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This paper presents a tactile sensor capable of measuring Young's modulus and shear modulus of elasticity for polymers, soft tissue, and other materials. The sensor is built by using polydimethylsiloxane as the structural material. A number of sensing cells of different stiffnesses are used in each sensor. The Young's modulus of the targeted object can be measured based on the relative deflections of adjacent sensing cells. The shear modulus measurement is enabled by using a quad-electrode structure in each sensing cell. Experimental results show sensing resolutions of 0.1 MPa for Young's modulus measurement in the range of 0.1-1 MPa and 0.05 MPa for shear modulus measurement ranging from 0.05 to 0.2 MPa. The flexible tactile sensor can be integrated on the end effector of robotic arms to achieve tactile sensing feedback. The proposed sensing technology can also be utilized for fast measurement of both Young's modulus and shear modulus for industrial applications that involve measuring mechanical properties of materials.