Strain-increment tensor

Corresponding system plastic-strain increments are also obtained at the atomic level from the displacement gradients between the four relevant neighboring corner atoms of Delaunay tetrahedra for each external distortion increment and are allocated subsequently as an atomic site average to each Voronoi polyhedral atom environment by a special procedure of double space tessellation developed by Mott et al. (1992) for this purpose, leading eventually to volume averages of strain-increment tensors of all Voronoi atom environments to attain the system-wide strain-inerement tensor. 

The simulated dilatations involved increasing steps of imposed dilatation on the simulation cell. To permit a detailed understanding of the dilatational response of the polymer at the atomic level the entire volume of the simulation cell was tessellated into Voronoi polyhedra at each atomic site, permitting determination of strain-increment tensor elements dcy for each site from local displacement gradients by a technique described by Mott et al. (1992). Such increments of imposed dilatation at a level of 3 x 10 were applied 100 times to obtain total system dilatations of 0.3 (Mott et al. 1993b). For eaeh dilatation increment the atomic site strain-tensor increments de were obtained for each site n. The two invariants, de", the atomic site dilatation increment, and the work-equivalent shear-strain increment, dy", were obtained from the individual increments as... 

For each external dilatation increment the system-wide strain-increment tensors... 

Levy [12] and von Mises [11] independently proposed that the principal components of the strain-increment tensor... 

Deviatoric stress tensor Norm of deviatoric stress Lode s angle for stress Second invariant of deviatoric stress Third invariant of deviatoric stress Strain increment tensor Elastic strain increment tensor Plastic strain increment tensor Volumetric plastic strain increment tensor... 

The Levy-Mises equations define one of a number of possible flow rules that can be derived via an argument that depends upon a concept known as the plastic potential. This idea has been discussed by Hill [ 15]. It is assumed that the components of the plastic strain increment tensor are proportional to the partial derivatives of the plastic potential, which is a scalar function of stress. The flow rule can thus be generated by this differentiation process. We may choose to assume, for a particular form of yield criterion, that the plastic potential has the same functional form as the yield criterion then, the derived flow rule is described as being associated with the yield criterion (or as an associative flow rule). However, this assumption is not obligatory and when it is not true we will be applying a yield criterion together with a non-associated flow rule. This is discussed further by de Souza Neto etal. [19],... 

The principle of maximum plastic work for granular materials is explained by using newly proposed decompositions of stress and strain increment tensors. In forming the decompositions both the condition of stress path and the stress-dilatancy equation are taken into account. The 3D stress-dilatancy equation in a tensorial form, which is a natural extension of the form in 2D, is proposed. The application of the modified associated flow rule for obtaining the strain increment tensor in 3D is explained by virtue of the proposed decompositions. 

To consider the principle of maximum plastic work in 2D, we introduce the following decompositions for stress and strain increment tensors ... 

In 3D, the stress dilatancy equation is not sufficient to determine the strain increment tensor and the flow rule becomes necessary. In this case, the stress-dilatancy equation is considered as a constraint condition for strain increments. Kanatani [2] proposed a modified associated flow rule having a constraint condition on the deformation, and he states that the differentiation in the associated flow rule is to be made in keeping the constraint force constant. The constraint force is a force to make work with the deformation, which must disappear by the given constraint condition. Thus, as is seen in Eq.(3l)> de = 0 is the constaint condition and p" is the corresponding constraint force, and by the condition of stress path, p is kept constant in the considered decomposition, as is shown in Eq.(l9) ... 

In this paper, a new decomposition of stress and strain increment tensors determined by the stress condition, i.e. the condition of stress path, and the strain condition, i.e. the stress-dilatancy equation, is proposed. Using the... 

F, G, H, L, M and N are the Hill parameters, ffij the components of the stress tensor. The strain increment tensor results in... 

As mentioned above, the calculatimi for a given strain increment is the key to calculate the backstress-tensor a for a defined strain path because... 

The return mapping techniques in inelastic solutions are a natural consequence of splitting the total strain into elastic and inelastic strains. Let tensor uy, an incremental field to describe the deformation, and its gradient, Vt/,y, show the deformation rate. The solution is implemented by the following steps. Step 1 introduces a loading condition such as F = (/, 4- V ,t)Fj." where ly is the unity second-rank tensor and the superscripts n and n 4-1 represent, respectively, the previous and current load steps. In step 2 the material is elastically stretched... 

H is known as the hardening or plastic modulus. To relate the stress increment directly to the strain increment via the tangent stiffness tensor, we substitute ... 

If the material is assumed to remain isotropic after yield, then there is no dependence on the deformation or stress history. Furthermore, if we assume that the yield behaviour is independent of the hydrostatic component of stress, then the principal axes of the strain increment are parallel to the principal axes of the deviatoric stress tensor. 

Note 2.8 (Solid and fluid). The term solid is used for the material body where the response is between the stress a and the strain e or between the stress increment da and the strain increment de. The term fluid is used for the material body where the response is between the stress a and the strain rate k (or the stretch tensor D). For a fluid we have to introduce a time-integration constant, which is referred to as the pressure ... 

The flow rule (2.302) implies that the direction of the plastic strain increment deP is normal to the surface g = constant, and coincides with the stress a. For isotropic materials this can be described as follows. We introduce the unit tensors (see Sect. 2.8.3) as... 

Since the plastic compliance tensor of (2.312), determined by the flow rule, is represented by a product of two second-order tensors, the determinant is identically zero (detC = 0, if we set the second-ordCT tensors as vectors as mentioned in (2.310)). Since it is not possible to obtain the inverse of Cp directly, we use the properties of the elastic compliance C, which has the inverse, along with the direct sum of the strain increment given by (2.293). That is. 

Eij is the infinitesimal strain tensor is the strain deviator Cij and the volumetric strain e = ea Ui denotes the material displacement in j-th direction g = T - and 0 = 2 - X / are the increments of temperature and liquid content with respect to reference values... 

Cartesian components of the 2nd Piola-Kirchhoff stress tensor Cartesian components of the Green-Lagrange strain tensor Components of the linear elasticity tensor Increment in the /th displacement component... 

The finite element procedures for the analysis of elastic>plastic solids at large strain have been given by Lee [9] and implemented by Chiou [11] and Chiou et al. [10]. In this work, only a few comments on the finite element procedures will be made. Equation (16), which links the Truesdell stress rate tensor and the deformation rate tensor, may be regarded as the stress-strain relation in rate fom with a being the "slope" at a particular point in stress space. However, in nonlinear finite element analysis, one has to have a stress-strain relation in incremental form which enables the increments in displacements, strains, and stresses not to be infinitesimally small. Therefore, it is proposed to adopt the following incremental stress-strain relation... 

So far we considered phases with sufficient atomic mobihties and vanishing pressure anisotropies such that we could use the term -pdV to describe the mechanical energy increment (see also footnote 6). Generally, for elastic deformations (i.e. usually small deformations), this increment has to be expressed in terms of the stress tensor components Sy and the differential strain tensor components dey ... 

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