 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| I. E. Keller and P. V. Trusov, "Fragmentation of a geometrically nonlinear crystal medium with torque stresses," Mech. Solids. 38 (2), 81-89 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
2 |
Pages |
81-89 |
| Title |
Fragmentation of a geometrically nonlinear crystal medium with torque stresses |
| Author(s) |
I. E. Keller (Perm)
P. V. Trusov (Perm) |
| Abstract |
In the presence of large plastic strains in quasi-statically loaded metal
bodies, one observes grains divided into subgrains (regions with uniformly
oriented crystal lattice separated by the boundaries of its misalignment),
i.e., there arises a new spatially modulated structure of
misalignments of the lattice [1, 2]. In this work, a model
of this phenomenon is proposed on the basis of the following assumptions:
(1) locally, the deformable crystal medium under investigation
is an interpenetrating system of a material and a lattice; the material is capable
of arbitrary affine change in shape due to crystallographic slip, and the
lattice admits only deflections and twists [3]; (2) the structural
transition in the stressed medium is connected with the material
instability in the sense of Hill [4, 5]; this instability is caused by the
effects of geometrical nonlinearity (geometrical softening) [6]; (3) at
the instant when the material loses stability, the load starts being
borne by the lattice which is subjected to deflection or torsion when
resisting, and that is why there appears a spatially modulated structure of
deflections and twists of the lattice.
On the basis of these assumptions, equilibrium equations are constructed
for the crystal medium as a geometrically nonlinear Cosserat continuum. The
Cosserat continuum [7] is a material whose particles possess spin. In the
present paper, this spin is identified with the spin of the lattice, and
the torque equilibrium equation serves an the equation for the
spin variable. We construct constitutive equations that take into account
mechanisms of local change in shape of the crystal medium. The constants
of flexural rigidity in these equations provide our model with a scale.
These equations serve for the investigation of equilibrium bifurcations of
the medium and are, therefore, stated for active loading (with possible
load relief neglected) [8]. Moreover, these equations are geometrically
nonlinear and contain a mechanism of destabilization of the deformation
process. The constitutive equations and the equilibrium equations are
written for velocities in terms of the current Lagrangian approach [9]. The
presence of the lattice corotational derivative, rather than the Jaumann
derivative, in the constitutive equations makes the system
non-conservative; the non-conservative system is studied by the dynamic
method.
A model problem is formulated for the uniaxial compression of a two-dimensional
unbounded homogeneous single-crystal body along the axis of cubic symmetry of
its properties. As the boundary conditions it is required that the displacement rates
and the lattice spin are bounded at infinity. We find a bifurcation point
beyond which the loss of stability occurs with the formation
of a rectangular cell structure or a wave structure (along the compression axis),
depending on the ratio of the shear moduli. In any two neighboring cells, the
spin is different in sign, and this allows us to consider such cells
as "nuclei" of subgrains whose boundaries have not yet been formed completely.
Thus, the model describes the tendencies eventually resulting in
a subgrain structure. In the limit case of a moment-free medium [10]
corresponding to zero flexural rigidity of the lattice, this effect
cannot be described at all. For fixed moduli and supercritical stresses,
the size of the cells grows with the increase of the flexural rigidity of the lattice.
Of principal importance for cell formation is the anisotropy of the medium, as well as its
stress state. Under a pure shear stress acting along the axis of symmetry of
the medium properties, there arises an oblique cellular structure or a
wave structure inclined at an angle of 45°, depending on
the ratio of the shear moduli. |
| References |
| 1. | V. V. Rybin,
Large Plastic Deformations and Fracture in Metals [in Russian], Metallurgia,
Moscow, 1986. |
| 2. | B. Bay, N. Hansen, D. A. Hughes, and D. Kuhlmann-Wilsdorf,
"Evolution of F.C.C. deformation structures in polyslip,"
Acta Metall. Mater., Vol. 40, No. 2, pp. 205-219, 1992. |
| 3. | S. Forest, G. Cailletaud, and R. Sievert,
"Cosserat theory for elastoviscoplastic single crystals at
finite deformation," Arch. Mech., Vol. 49, No. 4, pp. 705-736, 1997. |
| 4. | R. Hill, "A general theory of uniqueness and stability in elastic-plastic solids,"
J. Mech. Phys. Solids, Vol. 6. No. 3, pp. 236-249, 1958. |
| 5. | R. Hill, "Acceleration waves in solids," J. Mech. Phys. Solids,
Vol. 10, No. 1, pp. 1-16, 1962. |
| 6. | R. J. Asaro, "Geometrical effects in the inhomogeneous deformation
of ductile crystals," Acta Metall., Vol. 27, No. 3, pp. 445-453, 1979. |
| 7. | W. Nowacki, Theory of Elasticity [Russian translation], Mir, Moscow, 1975. |
| 8. | V. D. Klyushnikov, Lectures on the Stability of Deformable Systems [in Russian],
Izd-vo MGU, Moscow, 1986. |
| 9. | A. A. Pozdeev, P. V. Trusov, and Yu. I. Nyashin,
Large Elastic-Plastic Deformations [in Russian], Nauka, Moscow, 1986. |
| 10. | R. Hill and J. W. Hutchinson,
"Bifurcation phenomena in the plane tension test,"
J. Mech. Phys. Solids, Vol. 23, No. 4/5, pp. 239-264, 1975. |
| 11. | I. E. Keller and P. V. Trusov,
"A generalization of the Bishop-Hill theory of single-crystal plastic
forming," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 93-102, 1997. |
| 12. | E. Kröner, "On the physical reality of torque stresses in
continuum mechanics," Intern L. Engng. Sci., Vol. 1, No. 2, pp. 261-278,
1963. |
| 13. | H. Ziegler, Basic Principles of the Theory of Stability of
Structures[Russian translation], Mir, Moscow, 1971. |
| 14. | R. Hill, "On constitutive inequalities for simple materials,"
J. Mech. Phys. Solids, Vol. 16, No. 4, pp. 229-242, 1968. |
| 15. | D. C. Drucker, "A definition of stable inelastic material,"
J. Appl. Mech., Vol. 26, No. 1, pp. 101-106, 1959. |
|
| Received |
09 January 2001 |
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