Matti Ristinmaa

FE-formulation of a nonlocal plasticity theory

A nonlocal continuum plasticity theory is presented. The nonlocal field introduced here is defined as a certain weighted average of the corresponding local field, taken over all the material points in the body. Hereby, a quantity with the dimension of length occurs as a material parameter. When this so-called internal length is equal to zero, the local classical plasticity theory is regained. In the present model, the yield function will depend on a nonlocal field. The consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers. The integral equations are classified as Fredholm equations of the second kind and the existence of a solution will be commented upon. After discretization, a matrix equation is obtained, and an algorithm for finding the solution is proposed. For a generalized von Mises material, a plane boundary value problem is solved with a FE-method. Since the nonlocal quantities are integrals, C0-continuous elements are sufficient. The solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations. In the numerical simulations shear band formation is analysed and the results display mesh insensitivity.

Behaviour of the extensible elastica solution

The general form of the virtual work expression for the large strain Euler–Bernoulli beam theory is derived using the nominal strain (Biots) tensor. From the equilibrium equations, derived from the virtual work expression, it turns out that a linear relation between Biots stress tensor and the (Biot) nominal strain tensor forms the differential equation used to derive the elastica solution. Moreover, in the differential equation one additional term enters which is related to the extensibility of the beam axis. As a special application, the well-known problem of an axially loaded beam is analysed. Due to the extensibility of the beam axis, it is shown that the buckling load of the extensible elastica solution depends on the slenderness, and it is of interest that for small slenderness the bifurcation point becomes unstable. This means the bifurcation point changes from being supercritical, which always hold for the inextensible case, i.e. the classical elastica solution, to being a subcritical point. In addition, higher order singularities are found as well as nonbifurcating (isolated) branches.

Kinematic hardening in large strain plasticity

A finite strain hyper elasto-plastic constitutive model capable to describe non-linear kinematic hardening as well as nonlinear isotropic hardening is presented. In addition to the intermediate configuration and in order to model kinematic hardening, an additional configuration is introduced - the center configuration; both configurations are chosen to be isoclinic. The yield condition is formulated in terms of the Mandel stress and a back-stress with a structure similar to the Mandel stress. It is shown that the non-dissipative part of the plastic velocity gradient not governed by the thermodynamical framework and the corresponding quantity associated with the kinematic hardening influence the material behaviour to a large extent when kinematic hardening is present. However, for isotropic elasticity and isotropic hardening plasticity it is shown that the non-dissipative quantities have no influence upon the stress-strain relation. As an example, kinematic hardening von Mises plasticity is considered, which fulfils the plastic incompressibility condition and is independent of the hydrostatic pressure. To evaluate the response and to examine the influence of the non-dissipative quantities, simple shear is considered; no stress oscillations occur. (Less)

Consequences of Dynamic Yield Surface in Viscoplasticity

A theory of viscoplasticity is formulated within a thermodynamic concept. The key point is the postulate of a dynamic yield surface, which allows us to take advantage of the postulate of maximum dissipation to derive an associated formulation of the evolutions laws for the internal variables without using penalty techniques that only hold in the limit it when viscoplasticity degenerates to inviscid plasticity. Even a non-associated formulation is presented. Within this general formulation, a particular format of the dynamic yield function enables us to derive the static yield function in a consistent manner. Hardening, perfect and softening viscoplasticity is also defined in a consistent manner. The approach even includes associated and non-associated viscoplasticity where corners exist on the yield and potential surfaces.

Void growth in cyclic loaded porous plastic solid

In low-cycle fatigue, where plastic strains are of great importance, final ductile fracture depends upon the mechanisms of void growth and coalescence of voids. A cell model is used to simulate a periodic array of initial spherical voids and this model is subjected to different loads that include cyclic loading. Three different types of matrix material are simulated: elastic-perfectly plastic, isotropic hardening and kinematic hardening. The cell model results are compared with the approximate constitutive equations for a voided material suggested by Gurson. The simulations show that the unspecified parameters introduced by Tvergaard in the Gurson yield function depend on the hardening behavior of the matrix material. For a perfectly plastic matrix material, the parameters q1 = 1.5 and q2 = 1.02 provide very close predictions for a variety of loadings. However, for isotropic or kinematic hardening matrix materials these parameters result in an inferior agreement and a much closer accuracy is obtained by adopting q1 = 1.5 and q2 = 0.82. This suggests that the parameter q2 depends on the hardening behavior of the matrix material. For kinematic hardening of the Gurson model, it is shown that Zieglers hardening rule is superior to Pragers hardening rule. Finally, the void shape change due to loading is studied and it is found that this change has an insignificant effect on the response.

Towards an orientation-distribution-based multi-scale approach for remodelling biological tissues

The mechanical behaviour of soft biological tissues is governed by phenomena occurring on different scales of observation. From the computational modelling point of view, a vital aspect consists of the appropriate incorporation of micromechanical effects into macroscopic constitutive equations. In this work, particular emphasis is placed on the simulation of soft fibrous tissues with the orientation of the underlying fibres being determined by distribution functions. A straightforward but convenient Taylor-type homogenisation approach links the micro- or rather meso-level of fibres to the overall macro-level and allows to reflect macroscopically orthotropic response. As a key aspect of this work, evolution equations for the fibre orientations are accounted for so that physiological effects like turnover or rather remodelling are captured. Concerning numerical applications, the derived set of equations can be embedded into a non-linear finite element context so that first elementary simulations are finally addressed.

Use of couple-stress theory in elasto-plasticity

One way to avoid vanishing dissipative energy and localization to zero volume when examining localization and softening problems, is to introduce an internal length scale for the material by means of couple-stress theory. Here, the ‘constrained’ Cosserat theory is adopted where the displacement field also determines the rotation field. For this ‘constrained’ Cosserat theory an elasto-plastic theory is derived within a thermodynamic framework and it is shown that the evolution laws for the internal variables can be derived from the postulate of maximum dissipation. A generalization of the classical von Mises material is proposed; both the derivation of the model and the numerical treatment of the integration problem are discussed. The generalized von Mises model is used in finite element calculations where shear band formation is considered and the results turn out to be independent of the mesh spacing.

Rayleigh waves obtained by the indeterminate couple-stress theory

Rayleigh waves in a linear elastic couple-stress medium are investigated; the constitutive equations involve a length parameter l that characterizes the microstructure of the material. With cR=Rayleigh speed, cT=conventional transversal speed and q=wave number, an explicit expression is derived for the relation between cR/cT, lq and Poissons ratio ν. The Rayleigh speed turns out to be dispersive and always larger than the conventional Rayleigh speed. It is of interest that when lq=1 and ν≥0, it always holds that cR/cT=2. The displacement field is investigated and it is shown that no Rayleigh wave motions exist when lq→∞ and when lq=1, ν≥0. Moreover, a principal change of the displacement field occurs when lq passes unity. The peculiarity that no Rayleigh wave motions exist when lq=1, ν≥0 may support the criticism by Eringen (1968) against the couple-stress theory adopted here as well as in much recent literature.

Cyclic plasticity model using one yield surface only

A time-independent plasticity model using only one yield surface and capable of predicting cyclic loading is presented. Memory points are defined to monitor the loading history; these memory points have the property that they can be created as well as disappear during the load history. The generalized plastic modulus is defined from these memory points in such a way that it will be continuous and provide a smooth transition from elastic to elastic-plastic behavior. The model is formulated for general pressure-insensitive plasticity. Because only one yield surface is used, this framework allows all types of yield functions to be easily implemented. For piecewise linear response, the concept of memory points may be interpreted in terms of the classic Mroz model. Moreover, a generalization of the Mroz model for a smoothly varying response can be made. As an important example of pressure-insensitive plasticity, the von Mises criterion is incorporated into the model to illustrate the response for different load situations. Of particular interest is cyclic hardening, mean stress relaxation, and ratchetting. A simple law for controlling the ratchetting is also introduced.

Damage Evolution in Elasto-Plastic Materials - Material Response Due to Different Concepts

The purpose of this paper is to show that shortcomings exist in the plasticity induced damage theories. Existing phenomenological thermodynamic approaches used for describing elasto-plasticity coupled with damage are therefore evaluated. Within the concept of effective stress both the postulate of strain equivalence and the postulate of (complementary) energy equivalence, as well as extensions of the postulates, are considered. As a prototype model the von Mises plasticity model coupled with isotropic damage is considered. Simulations of a strain-controlled uniaxial model are also performed. The results reveal that a mapping, similar to that of the stress, of both the kinematic and isotropic hardening variables is to be preferred. More interesting is that, irrespective of the postulate employed, the elastic strain will not equal zero when failure takes place, i.e. the interpretation of elastic strain is lost. From the results it is also concluded that (complementary) energy equivalence have some undesirable properties.