Physics
Elastic Forces
Elastic forces refer to the restoring forces exerted by a material when it is deformed and then returns to its original shape. This behavior is described by Hooke's Law, which states that the force exerted by the material is directly proportional to the amount of deformation. Elastic forces are fundamental in understanding the behavior of materials under stress and strain.
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10 Key excerpts on "Elastic Forces"
eBook - PDF- E. Fjær, Per Horsrud, Arne Marius Raaen, R. Risnes, Rune Martin Holt, P. Horsrud, A.M. Raaen, R.M. Holt(Authors)
- 1992(Publication Date)
- Elsevier Science(Publisher)
Chapter 1 Elasticity Most materials have an ability to resist and recover from deformations produced by forces. This ability is called elasticity. It is the foundation for all aspects of rock me- chanics. The simplest type of response is one where there is a linear relation between the external forces and the corresponding deformations. When changes in the forces are sufficiently small, the response is (nearly) always linear. Thus the theory of linear elasticity is fundamental for all discussions on elasticity. Linear elasticity of solid ma- terials is described in Section 1 .l. The region of validity for linear elasticity is often exceeded in practical situations. Some general features of non-linear behaviour of rocks are described in Section 1.2. In petroleum related rock mechanics, much of the interest is furthermore focused on rocks with a significant porosity as well as permeability. The elastic theory for solid materials is not able to fully describe the behaviour of such materials, and the concept of poroelasticity has therefore to be taken into account. The elastic response of a rock material may also be time dependent, so that the deformation of the material changes with time, even when the external conditions are constant. The elastic properties of po- rous materials and time-dependent effects are described in Sections 1.3 and 1.4, re- spectively. 1.1 Linear elasticity The theory of elasticity rests on the two concepts stress and strain. These are de- fined in 8 1.1.1 and 8 1.1.2. The linear equations relating stresses and strains are dis- cussed in 8 1.1.3 for isotropic materials, and in Q 1.1.4 for anisotropic materials. 1.1.1 Stress Consider the situation shown in Fig. 1.1. A weight is resting on the top of a pillar. Due to the weight, a force is acting on the pillar, while the pillar reacts with an equal, but reversely directed force.
eBook - PDFIntroduction to Physics
Mechanics, Hydrodynamics Thermodynamics
- P. Frauenfelder, P. Huber(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
C H A P T E R 5 ELASTICITY 49. THE GENERAL CONCEPT OF STRESS The subject of statics, as presented in Chapter 2, was based on the con-cept of a rigid body. This idealization of actual bodies is necessary in order to describe the mutual interactions of two bodies by means of the concept of force. In fact, however, no truly rigid bodies exist. Every real object changes its shape when acted on by a force. A body is said to be completely elastic if it returns to its original shape when the forces are removed, and completely inelastic (or completely plastic) if the entire change of shape remains when the forces are removed. All actual bodies lie somewhere between these two extremes. We shall restrict the discussion in this chapter to isotropic bodies. A body is said to be isotropic if its properties are independent of direction when it is in a force-free state. Although most solid bodies (the metals, for example) are microcrystalline, volume elements which are large relative to the micro-structure can be considered isotropic. On the other hand, single crystals (such as crystals of rock salt or single crystals of metal) are anisotropic. The subject of elasticity deals with the relation between the forces on a quasi-elastic body and the deformations resulting from these forces. It also determines the limits of validity of the laws which will be developed. In studying the laws of elastic substances, it becomes evident that the actions of two systems on each other can no longer be described in terms of the elementary concept of force. In place of force, we introduce the con-cept of stress, a quantity which plays a major role in the mechanics of deformable bodies. This is done because the actions of the bodies on each other occur over extended surfaces. In contrast to rigid bodies, which make contact only at single points, elastic bodies always make contact over sur-faces of finite extent.
eBook - ePub- Roland Ennos(Author)
- 2011(Publication Date)
- Princeton University Press(Publisher)
If these sorts of materials are stretched or compressed, we are actually stretching or compressing the interatomic bonds (fig. 1.2b). They have an equilibrium length and strongly resist any such movement. In typically static situations, therefore, the applied force is not lost or dissipated or absorbed. Instead, it is opposed by the equal and opposite reaction force that results from the tendency of the material that has been deformed to return to its resting shape. No material is totally rigid; even blocks of the stiffest materials, such as metals and diamonds, deform when they are loaded. The reason that this deformation was such a hard discovery to make is that most structures are so rigid that their deflection is tiny; it is only when we use compliant structures such as springs or bend long thin beams that the deflection common to all structures is obvious. The greater the load that is applied, the more the structure is deflected, until failure occurs; we will then have exceeded the strength of our structure. In the case of the tree (fig. 1.1b), the trunk might break, or its roots pull out of the soil and the tree accelerate sideways and fall over. INVESTIGATING THE MECHANICAL PROPERTIES OF MATERIALS The science of elasticity seeks to understand the mechanical behavior of structures when they are loaded. It aims to predict just how much they should deflect under given loads and exactly when they should break. This will depend upon two things. The properties of the material are clearly important—a rod made of rubber will stretch much more easily than one made of steel. However, geometry will also affect the behavior: a long, thin length of rubber will stretch much more easily than a short fat one. Figure 1.2. When a tensile force is applied to a perfectly Hookean spring or material (a), it will stretch a distance proportional to the force applied
eBook - PDF- George C. King(Author)
- 2023(Publication Date)
- Wiley(Publisher)
9 The elastic properties of solids Solids are characterised by having a rigid shape. However, we can deform the shape of a solid object by applying an appropriate force to it. A spring provides a familiar example. When we pull on a spring, it extends. The distance x by which a spring extends is described by Hooke’s law: F αx , (9.1) where F is the applied force and the constant of proportionality α is the spring constant. This is an empirical law and applies so long as the extension x is small compared to the length of the spring. Moreover, when we release the spring, it returns to its original length. A solid that returns to its original shape when the applied force is removed is called elastic. The property of elasticity is generally observed in solids. It is usually the case that a solid returns to its original shape when an applied force is removed, so long as the deformation is relatively small, with the amount of deformation being proportional to the force. In this chapter, we discuss the behaviour of solids under the influence of external forces in various circumstances; how they may stretch, compress or twist. And we relate this behaviour to the interatomic forces and energies of the atoms in the solid. 9.1 Stress, strain, and elastic moduli For solids, there are three different forms of deformation, which are illustrated in Figure 9.1. The dashed lines represent the original shape of the solid, and note that the deformations are greatly exaggerated for the purpose of illustration. We will usually consider situations where the amount of deformation is very much less than the dimensions of the solid. In Figure 9.1a, a bar is stretched; in Figure 9.1b, a rectangular block is sheared, i.e. twisted; and in Figure 9.1c, a spherical object is compressed uniformly on all sides. What the three forms of deformation have in common is that a deforming force called stress produces a deformation called strain.
eBook - ePubStructural Biomaterials
Third Edition
- Julian Vincent(Author)
- 2012(Publication Date)
- Princeton University Press(Publisher)
• CHAPTER ONE • Basic Elasticity and Viscoelasticity In the physically stressful environment there are three ways in which a material can respond to external forces. It can add the load directly onto the forces that hold the constituent atoms or molecules together, as occurs in simple crystalline (including polymeric crystalline) and ceramic materials—such materials are typically very rigid; or it can feed the energy into large changes in shape (the main mechanism in noncrystalline polymers) and flow away from the force to deform either semipermanently (as with viscoelastic materials) or permanently (as with plastic materials). 1.1 Hookean Materials and Short-Range Forces The first class of materials is exemplified among biological materials by bone and shell (chapter 6), by the cellulose of plant cell walls (chapter 3), by the cell walls of diatoms, by the crystalline parts of a silk thread (chapter 2), and by the chitin of arthropod skeletons (chapter 5). All these materials have a well-ordered and tightly bonded structure and so broadly fall into the same class of material as metals and glasses. What happens when such materials are loaded, as when a muscle pulls on a bone, or when a shark crunches its way through its victim’s leg? In a material at equilibrium, in the unloaded state, the distance between adjacent atoms is 0.1 to 0.2 nm. At this interatomic distance the forces of repulsion between two adjacent atoms balance the forces of attraction. When the material is stretched or compressed the atoms are forced out of their equilibrium positions and are either parted or brought together until the forces generated between them, either of attraction or repulsion, respectively, balance the external force (figure 1.1). Note that the line is nearly straight for a fair distance on either side of the origin and that it eventually curves on the compression side (the repulsion forces obey an inverse square law) and on the extension side
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
However, as long as we are looking at small enough deformations, all materials are elastic. 7.2 Stress and Strain: Hooke’s Law While thinking microscopically in terms of atoms is useful conceptually to see where the properties of bodies come from, it is not very practical to always go down to the molecular 264 Continuum Mechanics level when looking at everyday forces acting on macroscopic bodies. Thus we will now want to have a description where we do not have to look at forces between individual atoms, and are rather trying to give a macroscopic description, which however is independent of the shape and size of the body. However, the force that gives leads to the deformation of an object does depend on its shape and size. A thick rubber eraser needs much more force to stretch than a thin rubber band, even if they are made from the same material. Thus we need another quantity to describe the force acting on a macroscopic object. For this purpose, let us consider a material that is composed of atoms (or molecules) that are regularly arranged and are connected to the neighbors by springs described by a spring constant. This is a reasonable model for a covalently or ionically bound solid. When we now exert a force F on the body, this force actually acts on single chains of springs. But because these springs are next to each other, all of these chains only incur a fraction of the macroscopic force. In fact, this fraction will depend on the number of such spring chains within the cross-section A, where the force attacks. This number can be obtained from the ration of the cross-section and the area occupied by a single atom, that is, by the square of the interatomic distance a. Each single microscopic spring chain thus senses a force f = Fa 2 /A.
- Georgy V. Kostin, Vasily V. Saurin(Authors)
- 2012(Publication Date)
- De Gruyter(Publisher)
Chapter 2 Basic concepts of the linear theory of elasticity 2.1 Stresses There are two types of forces in the linear theory of elasticity, namely, ones applied to the boundary of an elastic body or surface loads and others distributed within the body or volume forces . Surface loads arise if outside objects influence the body. Such forces can be either continuously distributed on the external boundary of the body, for example, as hy-drostatic pressure and wind load, or applied pointwise to the body, i.e., locally. Any concentrated force can be considered as a limiting case of surface force under assump-tion that the boundary fragment subject to this loading is negligible with respect to the whole body surface. Volume forces are continuously distributed in the domain occupied by the body. Gravitation is an example of such loads, which is especially often met in applications. The stress principle of Euler and Cauchy is one of the basic axioms that are used to describe mechanical phenomena in a deformed elastic body. It is read as follows: the interaction of forces taking place on any imaginary surface drawn inside the body is analogous to that appearing on the boundary [20, 21, 12, 13]. Some short historical comments are provided by Truesdell and Toupin in [73]. Firstly, this principle states that elementary surface forces exist on the boundaries of each subdomain arbitrarily drawn in the deformed body. Secondly, it claims that the variability of such loads at any internal point is only determined by the direction of the surface normal vector. It could be also assumed that this elementary force at the given point depends on other geometrical properties of the surface, e.g. its curvature. Nevertheless, as it has been shown by Noll in [52], it is possible to formulate a general theory of surface forces that allows us to leave out any dependencies on such supplementary geometrical characteristics (see also [24, 80]).
eBook - PDFConstitutive Equations for Engineering Materials
Elasticity and Modeling
- Wai-Fah Chen, Atef F. Saleeb(Authors)
- 2013(Publication Date)
- Elsevier(Publisher)
4.4 DEFINITIONS Elastic Material A material body is deformed when subjected to applied forces. If upon the release of the applied forces the body recovers its original shape and size, then the material body is called elastic. For such a material, the current state of stress depends only on the current state of deformation; that is, the stress is a function of strain. Mathematically, the constitutive equations for this material are given by fy = ^(e w ) (4-3) where the function F tJ is the elastic response function. Thus the elastic behavior described by Eq. (4.3) is both reversible and path independent in the sense that strains are uniquely determined from the current state of stress or vice versa. There is no dependence of the behavior on the stress or strain histories followed to reach the current state of stress or strain. The elastic material defined above is usually termed Cauchy elastic material (e.g., Eringen, 1962; 148 ELASTIC STRESS-STRAIN RELATIONS and Malvern, 1969). It can be shown that Cauchy elastic material may generate energy under certain loading-unloading cycles. Clearly, in such cases it violates the laws of thermodynamics. Therefore, the term hyperelastic or Green elastic material is used (e.g., Fung, 1965; Eringen, 1962; Green and Zerna, 1954; and Malvern, 1969) to indicate that the elastic response function in Eq. (4.3) is further restricted by the existence of an elastic strain energy function W, which is in general a function of strain components e iJ9 such that m (A A This ensures that no energy can be generated through load cycles and thermo-dynamic laws are always satisfied. Sometimes the term hypoelastic model is used to describe the incremental elastic constitutive relations (e.g., Malvern, 1969; and Truesdell, 1955). These models are often used to describe the behavior of a class of materials in which the state of stress is generally a function of the current state of strain as well as of the stress path followed to reach that state.
eBook - PDF- George Z. Voyiadjis, Peter I. Kattan(Authors)
- 2005(Publication Date)
- CRC Press(Publisher)
2 Elasticity Theory In this chapter we present a summary of the theory of elasticity including the concepts of force, deformation, stress, and strain. This is achieved us-ing both the material description and the spatial description. The theory of elasticity is reviewed in this chapter based on the book of Lai et al.(1984). 2.1 Motion of a Continuum Consider a body occupying a certain region of space at time t = t o . Let X denote the position vector of a particle in the body measured from an origin point O (see Figure 2.1). Then, the motion of every particle is described by the equation x = x ( X , t) X = x ( X , t o ) (2.1) where X denotes the position of the particle at t = t o . During the motion of a continuum, we describe it using either one of two approaches as follows: 1. By following the particles, i.e., we express the variables as functions of the particle material coordinates X and time t. This approach is known as the material description or the Lagrangian description . 2. By observing the changes at fixed locations, i.e., we express the vari-ables as functions of fixed position x and time t. This approach is known as the spatial description or the Eulerian description . Consequently, the coordinates X are called the material coordinates while the coordinates x are known as the spatial coordinates . Equation 2.1 relates these two types of coordinates and can be written explicitly as follows: 77 78 Damage Mechanics x X O P(t o ) P(t) FIGURE 2.1 A particle in a body at times t 0 and t with position vectors X and x , respectively.(Lai et al., 1984) x 1 = x 1 ( X 1 , X 2 , X 3 , t ) x 2 = x 2 ( X 1 , X 2 , X 3 , t ) x 3 = x 3 ( X 1 , X 2 , X 3 , t ) (2.2) The material description is defined as the time rate of change of a quan-tity of a material particle. Let D/Dt denote the material derivative. We next explore the material derivative using both the material description and the spatial description of motion.
eBook - PDF- Tribikram Kundu(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
1 Fundamentals of the Theory of Elasticity 1.1 Introduction It is necessary to have a good knowledge of the fundamentals of continuum mechanics and the theory of elasticity to understand fracture mechanics. This chapter is written with this in mind. The first part of the chapter (section 1.2) is devoted to the derivation of the basic equations of elasticity; in the second part (section 1.3), these basic equations are used to solve some classical boundary value problems of the theory of elasticity. It is very important to comprehend the first chapter fully before trying to understand the rest of the book. 1.2 Fundamentals of Continuum Mechanics and the Theory of Elasticity Relations among the displacement, strain, and stress in an elastic body are derived in this section. 1.2.1 Deformation and Strain Tensor Figure 1.1 shows the reference state R and the current deformed state D of a body in the Cartesian x 1 x 2 x 3 coordinate system. Deformation of the body and displacement of individual particles in the body are defined with respect to this reference state. As different points of the body move, due to applied force or change in temperature, the configuration of the body changes from the reference state to the current deformed state. After reaching equilibrium in one deformed state, if the applied force or temperature changes again, the deformed state also changes. The current deformed state of the body is the equilibrium position under current state of loads. Typically, the stress-free configuration of the body is considered as the reference state, but it is not necessary for the reference state to always be stress free. Any possible configuration of the body can be considered as the reference state. For sim-plicity, if it is not stated otherwise, the initial stress-free configuration of the body, before applying any external disturbance (force, temperature, etc.), will be considered as its reference state.
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