scholarly journals A finite element implementation of the stress gradient theory

Meccanica ◽  
2021 ◽  
Author(s):  
Tobias Kaiser ◽  
Samuel Forest ◽  
Andreas Menzel
Keyword(s):  
Finite Element ◽  
Displacement Field ◽  
Stress Gradient ◽  
Gradient Elasticity ◽  
Field Variable ◽  
Gradient Theory ◽  
Weak Formulation ◽  
Finite Element Implementation ◽  
Valued Field ◽  
Normal Projection

AbstractIn this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.

scholarly journals 1D HERMITE ELEMENTS FOR C1-CONTINUOUS SOLUTIONS IN SECOND GRADIENT ELASTICITY

2016 ◽  
Vol 7 ◽  
pp. 33-37 ◽  
Author(s):  
Christian Liebold ◽  
Wolfgang H. Müller
Keyword(s):  
Finite Element ◽  
Size Effect ◽  
Numerical Solution ◽  
Stiffness Matrix ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Additional Material ◽  
Hermite Finite Elements

We present a modified strain gradient theory of elasticity for linear isotropic materials in order to account for the so-called size effect. Additional material length scale parameters are introduced and the problem of static beam bending is analyzed. A numerical solution is derived by means of a finite element approach. A global C1-continuous displacement field is applied in finite element solutions because the higher-order strain energy density additionally depends on second gradients of displacements. So-called Hermite finite elements are used that allow for merging gradients between elements. The element stiffness matrix as well as the global stiffness matrix of the problem is developed. Convergence, C1-continuity and the size effect in the numerical solution is shown. Experiments on bending stiffnesses of different sized micro beams made of the polymer SU-8 are performed by using an atomic force microscope and the results are compared to the numerical solution.

scholarly journals Finite Element Modeling of Residual Stress at Joint Interface of Titanium Alloy and 17-4PH Stainless Steel

Metals ◽  
2021 ◽  
Vol 11 (4) ◽  
pp. 629
Author(s):  
Nana Kwabena Adomako ◽  
Sung Hoon Kim ◽  
Ji Hong Yoon ◽  
Se-Hwan Lee ◽  
Jeoung Han Kim
Keyword(s):  
Residual Stress ◽  
Finite Element ◽  
Stainless Steel ◽  
Finite Element Modeling ◽  
Equivalent Stress ◽  
Stress Gradient ◽  
Joint Interface ◽  
Element Modeling ◽  
Actual Experiment ◽  
Maximum Equivalent Stress

Residual stress is a crucial element in determining the integrity of parts and lifetime of additively manufactured structures. In stainless steel and Ti-6Al-4V fabricated joints, residual stress causes cracking and delamination of the brittle intermetallic joint interface. Knowledge of the degree of residual stress at the joint interface is, therefore, important; however, the available information is limited owing to the joint’s brittle nature and its high failure susceptibility. In this study, the residual stress distribution during the deposition of 17-4PH stainless steel on Ti-6Al-4V alloy was predicted using Simufact additive software based on the finite element modeling technique. A sharp stress gradient was revealed at the joint interface, with compressive stress on the Ti-6Al-4V side and tensile stress on the 17-4PH side. This distribution is attributed to the large difference in the coefficients of thermal expansion of the two metals. The 17-4PH side exhibited maximum equivalent stress of 500 MPa, which was twice that of the Ti-6Al-4V side (240 MPa). This showed good correlation with the thermal residual stress calculations of the alloys. The thermal history predicted via simulation at the joint interface was within the temperature range of 368–477 °C and was highly congruent with that obtained in the actual experiment, approximately 300–450 °C. In the actual experiment, joint delamination occurred, ascribable to the residual stress accumulation and multiple additive manufacturing (AM) thermal cycles on the brittle FeTi and Fe2Ti intermetallic joint interface. The build deflected to the side at an angle of 0.708° after the simulation. This study could serve as a valid reference for engineers to understand the residual stress development in 17-4PH and Ti-6Al-4V joints fabricated with AM.

Stress concentration factors for the torsion of curved beams of arbitrary cross-section

2006 ◽  
Vol 220 (12) ◽  
pp. 1709-1726 ◽  
Author(s):  
R E Cornwell
Keyword(s):  
Finite Element ◽  
Stress Concentration ◽  
Cross Section ◽  
Cross Sections ◽  
Shear Stresses ◽  
Curved Beams ◽  
Finite Element Implementation ◽  
Out Of Plane ◽  
Concentration Factors ◽  
Stress Concentration Factors

There are numerous situations in machine component design in which curved beams with cross-sections of arbitrary geometry are loaded in the plane of curvature, i.e. in flexure. However, there is little guidance in the technical literature concerning how the shear stresses resulting from out-of-plane loading of these same components are effected by the component's curvature. The current literature on out-of-plane loading of curved members relates almost exclusively to the circular and rectangular cross-sections used in springs. This article extends the range of applicability of stress concentration factors for curved beams with circular and rectangular cross-sections and greatly expands the types of cross-sections for which stress concentration factors are available. Wahl's stress concentration factor for circular cross-sections, usually assumed only valid for spring indices above 3.0, is shown to be applicable for spring indices as low as 1.2. The theory applicable to the torsion of curved beams and its finite-element implementation are outlined. Results developed using the finite-element implementation agree with previously available data for circular and rectangular cross-sections while providing stress concentration factors for a wider variety of cross-section geometries and spring indices.

Finite element implementation of an improved conservative level set method for two-phase flow

2014 ◽  
Vol 100 ◽  
pp. 138-154 ◽  
Author(s):  
Lanhao Zhao ◽  
Jia Mao ◽  
Xin Bai ◽  
Xiaoqing Liu ◽  
Tongchun Li ◽  
...  
Keyword(s):  
Finite Element ◽  
Level Set Method ◽  
Level Set ◽  
Two Phase Flow ◽  
Phase Flow ◽  
Two Phase ◽  
Finite Element Implementation ◽  
Conservative Level Set
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