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Elastomers are widely used in many industries. Their use requires thorough knowledge of their strength and stiffness parameters over a wide temperature range. However, determination of the parameters of such materials is still a challenge. Therefore, the paper presents research methodology allowing determination of the properties of rubber-like materials in a wide range of stretch and temperatures (from +50°C to −25°C) by using the example of styrene-butadiene rubber (SBR) and natural rubber (NR) elastomers. Additionally, two blends, chloroprene rubber/nitrile-butadiene rubber (CR/NBR) and NR/SBR blends, were also considered. Based on physical premises, a polynomial and Arruda–Boyce hyperelastic constitutive models parameters were determined using two different methods, namely curve-fitting and the successive response surface method.