Solution of Certain Problems in Quantum Mechanics (Dover Books on Physics)
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Author However, if the motion is steady, so that v is independent of t, then 4. Also determine the streamlines. The velocity at a point x in space in a body of fluid in steady flow is given by where U, V and a are constants. Clearly t n will depend on the position of P and the direction of n. Henceforth the symbol t n is used to denote the limit 5. The assumptions made above are plausible, but they are of a physical nature and can only be justified to the extent that conclusions based on them agree with observations of what happens to real materials.
It is possible for couples as well as forces to be transmitted across a surface. Such couples are of interest but are beyond the scope of this book. In practice their influence is restricted to rather special situations. Figure 5. The force transmitted across a surface does not necessarily act in the direction normal to the surface. In particular, given a system of rectangular cartesian coordinates with base vectors ei, there is such a vector associated with the direction of each of the base vectors.
Let t1 be the surface traction associated with the direction of e1, from the positive to the negative side that is, t1 is the force per unit area exerted on the negative side of a surface normal to the x1-axis by the material on the positive side of this surface; see Fig. Surface traction vectors t2 and t3 are similarly defined in relation to the directions of e2 and e3. Now resolve the vectors t1, t2 and t3 into components in the coordinate system with base vectors e i, as follows: 5.
The component T11, for example, is the component of t1 in the direction of e1. T11 is positive if the material on the x1-positive side of the surface on which t1 acts a surface normal to the x1-axis is pulling the material on the x1-negative side. The material is then in tension in the x1 direction. The material on the negative side of the surface is pulling in the opposite direction on the material on the positive side. The components T11, T22 and T33 are called normal or direct stress components. The remaining components T12, T13, etc.
All the stress components can be illustrated as the components of forces acting on the faces of a unit cube, as shown in Fig. We consider how we may determine the surface traction on an arbitrary surface through P. For this we examine the forces acting on the elementary tetrahedron illustrated in Fig. We wish to find the traction t n on a surface normal to n at P. In the tetrahedron shown in Fig. The minus signs arise because we wish to consider the forces acting on the tetrahedron, so that, for example,—t1 is the traction exerted on the surface PQ2Q3 by material to the left of this surface, on material to the right of the surface; that is, by the material outside the tetrahedron on the material inside the tetrahedron.
Similarly, t n denotes che mean surface traction on Q1Q2Q3 exerted by material on the side towards which n is directed the outside of the tetrahedron onto the other side. Then the areas of the other faces are 5. The most common example of a body force is a gravitational force, but there are other possibilities. We now assume that for any part of a body, and in particular for the elementary tetrahedron PQ1Q2Q3, the rate of change of momentum is proportional to the resultant force acting.
Moreover, it is an assumption which cannot be verified directly by experiment, for it is impossible to make direct measurements of internal surface tractions; their existence and magnitudes can only be inferred from observations of other quantities. Thus in this limit, 5. This gives the traction on any surface with unit normal n in terms of the stress components Tij. The components tj n of t n are given by 5.
The choice of a different coordinate system will lead to a different set of stress components. From 5. Then, from 5. The general relation is 5. T is called the Cauchy stress tensor, and it completely describes the state of stress of a body. Some other stress tensors will be considered briefly in Section 9. Equation 5. They are: a. A numerical example is given in Problem 1 in Section 5. Since it is now established that Tij are components of a tensor, equation 5. The notation of Section 5. In terms of components, 5.
It is assumed that Tij have continuous first derivatives. Then 5. Equations 5. In Section 7. However, it may happen that for certain special directions n the traction t n does act in the direction n. We investigate this possibility. Let the unit vectors in the directions of the principal axes be n1, n2 and n3. If these three orthogonal vectors are taken as base vectors at P then, referred to these axes, the matrix of the stress components is a diagonal matrix with diagonal elements T1, T2 and T3.
The principal components are the roots of the equation 5. In general, the principal directions vary from point to point, so that it is not usually possible to find a rectangular cartesian coordinate system in which the matrix of stress components is a diagonal matrix everywhere. It is shown in Example 5.
This property can be used to give an alternative definition of the principal stress components and principal axes of stress. Example 5. Hence the traction components on this plane are given by 5. Similarly, the direction ratios of the other two principal directions are— and —3 note that these directions are mutually orthogonal. Prove that as the orientation of a surface through P varies, T1 is the greatest, and T3 is the least, normal component of traction on the surface assume that T1, T2 and T3 are all different.
As T is a symmetric second-order tensor, the discussion of Section 3.
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This is called a pure hydrostatic state of stress, and p is the hydrostatic pressure. The negative sign arises because we conventionally regard pressure, which causes compression, as positive, but we define compressive stress as negative. The principal axes of S are the same as those of T. If the principal components of S are S1, S2, S3, then 5.
These are taken to be and , where 5. It is sometimes convenient to adopt J1, and as a set of basic invariants of T. The shear stress on this surface is the resultant of the other two components T12e2 and T13e3 of the traction on the surface. Hence the shear stress has magnitude , and acts in a direction which lies in the surface. Then it can be shown that, as n varies at point P, the magnitude of the shear stress on the surface normal to n reaches a maximum value T1 - T3 when n lies along either of the bisectors of the angle between n1 and n3.
The proof resembles that of Example 5. The stress components take the form 5. This is the state of stress in any fluid in equilibrium that is, in hydrostatics , or in an inviscid fluid whether it is in equilibrium or not. The pressure p is, in general, a function of position. In the remaining examples, body forces will be regarded as negligible and we seek stress states which satisfy the equilibrium equations 5. Nevertheless, they must be satisfied for any body in equilibrium, and it is of interest to examine some stress states which satisfy them.
When the body force is neglected, they are satisfied if the Tij are all constants, in which case the stress is homogeneous. The next two examples are in this category. The principal stress directions are the x1 direction and any two directions orthogonal to each other and to the x1 direction. The principal axes of stress have the directions of the x3-axis and the two bisectors of the x1 - and x2-axes.
Let 5. This approximates the stress in a prismatic beam, with generators parallel to the x1-axis, which is bent by end couples applied to its ends and acting about axes parallel to the x3-axis. The principal stress directions are as in b above. The x3 direction is a principal direction; the other two principal directions are in the plane of the plate.
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Suppose that 5. This corresponds to the state of stress in a circular cylindrical bar whose axis coincides with the x3-axis and which is twisted by couples acting about the axis of the cylinder and applied to the ends of the cylinder, with no forces acting on the curved surfaces. The principal directions are the radial direction and the bisectors of the tangential and axial directions. Verify that the principal axes of stress are mutually orthogonal. Use the answer to check the answers to d and e above.
If, in appropriate units find the principal components of stress, and show that the principal directions which correspond to the greatest and least principal components are both perpendicular to the x2-axis. The stress tensor has components where A, B and C are constants. The stress in the cantilever beam of Problem 4 is now given by where C and D are constants. Find the principal stress direction which corresponds to the intermediate principal stress component.
Let n be a unit vector, t n the traction on the surface normal to n, and S the magnitude of the shear stress on this surface, so that S is the component of t n perpendicular to n. Prove that as n varies, S has stationary values when n is perpendicular to one of the principal axes of stress, and bisects the angle between the other two.
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Prove also that the maximum and minimum values of S are. If at a later time t the particle XR has coordinates xi, then the equations 6. In Chapter 4 we were mainly concerned with the kinematics of individual particles. In this chapter we consider how a particle moves in relation to its neighbouring particles. A translation is a rigid-body motion of a body in which every particle undergoes the same displacement; thus the motion is described by the equations 6.
Thus, in Fig. Then by elementary geometry 6. We refer to Fig. In the rotation, the particle which is initially at P0 moves to P, with position vector x. Figure 6. Thus 6. Thus it follows from the results of Section 3. The components of any proper orthogonal tensor can be represented in the form 6. It can be shown that any rigid-body motion is a combination of a translation and a rotation about an axis through any point.
In particular, if the axis of rotation passes through O, then any rigid-body motion is described by equations of the form or 6. A motion in which a change of shape takes place is called a deformation; a body which can change its shape is deformable, in contrast to a rigid body which can only undergo rigid-body motions.
In a deformation, there are changes in distance between particles, whereas in a rigid-body motion there are no such changes. We therefore begin by examining the extension or stretch of a material line element. Thus if P0 has coordinates , then Q0 has coordinates. The motion is described by the relations 6. Hence 6. By squaring each side of 6. Details are left to the reader Problem 6. They are called the deformation gradients. It is clear that these quantities must be involved in the description of how a particle moves in relation to neighbouring particles, and so they are of importance in the analysis of deformation.
We denote 6. To confirm that FiR are components of a tensor, we introduce a new rectangular cartesian coordinate system by a rotation of the axes defined by the orthogonal matrix M. In general, F is not a symmetric tensor. By the results of Section 3. Since F—1 is the tensor whose components are where 6. Equation 6. If, in a fixed coordinate system, the components of A are written as a column matrix A, those of a as a column matrix a, those of F as a square matrix F, and those of F—1 as a square matrix F—1, then 6. The displacement gradients are 6.
This tensor is called the displacement gradient tensor. If there is no motion, then its components are all zero. Although the tensor F is important in the analysis of deformation, it is not itself a suitable measure of deformation. This is because a measure of deformation should have the property that it does not change when no deformation takes place; therefore it must be unchanged in a rigid-body motion. From 6. Consider an elementary material triangle bounded by three material line elements.
Knowledge of the stretch of these line elements completely determines the shape of the triangle though not its orientation in a deformed configuration. Hence the components CRS at a particle determine the local deformation in the neighbourhood of that particle. For the rigid-body motion 6. Thus C is essentially connected with the deformation, rather than the rigid motion, of a body, and is a suitable measure of the deformation.
C is called the right Cauchy-Green deformation tensor. C is not a unique measure of deformation. Trivially, any tensor function of C such as C2 or C—1 will serve as such a measure. It is sometimes convenient to employ the measure C—1, which is given in terms of F by 6. If we denote 6. If B and B—1 have components Bij and respectively, then 6. If the deformation is defined by 6. The components of these tensors will then be obtained as functions of the material coordinates XR, and so they describe the deformation in the neighbourhood of a given particle.
Consideration of these is deferred to Chapter 9. This defines a uniform extension in the x1 direction. If the body undergoes uniform extensions in all three coordinate directions, the deformation is described by the equations 6. Some special cases of 6. For the deformation 6. In this deformation, parallel planes are displaced relative to each other by an amount proportional to the distance between the planes and in a direction parallel to the planes.
For example, the simple shear deformation illustrated in Fig. Note that a simple shear involves no change in volume of any portion of the body. These are motions of the form or 6. Cases a and b above are special cases of 6. In the motion 6. The expressions for CRS, Bij, and so on, follow from 6. Homogeneous deformations have a number of properties including the following: i Material surfaces which form planes in the reference configuration deform into planes; two parallel planes deform into two parallel planes. The proof of these and other similar results is straightforward.
As an example we prove i. Particles which initially lie in a given deformation plane remain in that plane, and their displacement is independent of the X3 coordinate. Deformations which approximate to plane strain occur in many problems of practical interest. In this deformation, planes normal to the Z-axis rotate about the Z-axis by an amount which is proportional to Z. The deformation is most easily visualized in terms of the twisting of a circular cylindrical rod whose axis lies along the Z-axis.
There are no volume changes and the deformation is not homogeneous. The deformation illustrated in Fig. Such materials include the usual structural materials like metals, concrete and wood. In applications involving materials of this kind a great simplification can be achieved by approximating the finite and exact strain tensors introduced in Section 6. The approximation we introduce is that all components of the displacement gradient tensor which are dimensionless quantities are numerically small compared to one. Thus we assume 6. To this order of approximation, it follows from 6.
Since F is a second-order tensor, E is a second-order tensor, and clearly E is symmetric. The tensor E cannot be an exact measure of deformation, because it does not remain constant in a rigid-body rotation. To illustrate this, consider the rotation 6. Although the infinitesimal strain tensor is not an exact measure of deformation, it often provides an excellent approximation to such a measure. Typically, for deformations of structural materials, Eij are of order 0.
The classical theory of linear elasticity, with its numerous successful applications, is constructed on the basis of this approximation. This means that the techniques of linear analysis can be applied to the solution of boundary-value problems in, for example, the linear theory of elasticity.
The geometrical interpretation of E11 is illustrated in Fig. A similar geometrical interpretation of E23 is illustrated in Fig. The tensor E possesses the usual properties shared by all symmetric second-order tensors. It has an orthogonal triad of principal axes; if these are chosen as coordinate axes then the matrix of components of E has diagonal form. The corresponding diagonal elements E1, E2, E3 are principal components of infinitesimal strain. Symmetric functions of E1, E2 and E3 are invariants of the infinitesimal strain tensor. It can be verified by direct substitution from 6.
These six compatibility relations are themselves not completely independent, for it can be verified, again by direct substitution, that 6. Now consider a general infinitesimal motion with deformation gradient tensor F. The displacement gradient tensor F—I is now decomposed into its symmetric and anti-symmetric parts as follows: 6.
It is possible to envisage and to realize situations in which the strain components are everywhere small but some material elements undergo large rotations. This may occur, for example, in the bending of a long thin flexible rod. Individual elements of the rod change shape only slightly, but the rotations and displacements can be large. Such problems require careful formulation, and will not be discussed here.
This is especially the case in fluid mechanics, where it is usually required to find the fluid flow in a particular region of space, and the shape of the body of fluid at a reference time is rarely relevant. The starting point is equation 6. It is convenient to begin with 6. We differentiate 6. For any given direction a this extension rate is, from 6. We also observe that the right side of 6. The rate-of-deformation tensor D has properties which in almost every respect but with an important exception noted below are analogous to those of the infinitesimal strain tensor E. It is readily verified that D is a second-order symmetric tensor.
Referred to its principal axes as coordinate axes, the matrix of components of D has diagonal form with principal components D1, D2. The largest and smallest of the principal components are extremal values of the extension rate for variations of the direction a. Symmetric functions of D1, D2 and D3 are invariants of D. The components Dij obey compatibility relations which are precisely analogous to the relations 6.
The tensor D differs from the tensor E in that it is an exact measure of deformation rate, whereas it was emphasized in Section 6. The fact that Dij are linear in the velocity components is a fortunate circumstance which simplifies the solution of problems in fluid mechanics. The tensor W is called the spin or vorticity tensor, and it has properties analogous to those of the infinitesimal rotation tensor, except that no approximation is involved in its derivation or use. It is a measure of the rate of rotation of an element; the expressions 6. Moreover, if a general motion is modified by superposing on it the rigid-body rotation 6.
This confirms that D is unaffected by superposed rotations, and is therefore a suitable measure of the deformation rate. For this flow b Rectilinear flow. In rectilinear flow the material flows in parallel straight lines; this may but does not always occur in flow down a pipe of uniform cross-section, or in flow between parallel plates. Particles travel in circles around the x3-axis, with speed inversely proportional to the distance from the axis. The components of D and W are There is a singularity on the vortex line. The non-zero components of D are D11, D22 and D12, and these are functions of x1, x2 and t only.
The simple shearing and vortex flows defined above are special cases of plane flow. Prove the formulae 6. A body undergoes the homogeneous deformation Find: a the direction after the deformation of a line element with direction ratios in the reference configuration; b the stretch of this line element. Find the conditions on these constants for the deformation to be possible in an incompressible material. A body which in the reference configuration is a unit cube with its edges parallel to the coordinate axes undergoes this deformation.
Determine the lengths of its edges, and the angles between the edges, after the deformation. Sketch the deformed body. A circular cylinder in its reference configuration has radius A and its axis lies along the X3-axis. A line drawn on the surface of the cylinder has unit length and is parallel to the axis of the cylinder in the reference configuration. Find its length after the deformation. Find also the initial length of a line on the surface which has unit length and is parallel to the axis after the deformation. Find these directions and the corresponding stretches.
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Prove that in the homogeneous deformation 6. Find the components of infinitesimal strain and infinitesimal rotation. Show that one of the principal components of infinitesimal strain is always zero and find the other two principal components. Find also the principal axes of the infinitesimal strain tensor. Also find the principal values and principal axes of E. For the velocity fields given in Problems 2 and 3 of Chapter 4, find the components of the tensors L, D and W. Laws of this kind are general statements and are not restricted in their application to any particular material or class of materials.
The mathematical formulations of these laws are therefore equations which must be always satisfied. It is important to distinguish such equations from equations which we call constitutive equations which describe the properties of particular materials or classes of materials, and which are the subject of Chapters 8 and We note in passing that the second law of thermodynamics, although it is an important general law of physics, is rather different from the conservation laws mentioned above in that it is expressed as an inequality.
Continuum thermodynamics is outside the scope of this introductory text and we shall not discuss it. We first consider the effect of a finite deformation on a volume element. Deformation of a volume element. The notation of Sections 4. Consider an elementary tetrahedron in the reference configuration Fig. Hence 7. Then from 7. From 7. Conservation of mass — Lagrangian form. Conservation of mass requires that the mass of the material in the material volume element remains constant during the deformation. We note that 7.
Conservation of mass - Eulerian form. Equation 7. For many purposes it is more convenient to express the law in terms of the velocity components. The rate at which mass flows over an element of surface, of area dS, is p dS multiplied by the normal component of velocity. The negative sign on the right-hand side appears because n denotes Figure 7. By applying the divergence theorem to the surface integral, we obtain from 7. It therefore follows from 7. Naturally the results of doing this are valid only if the conditions for the theorem to be applicable are satisfied.
The most important of these is that the integrand of the surface integral should be differentiable, and therefore continuous. Problems do arise in continuum mechanics in which density, velocity, stress and other variables are discontinuous across certain surfaces, which may be stationary or in motion. This situation arises particularly in stress-wave propagation problems.
It is not difficult to extend the theory to deal with such cases, and for some problems it is essential to do so. However, in this text it is always assumed that necessary smoothness conditions are satisfied. The rate of change of the amount of D which is associated with the particles which instantaneously occupy 91 at t is called the material time derivative of the integral 7. Thus By applying the divergence theorem to the surface integral, and rearranging, we obtain 7.
Hence the integral on the right side of 7. Thus we again obtain the continuity equation in the form 7. Hence the law is expressed in the form 7. We now use 7. It reduces to the equilibrium equation 5. The generalization for a continuum, analogous to 7. This gives 7. It should be mentioned that in writing down 7. However, body and surface couples are of importance only in rather specialized applications, and we shall not consider them.
The kinetic energy of a continuum is only part of its energy. The remainder is called the internal energy E, which is expressed in terms of the internal energy density e by 7. The most important is energy due to heat flux across S. We shall consider only the heat flux.
The above statement of the law is not particularly helpful on its own because it can be regarded as being merely a definition of E. It really only becomes useful when some further properties of E or e are specified. To do this leads into the consideration of constitutive equations, which we defer until Chapters 8 and By employing 7.
Hence, after rearrangement, 7. Hence by 6. The term TijDij can be interpreted as the rate of working of the stress. To make further progress it is necessary to assign further properties to e and q. Such statements are not general laws, but are particular to certain materials, and are certainly not universally true. Although it is not a conservation law, it is convenient to introduce it here. Then, using 5. The relation 7.
Deduce that in this case the energy equation 7. A singular surface is a surface across which the stress, velocity and density may be discontinuous. By considering a thin cylindrical region which encloses part of a singular surface, show that in a body at rest in equilibrium, t n is continuous across a stationary singular surface, where n is the normal to the singular surface.
Suppose a singular surface propagates through a body with speed V relative to the body, in the direction of the normal to the surface. In themselves they are insufficient to describe the mechanical behaviour of any particular material. To complete the specification of the mechanical properties of a material we require additional equations, which are called constitutive equations. These are equations which are particular to individual materials, or classes of materials, and they serve to distinguish one material from another.
The mechanical constitutive equation of a material specifies the dependence of the stress in a body on kinematic variables such as a strain tensor or the rate-of-deformation tensor. Normally thermodynamic variables, especially temperature, will also be involved, but we shall make only brief references to these. Constitutive equations are also required in other branches of continuum physics, such as continuum thermodynamics and continuum electrodynamics, but these problems are outside the scope of this book, and we shall only discuss constitutive equations for the stress.
The mechanical behaviour of real materials is very diverse and complex and it would be impossible, even if it were desirable, to formulate equations which are capable of determining the stress in a body under all circumstances. Rather, we seek to establish equations which describe the most important features of the behaviour of a material in a given situation. Such equations can be regarded as defining ideal materials. It is unlikely that any real material will conform exactly to any such mathematical model, but if the ideal material is well chosen its behaviour may give an excellent approximation to that of the real material which it models.
The model should be selected with the application as well as the material in mind, and the same real material may be represented by different ideal materials in different circumstances.
For example, the theory of incompressible viscous fluids gives an excellent description of the behaviour of water flowing through pipes, but is useless for the study of the propagation of sound waves through water, because for sound-wave propagation a model which takes into account the compressibility of water is essential.
Historically, the constitutive equations which define the classical ideal materials linear elastic solids, Newtonian viscous fluids, etc. In applications of these theories this separation is natural. However, at the formulative stage there are advantages in a unified approach which clarifies relations between the different special theories.
Also it is possible to formulate some general principles which should be followed in the construction of constitutive equations. Since a constitutive equation always includes constants or functions which characterize the material under consideration, and these quantities have dimensions, the dimensional homogeneity requirement is usually not difficult to satisfy.
Constitutive equations should not depend on the choice of the coordinate system although they may be expressed in terms of components relative to any selected coordinate system. They therefore take the form of relations between scalars, vectors and tensors. An important restriction on mechanical constitutive equations is the requirement that the stress response of a body to a deformation is not affected by rigid-body motions, so that the stress in a body depends only on the change of shape of the body and is not affected except for the change in orientation of the stress field relative to fixed axes by a superposed motion in which the body moves as a whole.
To formalize this requirement we specify that if a body undergoes two time-dependent motions, which differ from each other by a time-dependent rigid-body motion, then the same stress results from each of these motions. This is essentially equivalent to saying that constitutive equations are invariant under translations and rotations of the frame of reference; two observers, even if they are in relative motion, will observe the same stress in a given body.
Materials are usually regarded as either solids or fluids, and fluids are subdivided into liquids and gases. We do not attempt a precise definition of this classification; the dividing lines are not always clear and there are materials which possess both solid-like and fluid-like properties.
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