Comparison of strength characteristics of engineering materials is usually based on comparison of data from quasi-static tests. However, under dynamic loads, strength characteristics of materials are unstable and the strength ratio of two materials can change dramatically. This paper presents a comparative study of the strain/stress rate dependencies of the critical stresses of various quasi-brittle and ductile engineering materials, as well as their various internal states. A case of continuously increasing load in compression and splitting tests is considered. Experimental results were obtained for bricks, mortar and limestone in dry, saturated and frozen states; copper and titanium in different structural states; and road bitumens with different modifiers. The results show that such dependencies can intersect, thereby realizing a strength competition effect. This means that one material can withstand higher stresses under a quasi-static load and lower stresses under a dynamic load in comparison with another material. Thus, such an intersection suggests that the choice of a material based on quasi-static parameters may be erroneous for structures intended for dynamic modes. The analysis of strain/stress rate dependencies of ultimate or yield strength is carried out on the basis of the incubation time approach. The introduction of dynamic strength characteristics with the dimension of time is considered. This parameter characterizes well the response of the materials to dynamic loading. The calculation results show the possibility of estimating the critical stresses and the strength competition effect at high rate loads using no more than three parameters, which can be easily used in engineering practice.