Non-Linear Behaviour

material non-linear, general:

general set of benchmarks:

a review of numerous potential benchmark problems, not worked up into formal benchmarks

Benchmark P15 .

creep;

2D single element tests subject to secondary creep (Norton law):

plane stress with constant uniaxial load
plane stress with constant applied uniaxial displacement
plane stress with constant biaxial load (3 different ratios)
plane stress with constant applied biaxial displacement (3 different ratios)
plane strain with constant unequi-biaxial load
plane strain with constant applied unequi-biaxial displacement

3D single element tests subject to secondary creep (Norton law):

with constant unequi-triaxial load
with constant applied unequi-triaxial displacement

axisymmetric multi-element test subject to secondary creep (Norton law):

internally pressurised thick hollow cylinder

2D single element tests subject to primary creep (Norton-Bailey law):

plane stress with constant uniaxial load
plane stress with constant uniaxial stepped load
plane stress with constant applied uniaxial displacement
plane stress with constant biaxial load (2 different ratios)
plane stress with constant biaxial stepped load (2 different ratios)
plane stress with constant applied biaxial displacement (2 different ratios)

3D single element test subject to primary creep (Norton-Bailey law):

with constant unequi-triaxial load

2D single element tests subject to combined primary and secondary creep:

plane stress with constant uniaxial load
plane stress with constant uniaxial stepped load
plane stress with constant applied uniaxial displacement

Benchmarks R0027 , R0049 (subset of above), R0080 (subset of above).

3D multi-element test subject to secondary Norton creep:

torsional creep of long prismatic circular shaft under uniform twist held constant with time
torsional creep of long prismatic circular shaft under uniform twist steadily increasing with time

3D multi-element test subject to primary creep (Norton-Bailey law) with time hardening:

torsional creep of a square shaft under constant twist
torsional creep of a square shaft under steady twist rate

2D plane strain thick cylinder under temperature transient followed by fluctuating internal pressure:

both elastic-plastic and creep behaviour over 4 stages; non-proportional loading

Axisymmetric multi-element test subject to temperature-dependent secondary creep (Norton law):

thick hollow cylinder with temperature gradients and thermally dependent creep laws

Benchmarks R0026 , R0030 , R0049 (subset of above), R0080 (subset of above).

plasticity:

2D single element tests:

plane strain with perfect plasticity
plane strain with isotropic hardening
plane strain comparing kinematic and isotropic hardening
plane stress with perfect plasticity
plane stress with isotropic hardening

3D single element tests:

with perfect plasticity
with isotropic hardening

Axisymmetric test with perfect plasticity
Axisymmetric test with isotropic hardening
Plane strain cylinder incompressibility test

Benchmarks P06 , R0049 (subset of above), R0072 (subset of above), R0080 (subset of above).

2D two-bar assembly under constant axial load and cyclic temperatures of differing values in each bar:

with kinematic hardening and thermal load in one bar only, to test elastic shakedown
with kinematic hardening and larger thermal load amplitude, to test alternating plasticity
with kinematic hardening and thermal load in both bars, to test different elastic shakedown in each bar
with perfect plasticity and thermal load in one bar only, to test thermal ratcheting

2D beam with constant end load and cyclic temperature gradient:

with kinematic hardening, to test elastic/plastic shakedown and cyclic plastic strain accumulation

2D plane strain rigid punch test under prescribed displacements, with many elements:

with perfect plasticity, testing the solution towards the limit load
with isotropic bilinear hardening

Thin square plate with uniform load and perfect plasticity:

with simply supported edges, testing the solution towards the limit load
with clamped edges, testing the solution towards the limit load

2D plane strain thick cylinder under temperature transient followed by fluctuating internal pressure:

with both elastic-plastic and creep behaviour over 4 stages; non-proportional loading

2D plane stress half of an eccentric tube under constant internal pressure and cyclically varying through-thickness temperature gradient:

with perfect plasticity, testing ratcheting and alternating plasticity

Benchmarks R0026 , R0030 , R0049 (subset of above), R0072 (subset of above), R0080 (subset of above).

geometric non-linearity:

Large displacements in truss elements:

the unrotated element (1)
limit point behaviour with one variable (2)
hardening with one variable (3)
bifurcation problem (4)
limit point with two variables (5)
hardening with two variables (6)
snap back (7)

Benchmarks P03 , R0065 (subset of above), R0072 (subset of above), R0080 (subset of above).

Large displacements and rotations in thin 2D beams and axisymmetric shells:

rigid body rotation (NLGB1)
straight cantilever with end moment (NLGB2)
curved cantilever with end moment (NLGB3)
straight cantilever with transverse free end point load (NLGB4)
straight cantilever with axial free end point load (NLGB5)
symmetric shallow pinned arch buckling (NLGB6)
unsymmetric deep arch buckling (NLGB7)
Lee’s frame buckling problem (NLGB8)
built-in circular plate with uniformly distributed load (NLGS1)
annular plate with inside edge load (NLGS2)
truncated cone with outside edge load (NLGS3)
built-in spherical cap with central point load (NLGS4)

Benchmarks P10 , R0010, R0065 (subset of above), R0072 (subset of above), R0080 (subset of above).

Assembly tests for 3D beams and shells:

elastic large deflection response of a Z-shaped cantilever beam under end load (3DNLG-1)
elastic large deflection response of a pear shaped cylinder under end shortening (3DNLG-2)
elastic lateral buckling of a right angle frame under in-plane end moments (3DNLG-3)
lateral-torsional buckling of an elastic cantilever subjected to a transverse follower force (3DNLG-4)
large deflection of a curved elastic cantilever under transverse end load (3DNLG-5)
buckling of a flat plate with an initial imperfection when subjected to in-plane shear (3DNLG-6)
large deflection elastic response of a hinged spherical shell with uniform pressure loading (3DNLG-7)
collapse of a straight pipe under pure bending (3DNLG-8)
large elastic deflection of a pinched hemispherical shell (3DNLG-8)
elasto-plastic behaviour of a stiffened cylindrical shell panel under compressive end load (3DNLG-10)

Benchmarks R0024 , R0029 , R0065 (subset of above), R0080 (subset of above).

A survey of possible benchmarks for 3D beams:

elastic lateral-torsional buckling of a cantilever (B1)
elastic lateral buckling of a right angle frame under in-plane end moments (B2)
lateral buckling of an I beam cantilever subjected to transverse force (B3)
elastic circular ring subjected to non-uniform pressure (B4)
elastic large deflection of a cantilever under follower transverse force (B5)
elasto-plastic large deflection of a cantilever under transverse end load (B6)
elastic large deflection of pinned deep arch subjected to central load (B7)
elastic large deflection of cantilever 45⁰ bend subject to end load (B8)
elastic large deflection analysis of a prestressed cable (B9)
elastic large deflection behaviour of a cable under gravity load (B10)
torsional buckling of end loaded cantilever (B11)
elastic large deflection behaviour of Lee’s frame (B12)
elastic large deflection of curved beam under non-conservative in-plane load (B13)
clamped-hinged deep circular arch subjected to a point load (B14)
large deflection of a diamond shaped frame (B15)
planar square frame loaded at the midpoints on opposite sides (B16)
cantilever under end moment (B17)
large deflection of a cantilever right angle frame under end load (B18)
elasto-plastic collapse of a 90⁰ pipe bend (B19)
elasto-plastic collapse of a thin walled elbow under in-plane loading and internal pressure (B20)
uniform collapse of a curved pipe under pure bending and internal pressure (B21)
elasto-plastic large deflection response of a Z-shaped cantilever beam (B22)
elasto-plastic analysis of I beams in torsion (B23)
finite rigid body rotation of a curved arch (B24)
collapse analysis of elasto-perfectly plastic triangular truss structure (B25)
elasto-plastic large deformation of a clamped beam under point load (B26)
plastic collapse of a two bay two story frame (B27)

Benchmark R0009 .

A survey of possible benchmarks for 3D shells:

elastic large deflection response of a centrally loaded spherical shell (S1)
large deflection elasto-plastic analysis of a cylindrical shell roof (S2)
thermal buckling of a simply supported rectangular plate (S3)
elastic post-buckling behaviour of a square plate subjected to shear (S4)
elastic large deflection of a cantilever plate with end moment (S5)
elasto-plastic buckling of a clamped imperfect spherical cap subjected to uniform pressure (S6)
elastic large deflection analysis of a pinched hemispherical shell (S7)
elastic large deflection of a hyperbolic paraboloid under concentrated moment at corners (S8)
elastic large deflection of a cylindrical shell under point load (S9)
elastic large deflection of a curved beam (S10)
finite rigid body rotation of a singly curved shell (S11)
elastic large deflection response of a pear shaped cylinder under end shortening (S12)
buckling and post-buckling response of axially compressed perfect and imperfect elliptical cylinders (S13)
elastic large deflection response of a cylindrical shell with two diametrically opposite cutouts (S14)
elastic large deflection response of a clamped square plate under uniform pressure (S15)
post buckling analysis of cylindrical shell under axial compression (S16)
post buckling behaviour of cylindrical panel under central point load (S17)
expansion bellows under axial end load (S18)
elasto-plastic response of a spherical cap under a central load (S19)
large deflection of a clamped cylindrical panel subjected to uniform pressure (S20)
snap through of hinged cylindrical shell (S21)
post buckling of a simply supported circular plate loaded in radial edge compression (S22)
elastic lateral buckling and large deflection response of a narrow cantilever beam (S23)
elasto-plastic lateral buckling of a narrow cantilever beam (S24)
large deflection elasto-plastic response of a circular plate (S25)

Benchmark R0009 .

A survey of possible benchmarks for 3D branched shells:

elasto-plastic behaviour of a stiffened cylindrical shell panel under compressive end load (BS1)
post buckling of stiffened panel under axial compression (BS2)
elastic buckling of T-ring stiffened cylinders under external pressure (BS3)
buckling of spherical shells ring supported at the edges (BS4)
elastic buckling of I-ring stiffened cylinders under external pressure (BS5)
3D finite rigid body rotation of stiffened shell connected to a hinged bar (BS6)
non-symmetric elasto-plastic bifurcation buckling of torispherical head with axisymmetric nozzle (BS7)
elasto-plastic large deflection response of a multi-mitre pipe bend (BS8)

Benchmark R0009 .

contact analysis:

2D multi-element plane strain tests:

contact patch test
rigid punch on deformable foundation
Hertzian contact
sliding wedge with linear springs
bending of a plate over a stiff cylinder
sliding and rolling of a ring on a rigid surface

2D multi-element plane stress tests:

cantilevered beam loaded against a rigid curvilinear surface

2D solid multi-element tests:

two contacting rings
buckling of a curved column with self-contact

2D axisymmetric multi-element tests:

interference between two cylinders

Benchmark R0081 .

Multi-element 2D/3D tests of a range of physical features:

2D contact of cylindrical roller
3D punch with rounded edges
3D sheet metal forming
3D loaded pin
3D steel roller on rubber

Benchmark R0094 .