Ultrasound-induced cavitation in the viscoelastic biological tissues has attracted considerable attention due to its mechanical bio-effects, such as cell sonoporation, hemolysis, vascular disruption and tissue erosion. Cavitation can exert strong mechanical stresses on the surrounding tissues during the rapid bubble growth and collapse. The occurrence of cavitation needs the ultrasound exposure exceeding a certain acoustic pressure threshold, and the cavitation threshold is very high in most tissues, probably causing undesirable side-effects. Introducing artificial cavitation nuclei, e.g., microbubbles and nanodroplets stabilized with a shell such as albumin, lipids or polymers, into the targeted region can effectively reduce the cavitation threshold and significantly enhance the cavitation effects. However, neither the cavitation dynamics of an encapsulated microbubble nor the cavitation-induced stress field around the bubble in a soft tissue is quite clear. In this study, a comprehensive numerical model is developed to describe the dynamics of a lipid-shelled microbubble in vivo and quantify the cavitation-induced mechanical stress in the tissue. Considering the nonlinear changes of both shell viscosity and elasticity, a Gilmore model that has been considered as the most developed and realistic cavitation model is coupled with the Zener viscoelastic model for precisely describing tissue viscoelastic behavior with both creep recovery and stress relaxation of tissue. The developed model has an advantage of accurately describing the bubble behaviors in different biological tissues at high ultrasound intensities, especially for the bubble collapse. Furthermore, the spatiotemporal evolution of mechanical stress in the surrounding tissue generated by the cavitation bubble is investigated. Finally, the effects of encapsulated shell, elasticity modulus and viscosity of tissue as well as ultrasound amplitude are examined. The results show that the viscoelasticity of encapsulated shell and tissue both inhibit the bubble oscillations, and the tissue viscoelasticity has a larger inhibition effect. During the bubble oscillation, the compressive (negative) stress is generated in the tissue with the bubble growing and it continuously increases until it reaches a maximum value at a maximum radius, while the tensile (positive) stress is generated at the stage of bubble collapse and initial stage of bubble rebound due to the restoration of deformed tissue. The stress magnitude is greatest near the bubble wall and decreases rapidly with depth extending into the surrounding tissue. By contrast, the tensile stress decreases at a higher rate than the compressive stress. The encapsulated bubble presents a smaller stress in the tissue, but the decrease of the stress can be ignored at large acoustic pressures. Moreover, the stress decreases with the increase of tissue elasticity modulus, whereas it first increases and then decreases with tissue viscosity increasing, showing a maximum at 15 mPa·s. The increasing of the ultrasound amplitude enhances the bubble oscillations and consequently increases the stress in the tissue. This study is helpful in understanding the bubble dynamics and cavitation-induced mechanical stress of an encapsulated microbubble in soft tissue, which is essential for a safe and precise ultrasound therapy.