弹性波动力学笔记(三) 应变张量简介上

Introduction of Deformation, Strain and Rotation Tensors

Elasticity theory, which lies at the core of seismology, is most generally studies as a branch of continuum mechanics. Although this general approach is not required in introductory courses, it is adopted here for two reasons. Firstly, it affords a number of insights either not provided, or not as easily derivable, when a more restricted approach is used. Secondly, it is essential to solve advanced seismology problems, as can be seen from the discussion in Box 8.5 of Aki-Richard (1980), and references therein, or from the book by Dahlen and Tromp (1998).

Continuum mechanics studies the deformation and motion of bodies ignoring the discrete nature of matter, and "confines itself to relations among gross phenomena, neglecting the structure of the material on a smaller scale". In this phenomenological approach a number of basic concepts and principles are used to derive general relations applicable, in principle, to all types of media (e.g., different types of solids and fluids). These relations, however, are not enough to solve specific problems. To do that, additional relations, known as constitutive equations , are needed. These equations, in turn, are used to define ideal materials, such as perfect fluids, viscous fluids, and perfect elastic bodies. Therefore, it can be argued that although continuum mechanics dose not take into consideration the real nature of matter, the latter ultimately expresses itself through the different constitutive laws, and the values that the parameters required to specify a particular material can take.

For elastic solids, for example, the constitutive equation is the relation between stress and strain. The result of combining this equation with general continuum mechanics relations is the elastic wave equation, but before this equation can be used to solve particular problems it is necessary to specify \(c_{ijkl}\). In the simplest case (isotropic media) two parameters are sufficient, but in the most general case 21 parameters are required. In either case, these parameters depend on the type of media under consideration. In the case of the Earth, in particular, they span a wide range, corresponding to the presence of different types of rocks and minerals, with physical properties that depend on factors such as chemical composition and structure， temperature and pressure.

2.1 Description of motion. Lagrangian and Eulerian points of view

Strain and rotation are two manifestations of deformation. when a body undergoes deformation, the relative of the particles of the body change. To see this, consider the following experiment. On the surface of an inflated balloon in the vicinity of the points. The result of this experiment is a change in the relative positions of the four points. Changes in the angles at the corners of the square are also possible. Furthermore, if the squeezing is performed slowly, it will also be observed that the positions of the four points change continuously as time progresses. If we want to describe the change in position of one of the four points mathematically, we have to find a function that describes the evolution of the coordinates of the point. This function will depend on the original location of the point and on the time of observation. Alternatively, we may be interested in finding the original position of one of the points from the knowledge of the position of the point some time after the deformation. The following discussion addresses these questions from a general point of view, although we will concentrate on small deformations because they constitute a basic element in the theory of wave propagation in the Earth.

Figure 2.1 shows a volume \(V_{0}\) inside a body before deformation begins, and an external coordinate system that remains fixed during the deformation. All the positions describe below are referred to this system. The use of upper- and lowercase symbols distinguishes undeformed and deformed states; it does not mean that two different coordinate systems are used. \(V(t)\) is the volume at time \(t_0+t\) occupied by the particles initially in the volume \(V_0\) occupied at time \(t_0\).

Let \(\mathbf{R}=(X_1,X_2,X_3)\) indicate the vector position corresponding to a particle in the undeformed body, and \(\mathbf{r}=(x_1,x_2,x_3)\) the vector position in the deformed body corresponding to the particle that originally was at \(\mathbf{R}\). Vector \(\mathbf{R}\) uniquely labels the particles of the body, while vector \(\mathbf{r}\) describes the motion of the particles. We will use uppercase letters when referring to particles. Since \(\mathbf{R}\) denotes a generic particle initially inside a volume \(V_0\) , the vector

\[\mathbf{r}=\mathbf{r}(\mathbf{R},t) \tag{2.1.1} \]

represents the motion (deformation) of all the particles that where in \(V_0\) before the deformation began. In component form (2.1.1) can be written as

\[ r_i=x_i=x_i(X_1,X_2,X_3,t), i=1,2,3 \tag{2.1.2} \]

Furthermore, it is a well-known result from calculus that given \(\mathbf{r}=\mathbf{r}(\mathbf{R},t)\) and assuming that the Jacobian:

\[J = \frac{{\partial \left( {{x_1},{x_2},{x_3}} \right)}}{{\partial \left( {{X_1},{X_2},{X_3}} \right)}} = \left| {\begin{array}{*{20}{c}} {\frac{{\partial {x_1}}}{{\partial {X_1}}}}&{\frac{{\partial {x_1}}}{{\partial {X_2}}}}&{\frac{{\partial {x_1}}}{{\partial {X_3}}}}\\ {\frac{{\partial {x_2}}}{{\partial {X_1}}}}&{\frac{{\partial {x_2}}}{{\partial {X_2}}}}&{\frac{{\partial {x_2}}}{{\partial {X_3}}}}\\ {\frac{{\partial {x_3}}}{{\partial {X_1}}}}&{\frac{{\partial {x_3}}}{{\partial {X_2}}}}&{\frac{{\partial {x_3}}}{{\partial {X_3}}}} \end{array}} \right| \tag{2.1.3} \]

is different from zero, it is possible to write \(\mathbf{R}\) in terms of \(\mathbf{r}\) :

\[\mathbf{R}=\mathbf{R}(\mathbf{r},t) \tag{2.1.4} \]

which in component form is written as

\[R_i=X_i=X_i(x_1,x_2,x_3,t) \tag{2.1.5} \]

Equation (2.1.1) corresponds to the so-called Lagrangian (or material) description of motion, while Equation(2.1.4) corresponds to the Eulerian (or spatial) description.

We will chose as reference time \(t_0=0\), so that

\[\mathbf{r}(\mathbf{R},t_0)=\mathbf{R} \tag{2.1.6} \]

and \(\mathbf{r}(\mathbf{R},t)\) is the current position of the particle that was at \(\mathbf{R}\) at time \(t=0\).

In the Lagrangian description we follow the motion of a specified particle \(\mathbf{R}\), while in the Eulerian description we are interested in particle that occupies a given point \(\mathbf{r}\) in space at a given time \(t\).

2.2 Finite Strain Tensors

Let \(\mathbf{dR}\) and \(\mathbf{dr}\) be vector line elements corresponding to \(\mathbf{R}\) and \(\mathbf{r}\)

\[d\mathbf{R}=(dX_1,dX_2,dX_3)\tag{2.2.1} \]

\[d\mathbf{r}=(dx_1,dx_2,dx_3) \tag{2.2.2} \]

Vectors \(d\mathbf{R}\) and \(d\mathbf{r}\) represent the same two close points before and after deformation, respectively. Their lengths, given by

\[dS=|d\mathbf{R}|=(d\mathbf{R}\cdot d\mathbf{R})^{1/2}=(dX_idX_i)^{1/2} \tag{2.2.3} \]

and

\[ds=|d\mathbf{r}|=(d\mathbf{r}\cdot d\mathbf{r})^{1/2}=(dx_idx_i)^{1/2} \tag{2.2.4} \]

will be used to quantify the deformation. Consider the difference

\[(ds)^2-(dS)^2=dx_idx_i-dX_kdX_k \tag{2.2.5} \]

Using the Lagrangian description (see (2.1.1)), equation (2.2.5) can be rewritten as

\[(ds)^2-(dS)^2=\frac{\part{x_i}}{\part{X_k}}dX_k\frac{\part{x_i}}{\part{Xl}}dX_l-dX_kdX_l\delta_{kl} \tag{2.2.6} \]

where the chain rule

\[dx_i=\frac{\part{x_i}}{\part{X_j}}dX_j=x_{i,j}dX_j \tag{2.2.7} \]

has been used, and the last term of equation (2.3.6) has been written so that the two terms can be combined into one:

\[(ds)^2-(dS)^2=(x_{i,k}x_{i,l}-\delta_{kl})dX_kdX_l=2L_{kl}dX_kdX_l \tag{2.2.8} \]

where

\[L_{kl}=\frac{1}{2}(x_{i,k}x_{i,l}-\delta_{kl}) \tag{2.2.9} \]

is known as Green （or Lagrangian）finite strain tensor.

In the Eulerian description (see (2.1.4)) the equivalent result is

\[ (ds)^2-(dS)^2=(\delta_{kl}-X_{i,k}X_{i,l})dx_kdx_l=2E_{kl}dx_kdx_l \tag{2.2.10} \]

where

\[E_{kl}=\frac{1}{2}(\delta_{kl}-X_{i,k}X_{i,l}) \tag{2.2.21} \]

is known as the Eulerian (or Almansi) finite strain tensor.

We now introduce the displacement vector \(\mathbf{u}\)

\[\mathbf{u}=\mathbf{r}-\mathbf{R} \tag{2.2.12} \]

An example of displacement is the ground motion felt in an earthquake. In component form (2.2.12) is written as

\[u_i=x_i-X_i \tag{2.2.13} \]

Therefore

\[x_i=u_i+X_i \tag{2.2.14} \]

so that

\[x_{i,k}=\frac{\part{x_i}}{\part{X_k}}=u_{i,k}+\delta_{ik} \tag{2.2.15} \]

and

\[L_{kl}=\frac{1}{2}[(u_{i,k}+\delta_{ik})(u_{i,l}+\delta_{il})-\delta_{kl}]=\frac{1}{2}(u_{k,l}+u_{l,k}+u_{k,l}u_{l,k}) \tag{2.2.16} \]

Alternatively in Euler's description, it is possible to write

\[X_i=x_i-u_i \tag{2.2.17} \]

so that

\[ X_{i,k}=\frac{\part{X_i}}{\part{x_k}}=\delta_{ik}-u_{i,k} \tag{2.2.18} \]

and

\[E_{kl}=\frac{1}{2}(u_{l,k}+u_{k,l}-u_{i,k}u_{i,l}) \tag{2.2.19} \]

2.3 The Infinitesimal Strain Tensor

Let us assume that the deformation is so small that the products of derivatives in equations (2.2.16) and (2.2.19) can be neglected. In this case we introduce the infinitesimal strain tensor \(\varepsilon_{kl}\)

\[\varepsilon_{kl}=\frac{1}{2}(u_{k,l}+u_{l,k}) \tag{2.3.1} \]

The assumption of small deformation has the effect of making the distinction between Lagrangian and Eulerian descriptions unnecessary, so that

\[\frac{\part}{\part{X_l}}=\frac{\part}{\part{x_l}} \tag{2.3.2} \]

to first order. However, we will keep the distinction in the following discussions, as it makes clear what quantities are involved in the deformation.

In the following, reference to the strain tensor will mean the tensor \(\varepsilon_{kl}\) . That this entity is, in fact, a tensor is shown below. From its definition it is clear that \(\varepsilon_{kl}\) is symmetric, with diagonal elements \(\varepsilon_{JJ}=u_{J,J}\) (no summation over uppercase indices). These elements are known as normal strains, while \(\varepsilon_{i,j}(i\neq j)\) are known as shearing strains. Because \(\varepsilon_{kl}\) is symmetric, it can be diagonalized. The eigenvectors of \(\varepsilon_{kl}\) are known as the directions of stain and the eigenvalues as the principal strains.

Equation(2.3.1) shows how to compute the strain tensor given a differentiable displacement field \(\mathbf{u}(\mathbf{R},t)\). On the other hand, equation (2.3.1) can also be viewed as a system of six equations in three unknowns, \(u_1,u_2,u_3\). Because the number of equations is greater than the number of unknowns, in general equation (2.3.1) will not have a solution for arbitrary values of the strain components. This raises the following equation: what restrictions must be placed on \(\varepsilon_{kl}(\mathbf{R},t)\) to ensure that \(\mathbf{u}(\mathbf{R},t)\) is a single-valued continuous field? The answer is that the strain components must satisfy the following equation:

\[\varepsilon_{ij,kl}+\varepsilon_{kl,ij}-\varepsilon_{ik,jl}-\varepsilon_{jl,ik}=0 \tag{2.3.3} \]

The system (2.4.3) represents \(3^4=81\) equations, but because some of them are satisfied identically and others are repeated because of the symmetry of \(\varepsilon_{jl}\) , equation (2.3.3) reduces to six equations, known as compatibility equations or conditions. Two representative equations are

\[2\varepsilon_{23,23}=\varepsilon_{22,33}+\varepsilon_{33,22} \tag{2.3.4} \]

\[ \varepsilon_{33,12}=-\varepsilon_{12,33}+\varepsilon_{23,13}+\varepsilon_{31,23} \tag{2.3.5} \]

2.4 Geometric meaning of \(\varepsilon_{ij}\)

Here it will be shown that the diagonal and off-diagonal elements of \(\varepsilon_{ij}\) are related to changes in lengths and angles, respectively, and that the trace of \(\varepsilon_{ij}\) is related to a change in volume.

We will consider the diagonal elements first. For small deformations, from (2.2.8),(2.2.10) and (2.4.1) we obtain

\[(ds)^2-(dS)^2=2\varepsilon_{kl}dX_kdX_l \tag{2.4.1} \]

Furthermore, \(ds=dS\), so that

\[(ds)^2-(dS)^2=(ds-dS)(ds+dS)=2dS(ds-dS) \tag{2.4.2} \]

Introducing this approximation in equation (2.4.2) and dividing by \((dS)^2\) gives

\[ \frac{ds-dS}{dS}=\varepsilon_{kl}\frac{dX_k}{dS}\frac{dX_l}{dS} \tag{2.4.3} \]

The left-hand side of (2.4.8) is the change in length per unit length of the original line element \(dS\). In addition, since \(dX_i\) is the element of \(d\mathbf{R}\) and \(dS\) is the length of \(d\mathbf{R}\), \(dX_i/dS\) represents the direction cosines of the line element.

No assume that the line element is along the \(X_1\) axis, so that

\[\frac{dX_1}{dS}=1, \space \frac{X_2}{dS}=0,\space \frac{dX_3}{dS}=0 \tag{2.4.4} \]

and

\[\frac{ds-dS}{dS}=\varepsilon_{11} \tag{2.4.5} \]

Similarly, if the line element is along the \(X_2\) or \(X_3\) axes, the fractional change is equal to \(\varepsilon_{22}\) , \(\varepsilon_{33}\). Therefore the diagonal elements of the strain tensor, known as normal strain, represent the fractional changes in the lengths of the line elements that prior to the deformation were along the coordinate axes.

To analyze the geometrical meaning of the off- diagonal elements, let us consider two line elements \(d\mathbf{R}^{(1)}=(dS_1,0,0)\) and \(d\mathbf{R}^{(2)}=(0,dS_2,0)\) along the \(X_1\) and \(X_2\) coordinate axes, respectively, and let \(d\mathbf{r}^{(1)}\) and \(d\mathbf{r}^{(2)}\) be the corresponding line elements in the deformed state. The components of the line elements before and after deformed state. The components of the line elements before and after deformation are related by equation (2.2.7)

\[dx_i^{(1)}=x_{i,k}dX_k^{(1)}=x_{i,1}dS_1;\space i=1,2,3 \tag{2.4.6} \]

and

\[dx_i^{(2)}=x_{i,k}dX_k^{(2)}=x_{i,2}dS_2; \space i=1,2,3 \tag{2.4.7} \]

Furthermore, using equation (2.2.15) we can gain

\[d\mathbf{r}^{(1)}=(dx_1^{(1)},dx_2^{(1)},dx_3^{(1)})=(1+u_{1,1},u_{2,1},u_{3,1})dS_1 \tag{2.4.8} \]

and

\[d\mathbf{r}^{(2)}=(dx_1^{(2)},dx_2^{(2)},dx_3^{(2)})=(u_{1,2},1+u_{2,2},u_{3,2})dS_2 \tag{2.4.9} \]

Neglecting second-order terms (infinitesimal deformation), the length of \(d\mathbf{r}^{(1)}\) is given by

\[ds_1=(d\mathbf{r}^{(1)}\cdot d\mathbf{r}^{(1)})^{1/2}=(1+2u_{1,1})^{1/2}dS_1=(1+u_{1,1})dS_1 \tag{2.4.10} \]

where in the last step only the first two terms in the series expansion of the square root have been retained. Similarly,

\[ds_2=(1+u_{2,2})dS_2. \tag{2.4.11} \]

Note that (2.4.10) can be rewritten as

\[\frac{ds_1-dS_1}{dS_1}=u_{1,1}=\varepsilon_{11} \tag{2.4.12} \]

which is the same as equation (2.4.5).

Also note that equations (2.4.8) and (2.4.9) show that

\[ds_{J}=dS_{J} \tag{2.4.13} \]

provided that \(1>>|u_{J,J}|\) (no summation over \(J\)).

Now consider the scalar product between \(d\mathbf{r}^{(1)}\) and \(d\mathbf{r}^{(2)}\) . Using equation (2.4.8) and (2.4.9), (2.4.13) and neglecting second-order terms we find

\[ d{{\bf{r}}^{(1)}}d{{\bf{r}}^{(2)}} = d{s_1}d{s_2}\cos \theta \approx \left( {{u_{1,2}} + {u_{2,1}}} \right)d{S_1}d{S_2} \approx 2{\varepsilon _{12}}d{s_1}d{s_2} \tag{2.4.14} \]

where \(\theta\) is the angle between the line elements. Furthermore, let \(\gamma=\pi/2-\theta\). Since the angle between the two line elements in the undeformed state was \(\pi/2\), \(\gamma\) represents the change in the angle between the line elements caused by the deformation. Introducing \(\gamma\) in equation (2.4.14) and using the approximation \(\sin\gamma\approx \gamma\) , valid for small deformations, we find

\[\varepsilon_{12}=\frac{1}{2}\gamma \tag{2.4.15} \]

Therefore, \(\varepsilon_{12}\) represents one-half the change in the angle between two line elements that were originally along the \(X_1\) and \(X_2\) axes. Similar interpretation apply to the other non-diagonal elements.

Finally, we will consider an infinitesimal parallelepiped with sides originally along the coordinate axes and lengths equal to \(dS_1\), \(dS_2\), and \(dS_3\). After deformation the corresponding lengths become \(ds_1\), \(ds_2\), and \(ds_3\), respectively. Using equation (2.4.11) and (2.4.10) and a similar relation for \(ds_3\), the volume of the deformed body is given by

\[d{s_1}d{s_2}d{s_3}=(1+u_{i,i})d{S_1}d{S_2}d{S_3} \tag{2.4.16} \]

If \(V_0\) and \(V_0+d{V}\) are the volumes of the small elements before and after deformation, then (2.4.16) can be written as

\[\frac{d{V}}{V_0}=u_{i,i}=\nabla\cdot \bf{u}=\varepsilon_{ii} \tag{2.4.17} \]

The last equality follows from (2.3.1). This relationship between the fractional change in volume and the divergence of the displacement vector \(\bf{u}\) is independent of the coordinate system used to derive it, because \(u_{i,i}\) is the trace of \(\varepsilon_{ij}\) , which is a tensor invariant.