Mechanics of Masonry Structures: Micro, Multiscale and Macro Models

Transcription

1 First French-Italian meeting on masonry Laboratoire de Mécanique et d Acoustique Marseille, October 24-25, 2013 Mechanics of Masonry Structures: Micro, Multiscale and Macro Models Elio Sacco Department of Civil & Mechanical Engineering University of Cassino & S.L. - Italy in collaboration with Daniela Addessi, Giulio Alfano, Sonia Marfia, Raimondo Luciano and Jessica Toti

2 Outline of the talk 2 1. MOTIVATIONS 2. MODELS FOR MASONRY micromechanical models phenomenological macro models multiscale models 3. CONCLUSIONS

3 3 MOTIVATIONS

4 4 Europe (and not only) is rich of historical buildings, monumental structures and heritage sites. The monumental heritage represents the history and the culture of the Country; it must be saved and preserved and (when necessary consolidated). Most of the heritage constructions are in masonry.

5 Lucca 2013 Modena Umbria-Marche 1997 L Aquila 2009 Molise 2002 map of the seismic risk in Italy Irpinia 1980

6 IRPINIA via palladino

7 palazzo nisivoccia 7

8 basilica interno 8

9 Umbria-Marche Foligno: Piazza della Repubblica October 14, ,23 pm bell tower of the Town Hall.

10 Molise 2002

11 L Aquila Church of S. Maria Paganica L Aquila

12 Masonry: heterogeneous material What is the masonry material? 12 mixture of (at least) two materials natural stones, bricks + lime or cement based mortar dry masonry

13 13

14 MODELS FOR MASONRY

15 15 Masonry modeling Micro-mechanics Phenomenological model Homogenization Discrete model Continuum model Macro-elements

16 16 Micromechanical models

17 Micromechanical models 17 different constitutive laws for units and the mortar; structural analysis performed considering each constituent of the masonry material; mortar joints modeled as interfaces and bricks characterized by a linear or nonlinear response; structural analyses characterized by great computational effort; FEM: unit blocks and the mortar beds discretized, high number of nodal unknowns. Lofti and Shing, 1994; Giambanco and Di Gati, 1997; Gambarotta and Lagomarsino, 1997; Lourenço and Rots, 1997; Giambanco et al., 2001; Oliveira and Lourenço, 2004; Alfano and Sacco, 2006; Fouchal et al. 2009;.

18 Analysis of masonry arch Geometry: R b =516 mm h=12 mm b=5.5 mm Mechanical properties: E=8300 N/mm 2 G=3400 N/mm 2 Marfia-Ricamato-Sacco Int. J. Comp. Methods in Engng Science and Mechanics, 2008 Cancelliere-Imbimbo-Sacco, Engineering Structures 2010

19 19 four hinges mechanism

20 Numerical simulation 20

21 21 Brick/mortar: linear elastic / elastoplastic Brick-mortar interface: cohesive model damage, unilateral contact, friction

22 partial decohesion total decohesion A B B C A C no decohesion T N mortar brick mortar brick mortar brick mortar brick partial decohesion total decohesion A B Microscale (REV) B C A C no decohesion T N mortar brick mortar brick mortar brick mortar brick Cohesive-zone model Representative Elementary Volume

23 Main idea for coupling interface damage and friction On the undamaged part elastic interface law On the damaged part unilateral-friction law 23 A u u σ (1-D) A σ d σ D A Kinematic compatibility: Additive decomposition of s d : inelastic relative displacement Equilibrium: 1 D stress on A u stress on A d A d u d σ σ D σ A u Undamaged part D [ 0, 1 ] : damage parameter u d s s s s s s d de di di s cp σ u Ks KN 0 σ d K s cp K 0 u s unilateral contact K T A d Damaged part d s di s friction de s s G. Alfano, E. Sacco, Int. J. Num. Meth. Engng, 2006

24 sn unilateral contact vector c hs ( N ) 0 inelastic slip: classical Coulomb yield function τ d d d d d N T N T damage ratios p N 1 N 0 h s if s 0 h s if s 0 d N N d T T d d d T T 0, 0, 0 d d σ σ sn sn N st st 1 T N, f T f sn 2GcN st 2G ct s damage evolution D maxmin1, D s 2 N N st T N 2 s 2 T o s o1 Area = G ci s c1 o Area = G cii -s c1i -s o1i s s o1i c1i history s II 24

25 25 N interphase interface h mortar brick h m h b T N T 2a 2b unilateral contact & friction Problem: assigned a history of average strain evaluate the average stress state accounting for the crack growth, the unilateral contact and the friction E 1/ h s, s T N Σ 1/ V NT, V T N T dv E. Sacco, F. Lebon Int. J. Solids & Structures, 2012

26 26 Linear elastic constitutive laws for the mortar and brick σ C ε, σ C ε m m m b b b Unilateral contact Friction d N 0, 0, d 0, d d p 0 0 0,, 0,, 0 N Crack growth: damage evolution Based on the relative displacement Fracture Mech dnt, dn, relative displs & stresses at the crack mouths

27 Three subproblems 27 s d e d c d f p1 p2 p3 p1) RVE subjected to s, i.e. to E; relative displacement at the crack d e. p2) relative displacement prescribed at the crack mouths d c =- d e. p3) RVE subjected to a frictional p sliding at the crack mouths T p. 0 Solution s1 s2 s3

28 28 By simple superposition of the three solutions, it is possible to recover any possible mechanical situation. Open crack s1 Closed crack with no-sliding s1+s2 Closed crack with sliding s1+s2+s3

29 Mechanical response 29 normal shear

30 Limit load, experimental results and numerical results F[N] Experimental result Limit Analysis Numerical result: n h =2; n b =5 ; n m =1; 2 Gauss points n h =2; n b =10; n m =1; 2 Gauss points n h =2; n b =20; n m =1; 2 Gauss points v [mm] 200 Sacco-Toti, Int. J. Comp. Methods in Engng Science and Mechanics, 2010

31 masonry wall loaded in compression and shear studied by Vermeltfoort and Raijmakers (1993) Lourenço (1996) 31 Loading: initial compression obtained by prescribing the vertical displacement of the top (vertical reactions 30KN); vertical displacements kept constant during the analysis; horizontal displacement of the top-right corner incremented left-ward. Alfano-Sacco, Int. J. Num. Meth. Engng, 2006

32 Each half brick discretized with noded, plane stress elements with enhanced strains. 32 Interface elements placed on the brick/mortar and on the brick/brick interfaces, to simulate the possible failure of a brick. Numerically computed horizontal reaction F plotted vs prescribed horizontal displacement; comparison with the experimental data provided by Lourenco.

33 33 Crack path

34 34 Phenomenological models

35 Macromechanical models 35 phenomenological constitutive laws for the masonry, derived performing tests on masonry, without distinguishing the blocks and the mortar behavior; unable to describe in detail some micro-mechanisms occurring in the damage evolution of masonry; very effective from a computational point of view for the structural analyses. NO-TENSION MODEL Heyman, 1966; Zienkiewicz et al., 1968; Di Pasquale, 1978; Romano and Romano, 1979; Baratta, 1982; Como and Grimaldi, 1982; Romano and Sacco, 1983; Giaquinta and Giusti, 1988; Del Piero, 1989; Sacco, 1990; Angelillo, 1993; Lucchesi et al. 1994; Luciano and Sacco, 1994; Alfano et al., 2000; Cuomo and Ventura, 2000; Marfia and Sacco, 2005;.

36 Damage-plastic (nonlocal) model stress-strain relation isotropic damage, uncoupled damage and plastic evolutions, Drucker-Prager plasticity with isotropic hardening. softening strain and damage localization strong mesh dependency nonlocal constitutive law Addessi Marfia-Sacco Comp. Methods Appl. Mech Engng, 2002

37 37 damage limit function: nonlocal term Y t eq. strain tension: Y c eq. strain compression: elastic strain total strain

38 38 damage evolution Consistency Kuhn-Tucker nonlocal term partial differential equation

39 39 plasticity limit function (Drucker-Prager): effective stress isotropic hardening force s t, s c compressive and tensile strengths I 1, J 2 1 st stress, 2 nd deviatoric invariants

40 40 Masonry walls 6000 mm 4000 mm 4000 mm 4000 mm 4000 mm 6000 mm 1400 Mesh independence (wall1) 2200 Structural response elements elements R [N] R [N] wall wall2 wall u [mm] u [mm]

41 41 damage distribution fracture fracture fracture minimum principal stress distribution

42 No-tension model 42 Masonry material characterized by very low tensile strength with respect to the compression strength. Assumptions: the tensile strength is zero, indefinitely elastic in compression. Consequences: convex strain energy function, reversible constitutive law, no-energy dissipation (unrealistic?), uniqueness of solution stress no-uniqueness of solution displacement apparent simplicity, not trivial numerical treatment. Material response No-tension model

43 No-tension model with limited compressive strenght Kinematics Stress-stress relation (isotropic) convex cone K of the admissible stresses: Marfia-Sacco Int. J. Comp. Methods in Engng Science and Mechanics, 2006

44 44 Fracture & plasticity for the Drucker-Prager type yield plastic locus fracture & plasticity fracture 2 plasticity 1 admissible stresses y

45 45

46 Unreinforced structure (right) 46 S T R E S S 4 S T R E S S E E E E E E E E+00 Current View Min = -4.54E+00 X = 6.40E+03 Y = 0.00E+00 Max = -2.13E-03 X = 6.13E+03 Y = 1.03E E E E E E E E E-02 Current View Min = 0.00E+00 X = 0.00E+00 Y = 0.00E+00 Max = 1.57E-02 X = 8.75E+03 Y = 8.25E+03 Time = 2.50E+01 Time = 2.50E+01

47 Unreinforced structure (left) 47 S T R E S S 4 S T R E S S E E E E E E E E+00 Current View Min = -7.07E+00 X = 0.00E+00 Y = 0.00E+00 Max = -3.04E-03 X = 4.78E+01 Y = 1.03E E E E E E E E E-02 Current View Min = 0.00E+00 X = 3.20E+03 Y = 0.00E+00 Max = 2.67E-02 X = 7.94E+03 Y = 9.69E+03 Time = 5.00E+01 Time = 5.00E+01 S T R E S S E E E E E E E E-04 Current View Min = 0.00E+00 X = 8.00E+02 Y = 0.00E+00 Max = 2.61E-04 X = 0.00E+00 Y = 0.00E+00 Time = 5.00E+01

48 Reinforced structure (left) 48 S T R E S S 4 S T R E S S E E E E E E E E+00 Current View Min = -7.37E+00 X = 0.00E+00 Y = 0.00E+00 Max = -2.41E-03 X = 1.00E+03 Y = 1.03E E E E E E E E E-02 Current View Min = 0.00E+00 X = 4.00E+03 Y = 0.00E+00 Max = 2.30E-02 X = 8.17E+03 Y = 8.92E+03 Time = 4.00E+01 Time = 4.00E+01 S T R E S S E E E E E E E E-04 Current View Min = 0.00E+00 X = 8.00E+02 Y = 0.00E+00 Max = 2.95E-04 X = 0.00E+00 Y = 0.00E+00 Time = 4.00E+01

49 Behavior of the structure P reinforced reinforced Multiplier of the horizontal loading unreiforced unreinforced Horizontal displacement of P ( mm)

50 50 The no-tension elasto-plastic model allows to catch the main features of the mechanical response of the masonry.

51 51 Multiscale models

52 Multiscale models different constitutive laws for the units and the mortar joints; homogenization procedure; very appealing, rational way to get the stress-strain relationship of the masonry, accounting for the failure micromechanisms of the masonry; effective models, with reduced computational effort; nonlinear homogenization procedure required to recover a macro-model could induce some theoretical or computational difficulties. 52 Kralj et al.,1991; Pietruszczak and Niu, 1992; Gambarotta and Lagomarsino, 1997; Luciano and Sacco, 1997; Cecchi and Sab, 2002; Cecchi et al. 2005; Uva and Salerno, 2006; Milani et al., 2006; Massart et al, 2007; Sacco, 2009;.

53 Santa Emerenziana Church, built in designed by architect Tullio Rossi

54 Motivations: micro-macro analysis FE 2 ε σ Cε p ε Main idea: to develop a procedure which avoids the nonlinear FE 2 (FEA at each iteration in each Gauss point of each element) Luciano-Sacco Int. J. Solids Struct., 1997; Sacco Eur. J. Solids A, 2009; Addessi-Sacco-Paolone Eur. J. Solids A, 2010; Marfia-Sacco Comp. Meth. Appl. Mech. 2011

55 UC for masonry material Simple damage model: the cracks occur only in the mortar material which behaves in a perfect elastic-brittle manner, the bricks are indefinitely elastic, the mortar thickness is small, so that the cracks can develop only vertically or horizontally, when a fracture starts to develop, a full failure of a mortar junction is supposed. 55 Luciano, Sacco Int. J. Solids Struct., 1997

56 Possible damaged states of the masonry 56 Paths S1 S2 S3 S8 S1 S2 S4 S8 S1 S5 S3 S8 S1 S5 S6 S8 S1 S7 S4 S8 S1 S7 S6 S8

57 57 Homogenization UC problem solved for each possible state averages of the local stresses evaluated in the UC averages of the local stresses evaluated in each mortar joint failure of the mortar joint evaluated using the Coulomb friction criterion su V c 0 c ncohesion n n n c t n friction Solved via finite element method c t t cohesion Periodicity continuity

58 58 Structural computation for different values of the cohesion for =1

59 New approach first order homogenization technique, Cauchy continuum at macro- and micro-scale disadvantages of the first order models: absolute size of the microstructure not incorporated, intrinsic assumption of uniformity of the macroscopic fields (not appropriate in critical regions characterized by high deformation gradients), mesh-dependency with softening. enhanced continua Cosserat model for masonry Cosserat continuum at the macro-scale, Cauchy continuum at the micro-scale. Kouznetsova et al. (2004); Forest and Sab (1998); van der Sluis et al. (1999); Masiani, Rizzi and Trovalusci (1995); Masiani and Trovalusci (1996, 2003); Casolo (2006); Brasile et al. (2007); De Bellis et al. (2008); Bacigalupo, Gambarotta (2010), Addessi et al. (2010). Sacco, Eur. J. Mech. A/Sol., 2009; Addessi, Sacco, Paolone, Eur. J. Mech. A/Sol., 2010; Marfia, Sacco, Comp. Meth. Appl. Mech., 2011; Addessi, Sacco, Int. J. Solids Struct, 2012

60 60 Macro-level BVP Displacement vector Strain vector Stress vector Governing equations

61 61 Micro-level BVP Displacement vector Strain vector Stress vector Governing equations i = mortar / brick

62 62 Micro-Macro link Displacement vector Kinematical map map perturbation Periodicity conditions

63 63 Constitutive laws brick model: linear elastic mortar joint model: plasticity-damage-friction crushing damage unilateral effect friction sliding (exponential law)

64 64 TFA: Transformation Field Analysis Composite material Nonlinear effect in a region Constant inelastic strain in the region (Dvorak, 1992) Piecewise constant inelastic strain in the region (Chaboche, 2000) Non-uniform TFA (Michel- Suquet, 2003) TFA for damage / unilateral contact / friction (Sacco, 2009) Non-uniform TFA (Sepe-Marfia- Sacco, 2013)

65 Homogenization procedure (TFA) sub-domains, where inelastic effects occur Assumptions: 1. the inelastic strain p in each subset is uniform 2. the nonlinear behavior of the mortar is governed by the average stress in the subset 4

66 Unit cell subjected to average elastic and to inelastic strains 66 elastic strain Ee localization of the strain e e x R x E e elastic strain in M j e M R j M j e E e average strain Ee average stress (Hill-Mandel principle) E Σ e σ E R C R R C R E E CE Σ CE overall elastic matrix j T T T T B T M e e d e e ed e ed e B j1 j M T e e e e

67 Unit cell subjected to average elastic and to inelastic strains 67 inelastic strain i π i localization of the strain i p x R x π elastic strain in η R I j M j im M i j i ij i π average strain i P 0 average stress 8 1 T j T j j Σ i i i d i i ij d R C R R C R I π S π π B j1 j π π π π M B B B M M M i i i

68 Strain field localization 68 E e e e i π i e x R x E i p x R x π i e e average stress Σ CE Σ i E e average strain π 0 i i S π total stress 8 i i Σ CE e S π j1 total strain E E e Overall stress-strain relationship Σ CEP overall inelastic strain P 8 j1 1 i i C S π

69 masonry wall loaded in compression and shear studied by Lourenço 69

70 70 Homogenization: Macro-Cosserat / Micro-Cauchy Damage-friction cohesive model for the mortar TFA technique FEM for periodic masonry arrangements Cosserat components strongly affect the nonlinear behavior by influencing the damage initiation and evolution and the friction plastic flow Relevance of the use of the micro-polar Cosserat continuum for developing accurate models for masonry.

71 71 enhanced TFA

72 inelastic strain h i i i kk k i1 i i i π0 x1π1x2π2 x1x2π3 π π x Non-uniform TFA t=0 t=1 t=2 t= ,00E+00 3,00E+00 2,00E+00 1,00E+00 N/mm 2 0,00E+00-5,00E-04 0,00E+00 5,00E-04 1,00E-03 1,50E-03 2,00E-03 2,50E-03 3,00E-03-1,00E+00-2,00E+00-3,00E+00-4,00E+00 uniform TFA nonuniform TFA -5,00E Marfia, Sacco, Comp. Meth. Appl. Mech., 2011 Sepe, Marfia, Sacco Int. J. Solids Struct., 2012

73 73 CONCLUSIONS

74 Micromechanics Discrete model

75 Micromechanics Homogenization Continuum model

76 Phenomenological model Continuum model

77 Masonry modeling Micro-mechanics Phenomenological model Homogenization Discrete model Continuum model Macro-elements

78 Thanks for your attention