A Simple Theory of Asymmetric Linear Elasticity

Rotation is antisymmetric and therefore is not a coherent element of the classical elastic theory, which is characterized by symmetry. A new theory of linear elasticity is developed from the concept of asymmetric strain, which is defined as the transpose of the deformation gradient tensor to involve rotation as well as symmetric strain. The new theory basically differs from the prevailing micropolar theory or couple stress theory in that it maintains the same basis as the classical theory of linear elasticity and does not need extra concepts, such as “microrotation” and “couple stresses”. The constitutive relation of the new theory, the three-parameter Hooke’s law, comes from the theorem about isotropic asymmetric linear elastic materials. Concise differential equations of translational motion are derived consequently giving the same velocity formula for P-wave and a different one for S-wave. Differential equations of rotational motion are derived with the introduction of spin, which has an intrinsic connection with rotation. According to the new theory, S-wave essentially has rotation as large as deviatoric strain and should be referred to as “shear wave” in the context of asymmetric strain. There are nine partial differential equations for the deformation harmony condition in the new theory; these are given with the first spatial differentiations of asymmetric strain. Formulas for rotation energy, in addition to those for (symmetric) strain energy, are derived to form a complete set of formulas for the total mechanical energy.


Introduction
Modern seismology is developed based on the classical theory of linear elasticity grade that incorporates rotation as a coherent part from the pure classical linear elastic theory, independent of nonlinearity, finite deformation, and source process. S-wave has a valid identity in the new theory, but its velocity formula has a different explanation.

Stress and Strain
A theory of linear elasticity studies only small deformation of ideal homogeneous elastic continuum, in which stress tensor and strain tensor are two fundamental concepts. In the classical linear elastic theory, stress and strain are symmetric; thus, they are here referred to as symmetric stress and symmetric strain, respectively. Correspondingly, in the new theory, stress and strain, which are asymmetric, are referred to as asymmetric stress and asymmetric strain, respectively.
It is important to note that in (4) the order of the subscripts of u is "j, i" and not "i, j". Only in this way do the shear strains match the corresponding shear stresses. As shown in Figure 1, for instance, 21 p and 12 p are different and have different effects on the infinitesimal block. They act in perpendicular directions and can produce unequal  The asymmetric strain tensor can always be uniquely divided into the symmetric strain tensor and an antisymmetric rotation tensor: where rotation Rotation can also be written as a vector (Bullen, 1963, p. 14): According to (7), rotation ω is half the curl of displacement u . Therefore, the term "rigid rotation" in the classical theory is misleading because the curl cannot be regarded as rigid unless it is spatially constant. A nonzero constant distribution of the displacement curl corresponds to the rotation of a finite rigid, not elastic, body around a fixed axis. In the case of elasticity, there is no room for such constant curl of displacement.

Hooke's Law
Hooke's law defines the linear relation between stress and strain. There are only six independent components in a symmetric stress or strain tensor; thus, the generalized Hooke's law in the classical theory can be given as 11  11  11  12  13  14  15  16   12  12  21  22  23  24  25  26   13  13  31  32  33  34  35  36   22  22  41  42  43  44  45  46   23  23  51  52  53  54  55  56   61  62  63  64  65  66  33 33 There are 36 elastic constants in this symmetric relation. When the symmetric medium is also isotropic, the number of constants is decreased to two, and the symmetric Hooke's law is obtained [23]:  (10) indicates dilatation, λ and µ are Lamé's constants, and δ is the Kronecker delta. One remarkable but usually ignored point of the Hooke's law (9) is that it formally excludes rotation from the classical theory, which is effectively equivalent to assuming 0 ij ω = .
In the new asymmetric theory, the generalized Hooke's law is written as where the 81 constants produces an entire fourth-order tensor ijmn c ( , , , 1, 2, 3 i j m n = ). Relation (11) can be concisely rewritten as When the asymmetric medium is isotropic, the number of independent elastic constants is decreased to three, and the asymmetric Hooke's law can be written as [24]: λδ θ µ µ = + + (13) or ( ) ( ) where kk kk q θ ε 1 µ and 2 µ are two independent constants, here termed the parallel and the perpendicular shear modulus, respectively. When there is symmetry, ij p be- The asymmetric stress tensor can be further divided into the symmetric stress tensor and an antisymmetric tensor ij η : Here, is termed stress torque, which can also be written as a vector. By introducing (9) into (17), which is then compared with (14), the cause of rotation is eventually found: ( )  (14) reverts to (9), and an asymmetric medium turns back to a symmetric medium. In a symmetric medium, even if rotation is produced at a source, it will not be able to spread out.
Other elastic constants for the new asymmetric theory can be easily rewritten with the substitution in (16). For example: and Poisson's ratio ( )

Differential Equations of Translational Motion
Differential equations of translational motion are originally derived from the momentum principle or Newton's second law. In seismology, the term of body force in the equations is often ignored to obtain the equations of body waves.
In the classical theory of linear elasticity, the differential equation of translational motion in terms of stress can be written in indicial notation as where ρ indicates density. By introducing the symmetric Hooke's law (9), stress is replaced with strain, and the equation becomes Further plugging (2) into (22) for displacement to substitute for strain finally obtains the equation in vector notation [23]: Equation (23) can be rewritten by using the general relation When rotation is absent, 0 ∇ × ≡ ω , the following equation for P-wave is obtained from (25): (26) which yields P-wave velocity as 2 λ µ α ρ This is why P-wave is termed "irrotational wave" [1] [2].
In the classical theory, when there is no change in dilatation, 0 and S-wave velocity is expressed as This is why S-wave is called "equivoluminal wave" [2].
However, (28) and (29) may not be valid. As concluded in the previous chapter, 0 ij ω = is implicitly assumed in the establishment of the symmetric Hooke's law (9) in the classical theory. Therefore, the relation (24) becomes By substituting (30) into (23), Equation (26) is again obtained, but not (28).
This implies that there is only one type of body wave, namely, P-wave, which propagates in a symmetric medium.
In the new asymmetric theory, corresponding to the translational Equations then, by applying the asymmetric Hooke's law (13), the equation in terms of and, by further applying (4), the equation in terms of displacement By using the general relation (24), Equation (33) can be rewritten as For "irrotational" P-wave, 0 ∇ × ≡ ω ; thus, from (34) are obtained P-wave eq-   (35) and the formula for P-wave velocity 1 2 λ µ µ α ρ Equation (36) is in fact the same as (27).
Regarding "equivoluminal" S-wave, 0 θ ∇ ≡ ; thus from (33) are obtained S-wave equation and the formula for S-wave velocity Interestingly, S-wave velocity is determined only by 1 The velocity formulas for P-and S-waves can also be derived by introducing a scalar potential ( ) ,t φ x and a vector potential ( ) ,t x ψ to separate the displacement field into two exclusively different parts, one with no rotation and the other with no dilatation [23]: This can be proven to achieve the same results (see Appendix). Equation (39) clarifies that there are only two kinds of body waves, P-wave corresponding to

Deformation Constitution of P-and S-Waves
The entire asymmetric strain tensor can be divided into three portions, including the mean strain tensor 3 ij θδ , the strain deviator tensor D ij ε , and the rotation tensor: where the strain deviator P-and S-waves are also reasonably called "dilatational" and "rotational" waves, respectively [2]. Interestingly, according to (39), whereas dilatation must go with P-wave, and rotation with S-wave, deviatoric strain is left to be explained.
Based on the above discussions, P-wave has dilatation, and S-wave has rotation.
However, it is not that merely dilatation travels with P-wave and rotation alone , Figure 2 shows how, in the ( ) 1 2 , x x plane, elastic material deforms in P-wave (longitudinal wave) and S-wave (transverse wave), respectively, which proceed in the 1 x direction as plane waves. For P-wave, the strain tensor is where 1,1 u θ = . For S-wave, the strain tensor is Note that (45) is an asymmetric tensor and 0 θ = . This is the simplest expression of S-wave to prove its asymmetry.
The symmetric strain of P-wave involves two parts: mean strain and deviatoric strain:  The asymmetric strain of S-wave also contains two parts: deviatoric strain and rotation: As shown in Figure 2(b), rotation must be considered together with deviatoric strain for S-wave. Note that, according to (47), the amount of rotation is not smaller than but just as large as that of the deviatoric strain in S-wave. In fact, rotation and deviatoric strain occur as one inseparable identity in S-wave, as indicated by (45).
In general, the real deformation constitution of a certain wave should satisfy the condition of compatibility. This requires that continuous single-valued displacements can be obtained by integrating the strains.
In the classical theory, there are six strain-displacement relations given by (2) In three dimensions, a total of nine equations of the compatibility condition are obtained for the asymmetric strain field to exist: Assuming that the 1 x -axis directs the proceeding of the wave so that

Differential Equations of Rotational Motion
Differential equations of rotational motion are originally derived from the moment of momentum principle. In the classical theory, there is no such equation because there is no rotation. Instead, the symmetry of stress occurs in the static equilibrium of the moment of momentum [23]. In the new asymmetric theory, the differential equations can be easily deduced by comparison of the integral equation of the theory of micropolar elasticity with that of the new theory regarding the moment of momentum principle.
In the classical theory, spin is not considered when the moment of momentum principle is applied to a body of volume V and surface S to establish the following equation in the integral form [3] [15] [25]: x denotes the position of the reference point of the moments, b represents the body force, and the surface traction t can be written in indicial notation as where i n is the normal vector of S. The term on the left-hand side of (53) in- where s stands for spin, that is, the spin angular momentum per unit mass of an infinitesimal volume of the body. In order to deduce the differential equation of rotational motion from Equation (56), the integral equation of angular momentum of the theory of micropolar elasticity is first considered [15] [17] [21].
In the theory of micropolar elasticity, spin can be produced by couple stresses, including surface couples and body couples. The integral equation of the theory of micropolar elasticity takes the form [15] ( where m and c are, respectively, the surface and body couples to produce spin. After a difficult derivation, an inference from (57) takes the form [15] li or written in vector notation: Equation (61) shows that spin can be caused by stress torque or, put the other way around, stress torque is induced by spin.
By comparing (18) and (61), the intrinsic connection between rotation and spin can be easily determined: The time-dependent property of s , which is a quantity of motion, indicates that rotation should be related only to dynamic problems and does not exist stat- ically. Thus, we clarify that without rotation, the theory of symmetric linear elasticity can be perfectly valid in the study of static problems.
Note that spin is here in the new theory referred to an infinitesimal block of deformable medium, but not to a rigid particle as in the theory of micropolar elasticity. The former can bear shear stress, whereas the latter cannot.

Rotation Energy
For simplicity, in the following discussion, the term "strain" is used for symmetric strain, and "deformation" for asymmetric strain. Accordingly, strain energy refers to symmetric strain energy, and deformation energy denotes asymmetric strain energy, which additionally involves rotation energy.
Disregarding body force and dissipation, the total mechanical energy U of an elastic body is equal to the sum of the work done by the surface traction: In the symmetric theory, the surface traction In the new theory, the surface traction i ji j t p n = . (68) Accordingly, by a similar derivation to (65), the total mechanical energy is obtained as ( ) and deformation energy Rotation energy can also be written in the form of a surface integral as ( ) The term "rigid rotation" may mislead one into thinking that there is no deformation energy associated with rotation. In fact, however, there is. Deformation should not be related merely to an infinitesimal block itself. Deformation occurs due to the spatial variation of displacement or, in other words, due to deformational interactions between adjacent infinitesimal blocks. Figure 3 shows a deformational interaction between an infinitesimal area and its neighborhood, in which either strain or rotation of the infinitesimal area appears. When strain occurs to the imaginary circle, it becomes an oval. However, only such transformation of the area shape is not the real meaning of deformation. What's important is that its neighborhood is affected to deform. When rotation occurs to the circle, although the circle itself does not seem to deform, the neighborhood is affected to deform, too.

Discussion
The fact that rotation is not incorporated in the classical theory has already been recognized by many researchers but may still be unclear to others. Some consider rotation to be implicitly involved in, but not excluded from, the equations of the classical theory, a view that is wrong because it is impossible. Mathematically, when rotation is considered, it appears as nonzero. Otherwise, it is zero and hence not involved. In fact, rotation is treated inconsistently in the classical theory, being excluded from the core but included in the inferences, for example, on the S-wave velocity.
Given the study results on the fourth-order tensor, it is not difficult to figure out the asymmetric Hooke's law (13), which results in the definition of stress torque and its relation to rotation (18). Also, given the equations of rotational motion of the theory of micropolar elasticity (57) and (58), it is easy to draw the conclusion of the new theory regarding the relation between stress torque and spin (61). Both relations in (18) and (61) confirm the asymmetry of stress and the association between rotation and spin.
The focus is on rotation, which is essentially excluded from the core of the classical theory. The prevailing asymmetric theories that address this problem originated from the Cosserat brothers' theory, which uses the concept of particle spin, followed by couple stresses, to explain rotation. However, particle spin is not an appropriate counterpart. The rotation involved is initially defined in terms of a deformation gradient; thus, the problem must be tackled by using relevant concepts. The introduction of the concept of particle spin is misleading and renders the asymmetric theories too sophisticated to be applicable.
The introduction of spin adds three more degrees of freedom to the infinitesimal block in the classical theory to make up a total of six, as required by Newton's laws of mechanics. However, the relation (62) links spin and rotation together to reduce the degrees of freedom from six to three. Therefore, solutions of the problems in the new theory can also be given in terms of only displacements, without spin.
Because S-wave is intrinsically rotational, there would be no S-wave without rotation. Factually, however, S-waves are always observed in seismic recordings, so there must be rotation although usually not directly observable due to the lack of proper instruments.
The new theory has shown for the first time the deformation constitution of S-wave (45) and has clarified that when S-wave is called "shear wave," it should be in the context of asymmetric strain, which includes both symmetric strain and rotation (47). Moreover, according to (47), rotation is not as small as "negligible" [28] but generally is as large as the deviatoric strain in S waves.

Conclusions
The new theory of asymmetric linear elasticity is established self-consistently in logic. It provides a better description of the mechanical behavior of the ideal me-  (13), is established based on the definition of asymmetric strain (4) and the theorem about isotropic asymmetric linear elastic materials. The new Hooke's law (13) reveals that rotation is caused by stress torque, the difference between unequal conjugate shear stresses.  Concise differential equations of translational motion are derived consequently giving the same velocity formula for P-wave and a different one for S-wave (38). For instance, the new formula for S-wave velocity provides the key to solving the problem of why the "static moduli" of rocks are usually significantly smaller than the "dynamic moduli" [29] [30]. More importantly, the prevailing earth model has to be modified for those elastic parameters calculated with the old formulas of S-wave velocity.  Differential equations of rotational motion are developed with the introduction of spin, which has an intrinsic connection with rotation. It further unveils the relation between rotation and stress torque.  There are nine partial differential equations for the deformation harmony condition in the new theory; these are given with the first spatial differentiations of the asymmetric strain. For instance, according to the new condition, S-wave cannot travel without either symmetric shear strain or rotation. Similarly, some theoretical conclusions in modern seismology should be reexamined with this condition.  Formulas for rotation energy, in addition to those for strain energy, are derived to form a complete set of formulas for the total mechanical energy.