Microplane Model formulation ANSYS, Inc. All rights reserved. 1 ANSYS, Inc. Proprietary

Table of Content Engineering relevance Theory Material model input in ANSYS Difference with current concrete model Test cases Q & A Appendix 2010 ANSYS, Inc. All rights reserved. 2 ANSYS, Inc. Proprietary

Engineering Relevance The roplane model is well suited for simulating engineering materials consisting of various aggregate compositions with differing properties (for example, concrete modeling, in which rock and sand are embedded in a weak matrix of cements). This material model can be used to include Damage in the form of directional-dependent stiffness degradation on individual potential failure planes, leading to a macroscopic damage. This is unique ability of this material model and currently no other model in ANSYS can do it. Thus with the help of this model strain-softening effect (Fig.1) can be very effectively captured which has eluded the ANSYS users in the past. Figure1. Strain-softening effect due to gradual damage of the material 2010 ANSYS, Inc. All rights reserved. 4 ANSYS, Inc. Proprietary

Theory The modeling of anisotropic damage leads to complex 3D material formulations that includes higher order damage variables which can be very difficult to handle in an efficient way. Microplane model provides an alternative and efficient approach to model anisotropic damage without the complications involved with macroscopic formulations. The roplane model can be summarized in three main steps: 1) Projection Step: Define kinematic constraint to relate the macroscopic strain tensor to their roplane counterparts. Thus the roplane strains can be derived as projections of the overall strain tensor ε. 2) Microlevel material model: Second step is to define constitutive law on the roplane level. This will be discuss in detail in later slides. 3) Homogenization step: This step relates to the homogenization process on the material point level to derive the overall response. This step is based on the principle of energy equivalence which thermodynaally consistent. The assumption is that a roscopic free energy on the roplane level ψ exists and the integral of ψ over all roplane is equivalent to a macroscopic free Helmholtz energy ψ mac. mac 3 4 d 2010 ANSYS, Inc. All rights reserved. 6 ANSYS, Inc. Proprietary

Theory The strains and stresses at roplanes are additively decomposed into volumetric and deviatoric parts, respectively, based on the volumetric-deviatoric (V-D) split. ε V = V: ε ε D =Dev: ε V and Dev are the projection tensors which is used to project macro strains on roplanes. For each roplane we can divide the free energy in its volumetric and deviatoric parts. Ψ (ε V,ε D ) = Ψ vol(ε V ) + Ψ dev(ε D )= ½ K ε 2 V + G ε D : ε D K and G represents roscopic bulk and shear modulus. We can define roplane stresses as the derivatives of the roscopic free energy for the associated strain comp. v : K v, D : 2G D v D Similarly, macroscopic stress σ can be obtained as thermodynaally conjugate variables to the macroscopic strains ε : mac mac 3 3 T Using the free energy assumption we get: d K Vv 2G.Dev. D d 4 4 The integrals of the macroscopic strain equation and the derived stresses equation are solved via numerical integration: N 3 P 3 i i (.) d (. ).w 4 4 N1 2010 ANSYS, Inc. All rights reserved. 7 ANSYS, Inc. Proprietary

Theory To do this numerical integration Discretization of the sphere is required. This process results in the approximation of rosphere with many roplanes. In ANSYS forty two roplanes are used for the numerical integration (Fig.2). Due to symmetry of the roplanes, 21 roplanes are considered and summarized. In this case the damage roplane free energy is defined as: Ψ (ε V,ε D,d ) = ½ (1-d ).K ε 2 V + (1-d ).G ε D : ε D d is the normalized damage variable (0 d 1) We can define roplane stresses as the derivatives of the roscopic free energy for the associated strain comp. v : (1 d ) K v, D : 2(1 d ) G v The state of damage can be characterize by a loading function of the form: D Figure.2 Sphere Discretization by 21 Symmetric Microplanes ( ) d 0; In the above relations η is called equivalent strain energy which characterizes the damage evolution law. It is generally define in terms of strain invariants. Macroscopic stress σ can be obtained as thermodynaally conjugate variables to the macroscopic strains ε : mac 4 3 d 4 [1 d 2010 ANSYS, Inc. All rights reserved. 8 ANSYS, Inc. Proprietary D K V V 2G.Dev.Devd : 3 T ]

Material Model Input in ANSYS In ANSYS equivalent strain energy (η ) which characterizes the damage evolution is define as the function of strain invariants in general. The function is given as: k 0 I 1 k I 2 2 1 1 k 2 J 2 And the damage evolution is modeled by the following function: 0 d 1 1.exp 0 Where α determines the maximum degradation, β determines rate of degradation and represent the equivalent strain energy at which the material starts degrading. The parameters k 0, k 1, k 2, α, β and TB,MPLANE command. Syntax: 0 are defined via 0 TB,MPLAN,mat,ntemp,npts,tbopt TBDATA,1,C1,C2,C3,C4,C5,C6 Constant Meaning Property C1 k 0 Damage function constant C2 k 1 Damage function constant C3 k 2 Damage function constant C4 γ 0 Critical Equivalent strain energy C5 α Maximum degradation C6 β Rate of degradation 2010 ANSYS, Inc. All rights reserved. 10 ANSYS, Inc. Proprietary

Difference with current concrete model Microplane model do not have cracking and crushing post processing which is available with current concrete model via PLCRACK command. Failure criterion for current concrete model is stress based whereas roplane model assumes energy based criteria. In current model cracking is permitted in three orthogonal direction only at each integration point whereas roplane computes damages from 21 symmetric planes at each integration point providing better estimation of damage. In current model the stress-strain matrix needs to be adjusted for each failure mode. For cracking adjustment differs from the crushing case. However for roplane model no separate modification is needed for different modes of failure. Current concrete model may give erroneous results in cases where large rigid body rotations are involved. Current concrete model can only be used with SOLID65 element whereas roplane model can be used with PLANE182, PLANE183, SOLID185, SOLID186 and SOLID187 elements. 2010 ANSYS, Inc. All rights reserved. 12 ANSYS, Inc. Proprietary

Example 1 One element cube of size 1x1x1mm is used to test the material coefficients of roplane model. Following material properties were chosen: E1=31622Mpa v=0.18 fc = 40 Mpa ft = 4 Mpa k0= 0.273 k1= 0 k2= 1 γ 0 = 3.84e-2 α = 0. 96 β = 300 Value of k0, k1 and k2 was determined assuming drucker prager criteria. Thus, total equivalent strain energy is given as: η = α I 1 + (J 2 ) 0.5 γ 0 was calculated based on the description given in help document for roplane model i.e. γ 0 characterizes the equivalent strain energy on which material damaging starts. It is assumed that in uniaxial compression the material will show damage after the axial strain (ε x ) of 1.265e-3 (=fc/e1). For this loading condition, strain matrix will be: ε = ε x 0 0 0 -vε x 0 0 0 -vε x From the strain tensor, strain invariants were calculated as: I 1 = ε x (1-2v) and J 2 = -ε x (1+v). For ε x = - 1.265e-3; I 1 = -0.8096e-3 and J 2 = 1.4927e-3. Putting these values in the expression for η we get γ 0 = 3.84e-2. Thus when η is equal to 3.84e-2, damaging will start and we should see strain softening when ε x = - 1.265e-3. 2010 ANSYS, Inc. All rights reserved. 14 ANSYS, Inc. Proprietary

Example 1 The cube was held at one end and displacement of -0.1 was applied at the other end. Graph between Sx and ex was plotted and is shown below. 2010 ANSYS, Inc. All rights reserved. 15 ANSYS, Inc. Proprietary

Example 2 Concrete Compressive lab test was simulated in ANSYS to test the new Micro Plane model for concrete and other brittle material. Model consists of a 150mm dia cylinder, 300mm long without any reinforcements subjected to compression using displacement boundary condition Material constants: E = 29725 MPa v = 0.18 k=10 (approx fc/ft) k0=(k-1)/(2*k*(1-2*v))/3 k1=k0 k2=2/k/(1+v)/(1+v) α = 0.1,0.25,0.5,0.96,1 β = 400 γ = 1e-3 Rigid Surfaces Concrete Cylinder 2010 ANSYS, Inc. All rights reserved. 16 ANSYS, Inc. Proprietary

Stress (MPa) Example 2 Several tests were run to verify the effect of damage parameter α to demonstrate strain softening effect. Effect of damage parameter 200 180 160 140 120 a=1 a = 0.96 a = 0.5 a = 0.25 a = 0.1 100 80 60 40 20 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 strain (mm/mm) 2010 ANSYS, Inc. All rights reserved. 17 ANSYS, Inc. Proprietary

Appendix - Introduction The modeling of anisotropic damage leads to complex 3D material formulations that includes higher order damage variables which can be very difficult to handled in an efficient way. Microplane model provides an alternative and efficient approach to model anisotropic damage without the complications involved with macroscopic formulations. The roplane model can be summarized in three main steps: 1) Projection Step: Define kinematic constraint to relate the macroscopic strain tensor to their roplane counterparts. Thus the roplane strains can be derived as projections of the overall strain tensor ε. For many materials, the volumetric and deviatoric behavior is completely different, not only on macroscopic but also on the roscopic level. Thus there is need to define different constitutive models for deviatoric and volumetric parts and thus the kinematic constraints should be define between the macroscopic strain and rosopic volumetric and deviatoric parts. t ε = ε.n = [ε vol + ε dev ].n = 1/3 [I: ε].n + ε dev.n = V: ε.n + ε dev.n = ε V n + ε D ε V = V: ε ε D = t ε ε V n = Dev: ε V and Dev are the projection tensors 2010 ANSYS, Inc. All rights reserved. 20 ANSYS, Inc. Proprietary

Appendix - Introduction 2) Microlevel material model: Second step is to define constitutive law on the roplane level. This will be discuss in detail in later slides. 3) Homogenization step: This step relates to the homogenization process on the material point level to derive the overall response. This step is based on the principle of energy equivalence which thermodynaally consistent. The assumption is that a roscopic free energy on the roplane level ψ exists and the integral of ψ over all roplane is equivalent to a macroscopic free Helmholtz energy ψ mac. mac 3 4 d 2010 ANSYS, Inc. All rights reserved. 21 ANSYS, Inc. Proprietary

Appendix - Microplane Elasticity In this section we will develop roscopic elastic parameters and establish its relationship with its macroscopic counterparts. For each roplane we can divide the free energy in its volumetric and deviatoric parts. Ψ (ε V,ε D ) = Ψ vol(ε V ) + Ψ dev(ε D ) = ½ K ε 2 V + G ε D : ε D K and G represents roscopic bulk and shear modulus. We can define roplane stresses as the derivatives of the roscopic free energy for the associated strain comp. v : K v, D : 2G D v D Similarly, macroscopic stress σ can be obtained as thermodynaally conjugate variables to the macroscopic strains ε : mac Using the free energy assumption we get: mac 3 4 3 d 4 K V v 2G.Dev T. D d 2010 ANSYS, Inc. All rights reserved. 22 ANSYS, Inc. Proprietary

Appendix - Microplane Elasticity ε V and ε D can be related to the macroscopic strain tensor using kinematic constraints and thus can write macroscopic stress-strain relationship in terms of roscopic parameters. σ = [C]ε, where [C] is given as: The analytical integration properties of the fourth order product of the projection tensor (V,dev) are: 3 vol 3 T dev V Vd ; Dev.Devd 4 4 vol dev We can write above equations as: [ C] K 2G ; comparing this equation with vol dev Hookes law: [ C] 3K 2G, we can write ro elastic properties in terms of macro properites: K = 3K ; G = G 3 [C] 4 K V V 2G.Dev T.Dev d Above relationship represent that the corresponding roscopic volumetric or shear resistance depends on there macro counterparts. In other words we can say that we can independently obtain volumetric and deviatoric (shear) response not only on the macro but on the ro level too! 2010 ANSYS, Inc. All rights reserved. 23 ANSYS, Inc. Proprietary

Appendix - Microplane Damage In this section we will develop roscopic formulation to include dissipative damage mechanisms which can descibe progressive material degradation. In this case the damage roplane free energy is defined as: Ψ (ε V,ε D,d ) = ½ (1-d ).K ε 2 V + (1-d ).G ε D : ε D Where d V and d D are the damage variable for volumetric and deviatoric parts. We can define roplane stresses as the derivatives of the roscopic free energy for the associated strain comp. v : (1 d ) K v, D : 2(1 d ) G D v In addition to stress definitions, roscopic free energy gives the energy release rates as the thermodynaally conjugate variables to the variables d V and d D : Y 1 2 V : K V, YD : G D. d v 2 d D We can define the state of damage using the definitions of energy release: D (Y Y )d 0 V D D D 2010 ANSYS, Inc. All rights reserved. 24 ANSYS, Inc. Proprietary

Appendix - Microplane Damage The state of damage can be characterize by two loading function of the form: ( ) d 0; In the above relations η depends on the energy release rates. The above relationship holds the same significance and behavior to plastic potential for rate independent plasticity. Macroscopic stress σ can be obtained as thermodynaally conjugate variables to the macroscopic strains ε : mac mac 3 Using the free energy assumption we get: d 4 3 4 [1 d V V 2G.Devd}: Thus can write macroscopic stress-strain relationship in terms of roscopic parameters. σ = [C] sec ε. And the time differential of the above relationship will provide us the tangent modulus. ]{K.Dev T 2010 ANSYS, Inc. All rights reserved. 25 ANSYS, Inc. Proprietary

Appendix - Discretization The integrals of the macroscopic strain equation and the derived stresses equation are solved via numerical integration: N 3 P 3 i i (.) d (. ).w 4 4 N1 To do this numerical integration Discretization of the sphere is required. This process results in the approximation of rosphere with many roplanes. In ANSYS forty two roplanes are used for the numerical integration. Due to symmetry of the roplanes, 21 roplanes are considered and summarized. 2010 ANSYS, Inc. All rights reserved. 26 ANSYS, Inc. Proprietary