Deformation rigid body

V. Mises R (1913) Mechanik der festen Korper im plastisch-deformablen Zustand [Mechanics of Ductile Deformable Rigid Bodies]. Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen [News of the Academic Society of Gottingen], Mathematische-Physikalische Klasse, Vol. 1913 582-592 (in German)... 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

From this kind of continuum mechanics one can move further towards the domain of almost pure mathematics until one reaches the field of rational mechanics, which harks back to Joseph Lagrange s (1736-1813) mechanics of rigid bodies and to earlier mathematicians such as Leonhard Euler (1707-1783) and later ones such as Augustin Cauchy (1789-1857), who developed the mechanics of deformable bodies. The preeminent exponent of this kind of continuum mechanics was probably Clifford Truesdell in Baltimore. An example of his extensive writings is A First Course in... 

Mechanics is the physical science that deals with the effects of forces on the state of motion or rest of solid, liquid, or gaseous bodies. The field may he divided into the mechanics of rigid bodies, the mechanics of deformable bodies, and the mechanics of fluids. 

A rigid body is one that does not deform. True rigid bodies do not exist in nature however, the assumption of rigid body behavior is usually an acceptable accurate simplification for examining the state of motion or rest of structures and elements of structures. The rigid body assumption is not useful in the study of structural failure. Rigid body mechanics is further subdivided into the study of bodies at rest, stalks, and the study of bodies in motion, dynamics. 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

Figure 5.8 Illustration of (a) ideal elastic deformation followed by ideal plastic deformation and (b) typical elastic and plastic deformation in rigid bodies. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed.. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

These operations do not occur separately and in any particular sequence but are simply a convenient way to conceptualize the transformation as a series of operations, each of which can be analyzed separately, but which working together produce a martensitic structure containing an invariant plane. As such, they can be imagined to occur in any sequence. For purposes of analysis, it is convenient to imagine that the lattice-invariant deformation occurs first, followed by the lattice deformation, followed finally by the rigid-body rotation. We now show that a lattice-invariant shear by slip followed by the lattice deformation analyzed above can produce an undistorted plane. 

The plane containing a and c in Fig. 24.9 is the plane in the f.c.c. phase that initially contained the vectors a" and c". If the b.c.t. phase is now given a rigid-body rotation so that a" —> a and c" —> c, the undistorted plane in the b.c.t. phase will be returned to its original inclination in the f.c.c.-axis system and will therefore be an invariant plane of the overall deformation. In the present case, this can be achieved by a rotation around the axis indicated by u in Fig. 24.9 (see Exercise 24.3). The solution of the problem is now complete. The invariant plane is known, and the orientation relationship between the two phases and total shape change can be determined from the combined effects of the known lattice-invariant deformation, lattice deformation, and rigid-body rotation. 

If S is the lattice-invariant deformation tensor and R the rigid-body rotation tensor, the total shape deformation tensor, E, producing the invariant plane can be expressed as... 

Example of a Rotational Constant The ground-state rotational band of 152Gd is shown in Figure 6.11. Use the energy separation between the 2+ and 0+ levels to estimate the rotational constant in keV, the moment of inertia in amu-fm2, and then compare your result to that obtained to the rigid-body result with a deformation parameter of 3 = 0.2. Finally, evaluate the irrotational flow moment of inertia for this nucleus. 

Instead of an assumed contraction of fast moving objects, I have introduced the idea that the travelers spheres of observation by the velocity are transformed into ellipsoids of observation. One advantage is that this new concept is easier to visualize and that it makes possible a simple graphic derivation of distortions of time and space caused by relativistic velocities. Another advantage is that it is mentally easier to accept a deformation of spheres of observation than a real deformation of rigid bodies depending on the velocity of the observer. 

Conclusions We have established that the light Br and Rb isotopes presented here have very large quadrupole deformations of s 0.4 and moments of inertia close to the rigid body values. The odd proton in the 431 3/2+ Nilsson orbit polarizes and stabilizes the y-soft, shape coexistent Se and Kr cores into definite prolate triaxial shapes. This effect sets in at rather low spin and seems to be intimately connected with the suppression of pairing correlations near the N = Z = 38 gap developing at 82 = 0.4. We thus face a cumulative suppression of both proton and neutron pairing correlations in the same oscillator shell, a fairly unique feature in the periodic table. 

The simplest theory of impact, known as stereomechanics, deals with the impact between rigid bodies using the impulse-momentum law. This approach yields a quick estimation of the velocity after collision and the corresponding kinetic energy loss. However, it does not yield transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. Because of its simplicity, the stereomechanical impact theory has been extensively used in the treatment of collisional contributions in the particle momentum equations and in the particle velocity boundary conditions in connection with the computation of gas-solid flows. 

One physical restriction, translated into a mathematical requirement, must be satisfied that is that the simple fluid relation must be objective, which means that its predictions should not depend on whether the fluid rotates as a rigid body or deforms. This can be achieved by casting the constitutive equation (expressing its terms) in special frames. One is the co-rotational frame, which follows (translates with) each particle and rotates with it. The other is the co-deformational frame, which translates, rotates, and deforms with the flowing particles. In either frame, the observer is oblivious to rigid-body rotation. Thus, a constitutive equation cast in either frame is objective or, as it is commonly expressed, obeys the principle of material objectivity . Both can be transformed into fixed (laboratory) frame in which the balance equations appear and where experimental results are obtained. The transformations are similar to, but more complex than, those from the substantial frame to the fixed (see Chapter 2). Finally, a co-rotational constitutive equation can be transformed to a co-deformational one. 

Up to now we have seen how lattice distortions are detected and characterized. This does not provide a direct observation of the molecular translations, rotations, and deformations associated with the distortion. However, for a few compounds it has been possible to measure a large enough number of satellite or superstructure reflections so that the distorted structure can be parametrized and refined (rigid-body or full structural study). We consider below four examples, taken from materials selected in Section IV. A, which show that such studies are not easy and that the data collection requires special attention. Indeed, it is generally difficult to measure enough satellite reflections, especially if several kinds of the latter coexist (e.g., 2kp and 4kF satellites, high-order satellites, etc.). 

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

Fig. 2.3. Rigid body deformation involving a rotation through an angle 9 and a translation by the vector c.

Application of the definition of the deformation gradient introduced earlier to the case of rigid body deformation yields F = Q. Recalling that Q is orthogonal... 

In addition to the importance that attaches to rigid body motions, shearing deformations occupy a central position in the mechanics of solids. In particular, permanent deformation by either dislocation motion or twinning can be thought of as a shearing motion that can be captured kinematically in terms of a shear in a direction s on a plane with normal n. 

The representation of a staircase or a molecular chain in a BDS is very general, and embraces a family of recognizably similar systems. Operations on the BDS can effect transformations within this extensive set. Such operations fall into several distinct classes rigid-body motions, cylinder deformations, non-conservative transformations, and creation-annihilation operations. 

As demonstrated by the foregoing two formulations, some problems taken from mechanics can be formulated by using only Newton s laws of motion these are called mechanically determined problems. The dynamics of rigid bodies in the absence of friction, statically determined problems of rigid bodies, and mechanics of ideal fluids provide examples of this class. Some other mechanics problems, however, require knowledge beyond Newton s laws of motion. These are called mechanically undetermined problems. The dynamics of rigid bodies with friction and the mechanics of deformable bodies provide examples of this class. 

The constants in Eqns. (3.36) and (3.37) represent rigid-body translation and may be disregarded in considering deformation within the body. The functions fi(y) and fzix) may be examined through a consideration of the shearing strain yxy. From Eqns. (3.6), (3.7), and (3.24),... 

The unloading process relating the deformed state to the intermediate state is not uniquely defined since an arbitrary rigid body rotation of the intermediate state still leaves the state stress-free. The intermediate state can be made unique in different ways fi l one particularly convenient way is to prescribe... 

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