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 Lesson 1-2: introduction, linear problems in statics Lesson 3: dynamics Lesson 4: locking problems Lesson 5: geometrical non-linearities Lesson 6-7: small strain plasticity
Politecnico di Milano, February 17, 2017, Lesson 5 3 Lesson 5: Introduction to non linear analysis 1. Sources of non-linearities Unilateral contact Fracture propagation Non-linear constitutive laws Geometrical non-linearities 2. Methods of numerical solution Non linear equations: Newton-like iterative algorithms Example of iterative algorithm: large transformations
Example: Hertz contact Example of analytical solution: spherical rigid indenter against a deformable surface (Hertz solution, 1882) Radius of the contact region a and indentation depth depend non-linearly on force P The behaviour of a system of two solids in contact is a nonlinear function of external loading even if the deformable solid is linear elastic Politecnico di Milano, February 17, 2017, Lesson 5 4
Politecnico di Milano, February 17, 2017, Lesson 5 5 Unilateral contact (infinitesimal transformations) Unilateral contact without friction no tangential force compression! no compenetration complementarity Select a master surface and a slave surface. The nodes of the slave surface cannot penetrate the master surface : gap along normal direction master surface S c area of potential contact slave surface
Politecnico di Milano, February 17, 2017, Lesson 5 6 Lesson 5: Introduction to non linear analysis 1. Sources of non-linearities Unilateral contact Fracture propagation Non-linear constitutive laws Geometrical non-linearities 2. Methods of numerical solution Non linear equations: Newton-like iterative algorithms Example of iterative algorithm: large transformations
Fracture propagation Politecnico di Milano, February 17, 2017, Lesson 5 7
Politecnico di Milano, February 17, 2017, Lesson 5 8 Lesson 5: Introduction to non linear analysis 1. Sources of non-linearities Unilateral contact Fracture propagation Non-linear constitutive laws Geometrical non-linearities Plasticity; lessons 6-7 2. Methods of numerical solution Non linear equations: Newton-like iterative algorithms Example of iterative algorithm: large transformations
Politecnico di Milano, February 17, 2017, Lesson 5 9 Lesson 5: Introduction to non linear analysis 1. Sources of non-linearities Unilateral contact Fracture propagation Non-linear constitutive laws Geometrical non-linearities 2. Methods of numerical solution Non linear equations: Newton-like iterative algorithms Example of iterative algorithm: large transformations
Politecnico di Milano, February 17, 2017, Lesson 5 10 Geometrical non-linearities shear buckling of a membrane more than one possible solution. Transition between different solutions via buckling
Politecnico di Milano, February 17, 2017, Lesson 5 11 Geometrical non-linearities parachute instability
Politecnico di Milano, February 17, 2017, Lesson 5 12 Example of highly non-linear problem Tetra Pak contact with friction fracture with unknown path non-linear constitutive law (damage like) large displacements
Politecnico di Milano, February 17, 2017, Lesson 5 13 Geometrical non-linearities, very large transformations Tetra Pak lagrangian (typical of solid mechanics) vs eulerian (typical of fluid mechanics) approaches
Politecnico di Milano, February 17, 2017, Lesson 5 14 Geometrical non-linearities, very large transformations casting applications
Politecnico di Milano, February 17, 2017, Lesson 5 15 Lesson 5: Introduction to non linear analysis 1. Sources of non-linearities Unilateral contact Fracture propagation Non-linear constitutive laws Geometrical non-linearities 2. Methods of numerical solution Non linear equations: Newton-like iterative algorithms Example of iterative algorithm: large transformations
Politecnico di Milano, February 17, 2017, Lesson 5 16 Numerical solution of a non-linear scalar equation Find u such that r(u)=0 Iterative procedure: create a sequence u (k) u such that r(u)=0 Newton-Raphson method: truncated first order series expansion of r(u (k+1) ) around u (k) and solution of the associated linear equation u (0) u (1) u (2) r(u) u EXERCISE apply to r(u)=-3+(u+1) 2 1) compute r 2) linearization - expansion
Politecnico di Milano, February 17, 2017, Lesson 5 17 Quadratic convergence of the Newton-Raphson method Setting e (k) = u (k) - u (error w.r.t. solution) for any k, one has: Taylor expansion with remainder for r(u (k) ) and r (u (k) ) around the exact solution u Newton-Raphson method: quadratic convergence speed in the vicinity of the solution
Divergence examples of the Newton-Raphson method Politecnico di Milano, February 17, 2017, Lesson 5 18
Politecnico di Milano, February 17, 2017, Lesson 5 19 Convergence of the Newton-Raphson method r(u) = 0 r(u) u r(u (0) ) u 2 (0) u 2 (1) u 2 (3) r(u) = (N-i)/N r(u (0) ) i=1 N sequence of N sub-problems u (0) =0 the solution of i-th sub-problem is employed as initial guess for (i+1)-th sub-problem
Politecnico di Milano, February 17, 2017, Lesson 5 20 Modified Newton-Raphson method - 1 r (u) replaced with a constant K, which gives u (0) u (1) u (2) u (3) r(u) u K might be the tangent at the initial estimate The convergence is only linear near the solution
Politecnico di Milano, February 17, 2017, Lesson 5 21 Modified Newton-Raphson method - 2 Approximate tangent with segment passing through the two previous estimates r(u) u (0) u (1) u (2) u
Politecnico di Milano, February 17, 2017, Lesson 5 22 Non linear system: Newton-like iterative algorithms The analysis of structures often leads to the solution of a system of non-linear equations: total value of displacement or increment in an interative procedure (typically: weak enforcement of equilibrium with PPV + constitutive laws) see e.g. the examples in the sequel In the linear elastic case (lessons 1-3) one would have: Newton-like algorithms: iterative approaches for the numerical solution of a system of non-linear equations
Solution of a system of a non-linear system with NR technique Politecnico di Milano, February 17, 2017, Lesson 5 23
Politecnico di Milano, February 17, 2017, Lesson 5 24 Lesson 5: Introduction to non linear analysis 1. Sources of non-linearities Unilateral contact Fracture propagation Non-linear constitutive laws Geometrical non-linearities 2. Methods of numerical solution Non linear equations: Newton-like iterative algorithms Example of iterative algorithm: large transformations
Geometrical non-linearities: buckling of a beam in compression Politecnico di Milano, February 17, 2017, Lesson 5 25
Politecnico di Milano, February 17, 2017, Lesson 5 26 Buckling of a beam in compression
Politecnico di Milano, February 17, 2017, Lesson 5 27 Summary of background deformation gradient Green Lagrange tensor (second) Piola (Kirchhoff) first Piola Kirchhoff Saint Venant Kirchhoff model for large displacements but small strain case (typical of buckling analysis)
Politecnico di Milano, February 17, 2017, Lesson 5 28 Summary of background here assumed 0 here assumed given
Politecnico di Milano, February 17, 2017, Lesson 5 29 Geometrical non-linearities, finite transformations velocity strain tensor
Summary of background Politecnico di Milano, February 17, 2017, Lesson 5 30
Politecnico di Milano, February 17, 2017, Lesson 5 31 Newton iterative procedure given the current iterate find the new iterate through a linearization of the residuum:
Politecnico di Milano, February 17, 2017, Lesson 5 32 Newton iterative procedure Linearization of the residuum:
Newton iterative procedure Politecnico di Milano, February 17, 2017, Lesson 5 33
Politecnico di Milano, February 17, 2017, Lesson 5 34 Element procedures for a T6
Politecnico di Milano, February 17, 2017, Lesson 5 35
Politecnico di Milano, February 17, 2017, Lesson 5 36
Politecnico di Milano, February 17, 2017, Lesson 5 37
Politecnico di Milano, February 17, 2017, Lesson 5 38
Politecnico di Milano, February 17, 2017, Lesson 5 39 loading sequence initialisation of each step