Concrete　structures　are　often　in　complex　stress　state　for　a　long　time　in　practical　engineering.　This　study　presents　a　numerical　investigation　on　the　influence　of　strain　rate　and　lateral　stress　ratio　on　the　dynamic　biaxial　compressive　failure　and　the　corresponding　size　effect　of　concrete.　A　3D　meso-scale　modelling　approach　considering　concrete　heterogeneity　was　established.　The　mechanical　responses　of　cubic　concrete　specimens　having　different　sizes　under　dynamic　biaxial　compressive　conditions　with　different　strain　rates　and　lateral　stress　ratios　were　simulated.　The　simulation　results　indicate　that　under　the　same　lateral　stress　ratio,　the　size　effect　on　dynamic　biaxial　compressive　strength　is　gradually　weakened　with　the　increasing　strain　rate.　Under　the　same　strain　rate,　the　size　effect　is　weakened　first　and　then　strengthened　with　the　increasing　lateral　stress　ratio.　At　the　same　time,　the　increasing　strain　rate　would　weaken　the　influence　of　the　lateral　stress　ratio　on　the　size　effect.　In　addition,　the　strength　criteria　for　dynamic　biaxial　compressive　strength　of　concrete　under　medium　and　low　strain　rates　was　proposed　and　verified.　Finally,　a　semi-empirical　and　semi-theoretical　size　effect　formula　considering　the　quantitative　influence　of　strain　rate　and　lateral　stress　ratio　on　the　size　effect　was　established.