彈塑性力學(xué)第十三章

1、彈塑性力學(xué)第十三章 1 CHAPTER 13 DYNAMIC PROBLEMS 13. DYNAMIC PROBLEMS 13.1. Basic Concepts All the problems studied so far have been static or quasi-static. The solution of such a problem, the stress field satisfies the equilibrium equation with the prescribed body force and the static boundary conditions wi

2、th the prescribed surface tractions. The effects of inertia are neglected. When the elastic-plastic bodies or structures loaded by dynamically forces, such as impulsive loading, impact, earthquake loading, periodical machine loading etc., the effects inertia cannot be neglect. If not, we will imposs

3、ible obtain the correct solution. standard perfect plastic material, if a loading in excess of the limit loading is applied, then, obviously, the static problem has no solution, and inertia effects must be taken 13.1.1. Wave and Vibration (Traveling Wave and Standing Wave ) Physically, waves are a t

4、raveling disturbance and represent the transfer of energy from one point in a medium to some other point. Thus, there must be an initial disturbance of the medium, some forces must act to disturb the medium from its equilibrium position and thereby introduce new energy into the medium. If the medium

5、 is not elastic in its response to the energy introduction, it absorbs energy and only damped waves, plastic wave, and other kinds waves emanate from the disturbance area. There are several important aspects of the wave process. First of all, there is no transport of matter during wave motion. The c

6、onstituent particles of medium oscillate only about very space limited paths and do not go traveling off through the medium. CHAPTER 13 DYNAMIC PROBLEMS 2 c is commonly orders of magnitude larger than the particle velocity v. That is c >> v. Wave motion may be transient, periodic, or random. T

7、ransient motion is characteristic of the response of medium to a sudden, pulse-like excitation and dies out rapidly with increasing time. Periodic motion is repetitive in nature, reoccurring in exactly the same form at fixed form increments. Harmonic motion is the simplest form of periodic motion an

8、d is specified by the sinusoidal functions. For further details we give a simple example. Suppose there is a simple elastic rod, striking an end of the rod in the axial direction x generates longitudinal strain propagating along the rod: a longitudinal wave. Let the displacement u (x, t ), The equat

9、ion of motion for longitudinal wave in the rod is ?cuxx, c?E0, (13.1) u ?2u?t2,uxx?2ux2. One can easily confirm by substitution that where u an expression u?x,t?f?x?t?g?x?t? (13.2) When f and g are cosine functions, we have u?x,t?acos?kx?t?1?bcos?kx?t?2? (13.3) which is known as harmonic wave, and t

10、he constants a and b are called the wave amplitudes, ?1,?2 are also the constant initial phases of waves, ? is the circular frequency, and k?c the wave number. The later can be written as k?2?, where ?2?c?Tc is the wavelength and T is the period of wave. As the equation (13.2), the first term in (13

11、.3) describes a wave propagating in the positive x-direction, and the second term describes one propagating in the opposite direction. It is often convenient to write the waves in complex form: u?x,t?Aexp?i?kx?t?Bexp?i?kx?t? (13.4) where the constants A?aexp?i?1?,B?bexp?i?2? are termed complex wave

12、3 CHAPTER 13 DYNAMIC PROBLEMS amplitudes. As we know, in all linear operations with complex waves, the real part of the final result will be equal to the result of same operations applied to the real part of the original waves. After the action of an external force ceases, a rod of finite length con

13、tinues to oscillate with some resonant frequencies. The frequencies and form of oscillation can be found by solving the wave equation (13.2) or (13.4) for the corresponding boundary conditions. Consider, for example, a rod with fixed ends at x = 0 and x = l, i. e., u (o,t ) = u ( l, t ) = 0. Let us

14、look for a solution in the form u?x,t?x?exp?i?t? Substituting it into (13.2) we obtain for the function ?x? the ordinary differential equation where k?c If we take ?x?Asinkx, then the condition for x=0 is satisfied automatically. The boundary for x=l gives sinkl=0, whence we get a set permissible nu

15、mber kn: ?k2?0 kn?n,n?1,2,?, Since kn?nc, we obtain the resonant frequencies ?n?n?l, and the corresponding normal modes of vibration un?x,t?Ansin?n?x?exp?i?t? (13.5) Presenting the complex amplitude as An?anexp?i?n? and separating the real part, we have un?x,t?ansin?n?l?cos?nt?n? (13.6) where an,?n

16、remain arbitrary. The above equation (13.6) describes the longitudinal vibration. Fig. 13.1,a portrays the distribution of the vibration amplitudes sin?n?x? over the rod length for first three integers n. Fig. 13.1,b portrays the distribution of the vibration amplitudes of a rod with one end free. (

17、a) (b) Fig. 13.1 Amplitude distributions of vibration along rods for first three modes. (a) simple supported beam (b) cantilever beam CHAPTER 13 DYNAMIC PROBLEMS 4 Suppose we strike the rod in a transverse direction, then the problem becomes discussion of the bending waves in a rod (or commonly in a

18、 beam). The equation of motion is w?1?0 (13.7) w2a where a2?EI?0A. A is the rods section area and ?0is its density. In this case, we have some interesting new features because the bending wave equation is of fourth order. In particular, there must be more boundary conditions. Possible conditions are

19、: rigidly clamped end; free end; simple supported end and others. Therefore, we consider harmonic waves as the solution of Eq. (13.7) w?x,t?A?x?exp?i?t? (13.8) Substituting this expression into (13.7) leads to a ordinary differential equation for ?x?, solving and operating we have w?x,t?A?exp?i?kx?t

20、?A?exp?i?kx?t? ?B?exp?kx?i?t?B?exp?kx?i?t? (13.9) cph?k?a?4 (13.10) traveling waves. standing (or stationary ) waves. Note that the derivative ddk?cgis called the group velocity of the wave. Indeed we note that a narrow band disturbance resembles the modulated harmonic wave: f?x,t?F0?x?cgt?exp?i?k0x

21、?0t? with the envelope F0 (x - cgt ) propagating at the velocity cg without changing its form. The concept of a group velocity is valid for arbitrary nature. For longitudinal waves in the rod we have cg = c, and for bending waves cg?ddk?2?/k?2cph. 5 CHAPTER 13 DYNAMIC PROBLEMS It is useful to displa

22、y these relations in graphical form for easier interpretation. One is plot of frequency versus wave number and is called frequency spectrum of the system; the other is a plot of phase velocity versus wave number and is called dispersion curve of the system. 13.1.2. Reflection of Waves We confine our

23、selves to the simplest case of harmonic waves. Assume that the pulse f ( x ct ) can be represented by the Fourier integral: f?x?ct?A?exp?i?kx?t?d? (13.11) ? We consider a incident longitudinal harmonic wave u?Aexp?i?kx?t? propagating in a semi-infinite rod ?x?0 toward x = 0. Suppose that we have an

24、absolutely rigid boundary at x = 0. Obviously, u+ alone cannot satisfy the boundary condition of rigid ends, Hence, a wave propagating in the opposite direction (reflected wave ) of frequency, say, ?, must arise: u?A?exp?i?k?x?t?,k?c where ?is termed the reflection coefficient. The displacement of t

25、he rods particles is the sum of both waves u?x,t?Aexp?i?kx?t?A?exp?i?k?x?t? (13.12) Applying the boundary condition at x0 = 0 yields exp?i?t? (13.13) Therefore we have ?Hence k?k, too. For the reflection coefficient of fixed end condition we now have ?1. Thus the total wave field in the rod is u?x,t

26、?2iAsinkxexp?i?t? The reflection from a free end can be considered in a similar manner, leading to the reflection coefficient ?1. We can also be shown that the corresponding stress pulse situation. It is seen that the interaction at the boundary occurs, and the stress CHAPTER 13 DYNAMIC PROBLEMS 6 s

27、uperimpose to give double peak value at the fixed boundary. In a similar manner, for free-end boundary condition, we can get the reflection stress pulse is opposite to incident pulse; thus compression has reflected as tension and 13.2. Dynamic Constitutive Relations As mentioned in the beginning of

28、this chapter, for the perfect elastic body or structures subjected dynamic loading, the Hookes law is still available, because many experiments verify a fact that the loading speed, or strain rate, do not influenced the Youngs modulus E and yield limit. But for the elastic-plastic materials there ar

29、e some new physical phenomena discovered in many dynamically experiments. Therefore, for the dynamic problems of plasticity it is necessary to take into account plastic properties of materials and necessary to take new dynamic plastic constitutive relation. 13.2.1. Dynamic Properties of Material Som

30、e of characteristic features of stress-strain diagrams of ductile solids will now described. Such diagrams are characterized by a range of stress, extending from zero to a stress state past the elastic limit. For examples, the experimental results given by Nageeva, R.E. (1953), (Fig.13.2,a) Klepaczk

31、o, J.J. (1968) (Fig. 13.2,b ), and Clark-Duwez (1950) and Manjonie (1944) (Fig. 13.2,c) (a) (b) (c ) Fig. 13.2 Stress strain-rate diagrams 7 CHAPTER 13 DYNAMIC PROBLEMS (a) The experimental results showed that the stress- strain relations were markedly different with static one. It will be concluded

32、: (b) The yield limit increases, obviously, under dynamic loading; (c) The instantaneous stress increases with increasing the strain-rate; It can be treated the material as visco-plastic 13.2.2. Dynamic Stress-Strain Relation. The main characteristic of dynamic plastic constitutive relation is how t

33、o describe the strain-rate effect. We now introduce the practical dynamic plastic constitutive relations. (a) Overstress Model Theory Sokolovskii, V.V. (1948) and Malvern, L.E. (1951) suggested the following constitutive relation. For one-dimensional problems ? ?1?g?,? (13.14) ?E where the function

34、g?,? vanishes for stress whose magnitudes are below those given by the static stress-strain curve. For example, if the static relation is given by ?F? then in Sokolovskii model g is g?,?kf?|?|?s? (13.15) where ?s is the static yield stress. In Malvern model g is 1?F?g?,?exp?1? (13.16) b?a? where a,

35、b are constants of materials, or the first order approximately formula 1?F?g?,?1? (13.17) b?a? or simplified g?,?c?F? (13.18) where constant c = 106 s-1 for metals. CHAPTER 13 DYNAMIC PROBLEMS 8 In Bodner-Symonds model the constitutive relation is ?p?1? ?D? (13.19) ?s? where D and ? are constants of

36、 materials, and D= 40, ?5 for steel; D=6500, ?4 for aluminum alloy. (b) Viscoplasticity Model Theory Hohenemser,K. and Prager, W. (1932) proposed the following visco-plastic constitutive relation ? ?ij?2k?Q? 2?p?Q (13.20) ?ij where k is the pure shearing stress, and function Q is Q?(J2/k)?1 (13.21)

37、More generally, Perzyna P. (1963) developed the Hohenemser-Prager model and given ?ij?11?2?f?ij?ij?Q? (13.22) s?2GE?ij pwhere ? is constant of materials f?ij,?ij is the dynamic loading function, ? ?1?ii and ?3 ?Q?0?;?0,?Q? ?Q?,?Q?0?. 1Q?ij,?ijp,k?f?ij,?ijp?1 (13.23) k Discussion detail on Perzynas e

38、quation see, for example, references 3 13.3. General Dynamic Principles 13.3.1. Hamilton principle and generalized Hamilton principle Suppose V is the volume of a body, ?i?t? is the function described the state of motion at time t, then the kinematically energy is 9 CHAPTER 13 DYNAMIC PROBLEMS Ek? T

39、he total potential energy is 1?iu?idV ?0u?V2 ?i?dVi?iu?idS Et?U?FbiuVS? where U = U(ui) is the strain energy of the body. Let L (= Ek -Et ) be the Lagrange function, and JH?Ldt (13.24) t0t1 JH is called Hamilton action quantity. Thus, ?JH?Ldt?0. (13.25) t0t1 Generalized Hamilton Principle For non-co

40、nservative systems we have the generalized Hamilton principle: ?Ldt?Dpdt (13.26) ?JHt0t0t1t1 where Dp is the rate of plastic dissipation and is called function of dissipation. Let p is the plastic dissipation per unit volume, then we have ?jdV (13.27) Dp?pdV?QjqVV ?j is the generalized strain rate c

41、orresponded where Qj is the generalized forces and q with Qj . 13,3.2. Displacement Bound Theorems The deformation time t of a rigid-plastic body subjected of impulsive load has the lower bound ?i0u?i?dV/Dp?tf?0uij (13.28) V? and ?Dp?ij?ij?ijdV (13.29) V? ?i? is a kinematically admissible velocity field, independent of time, ?ij are where u? ?ij in accordance with the the stres