homework and exercises - Forge-flattening

Is there an easy way to estimate how much energy is needed to forge a cube of iron into a thin square sheet with a hammer? I suppose i should integrate the force $$F = \text{yield_strength} \times \text{area(h)}$$ through the height, $h_0 = \text{height of the initial cube}$, $h_1$ -- resulting sheet's thickness. Am I right? Also, where to look for compressive yield strength of iron at high temperatures? I've read its better to forge it at $1000-1100 °\text{C}$

Answer

Your suggestion is reasonable as an estimate for pressing the metal; hammering would require significantly more energy (see below). The volume $V=Ah$ of metal is constant so as its thickness $h$ decreases its area $A$ increases. A graph of the temperature dependence of Yield Strength for several metals is given in Engineering ToolBox.

At room temperature the Yield Strength is about $p=50MPa$. The work required to compress a cube of side $h_0$ to a plate of thickness $h_1$ is $ph_0^3 \ln{\frac{h_0}{h_1}}$. For $h_0=100mm$ and $h_1=1mm$ this work is approx. $230kJ$.

An upper bound is the energy required to heat the metal to its melting point then change it from solid to liquid. Gravity would then re-shape the metal without further input of energy. For $h_1^3=0.001m^3$ of iron ($\approx 0.8kg$) this is about $375kJ$ when starting from $1100^{\circ}C$ and $760kJ$ when starting from room temperature $25^{\circ}C$. This is the same order of magnitude as the Yield Strength calculation.

For a lower bound estimate you could calculate the energy of the bonds which need to be broken in order to increase the surface area of the metal between its initial and final shapes. Bond energy for iron is about $120 kJ/mol$.

Hammering also dissipates energy through heating with a much smaller amount lost as sound. Whether this non-useful energy is significant depends on whether the hammer blows exceed the yield strength of $50MPa$ and by how much. The proportion of energy dissipated as sound increases as the surface area of the metal plate increases but remains small. Vibration energy is eventually lost mostly through heating of dense metal rather than radiation of sound in air.

For an interesting discussion of hammering see Can hammer blows increase workpiece temperature? which states that $90-95 \text{%}$ of the impact energy goes into raising temperature. It concludes that manual hammering does not cause any significant rise in temperature because the metal anvil quickly conducts away the heat generated. But power hammering can cause a significant increase in appropriate circumstances.