Modeling and simulation of ductile damage in forged steel used to encapsulate high-level radioactive waste

Andra, the French national radioactive waste management agency, is in charge of studying the disposal of high-level and long-lived intermediate-level waste (HLW and ILW-LL) in a deep geological repository. According to the reference concept, it is planned to encapsulate high-level waste in non-alloy P285NH steel overpacks before inserting them into horizontal steel cased micro-tunnels. This work is a part of the study about the long-term behavior of a welded steel overpack subjected to external hydrostatic pressure and several localized loading paths. Indeed, the main objective of this work is to develop the most suitable model for non-alloy steel P285NH to be used in the prediction of the long-term overpack behavior. Dealing with a ductile steel, elastoplastic constitutive equations accounting for mixed nonlinear isotropic and kinematic hardening strongly coupled with ductile isotropic damage are adopted. They are formulated based on the classical thermodynamics of irreversible processes framework with state variables at the macroscopic scale, (Germain, 1973) (Lemaitre 1985, Saanouni 2012). In this paper, a new coupling formulation between the scalar isotropic ductile damage and the deviatoric and spherical part of the Cauchy stress and elastic strain tensors is proposed. In order to calibrate the developed model on P285NH steel, multiple tensile tests are performed using classical cylindrical specimens, notched specimens and double notched specimens. In the last part, some experimental fields are measured using digital image correlation. Application is made to a simplified overpack represented by thick walled cylinder subject to compressive loading path. A FEM (Finite Element method) crushing operation of an overpack’s cylindrical part has simulated and analysed.

Abstract MP18: The Impact Of Perivascular Adipose Tissue On Arterial Stiffness

PVAT is increasingly recognized as an essential layer of the functional vasculature, producing vasoactive substances and assisting arterial stress relaxation. We test the hypothesis that PVAT reduces arterial stiffness. Our model was the thoracic aorta of the male Sprague Dawley rat. Uniaxial mechanical tests for three tissue groups were performed: aorta +PVAT (+PVAT), aorta - PVAT (-PVAT), and PVAT ring separated from aorta (PVAT only) (N=5). Data are reported in the form of a Cauchy stress-stretch curve (fig 1a; line = mean;shaded = SDs). Low-stress stiffness ( E o ), high-stress stiffness ( E 1 ), and the stress corresponding to a stretch of 1.2 (sigma 1.2) were also measured (+PVAT sample in fig 1b) as metrics of distensibility (the higher the stress/stiffness, the less distensible). E 1 and sigma 1.2 for PVAT-only samples could not be quantified. The low-stress stiffness E o was the largest in the -PVAT samples and the smallest in PVAT-only samples (p < 0.05), while the +PVAT samples assumed values in the middle. Both the high-stress stiffness E 1 and the stress at 1.2 stretch (sigma 1.2 ) were significantly higher in -PVAT samples when compared to +PVAT samples (p < 0.05). Taken together these results suggest that -PVAT samples are stiffer both at low stress (not significant) as well as at high stress (significant) when compared to +PVAT samples. Moreover, -PVAT samples appear to also be less distensible (higher values of sigma 1.2 ) when compared to +PVAT samples. Thus, PVAT contributes significantly to decreasing the stiffness of the aortic wall. As such, PVAT should be considered as a target for improving vascular function in diseases with elevated aortic stiffness, including hypertension.

Finite element approximation of steady flows of colloidal solutions

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

A new type of constitutive equation for nonlinear elastic bodies. Fitting with experimental data for rubber-like materials

In this article, we develop a new implicit constitutive relation, which is based on a thermodynamic foundation that relates the Hencky strain to the Cauchy stress, by assuming a structure for the Gibbs potential based on the Cauchy stress. We study the tension/compression of a cylinder, biaxial stretching of a thin plate and simple shear within the context of our constitutive relation. We then compare the predictions of the constitutive relation that we develop and that of Ogden’s constitutive relation with the experiments of Treloar concerning tension/compression of a cylinder, and we show that the predictions of our constitutive relation provide a better description than Ogden’s model, with fewer material moduli.

Strain Gradient Theory Based Dynamic Mindlin-Reissner and Kirchhoff Micro-Plates with Microstructural and Micro-Inertial Effects

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect).

Field-Induced Transversely Isotropic Shear Response of Ellipsoidal Magnetoactive Elastomers

Magnetoactive elastomers (MAEs) claim a vital place in the class of field-controllable materials due to their tunable stiffness and the ability to change their macroscopic shape in the presence of an external magnetic field. In the present work, three principal geometries of shear deformation were investigated with respect to the applied magnetic field. The physical model that considers dipole-dipole interactions between magnetized particles was used to study the stress-strain behavior of ellipsoidal MAEs. The magneto-rheological effect for different shapes of the MAE sample ranging from disc-like (highly oblate) to rod-like (highly prolate) samples was investigated along and transverse to the field direction. The rotation of the MAE during the shear deformation leads to a non-symmetric Cauchy stress tensor due to a field-induced magnetic torque. We show that the external magnetic field induces a mechanical anisotropy along the field direction by determining the distinct magneto-mechanical behavior of MAEs with respect to the orientation of the magnetic field to shear deformation.

An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applications of Mathematics 2003; 28: 279–319), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as some classes of constitutive relations that imply limiting strain in tension, as special subclasses. Moreover, the constitutive relations that are developed allow the material moduli to depend on the density; thus, they can be used to describe the response of porous materials, such as porous metals, bone, rocks, and concrete undergoing small deformations.

Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations

The fundamental problem of dislocations in incompatible isotropic strain gradient elasticity theory of Mindlin type, unsolved for more than half a century, is solved in this work. Incompatible strain gradient elasticity of Mindlin type is the generalization of Mindlin’s compatible strain gradient elasticity including plastic fields providing in this way a proper eigenstrain framework for the study of defects like dislocations. Exact analytical solutions for the displacement fields, elastic distortions, Cauchy stresses, plastic distortions and dislocation densities of screw and edge dislocations are derived. For the numerical analysis of the dislocation fields, elastic constants and gradient elastic constants have been used taken from ab initio DFT calculations. The displacement, elastic distortion, plastic distortion and Cauchy stress fields of screw and edge dislocations are non-singular, finite, and smooth. The dislocation fields of a screw dislocation depend on one characteristic length, whereas the dislocation fields of an edge dislocation depend on up to three characteristic lengths. For a screw dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type agree with the corresponding ones in simplified incompatible strain gradient elasticity. In the case of an edge dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type are depicted more realistic than the corresponding ones in simplified incompatible strain gradient elasticity. Among others, the Cauchy stress of an edge dislocation obtained in incompatible isotropic strain gradient elasticity of Mindlin type looks more physical in the dislocation core region than the Cauchy stress obtained in simplified incompatible strain gradient elasticity and is in good agreement with the stress fields of an edge dislocation computed in atomistic simulations. Moreover, it is shown that the shape of the dislocation core of an edge dislocation has a more realistic asymmetric form due to its inherent asymmetry in incompatible isotropic strain gradient elasticity of Mindlin type than the dislocation core possessing a cylindrical symmetry in simplified incompatible strain gradient elasticity. It is revealed that the considered theory with the incorporation of three characteristic lengths offers a more realistic description of an edge dislocation than the simplified incompatible strain gradient elasticity with only one characteristic length.

STRESS-STRAIN BEHAVIOR OF ROCK MASS AROUND CYLINDRICAL EXCAVATIONS WITH THE PRESET CAUCHY STRESS STRESSES AND DISPLACEMENTS AT THE BOUNDARIES

The authors solve the problem on the stresses and strains of rock mass around a cylindrical excavation with the preset vectors of the Cauchy stresses and displacements at the boundary. It is assumed that the surrounding rock mass is elastic. Along the cylindrical excavation (free of stresses), displacements are measured as functions of two surface coordinates (polar angle and length along the symmetry axis of the excavation). These measurements are used to determine all components of tensors of stresses and strains at the boundary, and all coordinates of rotation vector. It is shown how this information can be used in the stress-strain analysis of rock mass farther from the excavation.

Nonuniform Stress Field Determination Based on Deformation Measurement

We demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same direction. The particular measures for strain and stress considered are the Lagrangian strain and the second Piola-Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola-Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola-Kirchhoff stress is determined. The Cauchy stress is related to the second Piola-Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The technique is then applied to the experimental case of nonuniform deformation of a PVC sheet with a circular hole and subject to tension. Limitations and restrictions of the technique and possible extensions will be discussed.