Home|Toolkits|Forums|Documents/Data|Links|About

« Previous

Joint Stiffness

The joint stiffness (i.e. dynamic compliance) will often decrease with the onset of damage within the joint (Figure 3). Ultimate failure is marked by a rapid reduction in joint stiffness and an increase in the loss or damping factor (tan δ). The loss factor tan δis the ratio between the tensile storage modulus E' and the tensile loss modulus E'' (i.e. tan δ = E'/E'') where is the phase angle between dynamic load/stress and the dynamic displacement/strain (see Figure 4) [1-2]. The storage modulus is proportional to the maximum energy stored during a loading cycle and represents the stiffness of the joint. The loss modulus is proportional to the energy dissipated (lost) during one loading cycle.

     Figure 3:  Normalised residual stiffness for fatigue loading of aluminium T-joints


Figure 4: Out of phase stress strain response for viscoelastic material, δ=20°


The stress and strain are given by the following relationships (see Figure 4):

σ = σ0sin(ωt + σ); ε = ε0sin(ωt)





Where ω is the period of strain oscillation, t is time and δ is the phase angle in radians.

The storage modulus E' and the loss modulus E'' are given by:

E' = 
σ0
ε
sinδ;    E' = 
σ0
ε
cosδ;





For a purely elastic material, the mechanical energy stored during loading is returned completely when the specimen is unloaded. This generates load/displacement or stress/strain curves for the material, which are completely in phase with each other (i.e. δ = 0°), producing no hysteresis (see Figure 5). At the other extreme a purely viscous material (i.e. δ = 90°), which exhibits no elasticity only damping, results in all the energy being dissipated and the load/displacement or stress/strain curves being out of phase with each other. Materials that fall between these two categories are classified as viscoelastic.

Figure 5: Hysteresis (stored energy): (left) elastic and (right) viscoelastic response


When viscoelastic materials are subjected to fatigue or cyclic motion a proportion of the mechanical energy per cycle is converted to thermal energy (heat) with the rest of the energy being stored. This produces load/displacement or stress/strain curves, which exhibit a phase lag (Figure 4); a hysteresis loop develops of which the area is equal to the dissipated energy per cycle (Figure 5). The stiffness and hysteresis can be derived mathematically.

When real data is analysed on specimens that have been fatigued to failure it should show that over time the stiffness of the material decreases towards failure with the energy loss of the material becoming greater as damage accumulates and friction causes material heating (see Figure 6). As damage accumulates within the bonded joint, the slope of the ellipse will decrease and the area within the ellipse will increase. For a considerable proportion of the fatigue life of the bonded joint (50-60%), stiffness and hystersis will remain fairly constant (see Figure 3). There will be a rapid change in stiffness and hysteresis within the last few hundred cycles.

Although measurement of global stiffness is relatively straightforward using a linear voltage displacement transducer (LVDT) or strain gauges, it is often difficult to obtain an accurate measurement of the hysteresis. Localised deformation measurements can be obtained using strain gauges bonded to the specimen surface. These gauges if strategically located at critical stress regions may indicate the onset of localised damage.

Figure 6: Storage and loss modulus changes


Useful Definitions (see also Cyclic Fatigue)



Fatigue Life is the number of cycles of fluctuating stress (load) or strain of a specified nature that a material will sustain before failure occurs. Fatigue life is a function of the magnitude of the fluctuating stress, specimen geometry and test conditions. Cyclic fatigue data is generally presented in the form of a S-N diagram (i.e. a plot of the fatigue life at various levels of fluctuating stress).

Fatigue (Endurance) Limit is the maximum fluctuating stress a material can endure for an infinite number of cycles. Under constant amplitude loading conditions, most materials or structures exhibit a plateau in the stress-cycle curve, which typically occurs at N > 106 cycles. The plateau level corresponds to the fatigue or endurance limit. Below this limit, the material or structure can be cycled indefinitely without causing failure.

Fatigue Strength is the magnitude of fluctuating stress required to cause failure in a fatigue test specimen after a specified number of cycles of loading. Usually determined directly from the S-N diagram.

References