Fatigue is a subjective feeling of tiredness which is distinct from weakness, and has a gradual onset. Unlike weakness, fatigue can be alleviated by periods of rest. Fatigue can have physical or mental causes. In materials science, fatigue is the weakening of a material caused by repeatedly applied loads. It is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The nominal maximum stress values that cause such damage may be much less than the strength of the material typically quoted as the ultimate tensile stress limit, or the yield stress limit.

Introduction:
When a material is subjected to the reversal of stresses, it experiences cyclic loading conditions. This means that the material undergoes alternating tension and compression forces repeatedly. In such scenarios, the material's ability to withstand fatigue becomes crucial. Fatigue failure occurs when a material fails under repeated or fluctuating stresses that are below its ultimate strength. However, there is a specific stress level at which the material does not fail, known as the endurance limit or fatigue limit. The endurance limit represents the maximum stress amplitude that a material can sustain indefinitely without failing, even under cyclic loading conditions.

Explanation:
To understand the concept of the material not failing at the endurance stress, let's examine the given options:

1) Fatigue stress: Fatigue stress is not the correct term. Fatigue is a phenomenon that leads to material failure under repeated or fluctuating stresses, and it is not a specific stress level.

2) Proof stress: The proof stress, also known as the yield strength, is the stress at which a material exhibits a specified amount of plastic deformation. It is not directly related to fatigue failure or the endurance limit.

3) Safe stress: Safe stress refers to the stress level at which a material is considered safe for use without experiencing failure. However, it does not specifically refer to the endurance limit under cyclic loading conditions.

4) Endurance stress: The correct term is the "endurance stress" or "endurance limit." This stress level represents the maximum stress amplitude that a material can withstand indefinitely without failing due to fatigue. It is below the ultimate tensile strength and is specific to each material.

Conclusion:
In summary, when a material is subjected to the reversal of stresses, it undergoes cyclic loading conditions. Fatigue failure can occur under repeated or fluctuating stresses, but there is a specific stress level called the endurance limit or endurance stress, below which the material does not fail. This endurance limit represents the maximum stress amplitude that the material can sustain indefinitely without experiencing fatigue failure. Thus, the correct option is 4) endurance stress.