NONLOCAL PLASTICITY APPLIED TO BENDING OF BEAMS

Transcription

1 IX International Conference on Computational Plasticity COMPLAS IX E. Oñate and D. R. J. Owen (Eds) CIMNE, Barcelona, 2007 NONLOCAL PLASTICITY APPLIED TO BENDING OF BEAMS L. Strömberg Div of Structural Mechanics, LTH, Se Lund, Sweden Key words: bending, size effects, nonlocal plasticity, boundary, brittleness, sagging, crack-tip layer 1 INTRODUCTION Bending of beams is discussed focussing on some properties and phenomena not predicted by conventional beam theory. Some experimental results are covered and in modeling, a nonlocal plasticity formulation is considered. This is able to capture non-uniform fields of strain and stress apparent at boundaries and where material is not homogeneous. 2 FIBRE REINFORCED CONCRETE, FRC FRC consists of a concrete matrix and stronger, more ductile fibres (of metal, glassfibre). In bending, the response after peak load is caused by micro-cracking, in combination with redistribution of stress to fibres. The behaviour for pure concrete matrix would be softening. If sufficiently reinforced, the beam may take higher loads after micro-crack initiation. Pure concrete is brittle in tension, and in compression von-mises-like condition is governing. At many engineering applications, strength properties at the tension side are improved by imposing pre-stress and further reinforcement with a steel wire. Figure 1. Load deflection curves at bending of FRC-beam, for two different reinforcements. Corresponding crack patterns at left, displaying global hardening and softening.

2 Consider a beam of radius R, loaded by a strain field given by ε(r)=ε 0 r/r, where 0 r R. Kinematics give that ε 0 /R is the radius of curvature inversely proportional to second derivative of deflection. Yielding is assumed to occur at r=r s. Applying a nonlocal plasticity model with a generalised expansion 1, give a differential equation in vicinity of plastic boundaries reading (E/h+1)λ(r)+l(r)λ''(r)= E/h ε 0 (r-r s )/R where E is the elastic modulus, h is softening/hardening, r s is the location of elastic-plastic boundary, λ(r) is plastic strain measure, and l(r) is a square-length field. The equation will be solved to obtain normalised bending stiffness, and bounds for softening will be deduced. It will be assumed that the equation holds in the entire plastic region r s r R, and posesses a polynomial solution, given by λ(r)=ar 2. Insertion into Hookes law for r s =0, give the expression for stress versus strain σ σ y =r(b-ar) where due to kinematic assumption of strain above b=eε 0 /R -Eε y /R, σ σ y =E(ε 0 ε y )r/r- ar 2 Subsequently, we consider rate properties denoted, t and hereby σ, t =E ε 0, t (r/r)-a,t r 2 Integrated properties from yield to present value will be denoted σ =σ σ y σ,t is assumed to have maximum inside the beam, giving the condition r max =b,t /(2a,t )<R, and thus a,t >Eε 0, t /(2R 2 ). To compare overall stiffness for different radius, the rate in bending moment (for upper half) is calculated and reads M,t =π(-a,t R 5 /5+b,t R 4 /4). Comparing with the local bending moment at initial yield M loc, y =π(σ y /R)R 4 /4 gives M,t / M loc, y =-a,t (4/5)R 2 /σ y +Eε 0, t /σ y (1) Integration gives the entire normalised moment when ε 0 >0 (M loc, y + M)/ M lin, y =-a(4/5)r 2 /σ y +Eε 1 /σ y (2) where ε 1 = ε 0 + ε y is the entire strain, and Eε y =σ y. To derive an explicit expression for a stiffness modulus S b such that M,t =S b ε 0,t the constant a,t should be expressed as a fraction of ε 0,t (and this will be done in the next section). At the limit a,t R 2 >5b,t /4 Eε 0,t, modulus will be negative, as seen in (1). Thus, for sufficiently large a,t and/or beam radius, the overall modulus is negative, resulting in global softening, 3 SIZE EFFECT IN THIN NI-BEAMS For thin nickel beams, experiments reported by 2 display an increase in normalised bending strength by a factor 3, when the wire diameter decreases from 100 to 12.5µm. The performance in simple tension was not affected, which lead to the conclusion that the size effects were caused by non-homogeneous strain, where gradients are significant. Due to its general format, the solution to the above differential equation will be exploited, also in this application. Insertion in expression for normalised bending moment (2), gives 3(-a(4/5)R 2 1 +Eε 1 )= -a(4/5)r 2 2 +Eε 1, where R 1 and R 2 are the radius 50µm and 6.25µm. When contribution from the smaller radius is neglected a is determined from a(4/5)r 2 1 =(2/3)Eε 1. Insertion into (1) give the normalised rate of bending moment 2

3 M,t /M lin,y =ε 0,t /ε y (1-2/3(R 2 /R 1 2 )), where it is seen that softening occurs only for larger radius than R 1, and thus may be out of bounds for model. Knowledge of a, admits calculation of λ surface =ar 2, and λ max =λ(r max ). If these quantities may be measured, there are options of further verification/calibration, eg. determination of σ y or E. 4 RECTANGULAR ALUMINIUM HOLLOW SECTIONS Forming of an aluminium hollow section with plastic bending, will be discussed. It is found that the lower flange may display buckling before plasticity 4. In a semi-simple model, it is assumed that a plastic stress- and strain- distribution act at the upper part, and a compressive force at the lower part. For pure bending, the resultant of stress equals the force. Results will be derived, and the implication, the harder the more brittle, is concluded. Bending of the section As in the previous examples, the kinematical assumption of strain rate is ε,t =w,t r, where w is curvature and r is the coordinate in cross-section. At the corners will be increased stiffness due to geometry, and more material. After plastic initiation, a layer of increased stress is assumed at the outer boundaries 3 (located towards the corners). Such a distribution may be solved from the differential-equation in Section 2. Layer width is denoted by l w, layer areas are l w t where t is thickness, and the stress rate in layer is assumed to be proportional to maximum strain rate (i.e strain rate at outer boundaries). For the remaining upper section, stress is determined from local plasticity. Hereby, the rate of bending moment may be written as the sum of a linear local distribution and the contribution from two upper stress layers, reading M,t =hi w,t (1+s l w /a), where h is the local plastic tangent modulus, I=16a 3 t in thin-walled cross-section, t is thickness, a is side length and s is a constant (of proportionality to stress raise in layer). To consider size effect, a thinner cross-section will be defined as t=τt o, a=αa o. Comparing with moment at initial yield M y =EI w y, where w y is the curvature at initial yield and assuming that l w depend on thickness l w =pt, (p is a constant), the thinner the harder is achieved for τ/α>1. For the section to be considered thinner, is at least required α<1 (3). The Euler buckling force for the lower compressed side reads, P k = π 2 EI k /L, I k =(2a)t 3 /12. Hereby for the thinner section P k (thinner)= ατ 3 I k,o. It is seen that P k decreases, ie. the response is more brittle, for ατ 3 <1 which due to (3) reduces to τ 3 =1, or 1<=τ 3 <1/α (if the socalled thinner section may have a larger thickness). In conclusion, for stress concentrations in layers, thinner the harder, implies harder the more brittle. Other measures of brittleness eg. P k /(2at), give other conditions. 4.1 Sagging Another phenomena that could emanate from the stiff corners, is a flange distortion of the upper side, known as sagging 4. It is found that an internal pressure reduces the effect. Here, only the qualitative experimental results from 4 will be used, and not the modeling. To model sagging, we assume that the stiff corners act as a distributed load for the beam 3

4 while bending. The vertical stress in cross-section, σ z is then determined from equilibrium for a plate with normal stress σ x from beam theory and boundary condition σ z (0)=p, where p is internal pressure. Hereby σ z may be solved (from a set of differential equations) to read σ z =-p+qz 3 /(6I), where q is distributed load and z is the vertical coordinate of cross-section. The value of q is not specified, and depends of bending radius, stiffness and thickness. Typical level 4 of the balancing pressure would be of the order p=3 bar for a=30mm, t=3mm, however in 4, section is rectangular 75x35mm. Remarks: In the above derivation, the framework of nonlocal plasticity was not used, merely equations of equilibrium. The result may also be used for non-hollow sections, where a heterogeneous material distribution makes the corners more dense and stiff. Such examples are laminae, and wood beams. For wood, since the strength perpendicular to grain fibre is low, results are important, and reduced corners, often used for square wood beams could be a design that decreases the effect. 4 SPECIMEN WITH INITIAL CRACK Conditions at a crack tip are governed by microstructural properties, geometry and external loading. In the well-known LEFM-approach, the stress field will have a singularity, and detailed modeling of plasticity is not considered. Here, a layer model will be used to achieve a bounded stress field, close to tip. The location of boundary layer is determined from an iterative boundary condition 1, that may result in multiple solutions. The change in layer location, may be regarded as a crack growth criterion. In brief, the BC for plastic strain at the vicinity of crack-tip reads 1 λ (x N )=(4/3)/x b (λ(x N )-λ(x N+1 )) and λ (x N+1 )= Α(4/3)/x b (λ(x N+1 )-λ 0 ), where x N is distance from the tip, x b is initial layer width, A is a parameter. When λ is a quadratic function as in the example above and λ 0 =0, the layer location is given by x n+1 =f(x n )=A x n (1-(2/3)x n ), where x n denotes x N /x b. Figure 2. Iteration graph for fix points to x n+1 =f(x n )=A x n (1-(2/3)x n ). Horizontal axis, increasing A, and vertical axis, the fixpoints. First bifurcations for A>3, and chaos at A=

5 Fix points to the iteration formula, as a function of A, constitutes a fractal as shown in the figure. For smaller A, the solution is unique, while for increasing A, bifurcations occur. The BC assumes an extension of boundary at micro-scale. Fix points (ie. solutions) to the iteration formula, correspond to coalescence, and thus elimination of points, such that fine structure at micro-scale is homogenised and information is exported to one point in the continuum. To further relate to micro-structural approaches, the format of BC 1 above, may be compared to the finite difference formula used to approximate gradients for a RVE 5. 5 CONCLUSION Due to its general format, the differential equation in section 2, has applications in many areas. Since there is a non-determinancy, conditions need to be added, to specify a certain example. Concepts of softening, size effects, hardening and brittleness were described for various applications. To model sagging, a structural model was proposed and to model crack growth, a layer model in conjunction with an iterative boundary condition 1, were discussed. 6 ACKNOWLEDGEMENTS To Evas S, Nybro Crystal, for valuable discussions, FyMaData who partly granted the report, and Swetech AB, for providing the figure of concrete. References [1] Strömberg L. Boundary layers in nonlocal plasticity. (2006) pp Proc. 9 th Int Conf on Material forming, Esaform, UK. [2] Stölken and Evans (1998). A microbend test method for measuring the plasticity length scale. Acta Mater., 46, [3] Engelen R.A.B. Fleck N.A., Peerlings R, Geers M, Int J of Solids and Structures 43, p.1865, ( ). An evaluation of higher-order plasticity theories for predicting size effects and localisation. (2006). [4] Moe, Baringbing, Lademo, Storen, Welo, (2006), A study of pure bending of hollow extrusions with internal pressure. pp Proc. 9 th Int Conf on Material forming, Esaform, UK. [5] Gao, H., and Huang, Y. (2001) Taylor-based nonl th of plast. Int. J. Sol Struct., 38, , as quoted by Bazant and Jirasek (2002). Nonl Integral Form of Plast and Damage: Survey of Progress. J of Engrg Mech, 128, No.11, p Parts 1-5 published in proceedings to: IX International Conference on Computational Plasticity COMPLAS IX E. Oñate and D. R. J. Owen (Eds) CIMNE, Barcelona,