Physically, this means that when you pull on the material in one direction it expands in all directions and vice versa :. Through Poisson's ratio, we now have an equation that relates strain in the y or z direction to strain in the z direction. We can in turn relate this back to stress through Hooke's law. This is an important note: pulling on an object in one direction causes stress in only that direction , and causes strain in all three directions.
Let's write out the strains in the y and z direction in terms of the stress in the x direction. Remember, up until this point, we've only considered uniaxial deformation. In reality, structures can be simultaneously loaded in multiple directions, causing stress in those directions. A helpful way to understand this is to imagine a very tiny "cube" of material within an object. That cube can have stresses that are normal to each surface , like this:. So, applying a load in the x direction causes a normal stress in that direction, and the same is true for normal stresses in the y and z directions.
And, as we now know, stress in one direction causes strain in all three directions. So now we incorporate this idea into Hooke's law, and write down equations for the strain in each direction as:. These equations look harder than they really are: strain in each direction or, each component of strain depends on the normal stress in that direction, and the Poisson's ratio times the strain in the other two directions. Now we have equations for how an object will change shape in three orthogonal directions.
Well, if an object changes shape in all three directions, that means it will change its volume. A simple measure for this volume change can be found by adding up the three normal components of strain:. Now that we have an equation for volume change, or dilation , in terms of normal strains, we can rewrite it in terms of normal stresses.
A very common type of stress that causes dilation is known as hydrostatic stress. This is just simply a pressure that acts equally on the entire material. Since it is acting equally, that means:. So, in the case of hydrostatic pressure we can reduce our final equation for dilation to the following:. This final relationship is important, because it is a constitutive relationship for how a material's volume changes under hydrostatic pressure.
The prefactor to p can be rewritten as a material's bulk modulus , K. Finally, let's get back to the idea of "incompressible" materials. What happens to K — the measure of how a material changes volume under a given pressure — if Poisson's ratio for the material is 0.
Hooke's Law in Shear In the previous section we developed the relationships between normal stress and normal strain. Thus, the bone in the top of the femur is arranged in thin sheets separated by marrow while, in other places, the bones can be cylindrical and filled with marrow or just solid. Overweight people have a tendency toward bone damage due to sustained compressions in bone joints and tendons. Privacy Policy. Skip to main content. Static Equilibrium, Elasticity, and Torque.
Search for:. Elasticity, Stress, Strain, and Fracture. Elasticity, Stress, and Strain Elasticity is a measure of how much an object deforms strain when a given stress force is applied. Learning Objectives Identify properties of elastic objects. Key Takeaways Key Points Elasticity is a measure of the deformation of an object when a force is applied. Objects that are very elastic like rubber have high elasticity and stretch easily.
Stress is force over area. Strain is change in length over original length. Just beyond the proportional limit is the elastic limit, at which point the material transitions from elastic behavior, where any deformation due to applied stress is reversed when the force is removed, to plastic behavior, where deformations caused by stress remain even after the stress is removed.
For many materials, the proportional limit and the elastic limit are the same or nearly equal. In the stress-strain curve shown here, the proportional limit and the elastic limit are assumed to be the same.
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.
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