Coupled Thermomechanical Contact Problems
|
|
- Preston Malone
- 5 years ago
- Views:
Transcription
1 Coupled Thermomechanical Contact Problems Computational Modeling of Solidification Processes C. Agelet de Saracibar, M. Chiumenti, M. Cervera ETS Ingenieros de Caminos, Canales y Puertos, Barcelona, UPC International Center for Numerical Methods in Engineering (CIMNE) COMPLAS Short Course, Barcelona, Spain, 4 September 2007
2 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
3 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
4 Motivation and Goals Truthful coupled thermomechanical simulation of solidification and cooling processes of industrial cast parts (usually meaning complex geometries), in particular for aluminium alloys in permanent moulds. Thermodynamically consistent material model allowing a continuous transition between the different states of the process, from an initial fully liquid state to a final fully solid state. Thermomechanical contact model allowing the simulation of the interaction among all casting tools as consequence of the thermal deformations generated by high temperature gradients and phase transformations during solidification and cooling processes. Suitable stable FE formulation for linear tetrahedral elements, which allow the FE discretization of complex industrial geometries, and for incompressible or quasi-incompressible problems.
5 Motivation and Goals
6 Motivation and Goals
7 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
8 Coupled Thermomechanical Problem Fractional Step Method, Product Formula Algorithm LOOP, TIME Solve the THERMAL PROBLEM at constant configuration Solve the discrete variational form of the energy balance equation C L, K, Q, Q, c Obtain the temperatures map Solve the MECHANICAL PROBLEM at constant temperature Solve the discrete variational form of the momentum balance equation s, u B, u t, u t, u Obtain the displacements map, and then the stresses map END LOOP, TIME Q mc tc
9 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
10 Mechanical Problem Solid and Liquid-like Phases
11 Mechanical Problem Solid and Liquid-like Phases
12 Mechanical Problem Solid and Liquid-like Phases Liquid-like phase Solid phase
13 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
14 Mechanical Problem Solid Phase 20º C T.D. Young Mod. T.D. Viscosity 500º C 700º C T.D. Yield Stress T.D. Hardening
15 Mechanical Problem Solid Phase J2 Elasto-Viscoplastic model, isotropic and kinematic hardening The thermoelastic constitutive behaviour in solid state written as, u p K e p s 2G dev ( S ) can be where the is the mean pressure, is the deviatoric part of the stress tensor, K p is the bulk modulus, G s is the shear modulus e and is the volumetric thermal deformation which takes also into account the thermal shrinkage due to phase-change, e L 1 3 ref 3 S S ref S
16 Mechanical Problem Solid Phase J2 Elasto-Viscoplastic model, isotropic and kinematic hardening The yield function can be written as, ( s, q, q, ) : s q R( q, ) where the yield radius takes the form, 2 R( q, ) 3 0 q
17 Mechanical Problem Solid Phase J2 Elasto-Viscoplastic model, isotropic and kinematic hardening The plastic evolution equations can be written as, p ε ( s, q, q, ) n ( s, q, q, ) 2 3 ξ ( s, q, q, ) n where the plastic parameter takes the form, q q σ 1 ( s, q, q, ) n and the unit normal to the yield surface takes the form, n s q s q
18 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
19 Viscous deviatoric model Mechanical Problem Liquid-like Phase In the liquid state, the thermal deformation is zero and the bulk modulus and shear modulus tends to infinity, characterizing an incompressible rigid-plastic material, such that e G ( L ) 0, K : p e, 0 A purely viscous deviatoric model can be obtained as a particular case of the J2 elasto-viscoplastic model, without considering neither isotropic or kinematic hardening and setting to zero the radius of the yield surface. Then the yield function takes the form, ( s) : s u 0
20 Mechanical Problem Liquid-like Phase Viscous deviatoric model The plastic evolution equation takes the form, p ε () s n 1 () s n s s and the plastic strain evolution takes the simple form, σ where the plastic parameter and unit normal to the yield surface take the form, ε p 1 s
21 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
22 Mechanical Problem Mushy Zone. Liquid-Solid Continuous Transition J2 Elasto-Viscoplastic model, isotropic and kinematic hardening The thermoelastic constitutive behaviour in mushy state can be written as, u p K e p s 2G dev ( ) where the is the mean pressure, is the deviatoric part of the stress tensor, K p is the bulk modulus, G s is the shear modulus e and is the volumetric thermal deformation due to the thermal shrinkage taking place during the liquid-solid phase-change, e V V 0 pc f S S L L S
23 Mechanical Problem Mushy Zone. Liquid-Solid Continuous Transition J2 Elasto-Viscoplastic model, isotropic and kinematic hardening The yield function can be written as, ( s, q, q, ) : s f q f R( q, ) where the solid fraction satisfies, f f s s The yield function for the solid phase is recovered for f 1 s 0 1 The yield function for the liquid-like phase is recovered for f 0 L S s s s
24 Mechanical Problem Mushy Zone. Liquid-Solid Continuous Transition J2 Elasto-Viscoplastic model, isotropic and kinematic hardening The plastic evolution equations can be written as, p ε ( s, q, q, ) n where the plastic parameter takes the form, ( s, q, q, ) 2 3 f q σ ξ ( s, q, q, ) n f q 1 ( s, q, q, ) n and the unit normal to the yield surface takes the form, f f n s q s q s s s s
25 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
26 Mechanical Problem Incompressibility or Quasi-incompressibility Problem Low order standard finite elements discretizations are very convenient for the numerical simulation of large scale complex industrial casting problems: Relatively easy to generate for real life complex geometries Computational efficiency Well suited for contact problems Low order standard finite elements lock for incompressible or quasiincompressible problems. Mixed u/p finite element formulations may avoid this locking behaviour, but they must satisfy a pressure stability restriction on the interpolation degree for the u/p fields given by the Babuska-Brezzi pressure stability condition. Finite elements based on equal u/p finite element interpolation order do not satisfy the BB stability condition. Standard mixed linear/linear u/p triangles or tetrahedra elements do not satisfy the BB stability condition, exhibiting spurious pressure oscillations.
27 Mechanical Problem Incompressibility or Quasi-incompressibility Problem Pressure stabilization goals To stabilize the pressure while using mixed equal order linear u/p triangles and tetrahedra Stabilization of incompressibility or quasi-incompressibility problems using Orthogonal Subgrid Scale Method, within the framework of Variational Multiscale Methods (Hughes, 1995; Garikipati & Hughes, 1998, 2000; Codina, 2000, 2002; Chiumenti et al. 2002, 2004; Cervera et al. 2003; Agelet de Saracibar et al. 2006)
28 Mechanical Problem Incompressibility or Quasi-incompressibility Problem Pressure map P1 Standard Galerkin linear u element P1/P1 Standard mixed linear/linear u/p element P1/P1 OSGS Stabilized mixed linear u/p element
29 Mechanical Problem Multiscale Stabilization Methods Strong form of the mixed quasi-incompressible problem Let us consider the following infinite-dimensional spaces for the displacements and pressure fields, respectively, 1 2 u H u u : L : on u u Find a displacement field and pressure field such that, s p B 0 in 1 u e p 0 K p Find U : such that, U F in
30 Mechanical Problem Multiscale Stabilization Methods Variational form of the mixed quasi-incompressible problem Let us consider the following infinite-dimensional spaces for the displacements, displacements variations and pressure fields, respectively, u H u u : H : L : on u u Find a displacement field and pressure field such that, s p 1 u Find p s, v, v B, v t, v 0 v in e, q p, q 0 q K U : such that, B U, V L V V : in
31 Mechanical Problem Multiscale Stabilization Methods Multiscale methods. Subgrid multiscale technique Let us consider the following split of the exact solution field, U : U U : h U h h h h where is the exact solution, is the finite element approximated solution (belonging to the finite-dimensional FE solution space) and is the subgrid scale solution, not captured by the finite element mesh, belonging to the infinite-dimensional subgrid scale space U U u uh U, h, p U p U h u 0
32 Mechanical Problem OSGS Stabilization Technique Orthogonal Subgrid Scale (OSGS) technique Within the OSGS technique, the infinite-dimensional subgrid scale space is approximated by a finite-dimensional space choosed to be orthogonal to the FE solution space (Codina, 2000) U h
33 Mechanical Problem OSGS Stabilization Technique Orthogonal Subgrid Scale (OSGS) technique The subgrid scale solution is then approximated, at the element level as, u n1 e, n1 Ph sh, n1 ph, n1 B where where p n1 0 en, 1 2 che en, 1 * * G is a (mesh size dependent) stabilization parameter defined as, 2G is the secant shear modulus.
34 Mechanical Problem OSGS Stabilization Technique Orthogonal Subgrid Scale (OSGS) technique Using a linear interpolation for displacements (triangles/tetrahedra) and assuming that body forces belong to the FE space, the subgrid scale solution is then approximated, at the element level as, where where u p n1 e, n1 h h, n1 n1 0 P p en, 1 2 che en, 1 * * G is a (mesh size dependent) stabilization parameter defined as, 2G is the secant shear modulus.
35 Mechanical Problem OSGS Stabilization Technique Discrete OSGS stabilized variational form Find U h h B such that, for any V 0, h U, V L V V, stab h h stab h h h where the stabilized variational forms can be written as, n elem U V U V, 1, 1 B, B, P p, q stab h h h h e1 e n h h n h L h V LV stab h h e
36 Mechanical Problem OSGS Stabilization Technique Discrete OSGS stabilized variational form : P p Let h, n1 h h, n1 the projection of the discrete pressure gradient onto the FE solution space and H 1, the pressure h gradient projection and FE associated spaces, respectively. Then the orthogonal projection of the discrete pressure gradient onto the FE soluctiuon space takes the form, P p p h h, n1 h, n1 h, n1
37 Mechanical Problem OSGS Stabilization Technique Discrete OSGS stabilized variational form Find u h, n1 h, n1 h, n1 h h h, p, such that, s s h, n1 n1 vh ph, n1 n1 vh B vh t vh 1 n elem u K,,,, 0, q p, q p, q 0 n1 h, n1 h h, n1 h e1 e, n1 n1 h, n1 h, n1 n1 h p, 0 n1 h, n1 h, n1 h e for any v h, qh, h h,0 h h,0
38 Mechanical Problem OSGS Stabilization Technique Strong form of the stabilized mixed quasi-incompressible problem Let us consider the following infinite-dimensional spaces for the displacements, pressure and pressure gradient projection fields, respectively, 1 2 u H u u : L : on u u p s p B p p u 0 in K p 0 Find a displacement field, a pressure field and a pressure gradient projection such that, 1 H
39 Mechanical Problem OSGS Stabilization Technique Staggered solution procedure. Problem 1 Given find h, n h uh, n1, ph, n1 h h such that, s s h, n1 n1 vh ph, n1 n1 vh B vh t vh,,,, 0 1 n elem 1 u, 1, 1,, 1,, q p, q p, q 0 n h n h K h n h e1 e n n h n h n n h e for any v,,0 q h h h h
40 Mechanical Problem OSGS Stabilization Technique Staggered solution procedure. Problem 2 Given u h, n1, ph, n1 h h find h, n1 h such that, for any n h h,0 elem e1 p, 0 e, n1 n1 h, n1 h, n1 h e
41 Mechanical Problem OSGS Stabilization Technique Matrix form of the staggered solution procedure Problem 1. Solve the algebraic system of equations to compute the incremental discrete displacements and pressures unknowns ( i) ( i) ( i) ( i) ( i) T n 1 n 1 n 1 u, n 1 n1 ( i) T ( i) ( i) ( i) n1 n1 p, n n1 p, n1 Problem 2. Compute the discrete continuous pressure gradient projections unknowns 1 ΠI n 1, n 1, n 1 n 1
42 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
43 Thermal Problem Variational enthalpy form of energy balance,,,, H K Q Q c where the enthalpy rate is given by, H C L where the latent heat release is given by, where the solid fraction satisfies, Energy Balance dfs dfl L L L d d f f s s 0 1 L S Q tc
44 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
45 Nonlinear Contact Kinematics Reference Configurations t 0 (2) (2) (1) (1) Master Body (2) X (2) N (1) X Slave Body (1) N (2) t (1) t Current Configurations t (2) (2) (1) n (1) (1) (2) x (2) n (1) x (2) (1), t ARGMIN t t x (2) X (1), t (2) (2) t X X X X X (2) (1) (1) (1) (2) (2) X (2) (2) 16/12/
46 Nonlinear Contact Kinematics Closest-Point-Projection onto the Master Surface, t argmin t t X X X X (2) (1) (1) (1) (2) (2) X (2) (2) (1) X (2) X (2) x, t, t t x X X X (2) (1) (2) (2) (1)... Coordinates of the slave particle at reference configuration... Coordinates of the master particle at reference configuration, defined as the closest-point-projection of the slave particle onto the master surface at the current configuration... Coordinates of the closest-point-projection of the slave particle onto the master surface at current configuration 16/12/
47 Nonlinear Contact Kinematics Normal Gap Function (1) (1) (1) (2) (2) (1) X, X X X, g t t N t t (2) (2) n x X (1), t, t g N... Normal gap on current configuration g N gn Contact constraint is violated... Contact constraint is satisfied... Outward unit normal at the closest-point-projection on the master surface at current configuration 16/12/
48 Nominal Contact Traction, Nominal Contact Pressure The nominal contact traction vector is defined as,, t, t T X P X N X (1) (1) (1) (1) (1) (1), t t,, N t t T X X n X (1) (1) (1) (1) (1) where the nominal contact pressure is defined as, t (1), (1) (1), (1) (1), N X t T X t n X t (1) (1) (2) (2) (1) Setting n X, t n X X, t, t (1) (1) (1) T X, N X, t t t (1) (1) (1) (2) (2) (1) (1) (1) tn X t T X t n X X t T X t,,,, 16/12/
49 Contact Kinematic/Pressure Constraints Kühn-Tucker contact/separation optimality conditions If (1) (1) (1) (1) X X X X t, t 0, g, t 0, t, t g, t 0 N N N N Contact consistency or persistency condition gn X (1), t 0 then (1) (1) (1) (1) X X X X t, t 0, g, t 0, t, t g, t 0 N N N N 16/12/
50 Penalty Regularized Contact Constraints The penalty regularization of the Kühn-Tucker contact/separation optimality conditions and contact persistency condition takes the form where N (1) (1) X, X, t t g t N N N is the contact penalty parameter. The penalty regularized contact constraints provides a constitutive-like equation for the nominal contact pressure as a function of the normal gap, allowing a displacement-driven formulation of the contact problem. 16/12/
51 Augmented Lagrangian Regularized Contact Constraints The augmented Lagrangian regularization of the Kühn-Tucker contact/ separation optimality conditions and contact persistency condition take the form N X (1), X (1), t t g t N N N N where is the contact penalty parameter and is the lagrange multiplier subjected to the following Kühn-Tucker optimality constraints N 0, g 0, g 0 N N N N 16/12/
52 Variational Contact Form Variational Contact Form The contact variational contribution to the variational form of the momentum balance equation takes the form,, 0 (1) G t g d c t N N c 16/12/
53 Mechanical Contact Problem Block-iterative Solution Scheme Arrow-shape block system. Block-iterative solution scheme Within the framework of penalty/augmented lagrangian methods, in order to get a better conditioned numerical problem, the resulting system of equations is arranged collecting cast, mould and interface variables, leading to a block arrow-shape system of equations of the form, cast 0 c, cast cast cast 0 mold c, mold mold mold c, cast c, mold c c c
54 Mechanical Contact Problem Block-iterative Solution Scheme Arrow-shape block system. Block-iterative solution scheme The arrow-shape block system of equations can be solved using a blockiterative solution scheme given by, ( i1) ( i) cast cast cast c, cast c ( i1) ( i) mold mold mold c, mold c ( i1) ( i1) ( i1) c c c c, cast cast c, mold mold
55 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
56 Thermal Contact Problem Heat Conduction Heat conduction flux is given by, q h cond cond c m where the heat conduction coefficient is considered to be a function of the contact pressure t N, thermal resistance by conduction R cond, Vickers hardness and an exponent coefficient, and is given by, H e h cond t N 1 t N Rcond He n n
57 Thermal Contact Problem Heat Conduction Thermal resistance by conduction depends on the air trapped between cast and mould due to the surfaces roughness and on the mould coating, and is given by, where by, k a R z R cond R 2k R cond z a c k is the mean peak-to-valley heights of rough surfaces, given R R R 2 2 z z, cast z, mould and is the thermal conductivity of the air trapped between the cast and mould surfaces roughness, is the coating thickness and is c kc the thermal conductivity of the coating c
58 Thermal Contact Problem Heat Convection Heat convection flux is given by, q h conv conv c m where the heat convection coefficient is considered to be a function of the contact gap g N, the mean peak-to-valley heights of rough surfaces R z, the thermal conductivity of the air trapped between the cast and mould surfaces roughness ka, the coating thickness c and the thermal conductivity of the coating, and is given by, h conv g N k c 1 max g, R k k N z a c c
59 Thermal Contact Problem Heat Radiation Heat radiation flux is given by, where, a c m c m q rad Boltzman constant cast temperature mould temperature cast emissivity mould emissivity c m 4 4 a c m 1 1 1
60 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
61 1. Computational Simulation Aluminum (AlSi7Mg) Part in a Steel (X40CrMoV5) Mould Mesh of cast, cooling system and mould: 380,000 tetrahedra
62 1. Computational Simulation Temperature
63 1. Computational Simulation Solid Fraction
64 1. Computational Simulation Temperature and Solid Fraction at Section 1
65 1. Computational Simulation Pressure and J2 von Mises Effective Stress at Section 1
66 1. Computational Simulation Convergence Behaviour at a Typical Step Penalty E E E E E E E-2 Augmented Lagrangian E E E E-3 5A E E-3 7A E-3 Block-iterative Penalty E E E E E E-3
67 2. Computational Simulation Cast Iron Part in a Sand Mould
68 2. Computational Simulation Cast Iron Part in a Sand Mould
69 2. Computational Simulation Solid Fraction at t=500 s.
70 2. Computational Simulation Temperature at t=1100 s.
71 2. Computational Simulation J2 von Mises Effective Stress at Section 1
72 2. Computational Simulation J2 von Mises Effective Stress at Section 2
73 2. Computational Simulation J2 von Mises Effective Stress at Section 3
74 3. Computational Simulation Geometry of the Core, Chiller and Part
75 3. Computational Simulation Geometry of the Core and Chiller
76 3. Computational Simulation Geometry of the Part
77 3. Computational Simulation Filling Process
78 3. Computational Simulation Temperature Map Evolution
79 3. Computational Simulation Temperature Map Evolution at Section A-B
80 3. Computational Simulation Solidification Time and Solid Fraction Map Evolution
81 3. Computational Simulation Temperature vs Time at Different Points
82 4. Computational Simulation Geometry of the Part
83 4. Computational Simulation Filling of the Part
84 4. Computational Simulation Temperature Map Evolution
85 4. Computational Simulation Solidification Times and Solid Fraction Evolution
86 4. Computational Simulation Temperature vs Time at Different Points
87 4. Computational Simulation Deformation of the Part
88 4. Computational Simulation Displacements maps
89 4. Computational Simulation Von Mises Residual Stresses
90 5. Computational Simulation Geometry of a Motor Block
91 5. Computational Simulation Temperature Distribution
92 5. Computational Simulation Equivalent Plastic Strain
93 5. Computational Simulation J2 Effectiove Stress
94 Outline Motivation and Goals Coupled Thermomechanical Problem Mechanical Problem Solid Phase Liquid Phase Mushy Zone. Liquid-Solid Continuous Transition Multigrid Scale Pressure Stabilization Thermal Problem Mechanical Contact Problem Thermal Contact Problem Computational Simulations Concluding Remarks
95 Concluding Remarks A suitable coupled thermomechanical model allowing the simulation of complex industrial casting processes has been introduced. A thermomechanical J2 viscoplastic material model allowing a continuous transition from an initial liquid-like state to a final solid state has been introduced. The thermomechanical J2 viscoplastic model reduces to a purely deviatoric viscous model for the liquid-like phase. An OSGS stabilized mixed formulation for tetrahedral elements has been used to avoid locking and spurious pressure oscillations due to the quasiincompressibility constraints arising in J2 viscoplasticity models. A thermomechanical contact model has been introduced. Ill conditioning induced by the penalty method used in the mechanical contact formulation has been smoothed introducing a block-iterative algorithm. The formulation developed has been implemented in the FE software VULCAN and a number of computational simulations have been shown.
VULCAN 2000: A FINITE ELEMENT SYSTEM FOR THE SIMULATION OF CASTING PROCESSES C. Agelet de Saracibar, M. Cervera & M. Chiumenti ETS Ingenieros de Camin
VULCAN 2000: A FINITE ELEMENT SYSTEM FOR THE SIMULATION OF CASTING PROCESSES C. AGELET DE SARACIBAR y & M. CERVERA z ETS Ingenieros de Caminos, Canales y Puertos International Center for Numerical Methods
More informationON THE CONSTITUTIVE MODELING OF THERMOPLASTIC PHASE-CHANGE PROBLEMS C. Agelet de Saracibar, M. Cervera & M. Chiumenti ETS Ingenieros de Caminos, Canal
On the Constitutive Modeling of Thermoplastic Phase-change Problems C. AGELET DE SARACIBAR y & M. CERVERA z ETS Ingenieros de Caminos, Canales y Puertos Edificio C1, Campus Norte, UPC, Jordi Girona 1-3,
More informationA Stabilized Formulation for Incompressible Elasticity Using Linear Displacement and Pressure Interpolations
A Stabilized Formulation for Incompressible Elasticity Using Linear Displacement and Pressure Interpolations M. Chiumenti y, Q. Valverde Λ, C. Agelet de Saracibar y and M. Cervera y ydept. de Resistència
More informationOn the constitutive modeling of coupled thermomechanical phase-change problems
International Journal of Plasticity 17 21) 1565±1622 www.elsevier.com/locate/ijplas On the constitutive modeling of coupled thermomechanical phase-change problems C. Agelet de Saracibar *, M. Cervera,
More informationComputational Solid Mechanics Computational Plasticity
Comutational Solid Mechanics Comutational Plasticity Chater 3. J Plasticity Models C. Agelet de Saracibar ETS Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Cataluña (UPC), Barcelona,
More informationACCESS Beiratssitzung November Dr. G. Laschet and L. Haas ACCESS e.v. Aachen, Germany. FENET Workshop Copenhagen
Coupling FE Contact and Heat Transfer Analysis in Casting Simulations ACCESS Beiratssitzung November 2001 Dr. G. Laschet and L. Haas ACCESS e.v. Aachen, Germany FENET Workshop Copenhagen 27.02.2002 About
More informationContinuum Mechanics and Theory of Materials
Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7
More informationLecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity
Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity by Borja Erice and Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling
More informationOn the Orthogonal Subgrid Scale Pressure Stabilization of Small and Finite Deformation J2 Plasticity
On the Orthogonal Subgrid Scale Pressure Stabilization of Small and Finite Deformation J2 Plasticity C. Agelet de Saracibar, M. Chiumenti,M.Cervera and Q. Valverde + International Center for Numerical
More informationSTABILIZED FINITE ELEMENT METHODS TO DEAL WITH INCOMPRESSIBILITY IN SOLID MECHANICS IN FINITE STRAINS
European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2012) J. Eberhardsteiner et.al. eds.) Vienna, Austria, September 10-14, 2012 STABILIZED FINITE ELEMENT METHODS TO
More informationFinite calculus formulation for incompressible solids using linear triangles and tetrahedra
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2004; 59:1473 1500 (DOI: 10.1002/nme.922) Finite calculus formulation for incompressible solids using linear triangles
More informationMODELING OF ELASTO-PLASTIC MATERIALS IN FINITE ELEMENT METHOD
MODELING OF ELASTO-PLASTIC MATERIALS IN FINITE ELEMENT METHOD Andrzej Skrzat, Rzeszow University of Technology, Powst. Warszawy 8, Rzeszow, Poland Abstract: User-defined material models which can be used
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationConstitutive models: Incremental plasticity Drücker s postulate
Constitutive models: Incremental plasticity Drücker s postulate if consistency condition associated plastic law, associated plasticity - plastic flow law associated with the limit (loading) surface Prager
More informationMODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Uniaxial Model: Strain-Driven Format of Elastoplasticity
MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #3: Elastoplastic Concrete Models Uniaxial Model: Strain-Driven Format of Elastoplasticity Triaxial
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY
More informationThe Finite Element Method II
[ 1 The Finite Element Method II Non-Linear finite element Use of Constitutive Relations Xinghong LIU Phd student 02.11.2007 [ 2 Finite element equilibrium equations: kinematic variables Displacement Strain-displacement
More informationMixed Stabilized Finite Element Methods in Nonlinear Solid Mechanics. Part I: Formulation
Mixed Stabilized Finite Element Methods in Nonlinear Solid Mechanics. Part I: Formulation M. Cervera, M. Chiumenti and R. Codina International Center for Numerical Methods in Engineering (CIMNE) Technical
More informationLocal-global strategy for the prediction of residual stresses in FSW processes
Local-global strategy for the prediction of residual stresses in FSW processes N. Dialami, M. Cervera, M. Chiumenti and C. Agelet de Saracibar International Center for Numerical Methods in Engineering
More informationNon-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, Politecnico di Milano, February 17, 2017, Lesson 5
Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, attilio.frangi@polimi.it Politecnico di Milano, February 17, 2017, Lesson 5 1 Politecnico di Milano, February 17, 2017, Lesson 5 2 Outline
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationOn the Numerical Modelling of Orthotropic Large Strain Elastoplasticity
63 Advances in 63 On the Numerical Modelling of Orthotropic Large Strain Elastoplasticity I. Karsaj, C. Sansour and J. Soric Summary A constitutive model for orthotropic yield function at large strain
More informationComputational Inelasticity FHLN05. Assignment A non-linear elasto-plastic problem
Computational Inelasticity FHLN05 Assignment 2017 A non-linear elasto-plastic problem General instructions A written report should be submitted to the Division of Solid Mechanics no later than October
More informationHSNV140 - Thermoplasticity with restoration of work hardening: test of blocked dilatometry
Titre : HSNV140 - Thermo-plasticité avec restauration d'éc[...] Date : 29/06/2015 Page : 1/11 HSNV140 - Thermoplasticity with restoration of work hardening: test of blocked dilatometry Summary: This test
More informationDEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS
DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS Mohsen Safaei, Wim De Waele Ghent University, Laboratory Soete, Belgium Abstract The present work relates to the
More informationINTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS
INTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS SHEN R. WU and LEI GU WILEY A JOHN WILEY & SONS, INC., PUBLICATION ! PREFACE xv PARTI FUNDAMENTALS 1 1 INTRODUCTION 3
More informationThe effect of natural convection on solidification in tall tapered feeders
ANZIAM J. 44 (E) ppc496 C511, 2003 C496 The effect of natural convection on solidification in tall tapered feeders C. H. Li D. R. Jenkins (Received 30 September 2002) Abstract Tall tapered feeders (ttfs)
More informationAbstract. 1 Introduction
Contact analysis for the modelling of anchors in concrete structures H. Walter*, L. Baillet** & M. Brunet* *Laboratoire de Mecanique des Solides **Laboratoire de Mecanique des Contacts-CNRS UMR 5514 Institut
More informationModeling of Thermo-Mechanical Stresses in Twin-Roll Casting of Aluminum Alloys
Materials Transactions, Vol. 43, No. 2 (2002) pp. 214 to 221 c 2002 The Japan Institute of Metals Modeling of Thermo-Mechanical Stresses in Twin-Roll Casting of Aluminum Alloys Amit Saxena 1 and Yogeshwar
More informationA Finite Element Study of Elastic-Plastic Hemispherical Contact Behavior against a Rigid Flat under Varying Modulus of Elasticity and Sphere Radius
Engineering, 2010, 2, 205-211 doi:10.4236/eng.2010.24030 Published Online April 2010 (http://www. SciRP.org/journal/eng) 205 A Finite Element Study of Elastic-Plastic Hemispherical Contact Behavior against
More informationModelling of bird strike on the engine fan blades using FE-SPH
Modelling of bird strike on the engine fan blades using FE-SPH Dr Nenad Djordjevic* Prof Rade Vignjevic Dr Tom De Vuyst Dr James Campbell Dr Kevin Hughes *nenad.djordjevic@brunel.ac.uk MAFELAP 2016, 17
More informationNonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess
Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable
More informationAnalysis of a Casted Control Surface using Bi-Linear Kinematic Hardening
Analysis of a Casted Control Surface using Bi-Linear Kinematic Hardening Abdul Manan Haroon A. Baluch AERO, P.O Box 91, Wah Cantt. 47040 Pakistan Abstract Control Surfaces or Fins are very essential parts
More informationBiomechanics. Soft Tissue Biomechanics
Biomechanics cross-bridges 3-D myocardium ventricles circulation Image Research Machines plc R* off k n k b Ca 2+ 0 R off Ca 2+ * k on R* on g f Ca 2+ R0 on Ca 2+ g Ca 2+ A* 1 A0 1 Ca 2+ Myofilament kinetic
More informationNon-linear and time-dependent material models in Mentat & MARC. Tutorial with Background and Exercises
Non-linear and time-dependent material models in Mentat & MARC Tutorial with Background and Exercises Eindhoven University of Technology Department of Mechanical Engineering Piet Schreurs July 7, 2009
More informationHERCULES-2 Project. Deliverable: D4.4. TMF model for new cylinder head. <Final> 28 February March 2018
HERCULES-2 Project Fuel Flexible, Near Zero Emissions, Adaptive Performance Marine Engine Deliverable: D4.4 TMF model for new cylinder head Nature of the Deliverable: Due date of the Deliverable:
More informationMECHANICS OF MATERIALS. EQUATIONS AND THEOREMS
1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal
More informationStress, Strain, and Viscosity. San Andreas Fault Palmdale
Stress, Strain, and Viscosity San Andreas Fault Palmdale Solids and Liquids Solid Behavior: Liquid Behavior: - elastic - fluid - rebound - no rebound - retain original shape - shape changes - small deformations
More informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
More informationFinite Element Simulation of the Liquid Silicon Oriented Crystallization in a Graphite Mold
Finite Element Simulation of the Liquid Silicon Oriented Crystallization in a Graphite Mold Alexey I. Borovkov Vadim I. Suprun Computational Mechanics Laboratory, St. Petersburg State Technical University,
More informationElmer :Heat transfert with phase change solid-solid in transient problem Application to silicon properties. SIF file : phasechange solid-solid
Elmer :Heat transfert with phase change solid-solid in transient problem Application to silicon properties 3 6 1. Tb=1750 [K] 2 & 5. q=-10000 [W/m²] 0,1 1 Ω1 4 Ω2 7 3 & 6. α=15 [W/(m²K)] Text=300 [K] 4.
More informationUniversity of Sheffield The development of finite elements for 3D structural analysis in fire
The development of finite elements for 3D structural analysis in fire Chaoming Yu, I. W. Burgess, Z. Huang, R. J. Plank Department of Civil and Structural Engineering StiFF 05/09/2006 3D composite structures
More informationAlternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE. University of Liège Aerospace & Mechanical Engineering
University of Liège Aerospace & Mechanical Engineering Alternative numerical method in continuum mechanics COMPUTATIONAL MULTISCALE Van Dung NGUYEN Innocent NIYONZIMA Aerospace & Mechanical engineering
More informationMeasurement of deformation. Measurement of elastic force. Constitutive law. Finite element method
Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation
More informationPLASTICITY FOR CRUSHABLE GRANULAR MATERIALS VIA DEM
Plasticity for crushable granular materials via DEM XIII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIII E. Oñate, D.R.J. Owen, D. Peric and M. Chiumenti
More informationCommon pitfalls while using FEM
Common pitfalls while using FEM J. Pamin Instytut Technologii Informatycznych w Inżynierii Lądowej Wydział Inżynierii Lądowej, Politechnika Krakowska e-mail: JPamin@L5.pk.edu.pl With thanks to: R. de Borst
More informationA FINITE ELEMENT STUDY OF ELASTIC-PLASTIC HEMISPHERICAL CONTACT BEHAVIOR AGAINST A RIGID FLAT UNDER VARYING MODULUS OF ELASTICITY AND SPHERE RADIUS
Proceedings of the International Conference on Mechanical Engineering 2009 (ICME2009) 26-28 December 2009, Dhaka, Bangladesh ICME09- A FINITE ELEMENT STUDY OF ELASTIC-PLASTIC HEMISPHERICAL CONTACT BEHAVIOR
More informationNotes. Multi-Dimensional Plasticity. Yielding. Multi-Dimensional Yield criteria. (so rest state includes plastic strain): #=#(!
Notes Multi-Dimensional Plasticity! I am back, but still catching up! Assignment is due today (or next time I!m in the dept following today)! Final project proposals: I haven!t sorted through my email,
More informationComparison of Models for Finite Plasticity
Comparison of Models for Finite Plasticity A numerical study Patrizio Neff and Christian Wieners California Institute of Technology (Universität Darmstadt) Universität Augsburg (Universität Heidelberg)
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems. Prof. Dr. Eleni Chatzi Lecture ST1-19 November, 2015
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture ST1-19 November, 2015 Institute of Structural Engineering Method of Finite Elements II 1 Constitutive
More informationFinal Project: Indentation Simulation Mohak Patel ENGN-2340 Fall 13
Final Project: Indentation Simulation Mohak Patel ENGN-2340 Fall 13 Aim The project requires a simulation of rigid spherical indenter indenting into a flat block of viscoelastic material. The results from
More informationConfigurational Forces as Basic Concepts of Continuum Physics
Morton E. Gurtin Configurational Forces as Basic Concepts of Continuum Physics Springer Contents 1. Introduction 1 a. Background 1 b. Variational definition of configurational forces 2 с Interfacial energy.
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More informationUsing MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,
Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
More informationELAS - Elasticity
Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2017 295 - EEBE - Barcelona East School of Engineering 737 - RMEE - Department of Strength of Materials and Structural Engineering
More informationModule-4. Mechanical Properties of Metals
Module-4 Mechanical Properties of Metals Contents ) Elastic deformation and Plastic deformation ) Interpretation of tensile stress-strain curves 3) Yielding under multi-axial stress, Yield criteria, Macroscopic
More informationChapter 2. General concepts. 2.1 The Navier-Stokes equations
Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work
More informationThe Finite Element Method for Solid and Structural Mechanics
The Finite Element Method for Solid and Structural Mechanics Sixth edition O.C. Zienkiewicz, CBE, FRS UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in
More informationConservation of mass. Continuum Mechanics. Conservation of Momentum. Cauchy s Fundamental Postulate. # f body
Continuum Mechanics We ll stick with the Lagrangian viewpoint for now Let s look at a deformable object World space: points x in the object as we see it Object space (or rest pose): points p in some reference
More informationDiscontinuous Galerkin methods for nonlinear elasticity
Discontinuous Galerkin methods for nonlinear elasticity Preprint submitted to lsevier Science 8 January 2008 The goal of this paper is to introduce Discontinuous Galerkin (DG) methods for nonlinear elasticity
More informationA FINITE ELEMENT FORMULATION FOR NONLINEAR 3D CONTACT PROBLEMS
A FINITE ELEMENT FORMULATION FOR NONLINEAR 3D CONTACT PROBLEMS Federico J. Cavalieri, Alberto Cardona, Víctor D. Fachinotti and José Risso Centro Internacional de Métodos Computacionales en Ingeniería
More informationBasic Principles of Weak Galerkin Finite Element Methods for PDEs
Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element
More informationEffect of Strain Hardening on Unloading of a Deformable Sphere Loaded against a Rigid Flat A Finite Element Study
Effect of Strain Hardening on Unloading of a Deformable Sphere Loaded against a Rigid Flat A Finite Element Study Biplab Chatterjee, Prasanta Sahoo 1 Department of Mechanical Engineering, Jadavpur University
More informationPerformance Evaluation of Various Smoothed Finite Element Methods with Tetrahedral Elements in Large Deformation Dynamic Analysis
Performance Evaluation of Various Smoothed Finite Element Methods with Tetrahedral Elements in Large Deformation Dynamic Analysis Ryoya IIDA, Yuki ONISHI, Kenji AMAYA Tokyo Institute of Technology, Japan
More informationEFFECT OF STRAIN HARDENING ON ELASTIC-PLASTIC CONTACT BEHAVIOUR OF A SPHERE AGAINST A RIGID FLAT A FINITE ELEMENT STUDY
Proceedings of the International Conference on Mechanical Engineering 2009 (ICME2009) 26-28 December 2009, Dhaka, Bangladesh ICME09- EFFECT OF STRAIN HARDENING ON ELASTIC-PLASTIC CONTACT BEHAVIOUR OF A
More informationSTUDY OF THE BARCELONA BASIC MODEL. INFLUENCE OF SUCTION ON SHEAR STRENGTH
STUDY OF TH BARCLONA BASIC MODL. INFLUNC OF SUCTION ON SHAR STRNGTH Carlos Pereira ABSTRACT The Barcelona Basic Model, BBM, is one of the most used elasto-plastic models for unsaturated soils. This summary
More informationPSEUDO ELASTIC ANALYSIS OF MATERIAL NON-LINEAR PROBLEMS USING ELEMENT FREE GALERKIN METHOD
Journal of the Chinese Institute of Engineers, Vol. 27, No. 4, pp. 505-516 (2004) 505 PSEUDO ELASTIC ANALYSIS OF MATERIAL NON-LINEAR PROBLEMS USING ELEMENT FREE GALERKIN METHOD Raju Sethuraman* and Cherku
More informationTransactions on Engineering Sciences vol 14, 1997 WIT Press, ISSN
On the Computation of Elastic Elastic Rolling Contact using Adaptive Finite Element Techniques B. Zastrau^, U. Nackenhorst*,J. Jarewski^ ^Institute of Mechanics and Informatics, Technical University Dresden,
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationSimulation in Computer Graphics Elastic Solids. Matthias Teschner
Simulation in Computer Graphics Elastic Solids Matthias Teschner Outline Introduction Elastic forces Miscellaneous Collision handling Visualization University of Freiburg Computer Science Department 2
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
More informationFEM for elastic-plastic problems
FEM for elastic-plastic problems Jerzy Pamin e-mail: JPamin@L5.pk.edu.pl With thanks to: P. Mika, A. Winnicki, A. Wosatko TNO DIANA http://www.tnodiana.com FEAP http://www.ce.berkeley.edu/feap Lecture
More informationTowards Efficient Finite Element Model Review Dr. Richard Witasse, Plaxis bv (based on the original presentation of Dr.
Towards Efficient Finite Element Model Review Dr. Richard Witasse, Plaxis bv (based on the original presentation of Dr. Brinkgreve) Journée Technique du CFMS, 16 Mars 2011, Paris 1/32 Topics FEA in geotechnical
More informationCHAPTER 7 FINITE ELEMENT ANALYSIS OF DEEP GROOVE BALL BEARING
113 CHAPTER 7 FINITE ELEMENT ANALYSIS OF DEEP GROOVE BALL BEARING 7. 1 INTRODUCTION Finite element computational methodology for rolling contact analysis of the bearing was proposed and it has several
More informationDiscrete Analysis for Plate Bending Problems by Using Hybrid-type Penalty Method
131 Bulletin of Research Center for Computing and Multimedia Studies, Hosei University, 21 (2008) Published online (http://hdl.handle.net/10114/1532) Discrete Analysis for Plate Bending Problems by Using
More informationCH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics
CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models
More informationUne méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé
Une méthode de pénalisation par face pour l approximation des équations de Navier-Stokes à nombre de Reynolds élevé CMCS/IACS Ecole Polytechnique Federale de Lausanne Erik.Burman@epfl.ch Méthodes Numériques
More informationA unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation
A unifying model for fluid flow and elastic solid deformation: a novel approach for fluid-structure interaction and wave propagation S. Bordère a and J.-P. Caltagirone b a. CNRS, Univ. Bordeaux, ICMCB,
More informationElements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004
Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic
More informationAn Energy Dissipative Constitutive Model for Multi-Surface Interfaces at Weld Defect Sites in Ultrasonic Consolidation
An Energy Dissipative Constitutive Model for Multi-Surface Interfaces at Weld Defect Sites in Ultrasonic Consolidation Nachiket Patil, Deepankar Pal and Brent E. Stucker Industrial Engineering, University
More informationMulti-Point Constraints
Multi-Point Constraints Multi-Point Constraints Multi-Point Constraints Single point constraint examples Multi-Point constraint examples linear, homogeneous linear, non-homogeneous linear, homogeneous
More informationHeat Transfer. V4 3June16
Heat Transfer V4 3June16 Heat Transfer Heat transfer occurs between two surfaces or bodies when there is a temperature difference Heat transfer depends on material properites of the object and the medium
More information1 Exercise: Linear, incompressible Stokes flow with FE
Figure 1: Pressure and velocity solution for a sinking, fluid slab impinging on viscosity contrast problem. 1 Exercise: Linear, incompressible Stokes flow with FE Reading Hughes (2000), sec. 4.2-4.4 Dabrowski
More informationSTABILIZED METHODS IN SOLID MECHANICS
FINITE ELEMENT METHODS: 1970 s AND BEYOND L.P. Franca (Eds.) ccimne, Barcelona, Spain 2003 STABILIZED METHODS IN SOLID MECHANICS Arif Masud Department of Civil & Materials Engineering The University of
More information1.050 Engineering Mechanics. Lecture 22: Isotropic elasticity
1.050 Engineering Mechanics Lecture 22: Isotropic elasticity 1.050 Content overview I. Dimensional analysis 1. On monsters, mice and mushrooms 2. Similarity relations: Important engineering tools II. Stresses
More informationLinear Cosserat elasticity, conformal curvature and bounded stiffness
1 Linear Cosserat elasticity, conformal curvature and bounded stiffness Patrizio Neff, Jena Jeong Chair of Nonlinear Analysis & Modelling, Uni Dui.-Essen Ecole Speciale des Travaux Publics, Cachan, Paris
More informationProject. First Saved Monday, June 27, 2011 Last Saved Wednesday, June 29, 2011 Product Version 13.0 Release
Project First Saved Monday, June 27, 2011 Last Saved Wednesday, June 29, 2011 Product Version 13.0 Release Contents Units Model (A4, B4) o Geometry! Solid Bodies! Parts! Parts! Body Groups! Parts! Parts
More informationMHA042 - Material mechanics: Duggafrågor
MHA042 - Material mechanics: Duggafrågor 1) For a static uniaxial bar problem at isothermal (Θ const.) conditions, state principle of energy conservation (first law of thermodynamics). On the basis of
More informationSequential simulation of thermal stresses in disc brakes for repeated braking
Sequential simulation of thermal stresses in disc brakes for repeated braking Asim Rashid and Niclas Strömberg Linköping University Post Print N.B.: When citing this work, cite the original article. Original
More informationLoading σ Stress. Strain
hapter 2 Material Non-linearity In this chapter an overview of material non-linearity with regard to solid mechanics is presented. Initially, a general description of the constitutive relationships associated
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationMOOSE Workshop MOOSE-PF/MARMOT Training Mechanics Part Nashville, February 14th Daniel Schwen
www.inl.gov MOOSE Workshop MOOSE-PF/MARMOT Training Mechanics Part Nashville, February 14th Daniel Schwen www.inl.gov III. Mechanics Mechanics Outline 1. Introduction 2. Tensor-based System 1. Compute
More informationA Simple and Accurate Elastoplastic Model Dependent on the Third Invariant and Applied to a Wide Range of Stress Triaxiality
A Simple and Accurate Elastoplastic Model Dependent on the Third Invariant and Applied to a Wide Range of Stress Triaxiality Lucival Malcher Department of Mechanical Engineering Faculty of Tecnology, University
More informationLingopti project: Semi-Continuous Casting Process of Copper-Nickel Alloys
Lingopti project: Semi-Continuous Casting Process of Copper-Nickel Alloys E.Pecquet 1, F.Pascon 1, J.Lecomte-Beckers 2, A-M.Habraken 1 University of Liège, 1 Department M&S, 2 Service MMS Chemin des Chevreuils
More informationENGN 2290: Plasticity Computational plasticity in Abaqus
ENGN 229: Plasticity Computational plasticity in Abaqus The purpose of these exercises is to build a familiarity with using user-material subroutines (UMATs) in Abaqus/Standard. Abaqus/Standard is a finite-element
More informationThe Particle Finite Element Method (PFEM) in thermo-mechanical problems
The Particle Finite Element Method (PFEM) in thermo-mechanical problems J.M. Rodriguez, J.M. Carbonell, J.C. Cante, J. Oliver June 30, 2015 Abstract In this work we present the extension of the Particle
More informationEffect offriction onthe Cutting Forces in High Speed Orthogonal Turning of Al 6061-T6
IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-1684,p-ISSN: 2320-334X, Volume 11, Issue 2 Ver. VII (Mar- Apr. 2014), PP 78-83 Effect offriction onthe Cutting Forces in High Speed
More informationTheoretical Manual Theoretical background to the Strand7 finite element analysis system
Theoretical Manual Theoretical background to the Strand7 finite element analysis system Edition 1 January 2005 Strand7 Release 2.3 2004-2005 Strand7 Pty Limited All rights reserved Contents Preface Chapter
More informationPlasticity R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur
Plasticity R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 9 Table of Contents 1. Plasticity:... 3 1.1 Plastic Deformation,
More informationIMPACT PROPERTY AND POST IMPACT VIBRATION BETWEEN TWO IDENTICAL SPHERES
ICSV4 Cairns Australia 9- July, 7 IMPACT PROPERTY AND POST IMPACT VIBRATION BETWEEN TWO IDENTICAL SPHERES Hirofumi MINAMOTO, Keisuke SAITOH and Shozo KAWAMURA Toyohashi University of Technology Dept. of
More information