1. PAT328, Section 3, March 2001 S10 - 1 MAR120, Lecture 4, March 2001MAR 120 – Procedures and Results Geometrically Nonlinear Framework

2. PAT328, Section 3, March 2001 S10 - 2 MAR120, Lecture 4, March 2001MAR 120 – Procedures and Results 0 1 2 3 Geometrically Nonlinear Framework Total Lagrangian  All integrals are evaluated with respect to the initial undeformed configuration Updated Lagrangian  All integrals are evaluated with respect to the last completed iteration of the current increment

3. PAT328, Section 3, March 2001 S10 - 3 MAR120, Lecture 4, March 2001MAR 120 – Procedures and Results  E = Engineering (infinitesimal) strain = (L – L 0 )/L 0 A measure preferred by structural engineers Work conjugate to the Engineering stress measure It is only applicable to small deformation and small strain analyses  L = Logarithmic (natural) strain =  L/L = ln(L/L 0 ) (update lagrangian) A measure that is incremental in form and preferred by metallurgists Work conjugate to the Cauchy stress measure This measure is typically used for large deformation, large strain analyses  G = Green-Lagrange strain = (L 2 – L 2 0 )/2L 2 0 (total lagrangian) This measure is typically used for large deformation, small strain analysis Additional work in the stress-strain relationship extends it to large strain work (Green’s strain accommodates finite rotations but not finite strains) It is work conjugate to the 2nd Piola-Kirchhoff stress measure   = Almansi strain = (L 2 – L 2 0 )/2L 2 This measure is also known as Eulerian strain It is work conjugate to the Almansi stress measure Not used in Marc Geometric Nonlinearity: Strain Measures

4. PAT328, Section 3, March 2001 S10 - 4 MAR120, Lecture 4, March 2001MAR 120 – Procedures and Results The salient property of the last three tensors is that their components are invariant under rigid body rotation of the material (strain=0 for 1D) Additional strain measures include Stretch and Biot Provided the strains remain small (say <3-4%), the different strain/stress measures will provide the same solutions ( 差異約 3~4%) Small strain examples include a fishing rod bending under the weight of a large fish, helicopter rotor blades under static dead load and the hair spring of a mechanical watch Conversions between the strain measures are readily possible as follows:  G =  E (1+  E /2)  L = ln(1+  E ) = ½ ln(1+ 2  G )  A = (2  E +  E 2 )/2(1 +  E ) 2 The strain measure used by MSC.Marc is determined mainly by the choice of a total or updated Lagrangian solution framework (discussed later) Geometric Nonlinearity: Strain Measures

5. PAT328, Section 3, March 2001 S10 - 5 MAR120, Lecture 4, March 2001MAR 120 – Procedures and Results  E = Engineering (nominal, conventional) = F/A o Defined in terms of the original area and the original geometry  C = Cauchy (true) stress = F/A Defined in terms of current area and current deformed geometry (force per unit deformed area) As a result, it is the most naturally understood stress measure, that is, it most naturally describes the material response  1 = 1st Piola-Kirchhoff stress Defined in terms of the original area and the current deformed geometry (current force per unit undeformed area)  = 2nd Piola-Kirchhoff stress Defined in terms of the initial area and current deformed geometry (transformed current force per unit undeformed area) It is work conjugate to the Green-Lagrange strain measure For small strain, the 2P-K stress can be interpreted as the Cauchy stress related to (local) axes that rotate with the material Without additional knowledge concerning the deformations, the 2P-K stresses are difficult to interpret 2P-K stresses are not uncommonly transformed into Cauchy stress to give a “true” stress of use to engineers Geometric Nonlinearity: Stress Measures

6. PAT328, Section 3, March 2001 S10 - 6 MAR120, Lecture 4, March 2001MAR 120 – Procedures and Results Stress conversions can be obtained via:  C = Cauchy (true) stress =  E (1 +  E )  = 2nd Piola-Kirchhoff stress =  E /(1 +  E ) The stress measure used by MSC.Marc is determined mainly by the choice of a total or updated Lagrangian solution framework (discussed later) MSC.Marc can be asked to provide either 2PK or Cauchy stress during the analysis setup. MSC.Marc will carry out a conversion as required during the analysis( 好像沒有 2PK ) Geometric Nonlinearity: Stress Measures