An expanding cavity model incorporating strain-hardening and indentation size effects

An expanding cavity model (ECM) for determining indentation hardness of elastic strain-hardening plastic materials is developed. The derivation is based on a strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic power-law hardening material. Closed-form formulas are provided for both conical and spherical indentations. The indentation radius enters these formulas with its own dimensional identity, unlike that in classical plasticity based ECMs where indentation geometrical parameters appear only in non-dimensional forms. As a result, the newly developed ECM can capture the indentation size effect. The formulas explicitly show that indentation hardness depends on Young’s modulus, yield stress, strain-hardening exponent and strain gradient coefficient of the indented material as well as on the geometry of the indenter. The new model reduces to existing classical plasticity based ECMs (including Johnson’s ECM for elastic–perfectly plastic materials) when the strain gradient effect is not considered. The numerical results obtained using the newly developed model reveal that the hardness is indeed indentation size dependent when the indentation radius is very small: the smaller the indentation, the larger the hardness. Also, the indentation hardness is seen to increase with the Young’s modulus and strain-hardening level of the indented material for both conical and spherical indentations. The strain-hardening effect on the hardness is observed to be significant for materials having strong strain-hardening characteristics. In addition, it is found that the indentation hardness increases with decreasing cone angle of the conical indenter or decreasing radius of the spherical indenter. These trends agree with existing experimental observations and model predictions.