the velocity of propagation – and the elastic constants.
Consider the propagation of a wave in a continuous medium. The elongation
We saw in Section 1.3.3.6
The position vector of the point under consideration is of the form where only depends on the time and defines the mean position. Equation (1.3.4.3)
Replacing u by its value in (1.3.4.1)
It can be seen that, for a given wavevector, appears as an eigenvalue
Let be the direction cosines of the wavevector q. The components of the wavevector are With this relation and (1.3.4.2)
On account of the intrinsic symmetry of the tensor of elastic stiffnesses
If we introduce into expression (1.3.4.7)
The expression for the effective value, , of the `stiffened' elastic stiffness in the case of piezoelectric crystals
Equation (1.3.4.7)
This shows that in a dynamic process only the sums can be measured and not and separately. On the contrary, can be measured directly. In the cubic system therefore, for instance, is determined from the measurement of on the one hand and from that of on the other hand.
The Christoffel determinant
We shall limit ourselves to cubic, hexagonal and tetragonal crystals and consider particular cases.
In hexagonal crystals, there are five independent elastic stiffnesses
In tetragonal crystals, there are six independent elastic stiffnesses
As mentioned in Section 1.3.4.1
The use of the resonance technique is a well established approach for determining the velocity of sound
The looseness of the bonding can be checked by the regularity of the arithmetic ratio, . On account of the elastooptic coupling
Pulseecho techniques are valid for transparent and opaque materials. They are currently used for measuring ultrasonic velocities in solids and can be used in very simple as well as in sophisticated versions according to the required precision (McSkimmin, 1964
In a solid, the elastic constants are temperature and pressure dependent. As examples, the temperature dependence of the elastic stiffnesses
We can observe the following trends, which are general for stable crystals:
These observations can be quantitatively justified on the basis of an equation of state of a solid: where represents the stress tensor
Different equations of state of solids have been proposed. They correspond to different degrees of approximation that can only be discussed and understood in a microscopic theory of lattice dynamics
Table 1.3.5.1
It is interesting to compare , the `elastic Debye temperature
In the case of temperatureinduced phase transitions, some elastic constants are softened in the vicinity and sometimes far from the critical temperature
As mentioned above, anharmonic potentials
Concerning the pressure dependence of the elastic constants
In a solid body, the relation between the stress tensor
For a solid under finite strain conditions, Hooke's law
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
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 where
The coordinates that identify the particles are called material coordinates. A description that, like (1.3.6.1)
The converse of (1.3.6.1)
A spatial description or Eulerian description uses the independent variables (t, , , ), the 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)
The displacement vector from the reference position of a particle to its new position has as components
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)
Let now the particle initially at (, , ) move to (, , ). The square of the initial distance to a neighbouring particle whose initial coordinates were is The square of the final distance to the same neighbouring particle is
In a material description, the strain components are defined by the following equations: Substituting (1.3.6.6)
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)
It is often useful to introduce the Jacobian matrix associated with the transformation (a, x). The components of this matrix are where
From the definition of matrix J, one has and where , and are the transpose matrices of da, dx and J, respectively, and δ is the Kronecker matrix.
The Lagrangian strain
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
If the Helmholtz free energy
Following Brugger (1964)
If the initial energy and the deformation of the body are both zero, the first two terms in (1.3.6.9)
More accurately, the isentropic and the isothermal elastic stiffnesses
From these definitions, it follows that the Brugger stiffness coefficients
The thirdorder stiffnesses
The three independent constants for isotropic materials are often taken as , and and denoted respectively by , , , the `thirdorder Lamé constants
The `thirdorder Murnaghan constants
Similarly, the fourthorder stiffnesses form an eighthrank tensor containing components, 126 of which are independent for a triclinic crystal and 11 for isotropic materials (the independent components of a sixthrank 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
In most experiments, the initial stress is small compared with the secondorder elastic constants
Now, in order to relate the properties at X to those at , we need to specify the strain from to X. Let Consequently, The secondorder elastic constants
The elastic strainenergy density has appeared in the literature in various forms. Most of the authors use the Murnaghan constants
The elastic strainenergy density
In recent years, the measurements of ultrasonic wave velocities as functions of stresses applied to the sample and the measurements of the amplitude of harmonics generated by the passage of an ultrasonic wave throughout the sample are in current use. These experiments and others, such as the interaction of two ultrasonic waves, are interpreted from the same theoretical basis, namely nonlinear dynamical elasticity.
A first step in the development of nonlinear dynamical elasticity is the derivation of the general equations of motion for elastic waves
Finally, the concept of natural velocity is introduced and the experiments that can be used to determine the third and higherorder elastic constants
For generality, these equations will be derived in the X configuration (initial state). It is convenient to obtain the equations of motion with the aid of Lagrange's equations. In the absence of body forces
For adiabatic motionwhere U is the internal energy
Combining (1.3.7.2)
Using now the equation of continuity or conservation of mass: and the identity of Euler, Piola and Jacobi: we get an expression of Newton's law of motion: with becomes since , the thermodynamic tensor conjugate to the variable , is generally denoted as the `second Piola–Kirchoff stress tensor
Using Φ, the strain energy per unit volume, Newton's law (1.3.7.4)
As an example, let us consider the case of a plane finite amplitude wave propagating along the axis. The displacement components in this case become Thus, the Jacobian matrix reduces to
The Lagrangian strain
In this case, the strainenergy density
From (1.3.7.5)
For the particular problem discussed here, the three components of the equation of motion are
If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion:
In this case, the strainenergy
If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion: which are identical to (1.3.7.7)
The coordinates in the medium free of stress are denoted either a or . The notation is used when we have to discriminate the natural configuration, , from the initial configuration X. Here, the process that we describe refers to the propagation of an elastic wave in a medium free of stress (natural state) and the coordinates will be denoted .
Let us first examine the case of a pure longitudinal mode, i.e.
The equations of motion, (1.3.7.7)
For both cases, we have a onedimensional problem; (1.3.7.7)
The same equation is also valid when a pure longitudinal mode is propagated along [110] and [111], with the following correspondence: Let us assume that ; a perturbation solution to (1.3.7.10)
If we substitute the trial solutions into (1.3.7.10)
If additional iterations are performed, higher harmonic terms will be obtained. A well known property of the firstorder nonlinear equation (1.3.7.10)
We now consider the propagation of smallamplitude elastic waves
Starting from (1.3.7.4)
The linearized stress derivatives become If we let , the equation of motion in the initial state is The coefficients do not present the symmetry of the coefficients except in the natural state where and are equal.
The simplest solutions of the equation of motion are plane waves. We now assume plane sinusoidal waves of the form where k is the wavevector.
Substitution of (1.3.7.14)
The quantities and A are, respectively, the eigenvalues
The main experimental procedures for determining the third and higherorder elastic constants are based on the measurement of stress derivatives of ultrasonic velocities and on harmonic generation
In order to interpret wavepropagation measurements in stressed crystals, Thurston (1964)
`According to equation of motion, the wave front is a material plane which has unit normal k in the natural state; a wave front moves from the plane to the plane in the time . Thus W, the natural velocity, is the wave speed referred to natural dimensions for propagation normal to a plane of natural normal k.
In a typical ultrasonic experiment, plane waves are reflected between opposite parallel faces of a specimen, the wave fronts being parallel to these faces. One ordinarily measures a repetition frequency F, which is the inverse of the time required for a round trip between the opposite faces.'
In most experiments, the thirdorder elastic constants
Displacement vector, .
Displacement vector,
Elongation, .
Elongation,
Quadric of elongations. The displacement vector, , at P in the deformed medium is parallel to the normal to the quadric at the intersection, p, of OP with the quadric. (a) The eigenvalues all have the same sign, the quadric is an ellipsoid. (b) The eigenvalues have mixed signs, the quadric is a hyperboloid with either one sheet (shaded in light grey) or two sheets (shaded in dark grey), depending on the sign of the constant [see equation (1.3.1.7)
Quadric of elongations

Geometrical interpretation of the components of the strain tensor. , , : axes before deformation; , , : axes after deformation. 
Geometrical interpretation of the components of the strain tensor. , , : axes before deformation; , , : axes after deformation.
Geometrical interpretation of the components of the strain tensor

Special deformations. The state after deformation is represented by a dashed line. (a) Simple elongation; (b) pure shear; (c) simple shear. 
Special deformations. The state after deformation is represented by a dashed line. (a) Simple elongation; (b) pure shear; (c) simple shear.
Special deformations
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.
Definition of stress
Stress, , applied to the surface of an internal volume.
Stress, , applied to the surface of an internal volume
Equilibrium of a small volume element.
Equilibrium of a small volume element

Symmetry of the stress tensor: the moments of the couples applied to a parallelepiped compensate each other. 
Symmetry of the stress tensor: the moments of the couples applied to a parallelepiped compensate each other.
Symmetry of the stress tensor

Special forms of the stress tensor. (a) Uniaxial stress: the stress tensor has only one component, ; (b) pure shear stress: ; (c) simple shear stress: . 
Special forms of the stress tensor. (a) Uniaxial stress: the stress tensor has only one component, ; (b) pure shear stress: ; (c) simple shear stress: .
Special forms of the stress tensor
Normal () and shearing () stress.
Normal () and shearing () stress
The stress quadric: application to the determination of the stress applied to a surface element. The surface of the medium is shaded in light grey and a small surface element, dσ, is shaded in medium grey. The stress at P is proportional to at the intersection of OP with the stress quadric.
The stress quadric
Determination of the energy density in a deformed medium. PP′ represents the displacement of the small parallelepiped during the deformation. The thick arrows represent the forces applied to the faces 1 and 1′.
Determination of the energy density in a deformed medium
Bar loaded in pure tension.
Bar loaded in pure tension

Schematic stress–strain curve. T: stress; : elastic limit; : elongation; the asterisk symbolizes the rupture. 
Schematic stress–strain curve. T: stress; : elastic limit; : elongation; the asterisk symbolizes the rupture.
Schematic stress–strain curve
Spherical coordinates.
Spherical coordinates
Representation surface of the inverse of Young's modulus. (a) Al, cubic, anisotropy factor ; (b) W, cubic, anisotropy factor ; (c) NaCl, cubic, anisotropy factor ; (d) Zn, hexagonal; (e) Sn, tetragonal; (f) calcite, trigonal.
Representation surface of the inverse of Young's modulus
Resonance technique: standing waves excited in a parallelepiped.
Resonance technique: standing waves excited in a parallelepiped
Block diagram of the pulseecho technique.
Block diagram of the pulseecho technique

Temperature dependence of the elastic stiffnesses of an aluminium single crystal (after Every & McCurdy, 1992 
Temperature dependence of the elastic stiffnesses of an aluminium single crystal (after Every & McCurdy, 1992
Temperature dependence of the elastic stiffnesses of an aluminium single crystal

Pressure dependence of the elastic stiffness of a KZnF_{3} crystal. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980 
Pressure dependence of the elastic stiffness of a KZnF_{3} crystal. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980
Pressure dependence of the elastic stiffness of a KZnF_{3} crystal
Temperature dependence of the elastic constant in RbCdF_{3}, CsCdF_{3} and TlCdF_{3} crystals; the crystals of RbCdF_{3} and TlCdF_{3} undergo structural phase transitions (after Rousseau et al., 1975
Temperature dependence of the elastic constant in RbCdF_{3}, CsCdF_{3} and TlCdF_{3} crystals
Temperature dependence of the elastic constant in KNiF_{3}, which undergoes a para–antiferromagnetic phase transition. Reprinted with permission from Appl. Phys. Lett. (Nouet et al., 1972
Temperature dependence of the elastic constant in KNiF_{3}

Temperature dependence of in DyVO_{4}, which undergoes a cooperative Jahn–Teller phase transition (after Melcher & Scott, 1972 
Temperature dependence of in DyVO_{4}, which undergoes a cooperative Jahn–Teller phase transition (after Melcher & Scott, 1972
Temperature dependence of in DyVO_{4}

Pressure dependence of the elastic constants in TlCdF_{3}. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980 
Pressure dependence of the elastic constants in TlCdF_{3}. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980
Pressure dependence of the elastic constants in TlCdF_{3}

















Figure 1.3.1.1
Displacement vector, . 

Figure 1.3.1.2
Elongation, . 

Figure 1.3.1.3
Quadric of elongations. The displacement vector, , at P in the deformed medium is parallel to the normal to the quadric at the intersection, p, of OP with the quadric. (a) The eigenvalues all have the same sign, the quadric is an ellipsoid. (b) The eigenvalues have mixed signs, the quadric is a hyperboloid with either one sheet (shaded in light grey) or two sheets (shaded in dark grey), depending on the sign of the constant [see equation (1.3.1.7) 

Figure 1.3.1.4
Geometrical interpretation of the components of the strain tensor. , , : axes before deformation; , , : axes after deformation. 

Figure 1.3.1.5
Special deformations. The state after deformation is represented by a dashed line. (a) Simple elongation; (b) pure shear; (c) simple shear. 

Figure 1.3.2.1
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. 

Figure 1.3.2.2
Stress, , applied to the surface of an internal volume. 

Figure 1.3.2.3
Equilibrium of a small volume element. 

Figure 1.3.2.4
Symmetry of the stress tensor: the moments of the couples applied to a parallelepiped compensate each other. 

Figure 1.3.2.5
Special forms of the stress tensor. (a) Uniaxial stress: the stress tensor has only one component, ; (b) pure shear stress: ; (c) simple shear stress: . 

Figure 1.3.2.6
Normal () and shearing () stress. 

Figure 1.3.2.7
The stress quadric: application to the determination of the stress applied to a surface element. The surface of the medium is shaded in light grey and a small surface element, dσ, is shaded in medium grey. The stress at P is proportional to at the intersection of OP with the stress quadric. 

Figure 1.3.2.8
Determination of the energy density in a deformed medium. PP′ represents the displacement of the small parallelepiped during the deformation. The thick arrows represent the forces applied to the faces 1 and 1′. 

Figure 1.3.3.1
Bar loaded in pure tension. 

Figure 1.3.3.2
Schematic stress–strain curve. T: stress; : elastic limit; : elongation; the asterisk symbolizes the rupture. 

Figure 1.3.3.3
Spherical coordinates. 

Figure 1.3.3.4
Representation surface of the inverse of Young's modulus. (a) Al, cubic, anisotropy factor ; (b) W, cubic, anisotropy factor ; (c) NaCl, cubic, anisotropy factor ; (d) Zn, hexagonal; (e) Sn, tetragonal; (f) calcite, trigonal. 

Figure 1.3.4.1
Resonance technique: standing waves excited in a parallelepiped. 

Figure 1.3.4.2
Block diagram of the pulseecho technique. 

Figure 1.3.5.1
Temperature dependence of the elastic stiffnesses of an aluminium single crystal (after Every & McCurdy, 1992 

Figure 1.3.5.2
Pressure dependence of the elastic stiffness of a KZnF_{3} crystal. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980 

Figure 1.3.5.3
Temperature dependence of the elastic constant in RbCdF_{3}, CsCdF_{3} and TlCdF_{3} crystals; the crystals of RbCdF_{3} and TlCdF_{3} undergo structural phase transitions (after Rousseau et al., 1975 

Figure 1.3.5.4
Temperature dependence of the elastic constant in KNiF_{3}, which undergoes a para–antiferromagnetic phase transition. Reprinted with permission from Appl. Phys. Lett. (Nouet et al., 1972 

Figure 1.3.5.5
Temperature dependence of in DyVO_{4}, which undergoes a cooperative Jahn–Teller phase transition (after Melcher & Scott, 1972 

Figure 1.3.5.6
Pressure dependence of the elastic constants in TlCdF_{3}. Reproduced with permission from Ultrasonics Symposium Proc. IEEE (Fischer et al., 1980 