Residual Stress (Forming)

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Residual Stress (Forming) Jun Yanagimoto Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

Definition Residual stress is the stress field inside a material that has been subjected to permanent deformation beyond its limit of plasticity or remaining elastic stresses in a workpiece, e.g., due to nonuniform plastic deformation and/or significant temperature gradients.

Theory and Application Introduction If the plastic deformation is uniaxial, such as that induced in a tensile test, and the material is isotropic without the initial stress field, the stress of a permanently strained material can recover to its initial state without residual stress. The stress–strain curve in Fig. 1 explains elastic recovery in such a case. The material after uniaxial tensile deformation can fully recover after unloading the imposed external force, because its stress is uniform without a gradient, and it can deform freely in the opposite direction to that of loading. However, in general, every point in a material after loading or forming has

a stress gradient and cannot deform freely during unloading to recover the stress field. Thus, the stress at every point in a material causes some of the stress before unloading or the stress state during forming to remain. This is the cause of residual stress, and it is closely related to elastic recovery. Residual stress is affected by the ease or difficulty of elastic deformation during unloading or elastic recovery; therefore, the residual stress at every point in a material is affected by the surrounding points, which limit deformation during unloading, while perfect elastic recovery without any constraints on deformation during unloading will result in zero residual stress. In summary, the amount of residual stress depends on the magnitude of the final stress and also on various factors influencing the constraints on deformation during unloading such as the stiffness of the material. Description of Residual Stress A material undergoing forming can be modeled as a deforming body under external force T with boundary Sf and constrained displacement U with boundary Su, as illustrated in Fig. 2. The stress s at every point in a deforming body should obey the following equilibrium equation or momentum equation: a ¼ r1 div sðrÞ;


where a is acceleration, r is a position vector, and

# CIRP 2015 The International Academy for Production Engineering et al. (eds.), CIRP Encyclopedia of Production Engineering, DOI 10.1007/978-3-642-35950-7_6678-3


(3) End of loading

(2) Elastic deformation

Plastic limit σy

(Slope givi ng Young’ s modulus )

(4) After unloading


Residual Stress (Forming), Fig. 1 Perfect elastic recovery after unloading a tensile force

Residual Stress (Forming)


(1) Initial state

Residual Stress (Forming), Fig. 2 Stress field after unloading

(Permanent strain)


Residual Stress (Forming)


Residual Stress (Forming), Fig. 3 Examples of residual stress

r is density. Generally, acceleration is negligibly small compared with the diver