The bulk modulus of the material is 10 to the 11th newtons per meter squared. What is the percent change in volume for a piece of this material when it is subjected to a bulk stress increase of 10 to the seventh newtons per meter squared? Assume that the force is applied uniformly over the surface.
Let’s start by highlighting some of the important information given. We’re told that the bulk modulus of the material under consideration is 10 to the 11th newtons per meter squared. Let’s call that 𝐵. We’re also told that this material is subjected to a stress increase of 10 to the seventh newtons per meter squared. We’ll call that Δ𝑃. we want to solve for the percent change in volume that the material experiences when it’s subjected to the stress. We’ll call that percent Δ𝑉.
To begin towards our solution, let’s recall the relationship for the bulk modulus of a material. Bulk modulus 𝐵 is equal to the volume of a material divided by its change in volume multiplied by the change in stress upon the material. Bulk modulus is essentially a measure of how much a material changes size when it is subjected to a certain stress. Applying this relationship to our scenario, we want to solve for the ratio Δ𝑉 divided by 𝑉.
To do that, we can rearrange this equation, starting by multiplying each side of the equation by Δ𝑉 divided by 𝑉. This cancels out the volume 𝑉 and change in volume, Δ𝑉, on the right side. If we then divide both sides of the equation by 𝐵, the bulk modulus, then that factor cancels out on the left side. We’re left with an equation for the ratio Δ𝑉 to 𝑉 is equal to Δ𝑃, the change in stress on the material, divided by the bulk modulus.
When we plug in our values for Δ𝑃 and 𝐵 and perform this division, we find the result of 0.0001. That’s the ratio of the change in volume to the original volume of the material. But we want to know the percent change. To find that, we take our result and multiply it by 100 percent.
When we do this, the decimal place shifts over to the right two spots, and we find that percent Δ𝑉 is equal to 0.01 percent. That’s the percent change in volume of the material due to the stress increase Δ𝑃.