A Deformation Theory of Plasticity
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International Journal of Plasticity, Vol. 9, pp. 907-920, 1993 0749-6419/93 $6.00 + .00 Printed in the USA Copyright 1993 Pergamon Press Ltd.
A DEFORMATION THEORY OF PLASTICITY BASED ON MINIMUM WORK PATHS
KWANSOO Cn-ONG and OWEN RICHMOND
Aluminum Company of America
(Communicated by Kenneth Neale, Universit6 de Sherbrooke)
Abst ract -A deformation theory of plasticity is proposed wherein the deformation paths for material elements are assumed and the plastic work becomes dependent on displacements. Among the infinite possible ways to assume deformation paths, one has been chosen that has several advantages when materials harden isotropically. Earlier, this path was shown to require the min- imum work path to achieve a desired strain. Here, a mathematical description of a constitu- tive law of deformation plasticity is developed based upon this path for rigid-plastic and for elastoplastic materials. The proposed deformation theory provides a convenient theoretical basis for FEM applications involving analysis, and especially design, of forming processes.
In the early development of the theory of plasticity of metals, two types of theories were proposed: flow and deformation. Flow theories eventually became dominant because they reflected the observed dependence of metal behavior on deformation path. Defor- mation theory, which did not reflect this dependence, was considered accurate only in limited cases where materials deform in special ways; therefore, use of the theory dwin- dled. In 1959, Budiansky introduced a new interpretation of the deformation theory for yield surfaces with edges (vertexes). After this work, major applications of the defor- mation theory developed in the area of instability analysis: the works of Hutchinson , St6ren and Rice , and Neale  among others.
Recently, deformation theory has been used in formulating finite element modeling (FEM) codes for a variety of engineering problems. Applications of deformation the- ory in FEM are so versatile that they include designing as well as analyzing forming pro- cesses. The range of application includes rigid-plastic and elastoplastic materials. In analysis codes the deformation theory is applied incrementally, and geometry and mate- rial properties are updated at discrete steps without the use of numerical integration (Wang ; Braudel et al. ; Yang & Kim ; Germain et al. ). This differs from the conventional flow formulations that use numerical integration to update geometry and material properties. In design codes, the deformation theory is applied in a single step. These codes provide information about formability of products and opti- mized process parameters in the preliminary design stages (Chung & Richmond [1992b, 1993]). There have been some efforts to apply the deformation theory in a single step to analyze forming processes. However, this approach might not be appropriate for gen- eral forming because the deformation theory with a single step does not properly reflect the dependence of metal behavior on deformation path (Levy et al. ; Chung & Lee ; Sklad ).
To provide a theoretical basis for application of deformation theory to FEM, a spe- cial deformation theory of plasticity is developed in this article for both rigid-plastic and
908 K. Cmm~ and O. RICHMOND
elastoplastic materials. The theory covers arbitrary anisotropic materials that harden iso- tropically. In this deformation theory, the deformation paths of material elements are specified, so that the plastic work becomes dependent on displacements. A mathemat- ical description of a constitutive law for the deformation theory is then derived from flow theory using the assumed deformation path. Among the infinite number of ways to assume a deformation path, one has been assumed that has several advantageous fea- tures. The specific assumed path is the minimum plastic work path in homogeneous deformation. The constitutive laws derived from the minimum work path are different from those of the conventional deformation theory. Earlier, Ponter and Martin  presented theoretical work that connects the deformation theory and the minimum work path, but without explicit details of the minimum work path or derivations of the con- stitutive laws.
Derivations of the deformation theory begin with a brief review of minimum work paths and their kinematics for rigid-plastic materials in Section II. In Section III, con- stitutive laws are derived for smooth yield conditions based on both the yield surface and its associated dissipation surface in rigid-plastic materials. In Section III, a one-step minimum work path is assumed in order to obtain results that are useful for design pur- pose FEM codes. In Section IV, the incremental deformation theory is obtained by dis- cretizing a deformation process into multistep minimum work paths. These results are useful for analysis purpose codes. In Section V, it is shown that the results derived for smooth yield conditions are also valid for yield conditions having sharp edges. The deformation theory is compared with hyperelasticity and extended to elastoplasticity in Sections VI and VII, respectively.
II. KINEMATICS OF MINIMUM WORK PATHS
Requirements for achieving minimum work paths in homogeneous deformations are well documented in the works by HILL , NADAI , D ~ , and HILL . In these works, it was found that the minimum work path is achieved if mate- rials deform in such a way as to satisfy two conditions: first, the set of three principal axes of stretching is fixed with respect to the material; second, the logarithms of the stretches remain in a fixed ratio. HILL  proved that these two conditions are valid for any convex yield condition with isotropic hardening. Recently, CnuN6 and RICH- MOND [1992a] showed that both of the conditions are required only when the yield sur- face is smooth enough to have a unique normal direction (in such a case, the minimum work path is uniquely defined). When the yield surface has sharp edges, no unique nor- mal direction is defined at these corresponding stress states, so that the two conditions are only partially required (in such a case, multiple minimum work paths are possible). For the Tresca yield condition, for example, only the maximum stretch axis is required to be fixed with respect to the material, and no other major conditions are necessary. For isotropic materials, the fixed material lines may be chosen arbitrarily; for anisotropic materials, they are more restricted. Because of the differences in the minimum work path, the deformation theory is discussed separately for smooth yield surfaces and for pointed yield surfaces.
The minimum plastic work paths can be quantified in equations. The deformation gra- dient tensor F describes a deformation between a final and an initial shape. F is defined, by polar decomposition, as
F(t) = R( t ) .U( t ) , (1)
Deformation theory of plasticity 909
where U is the right stretch tensor, which is symmetric, R is the orthogonal rotation ten- sor, and t is time. The three principal directions of U, because of its Lagrangian nature, refer to three material lines that are perpendicular to each other in the original config- uration. Eqn (1) represents a physical deformation process that is composed of an ini- tial stretch, U, of principal material lines followed by a rigid body rotation, R. It is a description of the total amount of deformation and the total rigid body rotation required by two prescribed geometries, but the history of deformation is only implicit.
When decomposed into principal values and directions, U and the logarithmic strain tensor eL are defined as
U(t) = Q(t) . , I ( t ) .Qt(t) , eL(t) = Q(t) . / j ( t ) -Qt(t ) , (2)
where the superscript t stands for transpose. In eqn (2), ,I and ~ are diagonal tensors consisting of principal values (,~i and ~i ~--- In Ai), and Q is an orthogonal rotation ten- sor consisting of the principal directions of U. Note that eL is also a Lagrangian quan- tity. When the same principal material lines are kept constant during deformation, the principal material lines of l~l and I~/~ coincide with those of U obtained for a final shape at a time tf; i.e.
U(t) = Q(tf) .A(t) .Qt(ty) , U(t) = Q(t f ) .A ( t ) .Qt ( ty ) 0 < t < tf (3)
~L(t) = Q(t f ) .~(t ) .Qt( t f ) , EL(t) = Q(tf) . / j ( t ) .Qt(t f ) 0 < t < tf. (4)
Let us consider a strain rate g, which is defined as
g = Rt .D .R = ( l ) -U - l ) s , (5)
where D is the rate of deformation tensor, and the subscript s stands for the symmet- ric part of a tensor. The quantity g is considered to be the value D measured with respect to the coordinate system that rotates by R(t) , so that the values are invariant with respect to R. (For this reason, ~ is called the "rotationless" strain rate by STOREN and RICE .) When the same principal material lines are kept constant during deforma- tion, the principal material lines of l~l and U coincide as shown in eqn (3). Consequently, in such cases, the principal directions of g are stationary and are aligned with the prin- cipal material lines of U(tf), regardless of arbitrary rotation R (t), whereas the princi- pal directions of D might vary continuously; i.e.
~(t) = gL(t) = Q(t I) .~(t) .Qt(tf) . (6)
In fact, when the same principal materials lines are kept constant, g also becomes a Lagrangian quantity whose principal directions refer to the fixed principal material lines and whose principal values mean the logarithmic (or true) strain rates of the principal material lines.
The minimum work path for a smooth yield function is achieved if