Impact force with unknown deformations or time duration?

I expected to find more documentation on analytical solutions to determine the contact force between bodies, where deformations ($\delta = \delta_{1} + \delta_{2}$) and impact time ($t = t_{f} - t_{i}$) are unknown.

The time interval begins at contact between bodies (with $m_{1}$, $v_{1,i}$, and $m_{2}$, $v_{2,i}$) and ends at the instant $v_{f} = v_{1,f} = v_{2,f}$. Without deformation, $d = 0$: $d = \frac {v'-v}{2}*t$ and $\mathrm {dP} = F\mathrm {dt}$ require the time interval $t = 0$ and $F \to \infty$.

Avoiding deformation analysis, is it possible to apply $\lim_{t \to 0}$ or $\lim_{d \to 0}$ to calculate a finite, average contact force?

If not, how is deformation ($\delta = \delta_{1} + \delta_{2}$) or time duration ($t$) calculated?

While accurate solutions are readily obtained by Finite Element Methods (continuum mechanics), as shown, do analytical solutions exist? Many engineering texts do contain partial solutions (for axial, bending, and torsional impact), where deformation is calculated for the impacted body only- they do not account for deformation in both bodies. This results is a conservative estimate of contact force and stress. Contact forces and stresses are well documented (ex. Contact Mechanics); however, they require that contact forces are known. Please explain how these might be introduced to solve for deformation in both bodies.

My incomplete solution follows. I am interested in critiques and/or practical, analytical approaches to this type of problem.

Conservation of Momentum: $$m_{1}(v_{f} - v_{1,i}) = m_{2}(v_{f} - v_{2,i}) \; \Rightarrow \; \therefore v_{f} = \frac {m_{1}v_{1,i}-m_{2}v_{2,i}}{m_{1}-m_{2}}$$

Without a known force, deformation is solved with respect to energy. For the ideal case, bodies are linear elastic (with stiffness $k = \frac{EA}{L}$ where $E$ is the Modulus of Elasticity, $A$ is contact area, and $L$ is the length perpendicular to contact area). Certainly, these large assumptions result in error; regardless, they allow for an estimate.

$$\Delta E_{Kinetic} = E_{Deform}$$

$$[\frac {1}{2}m_{1}v_{1,i}^2 + \frac {1}{2}m_{2}v_{2,i}^2]_\text{i} - [\frac {1}{2}(m_{1} + m_{2})v_{f}^2]_\text{f} = \frac {1}{2}k_{1}\delta_{1} + \frac {1}{2}k_{2}\delta_{2}$$

I am unable to complete the system of equations- I know that both bodies:

• Deform during the same time interval

• Have equal contact forces/surface stresses

Can this problem be solved analytically?