The most common way to analyze the relationship between stress and strain for a particular material is with a stress-strain diagram. 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. As long as the applied stresses are below the proportional limit, stress-strain relationships are the same whether the material is under tension or compression.
Offset yield strength is the stress that will cause a specified amount of permanent strain typically 0. It is found by drawing a line that crosses the X strain axis at 0. This property of a material is known as Poisson's ratio , and it is denoted by the Greek letter nu , and is defined as:. Or, more mathematically, using the axial load shown in the above image, we can write this out as an equation:.
Since Poisson's ratio is a ratio of two strains, and strain is dimensionless, Poisson's ratio is also unitless. Poisson's ratio is a material property.
Poisson's ratio can range from a value of -1 to 0. For most engineering materials, for example steel or aluminum have a Poisson's ratio around 0. Incompressible simply means that any amount you compress it in one direction, it will expand the same amount in it's other directions — hence, its volume will not change. 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. Now we have to talk about shear. Let's go back to that imaginary cube of material. In addition to external forces causing stresses that are normal to each surface of the cube, the forces can causes stresses that are parallel to each cube face.
And, as we know, stresses parallel to a cross section are shear stresses. Now that cube of material looks a lot more complicated, but it's really not too bad. On each surface there are two shear stresses, and the subscripts tell you which direction they point in and which surface they are parallel to.
For instance, take the right face of the cube. Stresses normal to this face are normal stresses in the x direction. There are two stresses parallel to this surface, one pointing in the y direction denoted tau xy and one pointing in the z direction denoted tau xz.
Therefore, there are now six stresses sigma x , sigma y , sigma z , tau xy, tau yz, tau xz that characterize the state of stress within a homogenous, isotropic, elastic material. So, how do these shear stresses relate to shear strains?
Hooke's law in shear looks very similar to the equation we saw for normal stress and strain:. In this equation, the proportionality between shear stress and shear strain is known as the shear modulus of a material. That's the equation in its general form, but we can rewrite it more explicitly in terms of its components of x,y , and z. Doing so will give us the generalized Hooke's law for homogenous, isotropic, elastic materials. In our generalized Hooke's law we have our six components of stress and strain, and three material properties.
A natural question to as is how do these three material properties relate to each other? That relationship is given by the following equation:. Summary We've introduced the concept of strain in this lecture. Strain is the deformation of a material from stress.
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