Non-linear and time-dependent material models in Mentat & MARC. Tutorial with Background and Exercises

Transcription

1 Non-linear and time-dependent material models in Mentat & MARC Tutorial with Background and Exercises Eindhoven University of Technology Department of Mechanical Engineering Piet Schreurs July 7, 2009

2 Contents 1 Nonlinear and time-dependent behaviour Background : Incremental loading and iterative solution Truss with elastomeric material behaviour Truss with elastoplastic material behaviour Truss with viscoelastic material behaviour Links

3 1 Nonlinear and time-dependent behaviour In the tutorial Staaf- en Balkconstructies (in Dutch) the deformation of truss structures has been calculated. In all cases it was assumed that the deformation of the trusses was very small and (as a consequence) their behavior was linearly elastic. In this chapter we will again focuss the attention on (simple) truss structures. Now, however, we will allow deformations to become large. Also the material behavior is not longer linearly elastic. In the examples we will consider elastomeric, elastoplastic and viscoelastic material behavior. Large deformations and nonlinear material behavior will render the problem the equations which must be solved nonlinear. The MARC program can solve these problems, but we have to incorporate the proper commands and parameters in the input with Mentat. 1.1 Background : Incremental loading and iterative solution When deformations are very small, there is virtually no difference between the undeformed and the deformed geometry. This implies that the stiffness matrix K can be formulated in the undeformed situation and does not need updating. If this is the case and if moreover the material behaviour is linear elastic and contact phenomena like friction are not considered, the deformation problem is completely linear. Two important characteristics hold for linear problems : the deformation is proportional with the load : when the external nodal forces are multiplied by a factor, say α, the nodal displacements ũ are also multiplied by α. superposition holds : when we determine the deformation ũ 1 and ũ 2 for two separate loadings, and respectively, the deformation for the combined loading + is the sum of the separate deformations : ũ 1 + ũ 2. For linear deformation problems the system of nodal equilibrium equations is linear and can be solved directly after implementation of boundary conditions by using the inverse of the stiffness matrix : f i = K ũ = ũ = ũ s = K 1 The solution is unique : we only get one ũ s for a given. f f i 0 ũ s ũ Deformations may be so large that the geometry changes considerably. This and/or nonlinear material behaviour and/or contact phenomena render the deformation problem nonlinear. Proportionality and superposition do not hold in that case. The system of nodal equilibrium equations is nonlinear. 3

4 f f i 0 ũ As can be seen from the figure, more than one solution may exist, some of which may be not realistic from a physical point of view. The real loading of the structure may be time dependent. To determine the associated deformation, the time is discretised : the load is prescribed at subsequent, discrete moments in time and deformation is determined at these moments. A time interval between two discrete moments is called a time increment and the time dependent loading is referred to as incremental loading. This incremental loading is also applied for cases where the real time (seconds, hours) is not important but where we want to prescribe the load gradually. One can than think of the time as a fictitious or virtual time. f f i f e 0 t t n t t n+1 ũ n ũ n+1 ũ Let us now assume that the load is prescribed and that the deformation is determined for the time t n, the beginning of time-increment n. Given the load at the end-increment time t n+1, we want to determine the associated deformation by solving the non-linear nodal point equations iteratively. In the figure below, the origin of the plot represents the beginning of the time increment (t n ). The nodal point displacements with relation to the undeformed situation are ũ n. The external load is incremented by = (t n+1) (t n). f f i ũ n ũ n+1 ũ 4

5 The equations for nodal equilibrium after implementation of boundary conditions can be written as follows : (ũ) = f i The solution ũ = ũ n+1, the intersection of and f i, cannot be determined directly, but has to be calculated iteratively. Starting from ũ = ũ n a number of approximations ũ are calculated in subsequent iteration steps. This iteration procedure is illustrated in the figure below. f K(ũ ) f i r f i (ũ ) ũ n ũ n+1 ũ ũ δũ ũ When we have the approximation ũ we can calculate the difference r = f i(ũ ). For the exact solution this so-called residual (load) is zero. When r is not zero but small enough, we are satisfied with the approximation ũ and if it is too large we start a new iterationstep. When the residual becomes smaller in subsequent iteration steps, the iterative procedure is said to converge. To decide whether the iteration process can be ended, the residual as a matter of fact a norm of the residual load column : r is compared to a convergence limit, which is a very small number. When a new approximation must be determined, the tangent stiffness matrix K(ũ ) is calculated and subsequently the iterative displacement δũ, leading to a new approximation ũ : K(ũ )δũ = r δũ = K 1 (ũ )r ũ = ũ + δũ Unfortunately the iteration process may show no convergence. After a maximum number of iteration steps the MARC program will stop the iteration procedure. In many cases deminishing the load increments will result in convergence, but problems may be very persistent. When the convergence criterion is satisfied, the displacement ũ n+1 will not satisfy the nodal equilibrium equations exactly, because the convergence limit is small but not zero. The difference between f i (ũ n+1) and is added to the load in the next increment, which is known as residual load correction. The iterative procedure described here is referred to as Newton-Raphson method. Results of the analysis vary with (pseudo)time due to the incremental loading. These time dependent results can be visualized as a plot of a certain variable against time or against another variable. In MARC this is done by making a HISTORY PLOT as is explained in the following examples. 5

6 1.2 Truss with elastomeric material behaviour The figure below shows one truss with dimensions and boundary conditions. y x A 0 l F u The dimensions are : l 0 = 100 mm = 0.1 m ; A 0 = 10 mm 2 = 10 5 m 2 We want to prescribe the elongation of the truss as a function of the pseudo time remember that the real time has no meaning in this case. The displacement u of node 2 is assumed to increase linearly with time t until it reaches a value u max = 400 mm = 0.4 m, so the elongation factor will finally be λ = 5. To prescribe the evolution of a variable in time in this case the displacement of a node we have to make a table. In the menu (MECHANICAL BC s) we find a button with this name. A table is represented in Mentat as a curve in a two-dimensional plane. We make a new table of the type time which indicates that the horizontal axis or independent variable of the plot represents time. (MAIN MENU) (PREPROCESSING) BOUNDARY CONDITIONS MECHANICAL TABLES NEW (NEW TABLE) 1 INDEPENDENT VARIABLE TABLE TYPE time (FORMULA) ENTER Enter formula : v1 FIT Mentat shows a plot representing the time-table : the horizontal v1-axis is the time and the vertical F-axis is a multiplication factor, which is used to prescribe a variable as a function of time. Because we want to prescribe the displacement of node 2 to increase linearly from 0 to 0.4 [m], we have to multiply the table factor (= F-axis value) with u max = 0.4 m. This is done in (BOUNDARY CONDITIONS) so we plot the model again with SHOW TABLE SHOW MODEL and go back to the (MAIN MENU). (MAIN MENU) (PREPROCESSING) BOUNDARY CONDITIONS MECHANICAL Prescribe the suppressed displacements in nodes 1 and 2. Prescribe the displacement in node 2. NEW FIXED DISPLACEMENT 6

7 DISPLACEMENT X Enter value for x : 0.4 This is u max in [m] TABLE table1 (NODES) ADD Select node 2. In this example we assume that the material of the truss is an elastomer (= rubber). Its behavior is described by a Neo-Hookean material model with one material parameter : C 10 = 0.5e6 Pa. This is modelled in the (MATERIAL PROPERTIES) menu. (MATERIAL PROPERTIES) MATERIAL PROPERTIES NEW MORE MOONEY (MOONEY PROPERTIES) C10 Enter value for c10 : 0.5e6 (ELEMENTS) ADD Enter add material element list : (ALL) EXISTING When load varies with time, a LOADCASE must be defined, where we can indicate the total analysis time and the number of incremental steps. (MAIN MENU) (ANALYSIS) LOADCASES NEW MECHANICAL STATIC TOTAL LOADCASE TIME : 1 (FIXED) # STEPS : 400 The loads which have to be applied in this loadcase, must be selected if this is not already done by default. LOADS Select all applies Although we do not have to change the presets, we can have a look in the submenus SOLUTION CONTROL and CONVERGENCE TESTING and see some parameter values. In the JOBS menu we now choose the element type (= 9) and indicate that we want to do a 3D analysis. After that we are going to define a MECHANICAL job and select the defined loadcase. Because all loading is defined in this loadcase, we do not do any loading in INITIAL LOADS. Only suppressed displacements are allowed there. We must also select the proper output variables. MECHANICAL (LOADCASES) Select lcase1 7

8 INITIAL LOADS Only select the boundary conditions which represent suppresed displacements. So do not prescribe any real loading here. You may also CLEAR all boundary conditions here. (ANALYSIS DIMENSION) JOB RESULTS 3D Select the output variables you want to see in the post-file. Nodal variables (forces and displacements) don t have to be selected. They are available by default. The model is complete now and we can save and run it in the usual way. In the submenu which is shown during running, we see that the solution is done iteratively and incrementally. When things proceed successfully, we get the exit number 3004 and are going to look at some results. Results After opening the post-file in the (POSTPROCESSING) RESULTS menu, we can MONITOR the deformation of the truss, but this is not very exciting. What we are interested in is a plot of the axial force as a function of time or, maybe better, as a function of the prescribed displacement. Such plots are referred to as a HISTORY PLOT. HISTORY PLOT SET NODES Select node 2. Don t forget End List. REWIND COLLECT DATA Enter first history increment : 0 Enter last history increment : Enter increment step size : 1 NODES/VARIABLES ADD 1-NODE CURVE Enter History-Plot node : select (lm) node in (NODES) Enter X-axis variable : Displacement X Enter Y-axis variable : Reaction Force X FIT There is the plot. The numbers, indicating the individual time steps, can be removed by setting SHOW IDS to 0. We can compare it to the analytical solution, which can be calculated for this case. 8

9 1.3 Truss with elastoplastic material behaviour The figure below shows one truss with dimensions and boundary conditions. y x A 0 l F u The dimensions are : l 0 = 100 mm = 0.1 m ; A 0 = 10 mm 2 = 10 5 m 2 Except for the suppressed displacements, which are indicated in the figure above, the displacement of node 2 is prescribed as a function of time according to the figure below. A proper time-table must be defined and used in (MECHANICAL BC s). The horizontal axis (= time) of the table must extent to a maximum of 4 seconds. u 0.01 [m] t TABLE NEW 1 INDEPENDENT VARIABLE TYPE time INDEPENDENT VARIABLE V1 MAX 4 STEPS 8 DATA POINTS ADD Click at the intersections of the lines in the grid to add points. SHOW TABLE SHOW MODEL In this example we assume the material of the truss to behave elastoplastically with isotropic hardening. The relevant material parameters are : Young s modulus E 200 GPa Poisson s ratio ν initial yield stress σ v0 250 MPa isotropic hardening parameter H 10 GPa 9

10 The material behaviour is specified in the MATERIAL PROPERTIES menu where we enter at a certain stage the ELASTIC-PLASTIC menu. (MAIN MENU) (PREPROCESSING) MATERIAL PROPERTIES MATERIAL PROPERTIES ISOTROPIC YOUNG S MODULUS : 200e9 POISSON S RATIO : 0.3 (PLASTICITY) ELASTIC-PLASTIC (YIELD SURFACE) VON MISES (HARDENING RULE) ISOTROPIC INITIAL YIELD STRESS Enter value for yield-stress : 250e6 In the ELASTIC-PLASTIC section we have selected the VON MISES yield surface or yield criterion, which was already selected by default. With this choice the so-called Von Mises effective stress σ vm will be calculated. It will be compared with the current yield stress σ v in the yield criterion to decide whether elastic or elastoplastic deformation occurs : f = σ vm 2 σv 2 < 0 elastic behaviour f = σ vm 2 σ2 v = 0 elastoplastic behaviour This Von Mises stress is defined as follows : 1 σ vm = 2 {(σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 } where σ 1, σ 2 and σ 3 are the principle stresses. For a truss the Von Mises stress equals the absolute value of the axial stress. The current yield stress σ v will increase as a function of the effective plastic strain ε p according to the isotropic yield criterion : σ v = σ v0 + H ε p This hardening law is modelled with a table in Mentat, where the x-axis represents ε p and the y-axis is the multiplication factor of the initial yield stress. We make the table : TABLES NEW 1 INDEPENDENT VARIABLE TYPE eq plastic strain This is the Equivalent Plastic Strain ε p INDEPENDENT VARIABLE V1 MAX Enter maximum value for V1 : 2 FORMULA ENTER Enter formula : 1 + (10000e6 * v1)/250e6 FIT 10

11 SHOW TABLE SHOW MODEL RETURN The table here table2 is selected in the ELASTIC-PLASTIC menu. ISOTROPIC (PLASTICITY) ELASTIC-PLASTIC TABLE (PLASTIC STRAIN) Select table2 Do not forget to add this material to the element. Now we define a loadcase as in the previous example. The total loadcase time is 4 seconds. The number of incremental steps is 400. In the JOBS menu we have to select the element type, the analysis dimension, the loadcase and the output variables. We also go into the menu ANALYSIS OPTIONS to set some parameters. ANALYSIS OPTIONS NONLINEAR PROCEDURE SMALL STRAIN Selection of the SMALL STRAIN (NONLINEAR PROCEDURE) indicates that we confine ourselves to small deformations. Because the prescribed elongation results in 10% axial strain, this requirement is satisfied. With the small strain procedure the hardening table is given in terms of engineering stress and strain. When strains are large, we must select the proper option : ANALYSIS OPTIONS NONLINEAR PROCEDURE LARGE STRAIN After saving the model we can run the analysis and take a look at results. Results After opening the post-file in the RESULTS menu, we make a HISTORY PLOT of the axial stress as a function of the axial strain. HISTORY PLOT SET NODES : 2 REWIND COLLECT DATA Enter first history increment : 0 Enter last history increment : Enter increment step size : 1 NODES/VARIABLES ADD 1-NODE CURVE Enter History-Plot node : select (lm) node in (NODES) 11

12 Enter X-axis variable : Comp 11 of Total Strain Enter Y-axis variable : Comp 11 of Stress FIT This result can also be calculated analytically. The analysis can now be repeated for the a material with the same parameters, but showing purely kinematic hardening. The Von Mises yield function is : with a hardening law for the shift stress q : f = ( σ vm q(ε p )) 2 σ 2 v0 q = Kε p where K is the constant kinematic hardening parameter, for which we take here K = 10 GPa. The kinematic hardening is modelled in Mentat with a table of type plastic strain in the same way as we did for isotropic hardening. In fact we draw the σ v ε p -curve for a monotonic tensile loading. The program knows that kinematic hardening occurs at unloading and load reversal. 12

13 1.4 Truss with viscoelastic material behaviour The figure below shows one truss with dimensions and boundary conditions. y x A 0 l F u The dimensions are : l 0 = 100 mm = 0.1 m ; A 0 = 10 mm 2 = 10 5 m 2 Exept for the suppressed displacements, which are indicated in the figure above, the displacement of node 2 is prescribed as a function of time according to the figure below. A proper time-table must be defined and used in BOUNDARY CONDITIONS. The x-axis (= time) of the table must extent to a maximum of 2 seconds. u 0.01 [m] t In this example we assume the material of the truss to behave linearly viscoelastic according to a generalised Maxwell model with tensile moduli E i [Pa], time constants τ i = ηi E i [s] η i are the viscosities and with an equilibrium modulus E [Pa]. Poisson s ratio is 0.3 [-] and parameter values of an 8-mode Maxwell model (n = 8), which have been fitted onto experimental data, are : E E i [Pa] τ i [s] E 0 mode e e-05 mode e e-04 mode e e-03 mode e e-03 mode e e-02 mode e e-01 mode e e+00 mode e e+01 E 1 E 2 E n η 1 η 2 η n σ ε In MARC the equilibrium modulus is not used as an input parameter. Instead we have to use the initial modulus E 0. Taking a look at the generalised Maxwell model learns that the initial modulus is the sum of all moduli, because initially all dashpots are locked, so we have : E 0 = E + i E i 13

14 This initial modulus is the YOUNG S MODULUS in the MATERIAL PROPERTIES menu. Now we enter the VISCOELASTIC menu and see that we are not asked for tensile moduli E i but for shear moduli G i and bulk moduli K i. These variables can be calculated according to : G = E 2(1 + ν) ; K = This results in the following table with viscoelastic data : E 3(1 2ν) G i [Pa] K i [Pa] τ i [s] mode e e e-05 mode e e e-04 mode e e e-03 mode e e e-03 mode e e e-02 mode e e e-01 mode e e e+00 mode e e e+01 These data find their way into Mentat as follows : (MAIN MENU) (PREPROCESSING) MATERIAL PROPERTIES MATERIAL PROPERTIES ISOTROPIC YOUNG S MODULUS : e+08 POISSON S RATIO : 0.3 (RATE EFFECTS) VISCOELASTIC # DEVIATORIC TERMS Enter value for ndeviatoric term : 8 Enter value for nvolumetric term : 8 (TIME) 1 Enter value for shear time1 : e-5 (SHEAR CONSTANT) 1 Enter value for shear1 : e7 (TIME) 1 Enter value for bulk time1 : e-5 (BULK CONSTANT) 1 Enter value for bulk1 : e8 Repeat the following for each mode. After entering these data, we can choose how much modes must be used in the actual analysis. Do not forget to appoint the material model to the element. Analyse the viscoelastic behaviour during 2 [s] and plot a HISTORY PLOT of stress agains time. 14

15 1.5 Links A LINK defines a relation between degrees of freedom (dof s) in different nodes. When dof s are defined to have the same value, the LINK is called a NODAL TIE. The dof s in the linked nodes are directly related to those in the retained nodes. In the appropriate submenu, we have to define the TIED NODE and the RETAINED NODES. In Mentat a LINK is indicated with a line between the linked nodes. The TYPE of the link indicates which dof s are linked. The dof s in the TIED NODES are no independent unknowns and are eliminated from the equation system. The dof s in the retained nodes are solved. Boundary conditions can only be applied in RETAINED NODES. Results can be extracted for all nodes. 15