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Many engineering and biological materials have been found to have tension–compression asymmetry properties. An extension of the linear thermoelastic behavior called a bi-modulus constitutive model is proposed to study the impact of thermo-mechanical coupling in such materials and structures. The well-posedness of the corresponding bi-modulus thermoelasticity problems is confirmed via a variational approach. An efficient numerical analysis method is developed for such nontrivial material behavior and is further employed for structural topology optimization. The importance of the proposed analysis and design framework is demonstrated by the significant impact of the tension–compression asymmetry and thermal effect on structural responses and optimum structural configurations. • A thermo-mechanical constitutive model considering TCA is constructed. • The well-posedness of the bi-modulus thermoelasticity theory is demonstrated. • An efficient analysis algorithm of thermoelastic structures with TCA is proposed. • MMC-based design optimization of thermoelastic structures with TCA is developed. [ABSTRACT FROM AUTHOR]