And for strength is an ability for resist the external force which directly act on the material. We can calculate strength of material by use the ratio of Stress / Strain at the same point. Stress is the force that resist the deformation by the external force, and Strain is the ratio of different length / original length.
So, we can conclude hardness is focus on getting scar on the surface of material and strength is focus on an ability for resist the deformation from external force.
Differences between strength and hardness
Tensile or Compressive Stress, Strain, and Young’s Modulus
Tension or compression occurs when two antiparallel forces of equal magnitude act on an object along only one of its dimensions, in such a way that the object does not move. One way to envision such a situation is illustrated in Figure 1. A rod segment is either stretched or squeezed by a pair of forces acting along its length and perpendicular to its cross-section. The net effect of such forces is that the rod changes its length from the original length L0 that it had before the forces appeared, to a new length L that it has under the action of the forces. This change in length ΔL=L−L0 may be either elongation (when L is larger than the original length L0) or contraction (when L is smaller than the original length L0). Tensile stress and strain occur when the forces are stretching an object, causing its elongation, and the length change ΔL is positive. Compressive stress and strain occur when the forces are contracting an object, causing its shortening, and the length change ΔL is negative.
In either of these situations, we define stress as the ratio of the deforming force F⊥ to the cross-sectional area A of the object being deformed. The symbol F⊥ that we reserve for the deforming force means that this force acts perpendicularly to the cross-section of the object. Forces that act parallel to the cross-section do not change the length of an object. The definition of the tensile stress is
Young’s modulus Y is the elastic modulus when deformation is caused by either tensile or compressive stress, and is defined by stress = (elastic modulus) × strain. Dividing this equation by tensile strain, we obtain the expression for Young’s modulus: