File Name: normal stress and strain .zip
Strain So far, we've focused on the stress within structural elements.
A model of a rigid body is an idealized example of an object that does not deform under the actions of external forces. It is very useful when analyzing mechanical systems—and many physical objects are indeed rigid to a great extent.
The extent to which an object can be perceived as rigid depends on the physical properties of the material from which it is made. For example, a ping-pong ball made of plastic is brittle, and a tennis ball made of rubber is elastic when acted upon by squashing forces. However, under other circumstances, both a ping-pong ball and a tennis ball may bounce well as rigid bodies.
Similarly, someone who designs prosthetic limbs may be able to approximate the mechanics of human limbs by modeling them as rigid bodies; however, the actual combination of bones and tissues is an elastic medium.
A change in shape due to the application of a force is known as a deformation. Even very small forces are known to cause some deformation. Deformation is experienced by objects or physical media under the action of external forces—for example, this may be squashing, squeezing, ripping, twisting, shearing, or pulling the objects apart.
In the language of physics, two terms describe the forces on objects undergoing deformation: stress and strain. Stress is a quantity that describes the magnitude of forces that cause deformation.
Stress is generally defined as force per unit area. When forces pull on an object and cause its elongation, like the stretching of an elastic band, we call such stress a tensile stress.
When forces cause a compression of an object, we call it a compressive stress. When an object is being squeezed from all sides, like a submarine in the depths of an ocean, we call this kind of stress a bulk stress or volume stress. In other situations, the acting forces may be neither tensile nor compressive, and still produce a noticeable deformation. For example, suppose you hold a book tightly between the palms of your hands, then with one hand you press-and-pull on the front cover away from you, while with the other hand you press-and-pull on the back cover toward you.
The SI unit of stress is the pascal Pa. When one newton of force presses on a unit surface area of one meter squared, the resulting stress is one pascal:. Another unit that is often used for bulk stress is the atm atmosphere. Conversion factors are. An object or medium under stress becomes deformed. The quantity that describes this deformation is called strain.
Strain is given as a fractional change in either length under tensile stress or volume under bulk stress or geometry under shear stress. Therefore, strain is a dimensionless number.
Strain under a tensile stress is called tensile strain , strain under bulk stress is called bulk strain or volume strain , and that caused by shear stress is called shear strain. The greater the stress, the greater the strain; however, the relation between strain and stress does not need to be linear.
Only when stress is sufficiently low is the deformation it causes in direct proportion to the stress value. The proportionality constant in this relation is called the elastic modulus.
In the linear limit of low stress values, the general relation between stress and strain is. As we can see from dimensional analysis of this relation, the elastic modulus has the same physical unit as stress because strain is dimensionless.
We can also see from Equation On the other hand, a small elastic modulus means that stress produces large strain and noticeable deformation. For example, a stress on a rubber band produces larger strain deformation than the same stress on a steel band of the same dimensions because the elastic modulus for rubber is two orders of magnitude smaller than the elastic modulus for steel.
Note that the relation between stress and strain is an observed relation, measured in the laboratory. Elastic moduli for various materials are measured under various physical conditions, such as varying temperature, and collected in engineering data tables for reference Table These tables are valuable references for industry and for anyone involved in engineering or construction. In the next section, we discuss strain-stress relations beyond the linear limit represented by Equation In the remainder of this section, we study the linear limit expressed by Equation Tension or compression occurs when two antiparallel forces of equal magnitude act on an object along only one of its dimensions, in such a way that the object does not move.
One way to envision such a situation is illustrated in Figure A rod segment is either stretched or squeezed by a pair of forces acting along its length and perpendicular to its cross-section.
The net effect of such forces is that the rod changes its length from the original length L 0 L 0 that it had before the forces appeared, to a new length L that it has under the action of the forces.
Forces that act parallel to the cross-section do not change the length of an object. The definition of the tensile stress is. Compressive stress and strain are defined by the same formulas, Equation The only difference from the tensile situation is that for compressive stress and strain, we take absolute values of the right-hand sides in Equation Therefore, the compressive strain at this position is.
What is the tensile strain in the wire? Objects can often experience both compressive stress and tensile stress simultaneously Figure One example is a long shelf loaded with heavy books that sags between the end supports under the weight of the books. The top surface of the shelf is in compressive stress and the bottom surface of the shelf is in tensile stress.
Similarly, long and heavy beams sag under their own weight. In modern building construction, such bending strains can be almost eliminated with the use of I-beams Figure A heavy box rests on a table supported by three columns. View this demonstration to move the box to see how the compression or tension in the columns is affected when the box changes its position.
When you dive into water, you feel a force pressing on every part of your body from all directions. What you are experiencing then is bulk stress, or in other words, pressure. Bulk stress always tends to decrease the volume enclosed by the surface of a submerged object. This kind of deformation is called bulk strain and is described by a change in volume relative to the original volume:.
This kind of physical quantity, or pressure p , is defined as. We will study pressure in fluids in greater detail in Fluid Mechanics. An important characteristic of pressure is that it is a scalar quantity and does not have any particular direction; that is, pressure acts equally in all possible directions.
When you submerge your hand in water, you sense the same amount of pressure acting on the top surface of your hand as on the bottom surface, or on the side surface, or on the surface of the skin between your fingers. What you feel when your hand is not submerged in the water is the normal pressure p 0 p 0 of one atmosphere, which serves as a reference point.
When the bulk stress increases, the bulk strain increases in response, in accordance with Equation The proportionality constant in this relation is called the bulk modulus, B , or. The minus sign that appears in Equation The reciprocal of the bulk modulus is called compressibility k , k , or.
Compressibility describes the change in the volume of a fluid per unit increase in pressure. Fluids characterized by a large compressibility are relatively easy to compress. For example, the compressibility of water is 4. This means that under a 1. If the normal force acting on each face of a cubical 1.
The concepts of shear stress and strain concern only solid objects or materials. Buildings and tectonic plates are examples of objects that may be subjected to shear stresses. In general, these concepts do not apply to fluids. Shear deformation occurs when two antiparallel forces of equal magnitude are applied tangentially to opposite surfaces of a solid object, causing no deformation in the transverse direction to the line of force, as in the typical example of shear stress illustrated in Figure Shear strain is caused by shear stress.
Shear stress is due to forces that act parallel to the surface. The shear modulus is the proportionality constant in Equation Shear modulus is commonly denoted by S :. As an Amazon Associate we earn from qualifying purchases. Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4. Skip to Content. University Physics Volume 1 My highlights.
Table of contents. Chapter Review. Waves and Acoustics. Answer Key. By the end of this section, you will be able to: Explain the concepts of stress and strain in describing elastic deformations of materials Describe the types of elastic deformation of objects and materials.
Figure In both cases, the deforming force acts along the length of the rod and perpendicular to its cross-section. In the linear range of low stress, the cross-sectional area of the rod does not change. Compressive Stress in a Pillar A sculpture weighing 10, N rests on a horizontal surface at the top of a 6.
Find the compressive stress at the cross-section located 3. Stretching a Rod A 2.
A model of a rigid body is an idealized example of an object that does not deform under the actions of external forces. It is very useful when analyzing mechanical systems—and many physical objects are indeed rigid to a great extent. The extent to which an object can be perceived as rigid depends on the physical properties of the material from which it is made. For example, a ping-pong ball made of plastic is brittle, and a tennis ball made of rubber is elastic when acted upon by squashing forces. However, under other circumstances, both a ping-pong ball and a tennis ball may bounce well as rigid bodies.
In continuum mechanics , stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight , each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure , each particle gets pushed against by all the surrounding particles. The container walls and the pressure -inducing surface such as a piston push against them in Newtonian reaction. These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules.
A model of a rigid body is an idealized example of an object that does not deform under the actions of external forces. It is very useful when analyzing mechanical systems—and many physical objects are indeed rigid to a great extent. The extent to which an object can be perceived as rigid depends on the physical properties of the material from which it is made. For example, a ping-pong ball made of plastic is brittle, and a tennis ball made of rubber is elastic when acted upon by squashing forces.
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