# Characterization of the Evolution of Energy Loss Rate in Cyclic Load Testing of Marine Soft Soil
## 3 Envelope Surfaces of Energy Dissipation Rates for Marine Soft Soil in Dynamic Triaxial Tests  
### Energy Calculation  
The input energy is the total energy applied to a marine soft soil specimen under external dynamic loading during the marine phase. It is calculated as follows:  
\[\mathrm{W}=\int\sigma d\epsilon \tag{1}\]  
where \(\sigma\) is stress, \(\epsilon\) is strain, and \(\int\sigma d\epsilon\) represents the area of the stress-strain curve, which can be calculated by the numerical integration method. Under dynamic or cyclic loading, marine soft soil behaves not as an elastic medium but exhibits significant viscoplastic characteristics. These characteristics lead to internal friction, pore pressure changes, structural damage, and other phenomena. As a result, the stress-strain (\(\sigma\)-\(\epsilon\)) relationship curve shows a distinct \"hysteresis\" behavior, where a closed or semi-closed hysteresis loop is formed during one or more cyclic loading-unloading cycles. The area of the hysteresis loop in the stress-strain coordinate system represents the energy input to the soil sample, which is consumed through friction, damage, and other factors (energy dissipation) during each cycle. The amount of energy absorbed and ultimately dissipated or transformed by the soil over a period of time (or cycle), denoted as \(\Delta W\), can be expressed as follows:  
\[\Delta W=\int\sigma d\epsilon \tag{2}\]  
In a dynamic triaxial test, for a single cycle (e.g., the \(n\) cycle, within the time range \(t_{1}\) to \(t_{2}\)), the energy dissipated can be expressed as an integral over time, as follows:  
\[\Delta W_{n}=\int_{t_{1}}^{t_{2}}\sigma(t)\frac{d\epsilon(t)}{dt}dt=\int_{ \epsilon(t_{1})}^{\epsilon(t_{2})}\sigma(\epsilon)d\epsilon \tag{3}\]  
If the test apparatus is capable of collecting continuous \(\sigma\)-\(\epsilon\) data, the energy dissipation for a single cycle can be obtained through direct numerical integration of the hysteresis loop (i.e., by calculating the area enclosed by the curve). The cumulative energy dissipation over multiple cycles is then given by the following equation:  
Figure 2: Relationship between dynamic stress amplitude and axial strain of marine soft soil. \[\Delta W_{L}=\sum_{n=1}^{N}\Delta W_{n} \tag{4}\]  
However, laboratory data are typically in the form of discrete points, where \(\sigma_{i}\) and \(\epsilon_{i}\) are obtained at specific sampling frequencies (\(i\) = 1, 2, \(\ldots\), \(M\)). To compute the energy dissipation for the n cycle or a specific time period, the trapezoidal rule for numerical integration can be applied:  
(1) A loading-unloading loop is partitioned into a series of points: (\(\epsilon_{1}\), \(\sigma_{1}\)), (\(\epsilon_{2}\), \(\sigma_{2}\)), \(\cdots,(\epsilon_{k}\), \(\sigma_{k}\)). (2) The area of this cyclic hysteresis loop can be approximated as follows:  
\[\Delta W\approx\sum_{j=1}^{k-1}\frac{\sigma_{j}+\sigma_{j+1}}{2}\left( \epsilon_{j+1}-\epsilon_{j}\right) \tag{5}\]  
(3) To calculate the total energy dissipation over multiple cycles or different stages, the \(\Delta W\) for each cycle (or each stage) is summed. To assess the energy evolution pattern of soft soil under cyclic loading, the energy dissipation rate can be defined as follows:  
\[D=\frac{\Delta W}{W} \tag{6}\]  
In summary, the core method for calculating energy dissipation based on dynamic triaxial test data of marine soft soil is to consider the area enclosed by the hysteresis loop formed by stress \(\sigma\) and strain \(\epsilon\), measured either in real time or in discrete steps, as the cyclic energy dissipation of marine soft soil. The specific calculation process can be summarized as follows:  
1. Determine the study period/cycle range: Extract \(\sigma(t)\) and \(\epsilon(t)\) data for the corresponding time period. 2. Numerical integration: Calculate \(\Delta W=\int\!\sigma\,\mathrm{d}\epsilon\), and use the trapezoidal method to approximate the area of the hysteresis loop in the discrete case. 3.