Stress Waves in Solids

ABOUT

Stress can be understood as the internal resistance of a body to cope with external forces acting upon it. Under normal circumstances, forces are gradually applied over time, allowing the stress and displacement fields within the body to adjust continuously. This process leads to an overall equilibrium as the load is steadily imposed.

However, when loads are suddenly applied within the volume of the body—such as during earthquakes, impacts from projectiles, or explosions over the surface—local stresses are generated. These stresses may not be immediately propagated throughout the entire object, and the rest of the body might remain unaware of these localized impacts at the moment of application. Instead, these localized stresses travel through the body as stress waves over a finite period of time. As they propagate, they convey the impact of the sudden displacement to other regions of the object, eventually affecting its overall structural integrity.

These disturbance waves created locally will be transmitted to remote areas through the body as well as over the surface. Naturally the former is called body waves and the latter surface waves. Body waves (push-pull) are compressive – tensile stress waves and surface waves are shear waves. Further classifications as spherical waves, longitudinal waves (both push-pull) and Rayleigh, Love, and Lamb waves (all surface) are defined.

Body waves

P and S waves

Surface waves

Rayleigh and Love waves

THEORY - BALL BAR IMPACT - HERTZ CONTACT THEORY

The stress wave propagation technique is a non-destructive testing method used to characterise material properties. The method involves impacting an elastic sphere or a rod against another material (preferably a longer rod of known length). The velocity of impact directly influences the stress produced in the test material.

Longitudinal stress pulses set up by the collinear impact of two bars are easiest to comprehend. The ball-bar impact produces a point contact between the two surfaces.

Deformation kinematics during impact of a ball on a rod

The elastic deformation of an impacting solid will be accounted for. The duration, shape, and amplitude of the stress pulse are deduced using the basic principles of mechanics and Hertzian contact theory. Wave propagation effects within the ball will be neglected.

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For a bearing ball impacting a homogenous, isotropic bar at a defined velocity, the pulse generated in the bar travels along as a compressive pulse from the impact end to get reflected as a tensile pulse from the free, distal end. A spherical stress wave starting at the point of impact but will become a plane wave within a few diameters of the bar. If the bar is long enough, the incident and reflected pulses do not overlap and the two can be seen separately. The contact time between the ball bar interface is given as the Pulse Width (τ) and the time taken to complete one cycle is defined as the Characteristic Time (T)

The stress pulses thus created will cause axial (longitudinal) and a diametrical (lateral) strain pulse, the latter due to Poisson’s effect. This ratio of the two strains is Poisson’s ratio of the material of the rod. Schematically, strain pulses are shown in the figure.

A rectangular stress pulse of amplitude σ and period τ travels along the long bar from the impact end to its distal end. If this end is unconstrained, a compressive stress pulse travels along the bar and gets reflected from the free end as a tensile pulse travels back to the struck face of the bar at a time T=2L/C after impact. The new free face as the striker bar has departed, the wave reflects as a compressive pulse again.

If the bar were to be fixed at the distal end, the reflected pulse would be compressive, owing to the displacement constraint. The reflected pulse travels back to the impact face and is re-reflected from this, now free end. Such notional stress pulse propagations, reflections and re-reflections occur in finite bars until it is damped out due to material hysteresis. The pulses are represented as (i) – (vi). The odd number pulses start from the impact face and the even number pulses start from the distal face.

EXPERIMENT

.A straight bar of 20mm diameter bar of 1.2m length suspended with nylon threads attached to a post at equal distances throughout the length of the bar. A 30mm steel ball bearing was used as the impactor, also suspended from the top of the ball with the help of nylon threads. The ball was aligned with the rod to