Brief Wave Equations Introduction
In the field of dynamic mechanical testing where the tests last under one millisecond, any measurement is difficult. Electronic components have frequency bands that can limit the data that can be measured, and physical components have limits on the speed of measurements due to the speed waves can travel through them. Both limitations must be considered when using Hopkinson bar systems to acquire the dynamic mechanical properties of materials.
When studying solid mechanics and physics, any force applied to a material behaves as a wave when it is first applied. At the onset of the force, the atoms on the surface of the material are pushed or pulled which then causes the next layer of atoms to either be pushed or pulled. This begins a chain reaction of forces that travels one layer of atoms to the next. This chain reaction of forces is called an elastic stress wave because the forces are not high enough to induce permanent, plastic deformation of the atoms within the material.
The speed that a stress wave within a material moves from one layer of particles to the next is governed by the atomic forces between neighboring atoms and the weight of those atoms. These two properties are directly correlated to the overall stiffness and density of the material. A material that is less dense but similarly stiff compared to another material will have waves that travel quicker through it. A material that is stiffer but similarly dense compared to another material will also have waves travel quicker through it.
In terms of material properties, a stiffer material has a higher Young’s modulus (E) and a denser material has a higher density (ρ). The correlation between these material properties and the wave speed of the material (c) is:
From one-dimensional wave equations, there is a relationship that uses the equation above with the equation for one-dimensional momentum to derive the following property:
v = c ⋅ ε
where v is the velocity of a particle within a medium and ε is the strain experienced at that location in the medium. The final equation needed is the equation for engineering strain during a mechanical test
ε = ΔL / L
where (ΔL) is the change in length of the specimen and L is the original length of the specimen.
Hopkinson Bar System Components and Data
A Hopkinson bar system is comprised of four components:
- Striker bar
- Incident bar
- Transmission bar
- Striker bar accelerator
The incident and transmission bars are each instrumented with a strain gauge to measure stress waves as they pass through them. A test begins by starting the striker bar accelerator which propels the striker bar towards the incident bar. Once the striker hits the incident bar, it generates a stress wave that propagates down the incident bar. This initial stress wave is called the incident wave, and it is measured by the strain gauge in the middle of the incident bar. The stress wave continues through the incident bar until it reaches the specimen. At this point, some of the energy from the incident wave deforms the specimen. This portion of the wave transmits through the specimen and into the transmission bar. This is called the transmitted wave. The rest of the energy is reflected into the incident bar, and this is called the reflected wave. All calculations from a Hopkinson bar test are done using these three waves.
Derivation of the Specimen Strain History
During a compression Hopkinson bar test, the specimen is sandwiched between the incident and transmission bars; therefore, the original length of the specimen for the purpose of calculating its strain can be assumed to be the original separation of the two bars.
It can also be assumed that for the duration of the compression test, the two bars are always in contact with the specimen, so the displacement of the incident bar is equal to the displacement of the left side of the specimen, and the displacement of the transmission bar is equal to the displacement of the right side of the specimen. This is equivalent to saying that change of position or velocity of the incident bar and transmission bar is the same as the velocity of the left and right sides of the specimen.
v = vIBar – vTBar
where v is the time rate of change for the length of the specimen and vIBar and vTBar are the velocities of the ends of the incident and transmission bars respectively. Since each of the bars is experiencing a stress wave while its ends are moving, the particle velocity to strain relation applies:
Substituting in those equations gives:
v = c ⋅ εIBar – c ⋅ εTBar
Assuming the material of the two bars is the same means the wave speeds for each bar are also the same:
v = c ⋅ (εIBar – εTBar)
From a Hopkinson bar test, there are three waves that each have strain-time data: the incident wave, the reflected wave, and the transmitted wave. Since only one of the three waves occurs in the transmission bar, the strain in it can only come from the transmitted wave strain. Both the incident and reflected waves occur in the incident bar. Looking back at the animations showing the properties that affect wave speed in a material, for a given input displacement of the particle on the left, the right most particle moves double the distance. This is how a wave interacts with a free boundary or a boundary that has no resistance. In a Hopkinson bar test, the specimen is typically much weaker than the incident bar. While this is not a free boundary it is a weaker boundary. In fact, the weaker this boundary is, the closer the displacement is to doubling. Because of this, the strain in the incident bar is the summation of the amplitudes of the incident and reflected waves. Since the reflected wave has an opposite sign compared to the incident wave (again because the specimen is weaker than the incident bar), the strain in the incident bar is the incident wave subtracted by the reflected wave which gives:
v = c ⋅ ((εI – εR) – εT)
Where εI is the incident wave strain, εR is the reflected wave strain, and εT is the transmitted wave strain. Both sides of this equation can be integrated with respect to time to give the change in length of the specimen:
ΔL = ∫ c((εI – εR) – εT)dt
Finally, using the equation for engineering strain and dividing each side by the original length of the specimen the engineering strain at a time t for a specimen during a Hopkinson bar test is:
ε(t)=c/L∫0t ((εI – εR) – εT)dt.
