Abstract: The stress state at a point in the material can be represented by three principal stresses \sigma_1, \sigma_2, \sigma_3. When it is specified that the principal stress is positive in pressure, the shrinkage deformation occurs along the direction of the maximum principal stress \sigma_1. If both the intermediate principal stress \sigma_2 and the minimum principal stress \sigma _3 are far less than \sigma_1, the lateral extending deformation will occur along the direction of \sigma_2 and \sigma_3. When the lateral extending deformation reaches a certain limit, the extending tension failure will occur in the direction parallel to \sigma_1. There is still a lack of research on how to establish the strength criterion of this kind of extending tension failure, the maximum tensile strain theory (the second strength theory) is sometimes used to explain the extending tension failure, but it is difficult to apply it to the triaxial stress state. In this paper, \varepsilon_1, \varepsilon_2 are used to represent the maximum tensile strain and the intermediate tensile strain respectively. Based on the maximum strain theory, the failure will occur if \varepsilon_1 reaches the uniaxial tensile yield strain. The extension failure criterion will be established herein when the sum of \varepsilon _1 +\varepsilon_2 reaches the critical value \varepsilon_u and it can be proved that \varepsilon_1 +\varepsilon_2 actually denotes the extension rate of the \sigma_1-plane. When \sigma_3 <\sigma_2 \ll \sigma_1, most rocks have the characteristics of brittle failure, so the rock material in prefailure stage can be assumed as linear elastic that satisfies the generalized Hooke's law. Thus, the strength criterion expressed by \varepsilon_1 and \varepsilon_2 can be expressed by \sigma_1, \sigma_2, \sigma_3. In this process, the rock's characteristics of different elastic parameters and strength under tension and compression can also be considered, and the failure state of uniaxial tension and uniaxial compression can be used to determine \varepsilon_u. Regardless of whether \sigma_1, \sigma_2, \sigma_3 are compressive stress or tensile stress, or there is tension and compression in \sigma_1, \sigma_2, \sigma_3, corresponding strength criteria can be established based on \varepsilon_1 +\varepsilon_2 =\varepsilon_u. The established criterion can reflect the effect of intermediate principal stress \sigma_2 on the strength. It can also be proved that: like yielding which will happen under the hydrostatic tension but not under the hydrostatic compression; compression failure can increase the plastic volume, and the results can better reflect the actual situation than Mohr-Coulomb criterion. The established strength criterion is verified by experimental data under tension-compression stress state and the theoretical calculation results are in good agreement with the existing test data. Through the proposed strength criterion and the test results of disc splitting, a more reliable uniaxial tensile strength of rock can be obtained.