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RADIOSS THEORY Version 14.0MATERIALSwhere the subscripts denote the orthotropy axes. As the matrix C is symmetric:υ12E11 υ 21EQ. 9.2.1.2E22Therefore, six independent material properties are the input of the material:E 11 Young's modulus in direction 1E 22 Young's modulus in direction 2υ 12 Poisson's ratioG 12 , G 23 , G 31 Shear moduli for each directionThe coordinates of a global vectorV is used to define direction 1 of the local coordinate system of orthotropy.The angle Φ is the angle between the local direction 1 (fiber direction) and the projection of the global vectoras shown in Figure 9.2.1.VFigure 9.2.1 Fiber Direction OrientationThe shell normal defines the positive direction for Φ . Since fabrics have different compression and tensionbehavior, an elastic modulus reduction factor, RE, is defined that changes the elastic properties of compression.The formulation for the fabric law has a σ 11 reduction if σ 11 0 as shown in Figure 9.2.2.Figure 9.2.2 Elastic Compression Modulus Reduction30-July-201510

RADIOSS THEORY Version 14.0MATERIALS9.2.1.2 Fabric nonlinear law for elastic anisotropic shells (law 58)This law is used with RADIOSS standard shell elements and anisotropic layered property (type 16). The fiberdirections (warp and weft) define the local axes of anisotropy. Material characteristics are determinedindependently in these axes. Fibers are nonlinear elastic and follow the equation:σ Eε Bε 2 , withdσ 0dεEQ. 9.2.1.3The shear in fabric material is only supposed to be function of the angle between current fiber directions (axes ofanisotropy):τ G0 tan(α ) τ 0ifτ G tan(α ) G A τ 0ifα αTEQ. 9.2.1.4α αTandG A (G0 G ) tan(αT ) ,G GT1 tan 2 (α T )withτ 0 G0 tan(α 0 )αT , and G0 is a shear modulus at αG0 0, the default value is calculated to avoid shear modulus discontinuity at αT : G0 G.WhereαTis a shear lock angle, GT is a tangent shear modulus at 0. IfFigure 9.2.4 Elastic Compression Modulus Reductionweftαwarpφα0is an initial angle between fibers defined in the shell property (type 16).The warp and weft fiber are coupled in tension and uncoupled in compression. But there is no discontinuitybetween tension and compression. In compression only fiber bending generates global stresses. Figure 9.2.5illustrates the mechanical behavior of the structure.30-July-201511

RADIOSS THEORY Version 14.0MATERIALSFigure 9.2.5 Local frame definitionWarp traction (free in weft direction)Warp compression (no traction in weft)A local micro model describes the material behavior (Figure 9.2.6). This model represents just ¼ of a warp fiberwave length and ¼ of the weft one. Each fiber is described as a nonlinear beam and the two fibers are connectedwith a contacting spring. These local nonlinear equations are solved with Newton iterations at membraneintegration point.Figure 9.2.6 Local frame definitionWeftWarpTraction(fiber coupling)Compression(no coupling)9.2.2 Nonlinear pseudo-plastic orthotropic solids (laws 28, 50 and68)9.2.2.1 Conventional nonlinear pseudo-plastic orthotropic solids (laws 28and 50)These laws are generally used to model honeycomb material structures as crushable foams. The microscopicbehavior of this kind of materials can be considered as a system of three independent orthogonal springs. Thenonlinear behavior in orthogonal directions can then be determined by experimental tests. The behavior curves areinjected directly in the definition of law. Therefore, the physical behavior of the material can be obtained by asimple law. However, the microscopic elasto-plastic behavior of a material point cannot be represented bydecoupled unidirectional curves. This is the major drawback of the constitutive laws based on this approach. Thecell direction is defined for each element by a local frame in the orthotropic solid property. If no property set isgiven, the global frame is used.30-July-201512

RADIOSS THEORY Version 14.0MATERIALSFigure 9.2.7 Local frame definitionThe Hooke matrix defining the relation between the stress and strain tensors is diagonal, as there is no Poisson'seffect: σ 11 E11 σ 0 22 σ 33 0 σ 12 0 σ 23 0 σ 31 00E22000E330000000000G1200G230000 ε11 ε 22 ε 33 0 ε12 0 ε 23 G31 ε 31 EQ. 9.2.2.1E112EQ. 9.2.2.2000An isotropic material may be obtained if:E11 E22 E33 and G12 G23 G31 Plasticity may be defined by a volumic strain or strain dependent yield curve (Figure 9.2.8). The input yield stressfunction is always positive. If the material undergoes plastic deformation, its behavior is always orthotropic, as allcurves are independent to each other.Figure 9.2.8 Honeycomb typical constitutive curveThe failure plastic strain may be input for each direction. If the failure plastic strain is reached in one direction,the element is deleted. The material law may include strain rate effects (law 50) or may not (law 28).30-July-201513

RADIOSS THEORY Version 14.0MATERIALS9.2.2.2 Cosserat medium for nonlinear pseudo-plastic orthotropic solids(law 68)Conventional continuum mechanics approaches cannot incorporate any material component length scale.However, a number of important length scales as grains, particles, fibers, and cellular structures must be taken intoaccount in a realistic model of some kinds of materials. To this end, the study of a microstructure material havingtranslational and rotational degrees-of-freedom is underlying. The idea of introducing couple stresses in thecontinuum modelling of solids is known as Cosserat theory which returns back to the works of brothers Cosseratin the beginning of 20th century [110]. A recent renewal of Cosserat mechanics is presented in several works ofForest et al [111], [112], [113], and [114]. A short summary of these publications is presented in this section.Cosserat effects can arise only if the material is subjected to non-homogeneous straining conditions. A Cosseratmedium is a continuous collection of particles that behave like rigid bodies. It is assumed that the transfer of theinteraction between two volume elements through surface element dS occurs not only by means of a traction andshear forces, but also by moment vector as shown in Figure 9.2.9.Figure 9.2.9 Equilibrium of Cosserat volume elementµ32σ 22 σ12σ 21µ3121σ 11L σ 11µ31Surface forces and couples are then represented by the generally non-symmetrical force-stress and couple-stresstensors σ ij and µij (units MPA and MPa-m):ti σ ij n j ;mi µ ij n jEQ. 9.2.2.3The force and couple stress tensors must satisfy the equilibrium of momentums:σ ij , j f i ρuɺɺiµij , j ε iklσ kl ci IφɺɺiWhere, f i are the volume forces, ci volume couples,the signature of the perturbation (i,k,l).EQ. 9.2.2.4ρ mass density, I the isotropic rotational inertia and ε iklIn the often used couple-stress, the Cosserat micro-rotation is constrained to follow the material rotation given bythe skew-symmetric part of the deformation gradient:12φi ε ijk u j ,kEQ. 9.2.2.5The associated torsion-curvature and couple stress tensors are then traceless. If a Timoshenko beam is regarded asa one-dimensional Cosserat medium, constraint EQ. 9.2.2.5 is then the counterpart of the Euler-Bernoulliconditions.30-July-201514

RADIOSS THEORY Version 14.0MATERIALSThe resolution of the previous boundary value problem requires constitutive relations linking the deformation andtorsion-curvature tensors to the force- and couple-stresses. In the case of linear isotropic elasticity, we have:σ ij λekk δ ij 2 µ eijSymm. 2 µc eijSkew Symm.EQ. 9.2.2.6µij α k kkδ ij 2β κ ijSymm. 2γκ ijSkew Symm.WhereeijSymm. and eijSkewSymm. are respectively the symmetric and skew-symmetric part of the Cosseratdeformation tensor. Four additional elasticity moduli appear in addition to the classical Lamé constants.Cosserat elastoplasticity theory is also well-established. von Mises classical plasticity can be extended tomicropolar continua in a straightforward manner. The yield criterion depends on both force- and couple-stresses:f (σ , µ ) 3(a1sij sij a2 sij s ji b1µij µij b2 µij µ ji ) R2EQ. 9.2.2.7Where, s denotes the stress deviator and ai, and bi are the material constants.Cosserat continuum theory can be applied to several classes of materials with microstructures as honeycombs,liquid crystals, rocks and granular media, cellular solids and dislocated crystals.9.2.3 Hill’s Law for Orthotropic Plastic ShellsHill’s law models an anisotropic yield behavior. It can be considered as a generalization of von Mises yield criteriafor anisotropic yield behavior. The yield surface defined by Hill can be written in a general form:2F (σ 22 σ 33 ) G (σ 33 σ 11 ) H (σ 11 σ 22 ) 2 Lσ 23 2Mσ 312 2 Nσ 122 1 0222EQ. 9.2.3.1Where, the coefficients F, G, H, L, M and N are the constants obtained by the material tests in different orientations.The stress components σ ij are expressed in the Cartesian reference parallel to the three planes of anisotropy.EQ.9.2.3.1 is equivalent to von Mises yield criteria if the material is isotropic.In a general case, the loading direction is not the orthotropic direction. In addition, we are concerned with the planestress assumption for shell structures. In planar anisotropy, the anisotropy is characterized by different strengthsin different directions in the plane of the sheet. The plane stress assumption will enable to simplify EQ. 9.2.3.1,and write the expression of equivalent stress σ eq as:σ eq A1σ 112 A2σ 22 2 A3σ 11σ 22 A12σ 12 2EQ. 9.2.3.2The coefficients Ai are determined using Lankford’s anisotropy parameter rα :R r00 2r45 r90;4 1 A2 H 1 ; r90 30-July-2015H R;1 RA3 2 H ; 1 A1 H 1 r00 EQ. 9.2.3.3 11 A12 2 H (r45 0.5) r00 r90 15

RADIOSS THEORY Version 14.0MATERIALSWhere the Lankford’s anisotropy parameters rα are determined by performing a simple tension test at angleto orthotropic direction 1:dεrα α π2dε 33H (2 N F G 4 H )Sin 2α Cos 2α F Sin 2α G Cos 2αThe equivalent stressstrain rateεɺσ eqis compared to the yield stressσyαEQ. 9.2.3.4which varies in function of plastic strainεpand the(law 32):σ y a (ε 0 ε p )n . max(εɺ, εɺ0 )mEQ. 9.2.3.5Therefore, the elastic limit is obtained by:σ 0 a (ε 0 )n .(εɺ0 )mEQ. 9.2.3.6The yield stress variation is shown in Figure 9.2.10.Figure 9.2.10 Yield stress variationThe strain rates are defined at integration points. The maximum value is taken into account: dε dε dεd max x , y ,2 (ε xy ) dtdt dt dt EQ. 9.2.3.7In RADIOSS, it is also possible to introduce the yield stress variation by a user-defined function (law 43). Then,several curves are defined to take into account the strain rate effect.It should be noted that as Hill’s law is an orthotropic law, it must be used for elements with orthotropy propertiesas Type 9 and Type 10 in RADIOSS.9.2.3.1 Anistropic Hill Material Law with MMC Fracture Model(Law 72)This material law uses an anistropic Hill yield function along with an associated flow rule. A simple isotropichardening model is used coupled with a modified Mohr fracture criteria. The yield condition is written asφ (σ ,σ y ) σ Hill σ y 030-July-201516

RADIOSS THEORY Version 14.0σ HillWhere,MATERIALSis the Equivalent Hill stress given as: For 3D model (Solid) For Shellσ Hill F (σ yy σ zz )2 G (σ zz σ xx )2 H (σ xx σ yy )2 2 Lσ yz2 2 Mσ zx2 2 Nσ xy2σ hill Fσ yy 2 Gσ xx 2 H (σ xx σ yy )2 2 Nσ xy2Where, F, G, H, N, M, and L are Six Hill anisotropic parameters.For the yield surface a modified swift law is employed to describe the isotropic hardening in the application ofthe plasticity models :σ y σ 0y (ε 0p ε p )nwithσ y0 is the initial yield stress, ε 0pis the initial equivalent plastic strain,εpis the equivalent plastic strainand n is a material constant.Modified Mohr fracture criteria:A damage accumulation is computed as:εpD ε0Where, dε pf(θ ,η )ε f is a plastic strain fracture for the modified Mohr fracture criteria is given by :Anisotropic 3D model σ 0εf y c2 3θπθπ1θπ 1 c ()ηc 1 csec() 1cos()csin() 3 3163636 2 3 21 1nwith: θ 1 2 arccos ξπ 27 J 3 ξ 32 σ vmises 1 3 (σ xx σ yy σ zz ) η σ vmises Where, J 3 is the third invariant of the deviatoric stress. 2D Anisotropic Model σ 0 1 c 2 f y1εf f 3 f1 c1 η 2 33 c 2 30-July-2015 1n17

RADIOSS THEORY Version 14.0MATERIALSWith: 1 27 2 1 f1 cos arcsin η η 3 2 3 1 27 2 1 f 2 sin arcsin η η 3 2 3 3(1 c3 ) 1 1 f 3 c3 2 3 f1 Where, C1 , C 2 and C3 are parameters for MMC fracture model.The fracture initiates when D 1.In order to represent realistic process of an element, a softening function β is introduced toreduce the deformation resistance. The yield surface is modified as:σ y β σ y0 (ε 0p ε p )n D D with β c Dc 1 mWhere, Dc is Critical damage.We have crack propagation when 1 D DC in this case 0 β 1 is considered to reduce theyield surface otherwise the β 1.The element is deleted if D Dc .9.2.4 Elastic-Plastic Orthotropic Composite ShellsTwo kinds of composite shells may be considered in the modeling: Composite shells with isotropic layers Composite shells with at least one orthotropic layerThe first case can be modeled by an isotropic material where the composite property is defined in element propertydefinition as explained in Chapter 5. However, in the case of composite shell with orthotropic layers the definitionof a convenient material law is needed. Two dedicated material laws for composite orthotropic shells exist inRADIOSS: Material law COMPSH (25) with orthotropic elasticity, two plasticity models and brittle tensile failure, Material law CHANG (15) with orthotropic elasticity, fully coupled plasticity and failure models.These laws are described here. The description of elastic-plastic orthotropic composite laws for solids is presentedin the next section.9.2.4.1 Tensile behaviorThe tensile behavior is shown in Figure 9.2.12. The behavior starts with an elastic phase. Then, reached to theyield state, the material may undergo an elastic-plastic work hardening with anisotropic Tsai-Wu yield criteria. Itis possible to take into account the material damage. The failure can occur in the elastic stage or after plastification.It is started by a damage phase then conducted by the formation of a crack. The maximum damage factor willallow these two phases to separate. The unloading can happen during the elastic, elastic-plastic or damage phase.The damage factor d varies during deformation as in the case of isotropic material laws (law 27). However, three30-July-201518

RADIOSS THEORY Version 14.0damage factors are computed; two damage factorsdelamination:MATERIALSd1 and d 2 for orthotropy directions and the other d 3 forσ 11 E11 (1 d1 )ε11σ 22 E22 (1 d 2 )ε 22EQ. 9.2.4.1σ 12 G12 (1 d1 )(1 d 2 )γ 12Figure 9.2.11 Shear strainwhere d1 and d 2 are the tensile damages factors. The damage and failure behavior is defined by introduction ofthe following input parameters:ε t1 Tensile failure strain in direction 1ε m1 Maximum strain in direction 1ε t 2 Tensile failure strain in direction 2ε m2 Maximum strain in direction 2d max Maximum damage (residual stiffness after failure)30-July-201519

RADIOSS THEORY Version 14.0MATERIALSFigure 9.2.12 Tensile behavior of composite shells(1-dmax) E9.2.4.2 DelaminationThe delamination equations are:σ 31 G31 (1 d 3 )γ 31EQ. 9.2.4.2σ 23 G23 (1 d 3 )γ 23EQ. 9.2.4.3where d 3 is the delamination damage factor. The damage evolution law is linear with respect to the shear strain.Letγ γ 312 γ 232then:ford3 0γ γ iniford3 1γ γ maxEQ. 9.2.4.49.2.4.3 Plastic behaviorThe plasticity model is based on the Tsai-Wu criterion, which enable to model the yield and failure phases. Thecriteria are given by [57]:F (σ ) F1σ 1 F2σ 2 F11σ 12 F22σ 22 F44σ 122 2 F12σ 1σ 230-July-2015EQ. 9.2.4.520

RADIOSS THEORY Version 14.0WhereF1 F11 F44 1σ c1y1σt1y1σ σc1yt1y1σ σc12 yt12 y;F2 MATERIALS1σc2y 1σ 2t y1;F22 ;F12 σ σ 2t yc2yα2F11 F22Where, α is the reduction factor. The six other parameters are the yield stresses in tension and compression forthe orthotropy directions which can be obtained uniaxial loading tests:σ1yt Tension in direction 1 of orthotropyσ 2 y t Tension in direction 2 of orthotropyσ1y c Compression in direction 1 of orthotropyσ 2 y c Compression in direction 2 of orthotropyσ 12 y c Compression in direction 12 of orthotropyσ 12 y t Tension in direction 12 of orthotropyThe Tsai-Wu criteria are used to determine the material behavior: F (σ ) 1: elastic state F (σ ) 1: plastic admissible state F (σ ) 1: plastically inadmissible stressesEQ. 9.2.4.6( )For F σ 1 the cross-sections of Tsai-Wu function with the planes of stresses in orthotropic directions is shownin Figure 9.2.13.30-July-201521

RADIOSS THEORY Version 14.0MATERIALSFigure 9.2.13 Cross-sections of Tsai-Wu yield surface forF (σ ) 1σ2σ 2t yσ 12t yσ1σ 1cyσ 1t yσ 12c yσ 2c yσ 12F (σ ) 1 , the stresses must be projected on the yield surface to satisfy the flow rule. F (σ ) is compared to amaximum value F (W p ) varying in function of the plastic work W p during work hardening phase:IfF (σ ) F (W p ) 1 bWpnEQ. 9.2.4.7Where, b is the hardening parameter and n is the hardening exponent.Therefore, the plasticity hardening is isotropic as illustrated in Figure 9.2.14.30-July-201522

RADIOSS THEORY Version 14.0MATERIALSFigure 9.2.14 Isotropic plasticity hardening9.2.4.4 Failure behaviorThe Tsai-Wu flow surface is also used to estimate the material rupture by means of two variables:W pmax , plastic work limit maximum value of yield function Fmax .If one of the two conditions is satisfied, the material is ruptured. The evolution of yield surface during workhardening of the material is shown in Figure 9.2.15.Figure 9.2.15 Evolution of Tsai-Wu yield surface30-July-201523

RADIOSS THEORY Version 14.0MATERIALSThe model will allow the simulation of the brittle failure by formation of cracks. The cracks can either be orientedparallel or perpendicular to the orthotropic reference frame (or fiber direction), as shown in Figure 9.2.16. Formaxplastic failure, if the plastic work WP is larger than the maxim

RADIOSS THEORY MANUAL 14.0 version - July 2015 Large Displacement Finite Element Analysis PART 2 Altair Engineering, Inc., World Headquarters: 1820 E. Big Beaver Rd., Troy MI 48083-2031 USA Phone: 1.248.614.2400 Fax: 1.248.614.2411 www.altair.com info@altair.com