THE NUMERICAL AND MATHEMATICAL MODELLING OF ELASTIC PLASTIC TRANSVERSELY ISOTROPIC MATERIALS

Transcription

1 THE NUMERICAL AND MATHEMATICAL MODELLING OF ELASTIC PLASTIC TRANSVERSELY ISOTROPIC MATERIALS Abduvaly.A.Khaldjigitov, Uchqun E.Adambaev, Ravshan Khudazarov Deartment of Comuter Science, National University of Uzbekistan, Vuzgorodok, Tashkent-74, Uzbekistan Deartment of Mathematics, Tashkent State edagogical Institute, Angren, Uzbekistan ABSTRACT A new strain sace and stress sace lasticity theories for a fiber reinforced comosite materials (transversely isotroic) are roosed. On the basis of comarison of exerimental and analytical curves is shown the validity of the offered theories for descrition of lastic deformations of fiber reinforced comosite materials. It is shown that the offered strain sace theory is valid not only for hardening, but also softening materials and is suitable for numerical realization of elastic-lastic roblems in dislacements. For the numerical solution of lasticity roblems initial stress and initial strain and different modifications of iterative methods are alied. New variant of the initial stress method is roosed. D and 3D lasticity roblems on equilibrium of transversely isotroic materials under action of different boundary conditions are solved.. INTRODUCTION Develoment of science and engineering is connected with the wide alication of comosite materials having brightly exressed anisotroic roerties, which cannot be neglected at structure calculation. The strength calculation roblem for such structures becomes more comlicate if a material is nonlinear or elastic-lastic comosite one. In this connection there is an actual roblem of construction of nonlinear relations between a stress and strain tensor for comosite materials. The anisotroic nature of the mechanical resonse of fibre reinforced comosites is now well understood. It is also generally recognized that a comosite consisting of an isotroic matrix reinforced in one or two directions by families of continuous fibres is not just anisotroic but highly anisotroic, with the extensional modulus in the fibredirections being much greate than any other extensional or shear module. Elastic-lastic behavior of fibrous comosite materials has been investigated in many theoretical [3,5, 4,, 7,4,6], exerimental [5,4,30, 3,3] and numerical [9,6,7,0,8] studies in recent years. Most recent develoments in the constitutive theory of the nonlinear resonse of fibrereinforced comosites is based on the micromechanical aroach including the models roosed by Dvorak and Bahei-El-Din [3], Aboudi [5], Vanin [7,4]. Usually comosite material on the basis of any averaging method may be considered as a homogeneous anisotroic material[4,5]. We shall remind, that fibrous and layered comosite materials at averaging may be relaced with transversely isotroic or orthotroic materials [6,4,8]. By Hill[ 9], Bakhvalov [5], Pobedria[4 ] and others are roosed lasticity theories for anisotroic materials. The given work is devoted to the develoment of the adequate modelling equations and effective numerical methods of elastic lastic roblems for transversely isotroic materials. The strain sace and stress sace lasticity theories for transversely isotroic materials are offered. On the basis of comarison of exerimental and theoretical curves are shown the validity of constitutive relationshis for transversely isotroic materials. Exerimental curves [5] received at extension (comression) of samles cut out from fiber reinforced comosites under various angles concerning a direction of fibres are comared with theoretical one and is shown a good agreement. The stain sace theory has a number of theoretical and numerical advantages as against stress sace theory, and allows unequivocally to describe a softening materials. Constitutive relations of strain sace lasticity theory is based on the Ilyushin s [] lasticity ostulate and allows to exress an increment of stress tensor through strain tensor and its increment. Such relationshis is very convenient for the formulation of lasticity roblems, and allows to study softening materials []. Strain sace theory is very convenient for the

2 numerical solution of elastic lastic roblems on FEM, as there is no necessity for calculation of stress tensor at each increment of the external loading forces. In connection with stain sace theory become ossible to modify a initial stress method and to classify other methods from the oint of view of convergence. Numerical test examles on equilibrium of D and 3D lasticity roblems for transversely isotroic rectangle and a aralleleied under various boundary conditions are solved.. STRAIN-SPACE PLASTICITY THEORY FOR TRANSVERSELY ISOTROPIC MATERIALS Study of elastic lastic deformations rocess of fibrous comosite materials can be reduced to the construction of constitutive relationshis for transversely isotroic materials. A constructed in this section a lasticity theory for transversely isotroic materials is based on the Ilyushin s lasticity ostulate []. This theory is valid not only for hardening and ideal lasticity materials, but also for softening ones too. For the construction of constitutive relationshis at first the alternative rule of normality is determined using the Ilyushin s lasticity ostulate. Let us reresent a constitutive relation in the following form[3] where and = C dε () - stress tensor relaxation, = C dε () C - tensor of elastic module, ε -lasticity deformation. There are materials as concrete, rocks, a ground and some comosite materials in a nature,for which the diagram of deformation has a descending art AC. (Fig ). It is known, that existing theories of lasticity do not allow unequivocally to describe the deformed condition of such materials. The diagram given in fig. corresonds to geometrical interretation of relation () in one-dimensional case Fig. Diagram of deformation for softening materials Fig. Geometrical interretations of rucker s and Ilyushin s lasticity ostulates In case of the one-dimensional diagram of deformation, to the work done by external forces in a closed cycle of stress (Drucker s ostulate) corresonds the area of a curvilinear triangle АВС (Fig.) i.e. WD = dε f 0 (3) The work done in a closed A- В-Д-А cycle of deformation (Ilyushin s ostulate) can be found adding the area of the triangle АСД- dε / to W D i. e. Wu = WD + dε = dε (4)

3 Thus, the work done by external forces on Drucker s and Ilyushin s ostulates differ from each other for work of the relaxation tensor in lastic deformations. Notice that, the Ilyushin s lasticity ostulate as against the Drucker s one is valid not only for hardening but also for softening materials. The last exression can be generalized and is written in the following form dε + dε = dε 0 (5) σ ε Let us assume that there exists a loading surface in the strain-sace F( ε, σ, K) = 0 (6) where K is the arameter characterizing the history of loading. From (5) we find the following normality rule F = dλ (7) ε where d λ is the differential arameter. A strain tensor which is invariant concerning a rotations around of axis OX 3 may be resented as the following orthogonal decomosition θ ε = ( δ δδ j3 ) + ε33δδ j3 + + q (8) where θ = ε + ( δδ j3 δ ) + ε33δδ j3 ( εδ j3 + ε3jδ ) (9) q = ε δ + ε δ ε δ δ, (0) 3j 3j 3i 33 j3 θ = ε + ε, ε 33, =, q = qq () In case of transversely isotroy, there are two loading surfaces in the strain sace. They are connected with the quadratic invariants of the strain tensor about rotation round OX 3 axis where tensors and = R (P) 0, Fq = qq R 4 (Q) = 0 () q are determined with the use of equations (9), (0); P, Q are the F 3 = functions characterizing the history of loading. According to the equation (9-0) P and Q have the following forms: P = dp, Q = q dq (3) In case of transversely isotroy we derive from the normality rule (7) F Fq dp = dλ3, dq 4 = dλ, q d σ = dp + dq (4) Differentiating eq.() and taking into consideration eq.(3) we obtain F Fq dλ 3 = H3 d, dλ 4 = H4 dq (5) q where H 3, H 4 are the exerimentally determined functions. In case of transversely isotroy taking into consideration relationshis (4-5) and (8-) one can reresent relationshi () in the following form d σ = d σ ( δ δδ j3 ) + 33δδ j3 + dp + dq, (6) where

4 d s = ( λ + λ 4 ) dθ + λ dε33 33 = λ dθ + dε33 dp = λ 4 d H 3 ( d ) (7) dq = λ 5 dq H 4 ( q dq ) q loading conditions in that case are following F F = 0, d Fq Fq = 0, dq (8) q Note that the relationshis (6)-(8) allows us to investigate not only the monotonically increasing brunch (hardening) but also the decreasing brunch (softening) the deformation diagram. It is known that stress sace lasticity are unaccetable to softening materials because Drucker s ostulate become uncorrected in the decreasing brunch and the loading conditions becomes not unique. Another advantage of strain sace lasticity theory (6)-(8) consists in their indeendence of stress tensor and gives a considerably facilitates in numerical solution of elastic-lastic roblems in dislacements because one can avoids the calculation of the stress tensor increment at every ste of external load. 3. STRESS-SPACE PLASTICITY THEORY FOR TRANSVERSELY ISOTROPIC MATERIALS A transversely isotroic tensor invariant concerning rotations around of axis OX 3 can be written down as the sum of orthogonal terms σ = σ ( δ - δ δ j3 )+ σ33 δ δ j3 + Ρ +Q (9) P = σ + σ δ δ δ + σ δ δ σ δ + σ δ (0) where ( ) ( ) j3 j3 Q = σδ j3 + σ3jδ σ33δδ j3, σ = ( σ + σ ) () Let us accet two yield conditions of Mises tye connected with the quadratic invariants of stress tensor (9) for construction of lastic theory for transversely isotroic materials, i.e. f = PP K3() = 0, fq = QQ K4 (q) = 0 () where K 3, K 4 are the hardening functions; = P d, q = Q d (3) Taking into consideration the associated flow rule of lasticity one can derives from () q d = dλ3, dq 4 P = dλ (4) Q where dλ 3, dλ 4 are the differential arameters of roortionality. Let us reresent dλ 3, dλ 4 in the following form using the consistency condition for loading surface () and equations (3-4): q dλ 3 = h3 dp, dλ 4 = h 4 dq (5) P Q where h 3, h 4 are the scalar multiliers. Reresenting the general increment of deformations as a sum of elastic and lastic arts j3 3j

5 where e dε = C dp, dε = dε e + dε (6) d ε = d + dq (7) C = λδδ + λ4 ( δikδ jl + δ jlδ jk ) + λ3δδ j3δ k3δl3 + ( δ δ δ + δ δ δ ) + λ ( δ δ δ + δ δ δ + δ δ δ + δ δ δ ) (8) + λ j3 k3 l3 5 il k3 j3 jk l3 ik l3 j3 jl k3 multilying relationshi (6) by Pmn C mn and taking into consideration equation (4) one can obtain d λ = ε + 3 h3 Cmnd h 3 C P mn P P (9) q q q d λ = ε + 4 h 4 Cmnd h 4 C Q mn Q Q (30) Then using the Hook s law and equations (6)-(8), (), (4) one can obtain the basic relationshis of lasticity theory for transversely isotroic solids in the stress sace where ( δδ j3 ) + 33δδ j3 + dp dq d σ = d σ δ + (3) = ( λ + λ 4 ) dθ + λ dε33 33 = λdθ + λ3dε33 4λ 4 Pd dp = λ4d P (3) h3 + 4λ4P 4λ 5 Qdq dq = λ5dq Q h 4 + 4λ5Q in that case the loading conditions have the following form f = 0, dp P q f q = 0, dq (33) Q For change to the lastic state it is sufficient to satisfy one of the loading conditions. Fig. 3 shows the comarison of theoretical and exerimental curves obtained in tension of a lane secimens cut out from unidirectional comosite material(glass/eoxy) at different angles to fibers [5]. Theoretical curves were constructed with the use of strain sace( ) and stress sace ( ) theories.

6 Fig.3. Exerimental and analytical stress-strain curves of unidirectional lamina under tension. 4. NUMERICAL METHODS OF PLASTICITY PROBLEMS Usually, for numerical solution of elastic - lastic (isotroic and anisotroic) boundary roblems an elastic solutions method, an initial stress method, an initial strain method, a variable elasticity arameters method and a different iterative methods[9, 0, 6, 7] are used. In connection with the roosed strain sace theory became ossible to modify an initial stress method. Thus the initial stress method converges not only at small hardenings and ideal lasticity, but also for softening materials. The elastic solution method has received its substantiation and develoing in the following works [4,8] and allows to reduce a nonlinear roblem of the theory of lasticity to a sequence of elastic roblems with a variable right art in the equations. The elastic solution method is usually alied to the solution of elastic lastic roblems formulated on the basis of deformation theories.[4] Usually, in a case of flow theories the initial strain method and initial stress method in combination with elastic solution method are used,accounting at each stage of increment of the external force the initial stress and strains accordingly. The initial strain method is based on the associated flow rule and strain is reresented as the sum of elastic and lastic arts. The lastic strain is considered as an initial strain and dislacements is found from the revious aroach. We shall notice, that on each increment of external loading it is necessary to check u loading and flow conditions i.e. it is necessary to kee u the end of the stress "vector" was on a loading surface. In Zienkiewicz work [6] the aroached method for satisfaction of these conditions, with introduction of auxiliary arameter is offered, but this arameter imoses restriction on value of an increment of external loading. On the other hand the convergence of the initial strain method at small hardenings and ideal lasticity is worsened. If a constitutive relation of the flow theory to resolve concerning an stress increment tensor it is ossible to find more convenient relation for the formulation of elastic lastic roblems. The right art of which deends from stress tensor and an increment of strain tensor and it is convenient for alication of initial stress method. In this case initial stress method it is ossible to aly for hardening, ideal lasticity and softening materials. But, solving the elastic lastic roblems in dislacement, necessity of calculation of stress tensor and its increments at continually loading, and restriction on value of an increment of external loading remains. In connection with strain sace lasticity theory it is ossible to offer the modified variant of the initial stress method, without above stated lacks. The right art of constitutive relationshis of strain sace theory deends only from strain tensor and its increment and it is convenient for numerical realization of lasticity roblems in dislacement on FEM. Satisfying the loading and flow conditions also are easy and no required an introduction of additional arameters, as against the stress sace lasticity theory.

7 Elastic lastic boundary roblem consists of the balance equation, constitutive relationshi of lasticity theory, Cauchy relation and corresonding boundary conditions. Constitutive relationshi for lasticity theories may be write down in a such form d σ =C d ε -d σ (34) o where, d σ o - an initial stress. The initial stress of stain sace and stress sace theories for isotroic materials has such forms, corresondingly: Ctu Cq σ o tu σq = dε (35) + Cmnq h σmn σq o F F = H dε (36) ε ε Last relations in case of bilinear stress-strain diagram corresondingly become ' o µ µ = ( s de ) s σu ' o µ µ = ( e de ) e εu The right art of the first equation of (37) deends from stress tensor and an increment of strain tensor, and the second relation deends only from strain tensor and its increment. As it was already soken the deendence of the initial stress from strain tensor and of its increment is convenient in numerical realization of elastic lastic roblems on FEM in dislacement. Thus when a loading surface considered in strain sace, the loading and flow conditions too deends only of strains. The loading and flow conditions concerning strains are satisfied with unequivocal image not only for hardening and ideally lastic materials, but also for softening materials. 5. RESULTS & DISCUSSION In this section, the above stated strain sace and stress sace constitutive equations are used for numerical solution of lasticity roblems on equilibrium of transversely isotroic aralleleieds and rectangulars. The discrete analogy of lasticity roblems for numerical integration are constructed by the variational-difference method. For the solving of obtained discrete non-linear equations corresonding to different constitutive relationshis an initial stress method, an iterative method [5], two-ste iterative method [4] are used. In section 4 also was mentioned, that the use of the strain sace lasticity theory overcomes many difficulties encountered by alication of stress-sace lasticity theories for numerical integration. Test examles on equilibrium of a rectangular and a aralleleied under various boundary conditions are solved. The roblem on equilibrium of a rectangular under action of regular distributed forces on oosite sides of a rectangular is solved. The roblem, when on oosite sides of a aralleleied under action of domes similar forces also is solved. In table are comared the numerical results of elastic-lastic roblems of equilibrium of transversely isotroic aralleleieds solving by aforementioned iterative methods. U(3,4,3) V(3,,) Iterative method 0,0479-0,0080 Two ste iterative method 0,0480-0,008 Initial stress method 0,0479-0,0080

8 P * = 0., q * ' ' = 0., λ 7 =. 6, λ 9 = кг C = 0.39 см The roblem on equilibrium of isotroic rectangular (Fig.5, l =l =, λ =, µ = 0. 5 ) under the regular distributed loading (S=) is solved. The diagram of deformation is aroximated by bilinear function (Fig.4.). In tables -3 is shown the dislacement and stress defined on initial stress method.. External loading was given in two increments. You can see from tables -3 that the given and calculated values σ are aroximately equal. Table. u,v and σ accounting in two increments for hardening material (µ`=0.) IS u ν σ σ u ε u µ I I K II Table 3. u,v and σ accounting in two increments in case of ideal lasticity (µ =0) IS u ν σ σ u ε u µ Y S= σ u µ` = * * * µ` =0 µ ` 0. µ = ε u S=- X Fig.4.Bilinear deformation diagram for hardening ' ' ( µ =0.) and ideal lasticity ( µ =0) cases Fig.5. Equilibrium of rectangular A fiber reinforced comosite materials using any averaging method is relaced by transversely isotroic materials. The strain sace and stress sace lasticity theories for transversely isotroic materials are roosed. On the basis of comarison of exerimental and analytical curves is shown the validity of the offered theories for descrition of lastic deformations of fiber reinforced comosite materials. For the numerical solution of lasticity roblems initial stress and initial strain methods and different modifications of iterative methods are alied. New variant of the initial stress method is roosed. D and 3D lasticity roblems on equilibrium of transversely isotroic materials under action of different boundary conditions are solved.

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