Two-Field Prismatic Finite Element Under Elasto-Plastic Deformation

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1. Introduction When real structures are loaded, stress concentration zones always appear, in which local stresses exceed the yield strength of the material and the physical relationships between stress and strain are nonlinear. Analysis of the stress-strain state of structures in the zones of elasto-plastic deformation of the material is an important engineering problem. The solution of such problems is performed using numerical methods for determining design values. Among numerical methods for determining the strength parameters of structures, the finite element method (FEM) in various formulations has become widespread. The dis- placement-based FEM has been widely used to analyze elasto-plastic deformation of structures [1-4]. In this formulation, the FEM was used in the analysis of thermoplastic and contact problems of continuum mechanics [5-9]. The finite element method has also been used in cases of finite strains in elasto-plastic deformation of bodies of various configurations [10-13]. Elasto-plastic deformation processes of plates and shells have also been investigated using the mixed FEM [14-18]. In this paper, a prismatic finite element in mixed formulation is developed, with strains and stresses as the nodal unknowns. A nonlinear mixed functional was used to obtain the stiffness matrix of the finite element. As physical equations, two versions of the constitutive equations of the theory of plastic flow were used. In the second version of the physical equations, the relations between the strain increments and stress increments obtained based on the hypothesis of proportionality of the components of the incremental strain deviator and the components of the incremental stress deviator were used. 2. Methods 2.1. Linear Geometric Relationships at a Load Step In Cartesian coordinate system x, y, z the components of the incremental strain tensor Δεij are related to the displacement increments Δui by relationships[2] Δεij =(Δuij +ΔUij ); (i,j = 1, 2, 3), (1) or in matrix form { }Δε = [ ]L { }Δv , (2) 6×1 6×3 3×1 where { }Δε Т = {Δε11 Δε22 Δε33 2Δε12 2Δε13 2Δε23} is the row of strain increments;{ }Δv Т = {Δu1 Δu2 Δu3} 1×6 1×3 is the row of displacement increments; [ ]L is the matrix of differential operators. 6×3 2.2. Physical Equations at a Load Step 2.2.1. Physical Equations of Plastic Flow Theory In the theory of plastic flow, it is assumed that the total strain increments are equal to the sum of elastic strain incrementsΔεije and plastic strain increments Δεijp . The increments of elastic strain are determined by the relations of the Hooke’s law1 Δε11e = 1 (Δσ11 - Δν σ22 - Δν σ33); E Δεe22 = 1 (Δσ22 - Δνσ11 - Δνσ33); E Δε33e = 1 (Δσ33 - Δνσ11 - Δνσ22); E 2Δε12e = 2(1+ν) Δσ12; 2Δε13e = 2(1+ν) Δσ13; 2Δεe23 = 2(1+ν) Δσ23, (3) E E E where E is the elastic modulus of the material in tension; νis the Poisson’s ratio. To determine the plastic strain increments, the flow theory uses the hypothesis of proportionality of the components of the incremental plastic strain tensor to the components of the total stress deviator, occurring before the considered load step. According to this hypothesis, Δ Δεijp = εip (σij -δ σij c ), (4) σi where σi is the stress intensity; δij is the Kronecker delta symbol; σc =1(σ11 +σ22 +σ33) is the first 3 invariant of the stress tensor. The value of the plastic strain intensity increment included in (4) is determined by the expression [19] Δεip = Δei - Δεie = Δσi - Δσi , (5) Eх Eн where Δεi is the strain intensity increment; Δεije is the elastic strain intensity increment; Eх is the chord modulus of the stress-strain diagram; Eн is the modulus of the initial region of the stress-strain diagram; Δσi is the stress intensity increment. Considering (5), relationships (4) can be expressed as Δ =Δεmnp σi 1 - 1 1 (σmn -δ σmn c ). (6) E Ec 0 σi In order to express Δεmnp according to (6) by stress increment functions Δσij , Δσi needs to be written in such general form i mn , (7) σmn where σi = 1 (σ σ11 - 22 )2 +(σ22 -σ33 )2 +(σ33 -v11)2 + 6(σ122 +σ132 +σ232 ) 1/2 . By summing (3) and (6) and taking into account (7), the matrix relationship of the constitutive equations of the flow theory is formed {Δ =ε} {С1Р}{Δσ}, (8) 6 1× 6 6× 6 1× where{ }Δε Т = {Δε11Δε22 Δε33 2Δε12 2Δε13 2Δε23};{Δσ}Т = {Δσ11Δσ22 Δσ33 Δσ12 Δσ13 Δσ23}. 1×6 1×6 2.2.2. Version of Physical Equations without Separation of Strain Increments into Elastic and Plastic Parts The hypothesis of proportionality of the components of the incremental strain deviator to the components of the incremental stress deviator was used in obtaining the constitutive equations of the specified version, leading to the following relations Δεmn -δmnΔ =εc 3 Δεi (Δσmn -δmnΔσc ), (9) 2 Δσi Δε where Δσii = E1x ; Δεc = 13(Δε11+Δε22 +Δε33); Δσc = 13(Δσ11+Δσ22 +Δσ33). Using the condition of incompressibility under plastic deformation assumed in the theory of plastic flow, according to which the following relation holds Δεс = 1- 2ν Δσ , (10) Е constitutive equations (9) can be expressed in matrix form { }Δε = [С2Р]{Δσ}. (11) 6×1 6×6 6×1 3. Stiffness Matrix of Prismatic Finite Element A prism with triangular bases is taken as the finite element. Nodal unknowns are displacement increments and stress increments. For integration over the finite element volume, rectangular prism is assumed, the height of which is determined by coordinate -1≤ ζ ≤1, and the bases are right triangles with coordinates 0 ≤ξ, η≤1. Approximation of Cartesian coordinates xi of displacement increments Δvi and stress increments Δσij is performed in terms of the corresponding nodal quantities by linear functions λ = (1-ξ η- )1-ζMξ1-2ζMη1-2ζM{1-ξ η- }1+2ζMξ1+2ζMη1+2ζ { }λ ={ f (ξ1 6×,η)}T { }λ6 1×y , (12) 2 where { }λy T = {λi λj λkλm λn λp} is the row of nodal values of λ ; symbol λ denotes the values of 1×6 xi , Δvi ,Δσij . On the basis of (12), the necessary approximating expressions are written in matrix form {Δ =v} [A]{Δvy}; {Δ =ε} [L] [A] {Δ =vy} [B] {Δvy}; {Δ =σ} [S]{Δσy}, (13) 3 1× 3 18× 18 1× 6 1× 6 3 3 18× × 18 1× 6 18× 18 1× 6 1× 6 36× 36 1× where {Δvy}T = {Δv1iΔv1jΔv1kΔv1mΔv1nΔv1p...v3iΔv3jΔv3kΔv3mΔv3nΔv3p}; 1×18 {Δσy}Т ={Δσ11i Δσi22Δσi33Δσ12i Δσ13i Δσi23... Δσ11p Δσ22p Δσ33p Δσ12p Δσ13p Δσ23p }. 1×36 To form the stiffness matrix of the prismatic finite element, a mixed functional for the considered load step is used [16] Ф=V { }1 6σ× T+ 12{Δ1 6×σ}T [ ]6 3L× { }Δ3 1×v dV -12V {Δ1 6×σ}T G6 6×μp -1{Δ6 1×σ}dV - - S{Δ1 3v}T { }3 1q + 12{Δ3 1×q} dS; (μ =1.2), (14) × × where {Δq} { }, q is the load applied at the considered load step and before the considered step; V is the volume of the finite element; S is the area of load application. Taking into account approximating relations (13), functional (14) will be expressed as Ф≡ Δ{ vу}T [ ]B T { }σ dV + Δ{ σу}T [ ] [ ]S T B dV{Δ -vy} 1 36× V 36 6× 6 1× 1 36× V 36 6 6 18× × 18 1× T S T G S dV v T A T q dS v T [ ]A T { }q dS . (15) 2 1 36× V 36 6× 6 6× 6 36× 36 1× 2 1 18× S 18 3× 3 1× 1 18× S 18 3× 3 1× As a result of minimization of functional (15) with respect to nodal unknowns, two systems of equations are obtained ∂Ф , (16) ∂ Δ{ σy}Т ∂Ф , (17) ∂ Δ{ vy}Т where [ ]Q = [ ]S T [ ]B dV ; [ ]H [ ]S G [ ]S dV; [Δfq]= [ ]A T { }Δq dS ; 36×18 V36×6 6×18 36 36× V 36 6× 6 6× 6 36× 18×1 S18×3 3×1 [ ]R = [ ]A T { }q dS - [ ]B T { }σ dV is the Raphson residual error. 18×1 S18×3 3×1 V18×6 6×1 From system of equations (16), the column of stress increments is determined in terms of displacement increments {Δσy}=[H]-1[Q]{Δvy} . (18) 36 1× 36 36 36 18× × 18 1× Taking into account (18), the stiffness matrix of the finite element is determined from system (17) [K]{Δ = Δ +vу} { fq} {R}, (19) 18 18× 18 1× 18 1× 18 1× where [ ]K = [ ]Q Т [ ]H -1 [ ]Q . 18×36 36×36 36×18 Using (19), the stiffness matrix of the considered structure is formed, by solving which the displacements of all nodes are determined. The values of the stress increments are determined by equations (18). 4. Results and Discussion 4.1. Example of Analysis The stress-strain state of a simply-supported beam (Figure 1) is considered given the following data: l = 200 cm; h = 10 cm; b = 1 cm; Р = 13 kN. The tensile and flexural stress-strain diagrams of the material are shown in Figures 2 and 3. Physical and mechanical parameters are assumed to be the following: Е = 2⋅105 MPa; ν=0.3; σТ =200 MPa; σk =300 МPа; εТ =0.001; εk =0.01; σiТ =200 МPа; σik =300 MPa; = 0.866667 10⋅ -3; εik = 0.866667 10⋅ -2 ; W = bh2/6 = 1×102/6 = 16.66667 cm3. Section 1-1 Figure 1. Simply-supported beam S o u r c e: made by R.Z. Kiseleva Figure 2. Tensile stress-strain diagram Figure 3. Flexural stress-strain diagram S o u r c e: made by R.Z. Kiseleva S o u r c e: made by R.Z. Kiseleva The strain hardening curve is described by the equation σi = aεi2 + bεi + c , where a, b, c are coefficients with the following numerical values: a = - 6612835.53 MPa; b = 242231.48 MPa; c = 1795.033 MPa. The model of the beam is presented in Figure 4 (Nm = 40; Nt = 11 number of nodes along the beam axis and along its height). The results of calculation using the considered versions of the constitutive equations turned out to be virtually identical. Figure 4. Finite element model of the beam S o u r c e: made by R.Z. Kiseleva According to the obtained values of normal stresses, the normal stress diagram in the beam cross-section, which is located at distance h from the middle one, is plotted (Figure 5). Figure 5. Diagram of normal stress σ11 in the beam section S o u r c e: made by R.Z. Kiseleva Equations of statics х=0 for the diagram in Figure 5 are satisfied with a discrepancy of δ ≈ 0.7%. The moment from external forces is equal to Мр =Р l - =h 6.5×(100-10)=585 kN×cm, and 2 2 the moment from the internal forces is equal to Мint = 597.4 kN×cm. The equations of statics have a discrepancy of δ ≈ 0.5% in terms of the equality of moments from external and internal forces. 5. Conclusion Based on the obtained calculation results, it can be concluded: 1. The developed version of the constitutive equations without separation of strain increments into elastic and plastic parts leads to the same results as when using the constitutive equations of the flow theory, where the separation into elastic and plastic parts is adopted. 2. The use of the hypothesis of proportionality of the components of the incremental strain and incremental stress deviators in the developed version of the constitutive equations allows to express the stress increments immediately. In flow theory, the strain increments are first expressed in terms of the stress intensity increments, and only after performing differentiation operations it is possible to obtain the required constitutive equations. The proposed version of the constitutive equations of the plasticity theory is more preferable for use in the design practice of engineering structures.

About the authors

Rumia Z. Kiseleva

Volgоgrad State Agrarian University

Author for correspondence.
Email: rumia1970@yandex.ru
ORCID iD: 0000-0002-3047-5256
SPIN-code: 1948-5390

Candidate of Technical Sciences, Associate Professor of the Department of Applied Geodesy, Environmental Management and Water Use

26 Universitetskiy Prospekt, Volgograd, 400002, Russian Federation

Vitaliy V. Ryabukha

Volgоgrad State Agrarian University

Email: vitalik30090@mail.ru
ORCID iD: 0000-0002-7394-8885
SPIN-code: 9596-2597

Postgraduate student of the Department of Mechanics

26 Universitetskiy Prospekt, Volgograd, 400002, Russian Federation

Natalia A. Kirsanova

Financial University under the Government of the Russian Federation

Email: nagureeve@fa.ru
ORCID iD: 0000-0003-3496-2008
SPIN-code: 8393-5900

Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematics

49 Leningradsky Prospekt, GSP-3, Moscow, 125993, Russian Federation

Yuriy V. Klochkov

Volgоgrad State Agrarian University

Email: klotchkov@bk.ru
ORCID iD: 0000-0002-1027-1811
SPIN-code: 9436-3693

Doctor of Technical Sciences, Professor, Head of the Department of Higher Mathematics

26 Universitetskiy Prospekt, Volgograd, 400002, Russian Federation

Anatoliy P. Nikolaev

Volgоgrad State Agrarian University

Email: anpetr40@yandex.ru
ORCID iD: 0000-0002-7098-5998
SPIN-code: 2653-5484

Doctor of Technical Sciences, Professor of the Department of Mechanics

26 Universitetskiy Prospekt, Volgograd, 400002, Russian Federation

References

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