Mechanics of breaking

Intro

With proper training, board breaking is an excellent way to demonstrate the power of martial arts. A scientific understanding of the mechanics behind board breaking is helpful to all board breakers.

Board breaking is a relatively simple task, but, as with all applications of force, it can be empirically analyzed. The following is an analysis of the speed and force required to break a board.

Breaking analysis

Board breaking consists of applying a large force to a piece of wood for a short period. Since both force and instantaneous striking velocity are important, this breaking analysis was undertaken relative to breaking energy.

The boards were assumed to be standard 12” x 12”x 1” white pine. The boards were held tightly at both ends (so all the striking energy goes into breaking the board). Boards were assumed to be free of defects (no high local stress concentrations) and modeled according to column loading. This means that the maximum energy that the board can store before fractures begin to propagate is given by:

Umax=V*sb/2E

Where sb is the breaking stress, V is the board volume, and E is Young’s Modulus. Since the boards break mostly in tension, sb can be replaced with Mr, the modulus of rupture. The modulus of rupture is the highest tensile stress a material can undergo in bending before fracturing.

Apparent mass is a function of the striking technique. In the case of a punch, someone could punch with only the mass their arm, or they could put their entire body mass behind the punch by snapping the shoulder and hip into the strike. Striking speed is defined as the instantaneous velocity of the fist at the point of contact with the board.

The standard case is a fresh pine board with a volume of 0.0017 m3. With Mr=0.061 GN/m2 and E=8.81 GN/m2, the breaking energy is 359 J. The energy required to break the board is shown as a horizontal plane with z=359. Any combination of striking speed and apparent mass above this plane will result in the fractured board, any combination below the plane will result in a fractured hand.

Several recognized limitations exist with this analysis. First, the breaking stress-energy is defined as the average energy per unit volume. During a strike, the stress-energy is highly localized, and therefore neither shape nor geometric variations are adequately addressed. Secondly, at least some energy from the strike is used to displace the board, and some is dissipated as heat and sound.

Overall, the predictions of the model are consistent with the experiential evidence and provide a meaningful estimate of parameters that can determine whether a given board will break or not break.

Non-idealized holding conditions

The above analysis describes the ideal holding conditions. However, even the best holders will let the board move backward slightly. Also, speed breaks are performed with a board held only at one end. How does this affect the required breaking energy?

One-Side Hold

In a one-side hold, the board is modeled as a cantilever beam since the board holder's effect on the break is negligible. The martial artist must deliver enough energy to cause any deflections of the board in addition to the energy required to break the board. This will be equal to the displacement of the beam multiplied by the force applied to cause that displacement. With a point load P applied in the middle of the board, the maximum deflection is given by:

D=5PL3/(24EI)

Assuming the deflection is small and linear, multiplying this by the force that caused this displacement will give a reasonable approximation to the energy ‘wasted.’ Since E= Force times Distance, now the energy delivered to a board during a strike is:

E=1/2mav2 - 5P2L3/(24EI)

But how do we find P? For this analysis, we consider the impulse applied to the board. Since its original momentum is zero, we have the momentum of the strike (mav) equal to the force of the strike multiplied by the time of application. Here the time of application is taken to be 0.25 seconds. This leads to the final equation for a one-sided hold:

E=1/2mav2 – 5(mav)L3/(6EI)

For the given parameters over a normal range of strikes, less than one joule is lost to move the board, which moves only 7 mm. However, a perfectly fixed cantilever beam is an unrealistic assumption. A loosely held board sweeps out a quarter circle with a radius of its length, so the average board particle moves pi*L/8 meters.

E=1/2mav2 – (mav)L*pi/2

Two-Side Hold

A two-side hold, in which the board holder’s ability to hold the board stationary may affect the break, can be modeled in several different ways. The amount the holders allow the board to move backward could be expressed as a fixed distance, or a fixed or percentage amount of energy or force absorbed. But a percentage of the force absorbed is the best option, because the holders are applying a force. This will let the distance that the board moves be a function of the strike. With F as a fraction of the force absorbed by the holders, the board now absorbs the net force Fnet=(1-F) (mav)/0.25 s, which gives acceleration to the board of Fnet/mb. Since the board has this acceleration for the same 0.25 seconds and W=Fnet*1/2ab*t2, the energy imparted to the board during a strike is:

E=1/2mav2 – ((1-F) (mav))2 /2 mb

In general, for F values greater than 0.9 the difference is small. For 0.90, the difference becomes significant and only a powerful strike can fracture the board. Below this range, the model breaks down as the board undergoes an acceleration that causes it to move away from the strike faster than the strike itself, so the board cannot be broken regardless of the strike.

It is interesting to note that this model can also handle F values slightly greater than one, such as when the holders push the board into the strike, such as when an adult pushes the board against a child’s strike to make it easier for the child to accomplish the break.

Sources

Pottle, Bill. ABEN 456, 10/9/2000
https://www.dontow.com/2008/06/the-physics-of-martial-arts-breaking-boards/

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