Small Scale Yielding Model for Fracture Mechanics J. Thomas, K. C. Koppenhoefer*, J. S. Crompton AltaSim Technologies, LLC *Corresponding author: 130 East Wilson Bridge Road, Suite 140, Columbus, OH 43085, USA [email protected] Abstract: The small scale yielding model is eventually interacts with the finite boundaries of used to compare published linear elastic fracture the structure in which it is contained thus mechanics (LEFM) and elastic-plastic fracture exceeding the SSY condition. mechanics (EPFM) analytical solutions to elastic To examine the effects of violating SSY and plastic finite-element solutions developed conditions on the displacement, strain and stress using COMSOL Multiphysics. The EPFM in structures, much research in the field of solutions include small-strain and large-strain fracture mechanics relies on computational solutions. In addition, these cases are also modeling. The COMSOL Multiphysics Model developed using Abaqus as a point of Gallery gives a model of an edge crack [1] where comparison. This work compares and contrasts calculated stress intensity factors are compared the solution methodology and results of with the classic, single parameter linear elastic COMSOL Multiphysics with finite element fracture mechanics (LEFM) solution developed analysis software with an extensive history of by Irwin [2]. This model gives confidence in solving fracture mechanics problems. COMSOL Multiphysics ability to accurately Keywords: fracture mechanics, crack, small- predict cleavage fracture in brittle materials scale yielding model, small-strain plasticity, where fracture toughness is low and SSY large-strain plasticity. conditions are met. On the opposite extreme, limit load analysis is used to determine failure 1. Introduction when fracture toughness is high and failure is governed by the plastic flow characteristics of Materials that experience cleavage fracture the material. due to pre-existing cracks fail due to the elevated Many structures are constructed using steels stress ahead of these cracks. Conventional with moderate fracture toughness values that do fracture mechanics has developed single not fit into either of these two extremes. parameters that seek to uniquely describe the For this important class of problems, the displacement, strain and stress ahead of a crack. field of non-linear fracture mechanics, For materials that do not undergo plastic specifically elastic-plastic fracture mechanics deformation, the single parameter is referred to (EPFM), has developed equations and models as K. When plastic strain develops at the crack for describing the displacement, strain and stress tip, the J-integral provides this single parameter. ahead of a crack where plasticity effects cannot If plasticity remains well contained to the crack be neglected [3]. EPFM relies upon SSY as an tip region and is not influenced by the size of the important benchmark. This SSY modeling surrounding structure, then K and J are uniquely assumption has given rise to a number of related such that analytical solutions including that developed by Rice and Rosengren [4] and Hutchinson [5] (1) known as the HRR singularity solution. Now, even the best published analytical stress results = 1 ν break down in a singular zone near the crack tip 퐸퐽 being based on small strain theory. For these 퐾 � 2 areas, finite-element analysis with a large-strain where E is the elastic− modulus and ν is theory that takes into account geometry changes Poisson’s ratio. This condition is referred to as due to crack blunting is more accurate. Small Scale Yielding (SSY) since the plastic zone is small in comparison to the relative dimensions of the structure in which it is contained. As material toughness increases, the plastic zone at the crack tip increases in size and Excerpt from the Proceedings of the 2014 COMSOL Conference in Boston Table 1: Ramberg-Osgood Power-Law Material Parameter Value Unit Young’s modulus, E 200 GPa Poisson’s ratio, ν 0.33 n/a Yield strength, σ0 250 MPa Yield offset, α 0.002 n/a Power law exponent, n 13 n/a Figure 1. Blunted crack with displacement boundary conditions. Initial radius of crack width, Equation ) gives the plastic flow rule for the b0=10µm. Specimen radius, R=150cm. (5 small-strain case. 2. COMSOL Model = ( ) (6) 2.1 Material Model 푺 퐂 ∶ ε − εp ( , ) 0, (7) The analyses conducted in this work use three distinct material models available within 퐹 휎 휎0 ≤Q (8) COMSOL Multiphysics: linear elastic, small- = λ strain plasticity, and large-strain plasticity. For 휕 the linear elastic model, a linear constitutive 휀ṗ model and linear strain-displacement The large-strain material휕휎 model eliminates the relationships are used. assumption of small displacements in the strain- displacement relationship and uses the Ramberg- Osgood constitutive model specified in Eq. (5). (2) = v, = J , (9) = (3) ∇ ⋅ σ 퐅 ( ) (10) = −ퟏ+ 퐓 , σ 퐅푺퐅 ( ) 푺 1퐂 ∶ ε (4) J = det = ( + ( ) ) 퐅 퐈 ∇퐮 2 = 1/2 [ + ( )^ (11) 퐓 + (퐅 )^ ] For the smallε -strain∇ 퐮plasticity∇퐮 model, the linear ε ∇퐮 ∇퐮 퐓 strain-displacement relationships are used as they 2.2 Crack Loading ∇퐮 퐓 ∇퐮 were in the linear elastic model. (This decision is implemented by choosing to leave the Figure 1 gives a schematic of the blunted crack “Geometric Non-linearity” option unchecked for the small scale yielding model. under the “Study Settings” section of the The loading is applied using displacement COMSOL Multiphysics Study Node.) The conditions as a crack tip displacement field for constitutive model is a Ramberg-Osgood power Mode I: law plasticity model defined as K r θ 2 θ (12) n uxR = cos κ +− 21 sin ε σ σ (5) G 22 π 2 2 += α K r θ θ = κ −+ 2 ε σ 00 σ 0 u yR sin 21 cos G 22 π 2 2 and implemented by a user-defined interpolation where function to define the post-yield hardening −= 43 νκ ()plane strain behavior of the material. Table 1 provides the material property parameters. Excerpt from the Proceedings of the 2014 COMSOL Conference in Boston The J-integral calculation in COMSOL Multiphysics uses path lines that are in line with the coordinate directions (see Figure 3). For curved paths (green path), the J-integral calculation within COMSOL Multiphysics may be path-dependent due to the normal vector for the traction integral calculation not pointing outwards. To ensure that the contour normal are Figure 2. 2D half symmetric small scale yielding pointing outwards for all node points, the normal model. Initial crack width, b0=10µm. A quad mesh is specified explicitly rather than using the is generated; isoparametric elements with solid.nx and solid.ny path normal operators. quadratic shape order discretization are specified. These path normal operators cannot guarantee to be pointed outward causing some negative energy components to give a J-integral less than 2.3 Meshing and Discretization the actual J-integral. A free quad mesh (with custom size constraints) 2.5 Solution procedure generates the mesh in the region near the crack tip. The mapped meshing functionality provides The non-linear stationary solver was used a more efficient mesh in the remaining regular with the load ramping technique to aid in shaped arcs (see Figure 2). convergence. The default Newton non-linear method was selected but the damping settings A quadratic discretization was used for both the were set to “Automatic highly nonlinear”. Total solution shape order and geometry shape order degrees of freedom equal approximately 60,000. (isoparametric elements). For the solid mechanics elements, COMSOL Multiphysics 3. Results and Discussion does not give the freedom for the user to control how to perform the element integration. The For comparison, the normal stress software constrains the integration order to twice (normalized by yield strength) along the length the order selected for the shape functions, and of the remaining ligament is plotted using the this constraint could introduce locking. line graph post-processing functionality. Also, the x-distance along the ligament is normalized 2.4 J-integral Calculations by the yield strength divided by the J-integral loading rate. With this normalization, a value of The J-integral is given by: 1 on the x-axis corresponds to approximately K 2 ∂u (13) twice the crack tip opening displacement. = = − i J ∫Ws dy T i ds E Γ ∂x ∂u = − i ∫Ws n x T i ds Γ ∂x where 1 W =(σ ε + σ ε + 2 σ ε ) s 2 x x y y xyxy is strain energy density Figure 3. 2D half symmetric small scale yielding model with 2 J-integral path definitions. The blue Tx=σ x n x + σ xy ny, T y =σxy nx + σ y n y path works well because the lines are in the are traction vector components direction of the x and y coordinate directions. Excerpt from the Proceedings of the 2014 COMSOL Conference in Boston contribution to the stiffness causing locking for near incompressible conditions. Additionally, it is worth noting that the Gauss point evaluation for normal stress (i.e. solid.sGpy) rather than the “legacy” stress evaluation (i.e. solid.sy) must be used for accurate evaluation especially in the region near the crack tip. The Gauss point evaluation interpolates directly from the integration points which are the only place where the force balance equation is guaranteed to be satisfied for non- linear material models. In contrast, the less- Figure 4. Normal stress on remaining ligament accurate legacy evaluation interpolates from the versus distance along ligament normalized by loading for linear elastic case. Lagrange points (node points) and is an evaluation based on the derivative of the displacements at these points. Figure 4 gives the comparison between the For the large plastic strain formulation, LEFM single parameter Irwin solution along the Figure 6 shows plastic strain contours (mirrored ligament given by across the symmetry plane) near the crack tip for (14) loading of K=22.1 MPa√m.