Formulation of constitutive and damage evolution equations of elastic-plastic damage materials are first discussed. In order to facilitate the description of the experimental damage surface in axialtorsional stress space, Gibbs thermodynamic potential is employed. The damage surface is expressed as a function of the damage conjugate force which is easily related to stress tensor. Then, the resulting equations are applied to the experimental results of the spheroidized graphite cast iron. The change in elastic moduli due to damage development and the initial and the subsequent damage surface expressed in stress space are described well by the proposed theory. Finally, experimental verification of the existence of a damage potential and the corresponding normality law is performed. By noting that the normality law must be discussed in the space of damage conjugate, the direction of the damage rate vector on each loading path is identified from the change in the elastic modulus due to damage. The resulting direction of the damage rate at each loading path coincides sufficiently well with the normal of the damage surface, and hence, the proposed damage surface is certified to be identical to the damage potential.

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The local approach of fracture based on continuum damage mechanics (CDM) combined with the finite element method (FEM) was applied to establish a more precise method of estimation of service life of cold forging tools by simulating the initiation and growth of a fatigue crack. After a brief explanation on the local approach of fracture based on CDM combined with FEM, modeling of elastic-plastic-damage behavior of tool steel was conducted in the framework of thermodynamics. In the modeling, the salient unilateral property of the tool material is considered by introducing a modified stress tensor. The proposed constitutive equations can be found to describe the experimental behaviors under uniaxial tension and compression properly. Then, the initiation and propagation of a fatigue crack in the cold forward extrusion die was analyzed by the local approach of fracture. The initiation of the fatigue crack in the vicinity of the die radius was found to occur when the number of extrusion increased. Then, the crack propagated along the direction perpendicular to the surface of the die radius. Furthermore, the proposed approach was found to estimate the actual behavior of the fatigue crack growth in good agreement with the calculated change of the rate of crack propagation to the number of extrusion. The calculated crack propagation rate also tended to decrease due to a decrease in tensile principal stress at the crack tip as the crack propagates.

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