Blue spaces have been gradually recognized due to their positive impact on human mental and physical well-being. However, existing works have greatly depended on subjective questionnaires and lack objective and quantitative evidence, especially regarding the effects of specific spatial scales of water environments on brain activity.

To address this gap, this study uses virtual reality (VR) technology to construct immersive waterbody environments with systematically varied visual and auditory properties, which enables controlled experimental exposure to different water space scales. A total of 52 healthy participants aged 18-36 from Tsinghua University were involved in experiencing 35 water scenarios characterized by five levels of visible water area (0%, 30%, 50%, 80%, and 100%) and seven flow velocity levels (0-3.0 m/s). During the VR exposure, two neurophysiological indicators—electroencephalogram (EEG) alpha power and heart rate variability (HRV)—were concurrently recorded to reflect the cognitive and autonomic brain states of the participants. EEG alpha activity, which is associated with relaxation and creative ideation, and HRV, which is an index of emotional regulation and adaptive capacity, served as core outcome measures. After signal preprocessing and normalization, Gaussian process regression was adopted to model the nonlinear coupling between water environment features and physiological responses.

Results revealed significant interindividual variability in responses to water scale. For the majority of participants (Class I), moderate visible water areas (10%-30%) combined with low flow velocities (0.5-1.5 m/s) exhibited the most favorable neurophysiological responses, with EEG alpha power and HRV values increasing beyond resting baseline levels. However, these values decreased substantially in scenes with excessively large water areas (>50%) or higher flow velocities, which suggests that overstimulation from water features may suppress cognitive readiness and emotional stability. By contrast, a minority group (Class II) displayed the opposite pattern, which exhibited stronger EEG and HRV responses under conditions of larger water areas and either very low or high flow speeds. This divergence emphasizes the important role of individual backgrounds in shaping responses to blue space. Furthermore, a support vector machine classification model was developed based on the demographic and environmental background data of participants (including birthplace precipitation, humidity, and surface water ratio), which accurately predicted individual response categories with an accuracy of 95%. In addition, a single-factor analysis of water sound levels disclosed that moderate auditory stimuli (~30 dB) improved EEG alpha activity even in the absence of visual water elements, which reinforces the cognitive benefits of natural soundscapes and implies potential for non-visual design interventions.

Overall, this study constructs a robust experimental and modeling framework to quantify the neurocognitive impact of waterbody scale, which offers new insights into the modulating mechanism of specific aquatic features on brain states in dynamic and individualized ways. The findings show that the restorative and creativity-enhancing effects of blue space are neither universal nor linear but rather depend on environmental parameters and individual characteristics. Thus, these outcomes challenge conventional assumptions and highlight the need for tailored blue space design. The proposed method provides valuable scientific evidence for optimizing urban water landscapes not only for aesthetic or ecological purposes but also as cognitive infrastructures that support mental health, emotional resilience, and innovation across diverse populations and geographic contexts.

Accurate assessment of concrete fatigue life under fatigue load is essential to ensure the safety and stability of structures, especially the fatigue failure behavior dominated by stress and strain.The fatigue loading surface function is established to describe the fatigue state of concrete based on the constraint relationship in the stress-strain fatigue criterion.The fatigue loading surface function of concrete exhibits a monotonic variation with fatigue cycles, enabling the establishment of an equivalent function to represent the concrete fatigue state.The fatigue loading surface function of concrete can be described as a linear equivalent expression, and the coefficients can be calibrated by the characteristic points in the fatigue loading process.

Based on the constraint relationship between fatigue stress-strain and fatigue cycles, the equivalent fatigue cycles can be calculated from the fatigue stress-strain data.The equivalent fatigue cycles can effectively express the fatigue stress-strain state of the material, and the fatigue life indirectly represents the fatigue failure stress-strain state in the fatigue failure criterion of materials with the static constitutive curve as the limit value.The degree of fatigue accumulation of materials can be quantified by comparing the equivalent fatigue cycles and fatigue life.The evaluation method based on equivalent fatigue cycles overcomes the shortcomings of the current evaluation methods based on the classic fatigue criteria and fatigue envelope lines.Therefore, in this work, the fatigue loading surface function is constructed, and its evolution law is studied through the analogical form of the fatigue failure criterion of materials with a static constitutive curve as the limit value, thus proposing a description method for equivalent calibration and solving the equivalent fatigue cycles.The fatigue loading surface function is proposed to describe the fatigue state and determine the constraint relationship between fatigue stress and strain and fatigue cycles based on the fatigue failure criterion of materials with a static constitutive curve as the limit value.The equivalent fatigue loading surface function and coefficients can be obtained by the equivalent description method of feature point calibration.The R-square is introduced to ensure an equivalent description, and the maximum R-square directly relates to the optimal equivalent description results.Therefore, the maximum R-square algorithm is proposed based on the evolution law of the fatigue loading surface function.The linear equivalent form of the fatigue loading surface function is proposed to meet the equivalent description and practical application requirements.

Therefore, equivalent calibration can be achieved by selecting the optimal maximum R-square, and the coefficients of the fatigue loading surface function can be determined from the experimental results of the fatigue loading process.The equivalent fatigue loading surface function, feature point calibration, and maximum determinable coefficient algorithms were developed to achieve the equivalent fatigue state description of materials.Through the equivalent calibration results, the equivalent fatigue cycles can be obtained using the corresponding fatigue stress-strain.Furthermore, the fatigue stress-strain state of concrete can be quantified by the equivalent fatigue cycles, and corresponding evaluation processes and indicators are obtained through further study.

The proposed method provides an effective approach for the fatigue life analysis of concrete.