####
J. Japan Inst. Metals, Vol. 49, No. 9 (1985),

pp. 714-722

##
Calculation of the Deformation Gradient Tensor for Slip along an Arbitrary Path

**
Kazuwo Imai**^{1}

^{1}Faculty of Engineering, Iwate University, Morioka

#### Abstract:

A general method of calculating the deformation gradient tensor **\itshape F** for deformation of single crystals along any path due to slip on N slip systems is presented, N being an arbitrary positive integer; this method enables us to obtain the value of **\itshape F** easily, if we know N active slip systems and the amount of glide shear of each system at every moment. Examples of the calculation are shown for several typical deformations.

A glide strain tensor **\itshape G** determined by the above knowledge about slip is introduced, and a differential equation relating **\itshape G** to **\itshape F** is derived. By solving the equation, **\itshape F** is given in a form of a tensor-valued functional defined over the function of time, **\itshape G**( • ); the value of **\itshape F** depends on the path of slip deformation in the eight-dimensional **\itshape G**-space. For a wide variety of paths satisfying a certain condition (piecewise-commutative paths), the functional is reduced to a product of the exponential functions of the glide strain tensor increments. In general, the functional can be represented by an infinite series and approximated by the above product. An analytical method for evaluating the exponential of an arbitrary increment of the glide strain tensor is given explicitly. Degrees of multiplicity of the slip is introduced in the evaluation.

(Received 1985/3/8)

#### Keywords:

deformation gradient, slip deformation, multiple slip, path of deformation, glide shear, glide strain, calculation

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