Một cách tiếp cận xấp xỉ và mô hình hóa phần tử hữu hạn hệ số dẫn và mô đun đàn hồi của vật liệu nhiều thành phần tt tiếng anh

1 MINISTRY OF EDUCATION AND VIETNAM ACADEMY OF TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY - DO QUOC HOANG AN APPROACH TO APPROXIMATE AND FEM-MODEL THE CONDUCTIVITY AND ELASTICITY OF MULTICOMPONENT MATERIAL Major: Engineering mechanics Code: 52 01 01 SUMMARY OF DOCTORAL THESIS IN MECHANICAL ENGINEERING AND ENGINEERING MECHANICS Hanoi – 2019 The thesis has been completed at: Graduate University Science and Technology – Vietnam Academy of Science and Technology Supervisors: Assoc Prof DrSc Pham Duc Chinh Assoc Prof Dr Tran Anh Binh Reviewer 1: Reviewer 2: Reviewer 3: Thesis is defended at Graduate University Science and Technology-Vietnam Academy of Science and Technology at … , on ……… Hardcopy of the thesis be found at : - Library of Graduate University Science and Technology - Vietnam national library INTRODUCTION Relevance of the thesis topic Most effective medium approximations for isotropic inhomogeneous materials are based on dilute solutions of some typical inclusions in an infinite matrix medium, while the simplest approximations are those for the composites with spherical and circular inclusions Practical particulate composites often involve inhomogeneities of more complicated geometry than that of the spherical (or circular) one In our approach, those inhomogeneities are supposed to be substituted by simple equivalent spherical (circular) inclusions from a comparison of their respective dilute solution results Then the available simple approximations for the equivalent spherical (circular) inclusion material can be used to estimate the effective conductivity of the original composite Numerical illustrations of the approach are performed on some 2D and 3D geometries involving elliptical and ellipsoidal inclusions Thesis objective Develop near interaction approximations for the conductivity and elasticity of multi-component materials with spherical (circular) form inclusions Equivalent inclusion approach is then developed to account for possible diversions, such as non-idealistic geometric forms of the inhomogeneities, imperfect matrix-inclusion contacts, filler dispersions, and when the particular values of the fillers’ properties are unspecified, using available numerical or experimental reference conductivity data for particular composites We use the eXtended Finite Elements Method (XFEM) to estimate the effective conductivity of 2D macroscopically-isotropic composites containing elliptic inclusions and the equivalent ones with circular inclusions for comparisons with the approximations Scope The thesis focuses on conductivity and elasticity of multicomponent materials, the Finite Element Method (FEM) and approximation schems 4 Research methods  Near interaction approximations has been constructed from the minimum energy for the macroscopic conductivity and elasticity of the multi-component matrix composites with spherical (circular) inclusions Equivalent replacement of complex-geometry inclusions by the equivalent spherical, circular, disk and needle ones with equivalent properties using polarization approximation, dilute solutions, and experimental referemce results  Numerical method: use Matlab program to homogenize some periodic material models in the framework of FEM method (XFEM) The results of FEM are considered as the accurate reference results for comparisons with the approximation ones The contributions of the thesis Beside Introduction section, the thesis contains Chapters, a Conclusion section and a list of publications relevant to the thesis References cited in the thesis are listed at the end of the thesis CHAPTER OVERVIEW 1.1 Opening Multi-component materials have complex structures, different individual mechanical properties Many authors offered different evaluation methods, including the effective medium approximations and the variational ones Geometric parameters have bên added to improve the étimates In this chapter, the author presents the concept of hômgenization and an overview of the constructions of approximation methods for complex multi-component materials The stress field  (x) by Hook’s law: is related to the strain field  ( x)  (x)  C(x) :  (x), (1.1) The average values of the stress and strain on V is defined as:    dx , V V    dx V V (1.2) Assume homogeneous boundary conditions for displacements: u(x)    x (1.3) Or the respective ones for the tractions  n   n (1.4) With the solutions σ, ε on V, the relationship between the averaged stress and strain on V is presented through the effective elastic tensor Ceff:   Ceff :  Ceff  T(k eff ,  eff ) , (1.5) k eff and  eff are effective elastic bulk and shear moduli Another approach is to determine the effective elasticity coefficients by finding the infimum of the energy function on V (the fields  need to be compatible):  : Ceff :   inf       : C :  dx , (1.6) V Or through the dual principle (the fields  need to be equilibrated):  : (Ceff )1 :   inf     : (C) 1 :  dx (1.7) V Similarly, the equations for the conductivity problem: The flux J must satisfy the equilibrium condition: ·J (x)  With the solutions J, E  T on V, the thermal conductivity coefficient (effective) ceff is determined as: J  c eff E  c eff T (1.8) The minimum energy principles are also the main tools to find the macroscopic conductivity: c eff E0 ·E0  inf  cE·Edx,  E  E and: V (1.9) (c eff ) 1 J ·J  inf  c1 J·Jdx,  J  J (1.10) V 1.2 Overview of approximation methods for multi-component materials 1.2.1 Dilute solutions The effective conductivity ceff of the dilute solution of ellipsoidal inclusions with axes ratio a: b: c, randomly oriented in a continuous matrix is expressed in the form: c eff  cM  vI (cI  cM ) Dc (cI , cM ) , vI  , Dc (cI , cM )  (1.11) cM 1 [ ],   cI A  cM (1  A) cI B  cM (1  B ) cI C  cM (1  C ) The general fomula of Dc (cI , cM ) for spherical (d=3) and circular (d=2) inclusions is: Dc (cI , cM )  dcM cI  (d  1)cM 1.2.2 Maxwell Approximation Maxwell approximation is built for 2-phase material from the matrix + spherical inclusions with any volume ratios, not limited by dilute distribution case (M - matrix symbol, I – inclusion symbol) 1 c eff  vI v     M    d  1 cM , c  d  c dc   M M   I 1 K eff  eff   vI vM 2(d  1)    M ,   K*M ; K*M  K  d  K K  K d  *M M *M   I  vI vM d K M  2(d  1)(d  2) M   MA  (  ) 1  *M ; *M  M  I  * M M  * M 2dK M  4d  M (1.12) 1.2.3 Differential Approximation - DA we obtain the following differential equations for the effective conductivity ceff = c(1) of the composite dc n   vI (cI  c) Dc (cI , c), dt  vI t  1 n (1.16a) c(0)  cM ,  t  , vI   vI ,  1 For elastic coefficient dK n   vI ( K I  K ) DK ( K I , I , K ,  ), dt  vI t  1 d n   vI ( I   ) DK ( K I ,  I , K ,  ), dt  vI t  1 (1.16b) n  t  , vI   vI  , K (0)  K M ,  (0)   M  1 1.2.4 Self-consistent approximations - SA The Self-consistent approximation method (SA) for composite materials n components, is cSA=c solution of the following equation: n v  (c   c ) D  (c  , c )    I I c I (1.17) 1 SA for the moduli of elasticity are the solutions KSA=K and   of a system of two equations SA= n v  ( K   K ) D  ( K  ,   , K ,  )  0,   I I K I I 1 n v  (    )D   I I M (1.18) ( K I ,  I , K ,  )  1 1.2.5 Mori-Tanaka Approximation (MTA) Mori-Tanaka type approximation (MTA), based on the assumption that the fields in an circular inclusion are determined as if it is embedded in the matrix with remote average strain of the matrix, yields [Le Quang] for two-phase composites c MTA  c M  vI (c I  c M )·{vM [I  p·c M1·(c I  c M )]  vI I}1 (1.19) While for the multi-phase ones (matrix + n inclusions) n c MTA  {vM c M   vI c I ·[I  p ·c M1·(c I  c M )]1}  1 (1.20) n 1 M 1 1 ·{vM I   vI [I  p ·c ·(c I  c M )] }  1 MTA for the effective conductivity of d-dimentional multicomponent isotropic materials with spherical inclusions (circular) has the following form n v  (c   c   I cMTA  cM  I M )dcM / [cI  (d  1)cM ] 1 (1.21) n vM   vI dcM / [cI  (d  1)cM ]  1 1.3 Three-point correlation estimates of Phạm ĐC Three-point correlation estimates of Phạm ĐC are for the effective thermal conductivity of the multi-component materials The bounds have been built from the minimum energy principles The general expression of the upper bound for ceff is c eff  Pc (2c0 )  c** , (1.22) where c0 is a positive parameter, 1  v  Pc (c** )       c** ,   c  c*  n v ) 2  (c  c0 )A X  X   1 c  2c0  ,  , 1 (1.23) n c**  3( (1.24) Similarly, the respective expression of the dual principle is written in the form: n n v )2  (c1  c01 )A X  X  (1.25)  1 c  2c0  ,  , 1 c**  3c02 (1  2c0  Where n v  c  2c0  1 c  2c0 X   n v X    c  2c0  1 c  2c0 (1.26) We choose the value c0 to eliminate the component c** , c** to make the inequality stronger to get the respective bounds 1.3 Finite element method for homogenization solution The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations The finite element method formulation of the problem results in a system of algebraic equations The method approximates the unknown function over the domain To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function A typical work out of the method involves dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by systematically recombining all sets of element equations into a global system of equations for the final calculation The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer 10 CHƯƠNG FINITE ELEMENT METHOD 2.1 Introduce 2.1.1 FEM for thermal solution The finite element method with fine meshes is now used for reference comparisons Let us consider a periodic cell Ω with the external boundary ∂Ω The strong form of the conductivity problem is written as: q  x  in , q  x  c  x E  x in E  x   T  x  T x in periodicin q  x  n , , (2.1) , antiperiodicin , The weak form associated with the above equations is given by:  cN N T d  i i i (2.2)  Using linear form functions for triangular elements with joint as: N  x, y   ax  by  c, (2.3)  E e ( x)    B e  T e  (2.4) Equation in matrix form: The effective thermal conductivity in x1 direction can be computed as: c1eff  L1 q1d  V T V The domain V is the periodic cell, so the effective thermal conductivity of RVE is given by cFeffEM  c1eff 14  q(T )d    rTd  0,   By substituting the temperature field defined in (8) into the weak form (16) we obtain the discrete system of linear ordinary equations Kd  Q, d  T T a where d are nodal unknowns and K and Q are the global stiffness matrix and external flux, respectively More precisely, the matrix K and vector Q are defined by K   BT c( x) Bd, Q   N T rd   where B and N are the matrices of shape function derivatives and shape functions associated with the approximation scheme (8) The periodic boundary condition is finally introduced to (17) by mean of multiplicator Lagrange 2.2.2 Results of thermal solution, using XFEM The calculation results indicate the temperature change in the calculation model Hình 2.5: Temperature distribution in the model CHAPTER POLARIZATION APPROXIMATION (PA) 3.1 Introdution Consider a representative volume element (RVE) of an isotropic n-icomponent material that occupies spherical region V of Euclidean space The center of the sphere is also the origin of the Cartesian system of coordinates {x} The RVE consists of n components occupying regions Va V of volumes va and having conductivities (thermal, electrical, etc.) ca(a = 1, .,n; the volume of V is assumed 15 to be the unity) Starting from the minimum energy principles and using Hashin–Shtrikman polarization trial fields, one derives the following three-point correlation bounds on the effective conductivity ceff of the composite (Le & Pham, 1991; Pham, 1993) Pc (2c0 )  c**  c eff  [ Pc1 (2c0 )  c** ]1 (3.1) where n v 1 )  c*  1 c  c* Pc (c* )  ( A   ,ij,ij dx  V v   , (3.2) (3.3) where conventional summation on repeating Latin indices (but not on the Greek indices) is assumed; Latin indices after comma designate differentiation with respective Cartesian coordinates; the arbitrary positive constant c0 is often referred to as the conductivity of a comparison material; the harmonic potentials ua(x) appear in the expressions of Hashin– Shtrikman polarization trial fields; the threepoint correlation parameters Ab ac relate the microgeometries of the three phases Va, Vb, Vc Simple property (polarization) functions P, being monotonously increasing functions of their arguments and sharing the same structure, shall take a central place in our bounds and estimates for an easy qualitative comparisons between them If one takes c0 = cmax = max{c1, .,cn} (or c0 = cmin = min{c1, .,cn}), then c**  (or c**  ) and can be neglected to strengthen the inequalities in (1), and subsequently one obtains Hashin–Shtrikman bounds Pc (2cmax )  c eff  Pc (2cmin ) (3.4) With (3.5) in hands, one substitutes c0 = cM into (3.7) to obtain c**  c**  Then, the bounds (1) converge to the unique value of the effective conductivity of the model c eff  Pc (2cM ) (3.5) 16 c c 0 ** Then taking c0 = cM, one finds that ** and deduces the polarization approximation (PA) for the effective conductivity of our n-component matrix-inclusion composite (generally in d dimensions) expressed through a property (polarization) function P: c eff  cPA  Pc ((d  1)cM ) (3.6) 3.2 Result Using Ansys softwate mesh model for FEM, and author build a program to calculate by Matlab, show results as graphs 3.2.1 Two-dimensional periodic three-component composites formed We examine two-dimensional periodic three-component composites formed from a continuous matrix phase and two inclusion phases, which have isotropic effective isotropic properties The first one is bodycentered square periodic cell as given in the figure and the second one is body-centered hexagonal microstructure as shown in the figure The diameter of inclusions of each phases is taken such that vI2=vI3 The effective thermal conductivity ceff is computed with theparameters shown in the table The obtained results are reported in the figure 3.a), 3.b), 3.c), 3.d) corresponding the data in the table 1.a) 1.b), 1.c), 1.d) respectively Figure 3.1: body-centered square and hexagonal periodic threecomponent microstructure The thermal conductivity of inclusions and matrix: cM c1 c2 10 17 Figure 3.2: approximations and finite element results for the effective conductivity of the three-phase matrix mixtures 3.2.2 Three-dimensional periodic three-component composites formed Figure 3.3: 3D cubic periodic three-component BCC Figure 3.4: 3D cubic periodic three-component FCC Example of calculation according to the data in the table (a) cM = c1 = c2 = 10 (b) cM = c1 = c2 = 10 (c) cM = c1 = 10 c2 = 18 (d) cM = 10 c1 = c2 = Figure 3.5: Grahp of results for 3D solution 3.2.1 Effective medium approximations for the elastic moduli Building models for elastic problems according to Body-Center Cubic Example (a) KM =  M=2 KI2 =  I2=0.4 KI3 = 20  I3=12 (b) KM =  M=2 KI2 = 20  I2=12 KI3 =  I3=0.4 (c) KM =  M=0.4 KI2 =  I2=2 KI3 = 20  I3=12 (d) KM = 20  M=12 KI2 =  I2=2 KI3 = 10  I3=0.4 19 Fig 3.6: Graph of effective elastic modulu results CHAPTER EQUIVALENT APPROXIMATION 4.1 Equivalent inclusion approach 4.1.1 Dilute solution for equivalent circle inclusions Presume one has particles of certain shapes from a particular component material, and the effective conductivity of a dilute suspension of those randomly oriented particles, having conductivity cα and volume proportion vα (α = 2, ,n) in a matrix of conductivity c1 = cM, has the form c eff  cM  v (c  cM ) D(c , cM ), v  (4.1) 20 In the meantime, the dilute suspension of d-dimensional spherical particles having conductivity c¯α and volume proportion vα in the matrix of the same conductivity cM has the particular expression c eff  cM  v (c  cM ) dcM ,v  c  (d  1)cM (4.2) Equalizing (1) and (2), one finds c  dcM2  (d  1)cM (c  cM ) D(c , cM ) dcM  (c  cM ) D(c , cM ) (4.3) In special case the anisotropic inclusions have the elipse shape, has the particular expression D(c , cM )  cM (c  cM )(1  r ) 2(c  r cM )(r c  cM ) (4.4) and inclusions have ellipsoid (3D) shape D(c , cM )  cM   (4.5) 1     c A  c (1  A ) c B  c (1  B ) c C  c (1  C )    M    M    M   aˆ bˆ cˆ  aˆ bˆ cˆ C      A  aˆ bˆ cˆ dt , B  0 (aˆ  t )(t )   (cˆ   (bˆ  dt ,  t )(t ) dt ,(t )  (aˆ  t )(b  t )(c  t )  t )(t ) 4.1.2 Materials with anisotropic inclusions We consider the two-component 2D square-periodic suspension of anisotropic inclusions having conductivity cI1 and cI2 in a matrix of conductivity cM cI  cM D  cI , , cId , cM  (d  1)  2  D  cI , , cId , cM  (4.6) 4.1.3 Equivalent inclusion approach spherical inclusions (platelet, fibrous) 21 Our spherical equivalent inclusion polarization approximation (SEIPA) for the effective conductivity of the composite would have particular expression: eff cSEIPA ( vI v  M )1  2cM cI  2cM 3cM (4.7) Our platelet equivalent inclusion polarization approximation (PEIPA) for the effective conductivity of the composite would have particular expression: eff cPEIPA ( vI vM  )   2c I 3cI cM  2cI (4.8) Our fibrous equivalent inclusion polarization approximation (FEIPA) for the effective conductivity of the composite would have particular expression: eff cFEIPA ( 2vI / 2vM )1  cI  3cM (4.9)  cI  cM 5cM  cI 4.2 Result 4.2.1 Examples in 2D As the first numerical example, we consider some two-component 2D square periodic suspensions of elliptic inclusions having conductivity cI in a matrix of conductivity cM (Figure 4.1) 22 Figure 4.1: Square periodic cell with elliptic inclusions Figure 4.2: Graphics of the effective conductivity Hashin– Shtrikman upper (lower) bound (a) cM = 1, cI = 10; (b) cM = 10, cI = 4.2.2 Random suspension of elliptic inclusions in a continuous matrix The space of material can be entirely filled by spheres and fibres distributed randomly with dimensions varying to infinitely small such that the inclusion volume proportion can approach Fìgure 4.3: (a) A random elliptic inclusion configuration; equivalent circular inclusion configuration, with the same inclusion volume 23 proportion 4.2.3 Example in 3D Hình 4.4: Cubic periodic suspensions of prolate and oblate spheroid inclusions Figure 4.5: Graphics of the effective conductivity 4.2.4 Anisotropic inclusions As a numerical example we consider the two-component 2D squareperiodic suspension of anisotropic inclusions having conductivity cI1 and cI2 in a matrix of conductivity cM 24 Hình 4.6: The periodic cells of disorderly anisotropic inclusions having circular shape (a) The square cell; (b) The hexagonal cell; (c) The random cell Figure 4.7: The graphics of the Hình 4.8: The graphics of the effective conductivity for square cell effective conductivity for hexagonal cell Hình 4.9: Graphics of the effective conductivity with random cell 25 4.2.5 Experimental (EXP) data Figure 4.10: Spherical equivalent inclusion polarization approximation (SEIPA) Figure 4.11: Platelet equivalent inclusion polarization approximation (PEIPA) Figure 4.12: Fibrous equivalent inclusion polarization approximation (FEIPA) CONCLUSION Main conclusions of the thesis are 1) The author has studied the extended FEM method, to overcome the difficulties of meshing with complex phase geometry (such as random distribution ellipse) to solve the problem of shifting 26 grid system, possibly There is no need to pay attention to the intersection between phases with different conductivity 2) Using Ansys software and building FEM programs with Matlab The author has successfully developed the computing programs according to the eXtended Finite Element Method 3) Construction of the near-interaction approximations for the conductivity and elastic moduli of macroscopically isotropic composites with spherical inclusions based on the polarization bounds of Pham (1995), in which the near-interactions between the particles and the surrounding matrix have been estimated exactly, while the far-interctions between the different particles are approximated The approximations are simplr, always satisfy HS bounds, close to the numerical and experimental results 4) The thesis has constructed equivalent inclusion approximatation Using the available approximations (such as the near interaction or the polarization ones), I determine the inclusion equivalent conductivity based on a comparison between the dilute solutions for the idealistic inclusion and real inclusion composites That would bring the complex problems to the equivalent simple ones for all the ranges of inclusions’ proportions I also extend the approximation to materials with inclusions of spherical, platelet, and fiber forms, using experimental reference at certain volume proportions of the inclusions FURTHER DEVELOPMENTs After this thesis, the author would like to have more time and conditions to continue research on anisotropic materials, as well as to solve more complex inclusion composites, close to the reality I mant to refine the XFEM method in applications to ensure that the results would be reliable in case no experimental results are available Meanwhile, the experimental results are still needed for comparisons, whenever they are available 27 PUBLICATIONS OF THE AUTHOR D.C Pham, A.B Tran, Q.H Do, On the effective medium approximations for the properties of isotropic multicomponent matrix-based composites, International Journal of Engineering Science 68: 75–85, 2013 Q.H Do, A.B Tran, D.C Pham, Equivalent inclusion approach and effective medium approximations for the effective conductivity of isotropic multicomponent materials, Acta Mechanica 227, 387-398 (2016) Trung Kiên Nguyen, Duc Chinh Pham, Quoc Hoang Do, Equivalent inclusion approach and approximations for conductivity of isotropic matrix composites with sphere-like, platelet, and fibrous fillers Journal of Reinforced Plastics and Composites, 2018, Vol 37(14) 968–980 Do Quoc Hoang, Pham Duc Chinh, Tran Anh Binh Equivalentinclusion approach for the conductivity of isotropic matrix composires with anisotropic inclusions Vietnam Journal of Mechanics, VAST, Vol 38, No (2016), pp 239 – 248 Q.H Do, A.B Tran, D.C Pham, Differential and effective medium approximations for conductivity of isotropic two-dimensional multicomponent composites International conference on Suistainable built environment for now and the future Hanoi, 467472, March 2013 Đỗ Quốc Hồng, Trần Anh Bình, Phạm Đức Chính Xấp xỉ phân cực hệ số dẫn vật liệu đẳng hướng nhiều thành phần dạng cốt liệu Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XI, TP Hồ Chí Minh, Tập 1, 479-486, 2013 Đỗ Quốc Hồng, Trần Anh Bình, Phạm Đức Chính Xấp xỉ phân cực cho hệ số dẫn ngang vật liệu đa thành phần cốt sợi dọc trục Hội nghị Cơ học kỹ thuật toàn quốc, TP Hà Nội, tập 2, 205-210, 2014 Đỗ Quốc Hoàng, Trần Anh Bình, Phạm Đức Chính Xấp xỉ tương đương hệ số dẫn vật liệu đẳng hướng có cốt liệu hình dạng phức tạp Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XII, Đại học Duy Tân, TP Đà Nẵng, 2015 A.B Tran, Q.H Do, D.C Pham Equivalent-inclusion approach and effective medium approximations for the effective conductivity of 28 matrix-particulate media Innovations in Construction, Cigos France, pp 01-07, 2015 10 Đỗ Quốc Hoàng, Phạm Đức Chính, Nguyễn Trung Kiên Tiếp cận cốt liệu tương đương xấp xỉ phân cực xác định hệ số dẫn nhiệt vĩ mô vật liệu composite chứa cốt liệu hình cầu Hội nghị Cơ học tồn quốc X, Hà Nội, Quyển 1, 447-452, 2017 ... March 2013 Đỗ Quốc Hồng, Trần Anh Bình, Phạm Đức Chính Xấp xỉ phân cực hệ số dẫn vật liệu đẳng hướng nhiều thành phần dạng cốt liệu Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XI,... Hồng, Phạm Đức Chính, Nguyễn Trung Kiên Tiếp cận cốt liệu tương đương xấp xỉ phân cực xác định hệ số dẫn nhiệt vĩ mơ vật liệu composite chứa cốt liệu hình cầu Hội nghị Cơ học toàn quốc X, Hà... Trần Anh Bình, Phạm Đức Chính Xấp xỉ phân cực cho hệ số dẫn ngang vật liệu đa thành phần cốt sợi dọc trục Hội nghị Cơ học kỹ thuật toàn quốc, TP Hà Nội, tập 2, 205-210, 2014 Đỗ Quốc Hoàng, Trần Anh