Case Of Large Strains Research Articles

An exact solution to the problem of biaxial loading of a micropolar elastic plate made by joining two prestrained arc-shaped layers under large strains

An exact analytical solution to the static problem of biaxial tension-compression of a composite elastic plate made from a junction of two preliminarily strained circular arc-shaped panels (sectors of hollow cylinders) is presented for the case of large strains. It is assumed that the axes of the panels are orthogonal. The problem is formulated using the theory of repeatedly superimposed large deformations. The solution is obtained for a class of micropolar physically linear elastic materials. Some numerical results are given and the nonlinear effects are analyzed. The obtained solution can be useful for the strength analysis of structural elements made from the junction of prestrained parts with micropolar and nonlinear effects accounted for.

Preprocessors for Finite Element Method meshes generation for the case of large strains

The description of algorithms and software for Finite Element Method data preprocessing, which is designed for solution the problems of Solid Mechanics, is presented. Three dimensional Finite Element models for prismatic solids are regarded. For the case of large strains the arbitrary Lagrange-Euler approach is used. Two approaches which are realized in separate program unit are presented. Both of them are based on preliminary two-dimensional triangulation of the areas with complex shape. The first approach bases on the figure triangulation with use of triangles with optional shape, the second demands uniform mesh with rectangular triangles. The algorithms for meshing of material and additional ‘air’ areas with simultaneous generation of elements are regarded. The procedures of the generation of elements groups and their integration into the model are presented. The code design strategy with description of entity-relationship diagram is discussed. The graphical user interface for the data preprocessing including the matrix of index and co-ordinates array is described, the procedure of user’s actions is presented. The program realization of above described algorithms was done by use of programming language Python. The examples of mesh building with information of each stage of meshing for models of different shape are presented.

Mechanical origin of power law scaling in fault zone rock

A nearest neighbor fragmentation model, previously developed to explain observations of power law particle distributions in 3D with mass dimension D3 ≈ 2.6 (D2 ≈ 2.6 in 2D section) in low‐strain fault gouge and breccia, is extended to the case of large strains to explain recent observations of D3 ≈ 3.0 (D2 ≈ 2.0 in 2D section) in the highly strained cores of many exhumed fault zones. At low strains, the elimination of same‐sized nearest neighbors has been shown to produce a power law distribution which is characterized by a mass dimension near D3 ≈ 2.6. With increasing shear strain these isolated same‐size neighbors can collide, in which case one of them fractures. The probability of two same size neighbors colliding and fragmenting in a simple shear flow is a function of the size and density of the two particles. Only for a power law distribution with D3 = 3.0 is this collision probability independent of the size of the particles.

Instability prevention at the modeling of large elasto‐plastic strains under cyclic loading

AbstractThe cyclic instability phenomenon is investigated at the modeling of large elasto‐plastic strains. The possible causes of the cyclic instability and conditions ensuring cyclical stability of elasto‐plastic models are analyzed for the case of large strains. Among the possible causes of the cyclic instability the following are considered: the method of strain decomposition on elastic and plastic parts; the constitutive law for the elastic deformation (hypo‐ and hyper‐elasticity); the constitutive equation for the plastic deformation; the constitutive relation for the plastic spin; kinematic hardening law, in particular, the type of the objective rate in the generalized Prager's law. Predictions of 50 various models of the elasto‐plastic material have been compared in order to find the causes of the cyclic instability. Two test problems are considered: cyclic simple shear, combined cyclic simple shear and tension‐compression. Results of numerical experiments for the various material models are presented and discussed. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Models, algorithms and error estimation for computational viscoelasticity

This article reviews numerical algorithms for problems in solid polymer viscoelasticity in both small and large deformation. For the linear (small strain) case we review both the quasistatic and the dynamic problem and give recent results on a posteriori error estimation. For the large strain case we focus on the formulation and computational modelling of constrained membrane inflation, the application of which is to the thermoforming process.

Vision-based sensing of forces in elastic objects

A minimally intrusive, vision-based, computational force sensor for elastically deformable objects is proposed in this paper. Estimating forces from the visually measured displacements is straightforward in the case of the linear problem of small displacements, but not in the case of the large displacements where geometric non-linearities must be taken into account. From the images of the object taken before and after the deformation, we compute the deformation gradients and logarithmic strains. Using the stress–strain relationships for the material, we compute the Cauchy’s stresses and from this we estimate the locations and magnitudes of the external forces that caused the deformation. A sensitivity analysis is performed to examine the effect of small deviations in the experimentally captured displacements on the estimated external forces. This analysis showed that the small-strain case is more sensitive and prone to numerical errors than the large-strain case. Additionally, a related method that is indirect and iterative is also presented in which we assume that we know the locations of the external forces. Numerical and experimental studies are presented for both micro- and macro-scale objects. The main conclusion of this work is that the vision-based force estimation is viable if the displacements of the deforming object can be captured accurately.

Large irreversible local slips of a loose free-flowing material under hard sign-variable quasistationary loading

The theory of slip for the case of large strains is considered. Irreversible distortion by slippage is represented as a Truesdell transformation of simple shear and is specified with respect to the unstrained configuration. It is proved that a small local slip leads decreases tangential stresses on the slip area and changes the strained stress over the entire region. The calculated orientations of the local slip systems are compared to results of classical experiments on sign–variable quasistationary deformation of loose free–flowing materials.

Elastoplastic materials with martensitic phase transition and twinning at finite strains: Numerical solution with the finite element method

A new quasistatic problem formulation and finite element (FE) algorithm for martensitic phase transition (PT) and twinning in elastoplastic materials at large strains, based on a recently proposed thermomechanical approach [1–2] are presented. The instantaneous occurrence of PT in some region based on thermodynamics, without introduction of volume fraction and prescribing the kinetic equations for it, is considered. Stress history dependence during the transformation process is a characteristic feature of the PT criterion. The deformation model is based on the multiplicative decomposition of the total deformation gradient into elastic, transformation and plastic parts and the generalization of Prandtl-Reuss equations to the case of large strains and PT. The case of small elastic, but large plastic and transformation strains is assumed. For numerical simulation of PT the ‘inverse’ problem is considered, i.e. the position and size of the PT region (nucleus) are prescribed in advance, and then the condition for PT is defined from the PT criterion. Such an approach includes a finite element solution of the elastoplastic problem with the prescribed transformation deformation gradient and the changing elastoplastic properties in the transforming region during PT. The usage of the current configuration and the true Cauchy stresses along with assumptions of small elastic strains and zero modified plastic spin allows us to use—with small modifications—the radial return algorithm and the consistent elastoplastic moduli for the case of small strains. Some modifications of the iterative algorithm related to the numerical integration of constitutive equations along with the radial return algorithm are suggested in order to improve the accuracy of solutions for large increments of external load (such modifications can be used for any elastoplastic problem without phase transition as well). The model problems of nucleation at shear-band intersection and appearance of a single martensitic plate and a single twin are solved and analyzed.

Constitutive approach to rate-independent properties of filled elastomers

Filled rubber material is used in many technical applications. These structures are often subjected to dynamic loading. Apart from the material's stiffness characteristics dissipative properties are of great importance. At high frequencies rubber material behaves mainly viscoelastically. But at low strain-rate the response is nearly rate-independent. This observation can be interpreted as “internal material friction” and is dealt with in the following article. A constitutive approach is presented, which can be motivated by investigations on the molecular level. This phenomenological description is composed of an elastic model—idealized as an elastic spring—and elasto-plastic Prandtl-elements in parallel. The formulation is derived for small and finite strains. The large strain case is based on a multiplicative decomposition of the deformation gradient. Experimental results and numerical simulations are presented utilizing a mixed finite element formulation to give a robust algorithm and reliable results.

Recent advances in the numerical modeling of constitutive relations

In this paper we present an overview of the recent developments in the area of numerical and finite element modeling of nonlinear constitutive relations. The paper discusses elastic, hyperelastic, elastoplastic and anisotropic plastic material models. In the hyperelastic model an emphasis is given to the method by which the incompressibility constraint is applied. A systematic and general procedure for the numerical treatment of hyperelastic model is presented. In the elastoplastic model both infinitesimal and large strain cases are discussed. Various concerns and implications in extending infinitesimal theories into large strain case are pointed out. In the anisotropic elastoplastic case, emphasis is given to the practicality of proposed theories and its feasible and economical use in the finite element environment.

Finite element procedure for solving contact thermoelastoplastic problems at large strains, normal and high pressures

The paper presents a finite element (FE) procedure for solving contact thermoelastoplastic problems at large strains. A rigorous derivation of the constitutive relations used and the structure of a tangent stiffness matrix and load vector at arbitrary thermomecanical loading are given. The deformation model [1,2] is used which is based on the multiplicative decomposition of the total deformation gradient to an elastic, temperature and plastic parts and the generalization of Prandtl-Reuss equations to the case of large strains, high pressures and temperatures. A method is proposed for the generalization of the elastic law at small strains to the case of high pressures which is based on the use of the existence of an elastic potential (hyperelastic material) at large elastic strains. The algorithms for solving thermoelastoplastic problems at large strains and contact elastic problems at small strains are considered individually and in combination for solving contact thermoelastoplastic problems at large strains. An algorithm for solving thermoelastoplastic problems at large strains is based on a modified method of initial stresses allowing the use of both the constant and the variable tangent stiffness matrices. An algorithm allowing for contact conditions (which are described using a friction surface, in the particular case, the Coulomb's law of friction is considered) involves the simultaneous consideration of an arbitrary number of deformable bodies in contact. The pairs of nodes with the same coordinates are introduced along the interface. Owing to transformation of the set of FEM equations, the contact conditions reduce to the usual boundary conditions in terms of stresses or displacement with iterative redetermination of their types (adhesion, slip, no contact) in each pair of nodes. Optimum variants of combinations of the iterative procedures allowing for plastic flow and contact interaction are studied. Particular problems are solved. An attractive feature of the present approach is the simplicity of allowing for the contact conditions for an arbitrary number of deformable bodies in contact and insignificant modifications of computer program which are necessary for the change from small strains to large ones.

Unstable fracture criterion in a power-law hardening material

A fracture criterion for a pre-cracked plate subjected to large deformation is presented in a power-law hardening material based on the energy conservation law. First Hencky equations are applied to a non-linear elastic model in a small strain to derive the criterion and the theory is extended to a large strain case with the aid of numerical analysis. The criterion is expressed as a relation between a crack length and a remote fracture strain, and it is identical with Griffith's criterion when the power exponent is unity. Large strain and stress fields near a crack tip are also assessed. Experiments with pre-cracked steel plates well verify the theory, that is, the criterion based on elastic analysis appears to apply to an actual plastic material. The transition from the brittle fracture to the present unstable ductile fracture and a physical implication of the reduction of area in tension test are discussed.

Fundamental equations of the theory of asymmetric elasticity

An exact analytical solution to the static problem of biaxial tension-compression of a composite elastic plate made from a junction of two preliminarily strained circular arc-shaped panels (sectors of hollow cylinders) is presented for the case of large strains. It is assumed that the axes of the panels are orthogonal. The problem is formulated using the theory of repeatedly superimposed large deformations. The solution is obtained for a class of micropolar physically linear elastic materials. Some numerical results are given and the nonlinear effects are analyzed. The obtained solution can be useful for the strength analysis of structural elements made from the junction of prestrained parts with micropolar and nonlinear effects accounted for.