Structural Mechanics Module

Structural Mechanics Module User s Guide VERSION 4.3b

Structural Mechanics Module User s Guide 1998 2013 COMSOL Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/tm. Version: May 2013 COMSOL 4.3b Contact Information Visit the Contact Us page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: Support Center: www.comsol.com/support Download COMSOL: www.comsol.com/support/download Product Updates: www.comsol.com/support/updates COMSOL Community: www.comsol.com/community Events: www.comsol.com/events COMSOL Video Center: www.comsol.com/video Support Knowledge Base: www.comsol.com/support/knowledgebase Part No. CM021101

Contents Chapter 1: Introduction About the Structural Mechanics Module 2 Why Structural Mechanics is Important for Modeling......... 2 What Problems Can It Solve?.................. 3 The Structural Mechanics Module Physics Guide........... 6 Available Study Types..................... 10 Geometry Levels for Study Capabilities.............. 13 Where Do I Access the Documentation and Model Library?...... 15 Overview of the User s Guide 18 Chapter 2: Structural Mechanics Modeling Applying Loads 22 Units, Orientation, and Visualization............... 22 Load Cases......................... 23 Singular Loads........................ 23 Moments in the Solid Mechanics User Interface........... 25 Pressure.......................... 25 Acceleration Loads...................... 26 Temperature Loads Thermal Expansion.............. 26 Total Loads......................... 27 Defining Constraints 28 Orientation......................... 28 Symmetry Constraints..................... 28 Kinematic Constraints..................... 30 Rotational Joints....................... 30 Calculating Reaction Forces 31 Using Predefined Variables to Evaluate Reaction Forces........ 31 CONTENTS i

Using Weak Constraints to Evaluate Reaction Forces......... 32 Using Surface Traction to Evaluate Reaction Forces.......... 33 Evaluating Surface Traction Forces on Internal Boundaries....... 34 Introduction to Material Models 35 Introduction to Linear Elastic Materials.............. 35 Introduction to Linear Viscoelastic Materials............ 36 Mixed Formulation...................... 36 About the Material Databases for the Structural Mechanics Module... 36 Defining Multiphysics Models 38 Thermal-Structural Interaction.................. 38 Acoustic-Structure Interaction.................. 38 Thermal-Electric-Structural Interaction.............. 39 Modeling Geometric Nonlinearity 40 Geometric Nonlinearity for the Solid Mechanics User Interface..... 41 Geometric Nonlinearity for the Shell, Plate, Membrane, and Truss User Interfaces....................... 42 Prestressed Structures..................... 42 Geometric Nonlinearity, Frames, and the ALE Method........ 43 Reference for Geometric Nonlinearity............... 46 Linearized Buckling Analysis 47 Introduction to Contact Modeling 49 Constraints......................... 49 Contact Pairs........................ 50 Boundary Settings for Contact Pairs............... 51 Time-Dependent Analysis................... 52 Multiphysics Contact..................... 52 Solver and Mesh Settings for Contact Modeling........... 52 Monitoring the Solution.................... 53 ii CONTENTS

Eigenfrequency Analysis 54 Using Modal Superposition 57 Modeling Damping and Losses 59 Overview of Damping and Loss................. 59 Linear Viscoelastic Materials.................. 63 Rayleigh Damping....................... 63 Equivalent Viscous Damping................... 64 Loss Factor Damping..................... 65 Explicit Damping....................... 65 Piezoelectric Losses 66 About Piezoelectric Materials.................. 66 Piezoelectric Material Orientation................ 67 Piezoelectric Losses...................... 73 No Damping........................ 76 References for Piezoelectric Damping............... 76 Springs and Dampers 78 Tips for Selecting the Correct Solver 80 Symmetric Matrices...................... 80 Selecting Iterative Solvers................... 81 Specifying Tolerances and Scaling for the Solution Components..... 82 Chapter 3: Solid Mechanics Solid Mechanics Geometry and Structural Mechanics Physics Symbols 84 3D Solid Geometry...................... 84 2D Geometry........................ 85 Axisymmetric Geometry.................... 86 Physics Symbols for Boundary Conditions............. 86 About Coordinate Systems and Physics Symbols........... 88 Displaying Physics Symbols in the Graphics Window An Example.... 89 CONTENTS iii

The Solid Mechanics User Interface 91 Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics... 94 Linear Elastic Material..................... 95 Change Thickness...................... 99 Damping.......................... 99 Initial Values........................ 102 About the Body, Boundary, Edge, and Point Loads......... 103 Body Load........................ 103 Boundary Load...................... 104 Edge Load........................ 105 Point Load........................ 105 Fixed Constraint...................... 106 Prescribed Displacement................... 107 Free........................... 108 Symmetry........................ 109 Antisymmetry....................... 109 Roller.......................... 110 Periodic Condition..................... 110 Linear Viscoelastic Material.................. 111 Thermal Expansion..................... 112 Thermal Effects...................... 114 Rigid Connector...................... 115 Rigid Domain....................... 118 Applied Force....................... 118 Applied Moment...................... 119 Mass and Moment of Inertia.................. 119 Contact......................... 120 Friction......................... 123 Bolt Pre-Tension...................... 125 Bolt Selection....................... 125 Initial Stress and Strain.................... 126 Phase.......................... 127 Prescribed Velocity..................... 128 Prescribed Acceleration................... 129 Spring Foundation..................... 129 Pre-Deformation...................... 131 Thin Elastic Layer...................... 132 Added Mass........................ 133 iv C ONTENTS

Low-Reflecting Boundary................... 134 Attachment........................ 135 Theory for the Solid Mechanics User Interface 136 Material and Spatial Coordinates................ 136 Coordinate Systems..................... 137 Lagrangian Formulation................... 138 About Linear Elastic Materials................. 139 Strain-Displacement Relationship................ 146 Stress-Strain Relationship................... 149 Plane Strain and Plane Stress Cases............... 149 Axial Symmetry...................... 150 Loads.......................... 155 Pressure Loads...................... 156 Equation Implementation................... 157 Setting up Equations for Different Studies............ 158 Damping Models...................... 161 Modeling Large Deformations................. 164 About Linear Viscoelastic Materials............... 165 About Contact Modeling................... 170 Theory for the Rigid Connector................ 173 Initial Stresses and Strains.................. 175 About Spring Foundations and Thin Elastic Layers......... 176 About Added Mass..................... 178 Using Pre-tensioned Bolts.................. 179 Geometric Nonlinearity Theory for the Solid Mechanics User Interface........................ 182 About the Low-Reflecting Boundary Condition.......... 186 Cyclic Symmetry and Floquet Periodic Conditions......... 187 Chapter 4: Shells and Plates The Shell and Plate User Interfaces 192 Domain, Boundary, Edge, Point, and Pair Nodes for the Shell and Plate User Interfaces................... 195 Linear Elastic Material.................... 196 CONTENTS v

Thermal Expansion..................... 198 Initial Stress and Strain.................... 199 Damping......................... 200 Change Thickness..................... 202 Initial Values........................ 203 About the Body Load, Face Load, Edge Load, and Point Load Nodes......................... 203 Body Load........................ 204 Face Load........................ 204 Edge Load........................ 205 Point Load........................ 207 Phase.......................... 207 Pinned.......................... 208 No Rotation....................... 208 Prescribed Displacement/Rotation............... 209 Prescribed Velocity..................... 211 Prescribed Acceleration................... 213 Symmetry........................ 215 Antisymmetry....................... 216 Rigid Connector...................... 217 Results Evaluation..................... 219 Theory for the Shell and Plate User Interfaces 220 About Shells and Plates................... 220 Theory Background for the Shell and Plate User Interfaces...... 221 Reference for the Shell User Interface.............. 231 Chapter 5: Beams The Beam User Interface 234 Boundary, Edge, Point, and Pair Nodes for the Beam User Interface.. 236 Linear Elastic Material.................... 237 Thermal Expansion..................... 238 Initial Stress and Strain.................... 239 Damping......................... 240 Initial Values........................ 241 vi C ONTENTS

Cross Section Data..................... 241 Section Orientation..................... 245 About the Edge Load and Point Load Nodes........... 247 Edge Load........................ 247 Point Load........................ 248 Phase.......................... 249 Prescribed Displacement/Rotation............... 249 Prescribed Velocity..................... 251 Prescribed Acceleration................... 252 Pinned.......................... 253 No Rotation....................... 253 Symmetry........................ 254 Antisymmetry....................... 254 Point Mass........................ 255 Point Mass Damping.................... 256 Theory for the Beam User Interface 257 About Beams....................... 257 In-Plane Euler Beams.................... 258 3D Euler Beam...................... 259 Strain-Displacement/Rotation Relation.............. 259 Stress-Strain Relation.................... 260 Thermal Strain....................... 260 Initial Load and Strain.................... 260 Implementation...................... 261 Stress Evaluation...................... 264 Thermal Coupling..................... 265 Coefficient of Thermal Expansion............... 266 Common Cross Sections................... 266 Chapter 6: Beam Cross Sections Using the Beam Cross Section User Interface 278 About Beams and Cross Section Data.............. 278 Using the Beam Cross Section User Interface........... 279 CONTENTS vii

The Beam Cross Section User Interface 284 Hole.......................... 285 Theory for the Beam Cross Section User Interface 286 Cross Section Properties................... 286 Computation of Stresses................... 300 Chapter 7: Trusses The Truss User Interface 304 Boundary, Edge, Point, and Pair Nodes for the Truss User Interface... 305 Linear Elastic Material.................... 306 Thermal Expansion..................... 308 Initial Stress and Strain.................... 308 Cross Section Data..................... 308 Initial Values........................ 309 About the Edge Load and Point Load.............. 309 Edge Load........................ 310 Phase.......................... 310 Straight Edge Constraint................... 311 Pinned.......................... 311 Prescribed Displacement................... 312 Prescribed Velocity..................... 312 Prescribed Acceleration................... 313 Symmetry........................ 314 Antisymmetry....................... 314 Point Mass........................ 315 Theory for the Truss User Interface 316 About Trusses....................... 316 Theory Background for the Truss User Interface.......... 317 viii CONTENTS

Chapter 8: Membranes The Membrane User Interface 324 Boundary, Edge, Point, and Pair Nodes for the Membrane User Interface........................ 326 Linear Elastic Material.................... 327 Initial Stress and Strain.................... 328 Initial Values........................ 329 Face Load........................ 329 Edge Load........................ 330 Prescribed Displacement................... 330 Theory for the Membrane User Interface 333 About Membranes..................... 333 Theory Background for the Membrane User Interface........ 334 Chapter 9: Multiphysics User Interfaces The Thermal Stress User Interface 338 Domain, Boundary, Edge, Point, and Pair Nodes for the Thermal Stress User Interface................... 342 Initial Values........................ 343 Thermal Linear Elastic Material................ 344 Thermal Hyperelastic Material................. 345 Thermal Linear Viscoelastic Material.............. 347 The Fluid-Structure Interaction User Interface 348 Domain, Boundary, Edge, Point, and Pair Nodes for the Fluid-Structure Interaction User Interface............ 351 Initial Values........................ 353 Fluid-Solid Interface Boundary................. 354 Basic Modeling Steps for Fluid-Structure Interaction........ 354 CONTENTS ix

Theory for the Fluid-Structure Interaction User Interface 357 The Joule Heating and Thermal Expansion User Interface 359 Initial Values........................ 363 Domain, Boundary, Edge, Point, and Pair Nodes for the Joule Heating and Thermal Expansion User Interface.......... 363 The Piezoelectric Devices User Interface 366 Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric Devices User Interface................... 369 Piezoelectric Material.................... 370 Electrical Material Model................... 371 Electrical Conductivity (Time-Harmonic)............. 373 Damping and Loss..................... 374 Remanent Electric Displacement................ 375 Dielectric Loss....................... 375 Initial Values........................ 376 Periodic Condition..................... 376 Theory for the Piezoelectric Devices User Interface 378 The Piezoelectric Effect................... 378 Piezoelectric Constitutive Relations............... 379 Piezoelectric Material.................... 381 Piezoelectric Dissipation................... 381 Initial Stress, Strain, and Electric Displacement........... 381 Geometric Nonlinearity for the Piezoelectric Devices User Interface.. 382 Damping and Losses Theory................. 384 References for the Piezoelectric Devices User Interface....... 387 Chapter 10: Glossary Glossary of Terms 390 x C ONTENTS

1 Introduction This guide describes the Structural Mechanics Module, an optional add-on package that extends the COMSOL Multiphysics modeling environment with customized physics interfaces that solve problems in the fields of structural and solid mechanics, including special physics interface for modeling of shells, membranes, beams, plates, and trusses. This chapter introduces you to the capabilities of this module and includes a summary of the physics interfaces as well as information about where you can find additional documentation and model examples. The last section is a brief overview with links to each chapter in this guide. About the Structural Mechanics Module Overview of the User s Guide 1

About the Structural Mechanics Module In this section: Why Structural Mechanics is Important for Modeling What Problems Can It Solve? The Structural Mechanics Module Physics Guide Available Study Types Geometry Levels for Study Capabilities Where Do I Access the Documentation and Model Library? Overview of the Physics and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual Why Structural Mechanics is Important for Modeling The Structural Mechanics Module solves problems in the fields of structural and solid mechanics, adding special physics interfaces for modeling shells and beams, for example. The physics interfaces in this module are fully multiphysics enabled, making it possible to couple them to any other physics interfaces in COMSOL Multiphysics or the other modules. Available physics interfaces include: Solid mechanics for 2D plane stress and plane strain, axial symmetry, and 3D solids Piezoelectric modeling Beams in 2D and 3D, Euler theory Truss and cable elements Shells and plates Membranes The module s study capabilities include static, eigenfrequency, time dependent (transient), frequency response, buckling, and parametric studies. There are also 2 CHAPTER 1: INTRODUCTION

predefined interfaces for linear elastic and viscoelastic materials. Materials can be isotropic, orthotropic, or fully anisotropic, and you can use local coordinate systems to specify material properties. Large deformations as well as contact and friction, can also be included in a study. Coupling structural analysis with thermal analysis is one example of multiphysics easily implemented with the module, which provides predefined multiphysics interfaces for thermal stress and other types of multiphysics. Piezoelectric materials, coupling the electric field and strain in both directions are fully supported inside the module through special multiphysics interfaces solving for both the electric potential and displacements. Piezoelectric materials can also be analyzed with the constitutive relations on either stress-charge or strain-charge form. Structural mechanics couplings are common in simulations done with COMSOL Multiphysics and occur in interaction with, for example, fluid flow (fluid-structure interaction, FSI), chemical reactions, acoustics, electric fields, magnetic fields, and optical wave propagation. What Problems Can It Solve? The Structural Mechanics Module contains a set of physics interfaces adapted to a broad category of structural-mechanics analysis. The module serves as an excellent tool for the professional engineer, researcher, and teacher. In education, the benefit of the short learning curve is especially useful because educators need not spend excessive time learning the software and can instead focus on the physics and the modeling process. The module is a collection of physics interfaces for COMSOL Multiphysics that handles static, eigenfrequency, transient, frequency response, parametric, transient thermal stress, and other analyses for applications in structural mechanics, solid mechanics, and piezoelectricity. STATIC ANALYSIS In a static analysis the load and constraints are fixed in time. EIGENFREQUENCY ANALYSIS An eigenfrequency analysis finds the damped or undamped eigenfrequencies and mode shapes of a model. Sometimes referred to as the free vibration of a structure. Pre-stress effects can be taken into account. ABOUT THE STRUCTURAL MECHANICS MODULE 3

TRANSIENT ANALYSIS A transient analysis finds the transient response for a time-dependent model, taking into account mass, mass moment of inertia. The transient analysis can be either direct, or using a modal solution. FREQUENCY RESPONSE ANALYSIS A frequency response analysis finds the steady-state response from harmonic loads. The frequency-response analysis can be either direct, or using a modal solution. Effects of pre-stress can be included. LINEAR BUCKLING STUDY A linear buckling analysis uses the stiffness coming from stresses and material to define an eigenvalue problem where the eigenvalue is a load factor that, when multiplied with the actual load, gives the critical load in a linear context. PARAMETRIC ANALYSIS A parametric analysis finds the solution dependence due to the variation of a specific parameter, which could be, for instance, a material property or the position of a load. THERMAL STRESS In a transient thermal stress study, the program neglects mass effects, assuming that the time scale in the structural mechanics problem is much smaller than the time scale in the thermal problem. LARGE DEFORMATIONS You can also enable the geometric nonlinearity for the Linear Elastic Material under all structural mechanics interfaces except the Beam interface. The engineering strain is then replaced with the Green-Lagrange strain and the stress with the second Piola-Kirchhoff stress. Such material is suitable to study deformations accompanied by possible large rotations but small to moderate strains in the material, and it is sometimes referred to as Saint Venant-Kirchhoff hyperelastic material. To solve the problem, the program uses a total Lagrangian formulation. Hyperelastic material and elastoplastic materials for small and large strains are available with the Nonlinear Structural Materials Module. 4 CHAPTER 1: INTRODUCTION

Additional functionality and material models for geomechanics and soil mechanics soil plasticity, concrete, and rock material models is available with the Geomechanics Module. ELASTOPLASTIC MATERIALS An elastoplastic analysis involves a nonlinear material with or without hardening. Three different hardening models are available: Isotropic hardening Kinematic hardening Perfectly plastic hardening The elastoplastic materials are available in the Solid Mechanics interface. CREEP AND VISCOPLASTIC MATERIALS A number of different material models for creep and viscoplascticity are available. In these materials the rate of elastic strain depends on the stress. HYPERELASTIC MATERIALS In hyperelastic materials the stresses are computed from a strain energy density function. They are often used to model rubber, but also used in acoustic elasticity. Many different models are available. The hyperelastic materials are available in the Solid Mechanics interface. The hyperelastic, elastoplastic and creep materials are available with the addition of the Nonlinear Structural Materials Module. LINEAR VISCOELASTIC MATERIALS Viscoelastic materials have a time-dependent response, even if the loading is constant. The Linear Viscoelastic Material is available in the Solid Mechanics interface. CONTACT MODELING You can model contact between parts of a structure. The Solid Mechanics interface supports contact with or without friction. The contact algorithm is implemented based on the augmented Lagrangian method. ABOUT THE STRUCTURAL MECHANICS MODULE 5

PHYSICS USER INTERFACES AND APPLICATIONS Examples of applications include thin plates loaded in a plane (plane stress), thick structures with no strain in the out-of-plane direction (plane strain), axisymmetric structures, thin-walled 3D structures (shells), and general 3D structures modeled using solid elements. The Structural Mechanics Module Physics Guide The Acoustic-Structure Interaction and Poroelasticity interfaces require, and couple with, the Structural Mechanics Module and are discussed in the applicable user guides. For details about the Acoustic-Structure Interaction interface, see the Acoustic Module User s Guide. For details about the Poroelasticity interface, see the Subsurface Flow Module User s Guide. At any time, a new model can be created or physics added. Right-click the Root (top) node and select Add Model to open the Model Wizard, or right-click a Model node and select Add Physics. Depending on the physics interface, specify parameters defining a problem on points, edges (3D), boundaries, and domains. It is possible to specify loads and constraints on all available geometry levels, but material properties can only be specified for the domains, except for shells, membranes, beams, and trusses, where they are defined on the boundary or edge level. Structural Mechanics Modeling In the COMSOL Multiphysics Reference Manual: Studies and the Study Nodes The Physics User Interfaces For a list of all the interfaces included with the COMSOL Multiphysics basic license, see Physics Guide. 6 CHAPTER 1: INTRODUCTION

PHYSICS USER INTERFACE Fluid Flow ICON TAG SPACE DIMENSION AVAILABLE PRESET STUDY TYPE Fluid-Structure Interaction fsi Structural Mechanics 3D, 2D, 2D axisymmetric stationary; stationary, one-way coupled; time dependent; time dependent, one-way coupled Solid Mechanics* solid 3D, 2D, 2D axisymmetric Thermal Stress ts 3D, 2D, 2D axisymmetric stationary; eigenfrequency; prestressed analysis, eigenfrequency; time dependent; time dependent modal; frequency domain; frequency-domain modal; prestressed analysis, frequency domain; modal reduced order model; linear buckling stationary; eigenfrequency; frequency domain; time dependent Shell shell 3D stationary; eigenfrequency; prestressed analysis, eigenfrequency; time dependent; time dependent modal; frequency domain; frequency-domain modal; prestressed analysis, frequency domain; modal reduced order model; linear buckling Plate plate 2D stationary; eigenfrequency; prestressed analysis, eigenfrequency; time dependent; time dependent modal; frequency domain; frequency-domain modal; prestressed analysis, frequency domain; modal reduced order model; linear buckling Beam beam 3D, 2D stationary; eigenfrequency; frequency domain; frequency-domain modal; time dependent; time dependent modal; modal reduced order model Beam Cross Section bcs 2D stationary ABOUT THE STRUCTURAL MECHANICS MODULE 7

PHYSICS USER INTERFACE Truss truss 3D, 2D stationary; eigenfrequency; prestressed analysis, eigenfrequency; time dependent; time dependent modal; frequency domain; frequency-domain modal; prestressed analysis, frequency domain; modal reduced order model; linear buckling Membrane mem 3D, 2D, 2D axisymmetric Joule Heating and Thermal Expansion Piezoelectric Devices ICON TAG SPACE DIMENSION tem pzd 3D, 2D, 2D axisymmetric 3D, 2D, 2D axisymmetric AVAILABLE PRESET STUDY TYPE stationary; eigenfrequency; prestressed analysis, eigenfrequency; time dependent; time dependent modal; frequency domain; frequency-domain modal; prestressed analysis, frequency domain stationary; eigenfrequency; time dependent stationary; eigenfrequency; time dependent; time-dependent modal; frequency domain; frequency domain modal; modal reduced order model * This is an enhanced interface, which is included with the base COMSOL package but has added functionality for this module. SHOW MORE PHYSICS OPTIONS There are several general options available for the physics user interfaces and for individual nodes. This section is a short overview of these options, and includes links to additional information when available. The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF, only from within the online help. To locate and search all the documentation for this information, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree. 8 CHAPTER 1: INTRODUCTION

To display additional options for the physics interfaces and other parts of the model tree, click the Show button ( ) on the Model Builder and then select the applicable option. After clicking the Show button ( ), additional sections get displayed on the settings window when a node is clicked and additional nodes are available from the context menu when a node is right-clicked. For each, the additional sections that can be displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, and Inconsistent Stabilization. You can also click the Expand Sections button ( ) in the Model Builder to always show some sections or click the Show button ( ) and select Reset to Default to reset to display only the Equation and Override and Contribution sections. For most nodes, both the Equation and Override and Contribution sections are always available. Click the Show button ( ) and then select Equation View to display the Equation View node under all nodes in the Model Builder. Availability of each node, and whether it is described for a particular node, is based on the individual selected. For example, the Discretization, Advanced Settings, Consistent Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings. SECTION Show More Options and Expand Sections Discretization Discretization Splitting of complex variables Consistent and Inconsistent Stabilization Constraint Settings Override and Contribution CROSS REFERENCE Advanced Physics Sections The Model Wizard and Model Builder Show Discretization Discretization (Node) Compile Equations Show Stabilization Numerical Stabilization Weak Constraints and Constraint Settings Physics Exclusive and Contributing Node Types OTHER COMMON SETTINGS At the main level, some of the common settings found (in addition to the Show options) are the Interface Identifier, Domain, Boundary, or Edge Selection, and Dependent Variables. ABOUT THE STRUCTURAL MECHANICS MODULE 9

At the nodes level, some of the common settings found (in addition to the Show options) are Domain, Boundary, Edge, or Point Selection, Material Type, Coordinate System Selection, and Model Inputs. Other sections are common based on application area and are not included here. SECTION Coordinate System Selection Domain, Boundary, Edge, and Point Selection Interface Identifier Material Type Model Inputs Pair Selection CROSS REFERENCE Coordinate Systems About Geometric Entities About Selecting Geometric Entities Predefined Physics Variables Variable Naming Convention and Scope Viewing Node Names, Identifiers, Types, and Tags Materials About Materials and Material Properties Selecting Physics Adding Multiphysics Couplings Identity and Contact Pairs Continuity on Interior Boundaries Available Study Types The Structural Mechanics Module performs stationary, eigenfrequency, time-dependent, frequency domain (frequency response), linear buckling, parametric, quasi-static, and viscoelastic transient initialization studies. The different study types require different solvers and equations. STATIONARY STUDY A static analysis solves for stationary displacements, rotations, and temperature (depending on the type of physics interface). All loads and constraints are constant in time. The equations include neither mass nor mass moment of inertia. EIGENFREQUENCY STUDY An eigenfrequency study solves for the eigenfrequencies and the shape of the eigenmodes. When performing an eigenfrequency analysis, specify whether to look at the mathematically more fundamental eigenvalue,, or the eigenfrequency, f, which is more commonly used in a structural mechanics context. 10 CHAPTER 1: INTRODUCTION

f = -------- 2i If damping is included in the model, an eigenfrequency solution returns the damped eigenvalues. In this case, the eigenfrequencies and mode shapes are complex. TIME-DEPENDENT STUDY A time-dependent (transient) study solves a time-dependent problem where loads and constraints can vary in time. Time dependent studies can be performed using either a direct or a modal method. FREQUENCY DOMAIN STUDY A frequency domain study (frequency response analysis) solves for the linear response from harmonic loads. Frequency domain studies can be performed using either a direct or a modal method. For a frequency domain study, the harmonic loads is divided into two parts: The amplitude, F The phase (F Ph ) Together they define a harmonic load whose amplitude and phase shift can depend on the excitation angular frequency or excitation frequency f. F freq = 2f = F cos t+ F Ph --------- 180 The result of a frequency response analysis is a complex time-dependent displacement field, which can be interpreted as an amplitude u amp and a phase angle u phase. The actual displacement at any point in time is the real part of the solution: u = u amp cos2f t + u phase COMSOL Multiphysics allows the visualization of the amplitudes and phases as well as the solution at a specific angle (time). The solution at angle parameter makes this task easy. When plotting the solution, the program multiplies it by e i, where is the angle in radians that corresponds to the angle (specified in degrees) in the Solution at angle field. COMSOL Multiphysics plots the real part of the evaluated expression: u = u amp cos + u phase ABOUT THE STRUCTURAL MECHANICS MODULE 11

The angle is available as the variable phase (in radians) and is allowed in plotting expressions. Both freq and omega are available variables. In a frequency response analysis, everything is treated as harmonic: prescribed displacements, velocities, accelerations, thermal strains, and initial stress and strains; not only the forces. LINEAR BUCKLING STUDY A linear buckling study includes the stiffening effects from stresses coming from nonlinear strain terms. The two stiffnesses from stresses and material define an eigenvalue problem where the eigenvalue is a load factor that, when multiplied with the actual load, gives the critical load the value of a given load that causes the structure to become unstable in a linear context. The linear buckling study step uses the eigenvalue solver. Another way to calculate the critical load is to include large deformation effects and increase the load until the load has reached its critical value. Linear buckling is available in the Solid Mechanics, Shell, Plate, and Truss interfaces. PRESTRESSED ANALYSIS, EIGENFREQUENCY AND FREQUENCY DOMAIN The Prestressed Analysis, Eigenfrequency, and Prestress Analysis, Frequency Domain study types make it possible to compute the eigenfrequencies and the response to harmonic loads that are affected by a static preload. These studies involve two study steps for the solver (a Stationary study step plus an Eigenfrequency or Frequency Domain study step). You need to add a new study to the model to get access to such combined study types, and they cannot be added directly as new study steps to the existing study (solver sequence). You also have them available when starting a new model. VISCOELASTIC TRANSIENT INITIALIZATION A viscoelastic transient initialization precomputes initial states for transient and quasi-static transient analyses when the Linear Viscoelastic Material is used. It is a regime of instantaneous deformation and/or loading. Viscoelastic transient initialization is available only in the Solid Mechanics interfaces. 12 CHAPTER 1: INTRODUCTION

THERMAL COUPLINGS Solids expand with temperature, which causes thermal strains to develop in the material if the deformation is constrained. These thermal strains combine with the elastic strains from structural loads to form the total strain: = el + th Thermal strain depends on the temperature, T, the stress-free reference temperature, T ref, and the coefficient of thermal expansion, : th = T T ref Thermal expansion affects displacements, stresses, and strains. This effect is added automatically in The Thermal Stress User Interface and The Joule Heating and Thermal Expansion User Interface. Also add thermal expansion to the other interfaces. Only the coefficient of thermal expansion needs to be specified and the two temperature fields, T and T ref. The temperature field is a model input that typically is computed by a Heat Transfer user interface. Temperature coupling can be used in any type of study. Linearized Buckling Analysis Eigenfrequency Analysis Modeling Geometric Nonlinearity Tips for Selecting the Correct Solver Studies and Solvers in the COMSOL Multiphysics Reference Manual For an example of a Prestressed Analysis, Eigenfrequency study, see Eigenfrequency Analysis of a Rotating Blade: Model Library path Structural_Mechanics_Module/Dynamics_and_Vibration/rotating_blade. Geometry Levels for Study Capabilities The column for the dependent variables shows the field variables that formulate the underlying equations. Depending on the engineering assumptions and the geometry dimension, these variables include a subset of the displacement field u, v, and w in the global coordinate system, pressure, and temperature. The Piezoelectric Devices interface also includes the electric potential V. The Shell and Plate interfaces use as ABOUT THE STRUCTURAL MECHANICS MODULE 13

dependent variables the variables a x, a y, and a z, which are the displacements of the shell normals in the global x, y, and z directions, respectively. Such variables can be expressed in terms of customary rotations x, y, and z about the global axes. For each physics interface, the table indicates dependent variables and the geometry levels (where data such as material properties, loads, and constraints are specified). Edges exist only in 3D geometries. Studies are listed in a separate table in The Structural Mechanics Module Physics Guide section. Studies and Solvers in the COMSOL Multiphysics Reference Manual PHYSICS GEOMETRY LEVEL DEFAULT NAME DEPENDENT VARIABLES POINTS EDGES BOUNDARIES DOMAINS STRUCTURAL MECHANICS Solid Mechanics solid u, (p) Shell shell u, a Plate (3 DOF) plate w, a x, a y Plate (6 DOF) u, a Beam beam u, Truss truss u Membrane mem u Thermal Stress ts u, (p), T Joule Heating and tem u, (p), T, Thermal Expansion V Piezoelectric pzd u,v Devices Fluid Flow Fluid-Structure Interaction fsi u solid, u fluid, p 14 CHAPTER 1: INTRODUCTION

Where Do I Access the Documentation and Model Library? A number of Internet resources provide more information about COMSOL, including licensing and technical information. The electronic documentation, context help, and the Model Library are all accessed through the COMSOL Desktop. If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different guide. However, if you are using the online help in COMSOL Multiphysics, these links work to other modules, model examples, and documentation sets. THE DOCUMENTATION The COMSOL Multiphysics Reference Manual describes all user interfaces and functionality included with the basic COMSOL Multiphysics license. This book also has instructions about how to use COMSOL and how to access the documentation electronically through the COMSOL Help Desk. To locate and search all the documentation, in COMSOL Multiphysics: Press F1 or select Help>Help ( ) from the main menu for context help. Press Ctrl+F1 or select Help>Documentation ( ) from the main menu for opening the main documentation window with access to all COMSOL documentation. Click the corresponding buttons ( or ) on the main toolbar. and then either enter a search term or look under a specific module in the documentation tree. If you have added a node to a model you are working on, click the Help button ( ) in the node s settings window or press F1 to learn more about it. Under More results in the Help window there is a link with a search string for the node s name. Click the link to find all occurrences of the node s name in the documentation, including model documentation and the external COMSOL website. This can help you find more information about the use of the node s functionality as well as model examples where the node is used. ABOUT THE STRUCTURAL MECHANICS MODULE 15

THE MODEL LIBRARY Each model comes with documentation that includes a theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. You can use the step-by-step instructions and the actual models as a template for your own modeling and applications. In most models, SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available. To open the Model Library, select View>Model Library ( ) from the main menu, and then search by model name or browse under a module folder name. Click to highlight any model of interest, and select Open Model and PDF to open both the model and the documentation explaining how to build the model. Alternatively, click the Help button ( ) or select Help>Documentation in COMSOL to search by name or browse by module. The model libraries are updated on a regular basis by COMSOL in order to add new models and to improve existing models. Choose View>Model Library Update ( ) to update your model library to include the latest versions of the model examples. If you have any feedback or suggestions for additional models for the library (including those developed by you), feel free to contact us at info@comsol.com. CONTACTING COMSOL BY EMAIL For general product information, contact COMSOL at info@comsol.com. To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to support@comsol.com. An automatic notification and case number is sent to you by email. 16 CHAPTER 1: INTRODUCTION

COMSOL WEBSITES COMSOL website Contact COMSOL Support Center Download COMSOL Support Knowledge Base Product Updates COMSOL Community www.comsol.com www.comsol.com/contact www.comsol.com/support www.comsol.com/support/download www.comsol.com/support/knowledgebase www.comsol.com/support/updates www.comsol.com/community ABOUT THE STRUCTURAL MECHANICS MODULE 17

Overview of the User s Guide The Structural Mechanics Module User s Guide gets you started with modeling using COMSOL Multiphysics. The information in this guide is specific to this module. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual. As detailed in the section Where Do I Access the Documentation and Model Library? this information can also be searched from the Help menu in COMSOL Multiphysics. TABLE OF CONTENTS, GLOSSARY, AND INDEX To help you navigate through this guide, see the Contents, Glossary, and Index. MODELING WITH THE STRUCTURAL MECHANICS MODULE The Structural Mechanics Modeling chapter gives you an insight on how to approach the modeling of various structural mechanics problems. The contents cover subjects including topics such as Applying Loads, Defining Constraints, Calculating Reaction Forces, and Eigenfrequency Analysis. It also provides you with an Introduction to Material Models helps you start Defining Multiphysics Models and Modeling Geometric Nonlinearity. THE SOLID MECHANICS USER INTERFACE The Solid Mechanics chapter describes The Solid Mechanics User Interface, which is used to model 3D solids, plane strain and plane stress 2D models, and axisymmetric models. An overview of Solid Mechanics Geometry and Structural Mechanics Physics Symbols and the Theory for the Solid Mechanics User Interface is also included. THE SHELL AND PLATE USER INTERFACES The Shells and Plates chapter describes The Shell and Plate User Interfaces, which are used to model thin 3D structures (shell) and out-of-plane loaded plates (plate). The underlying theory for each interface is also included at the end of the chapter. THE BEAM USER INTERFACE The Beams chapter describes The Beam User Interface, which models Euler (Euler-Bernoulli) beams for modeling slender 3D and 2D structures. Typical examples 18 CHAPTER 1: INTRODUCTION

are frameworks and latticeworks. The underlying theory for the interface is also included at the end of the chapter. THE BEAM CROSS SECTION USER INTERFACE The Beam Cross Sections chapter describes The Beam Cross Section User Interface, which is used for computing cross section properties for beams. It can also be used for a detailed evaluation of stresses in a beam when the section forces to which it is subjected are known. The first section discusses Using the Beam Cross Section User Interface, and the last section is the underlying theory for the interface. THE TRUSS USER INTERFACE The Trusses chapter describes The Truss User Interface, which models slender 3D and 2D structures with components capable to withstand axial forces only. Typical applications are latticeworks, but it can also be used for modeling cables. The underlying theory for the interface is also included at the end of the chapter. THE MEMBRANE USER INTERFACE The Membranes describes The Membrane User Interface, which can be used for prestressed membranes, cladding on solids, and balloons, for example. The underlying theory for the interface is also included at the end of the chapter. THE MULTIPHYSICS USER INTERFACES The Multiphysics User Interfaces chapter describes these interfaces found under the Structural Mechanics branch of the Model Wizard: The Thermal Stress User Interface combines a Solid Mechanics interface with a Heat Transfer interface. The coupling appears on the domain level, where the temperature from the Heat Transfer interface acts as a thermal load for the Solid Mechanics interface, causing thermal expansion. The Joule Heating and Thermal Expansion User Interface combines solid mechanics using a thermal linear elastic material with an electromagnetic Joule heating model. This is a multiphysics combination of solid mechanics, electric currents, and heat transfer for modeling of, for example, thermoelectromechanical (TEM) applications. The Piezoelectric Devices User Interface include a piezoelectric material but also full functionality for Solid Mechanics and Electrostatics. Piezoelectric materials in 3D, 2D plane strain and plane stress, and axial symmetry, optionally combined with other solids and air, for example. OVERVIEW OF THE USER S GUIDE 19

20 CHAPTER 1: INTRODUCTION The Fluid-Structure Interaction User Interface, which is found under the Fluid Flow branch of the Model Wizard, is also described in this chapter. The interface combines fluid flow with solid mechanics to capture the interaction between the fluid and the solid structure. A Solid Mechanics interface and a single-phase flow interface model the solid and the fluid, respectively. The fluid-structure interactions appear on the boundaries between the fluid and the solid.

2 Structural Mechanics Modeling The goal of this chapter is to give you an insight on how to approach the modeling of various structural mechanics problems. In this chapter: Applying Loads Defining Constraints Calculating Reaction Forces Introduction to Material Models Defining Multiphysics Models Modeling Geometric Nonlinearity Linearized Buckling Analysis Introduction to Contact Modeling Eigenfrequency Analysis Using Modal Superposition Modeling Damping and Losses Piezoelectric Losses Springs and Dampers Tips for Selecting the Correct Solver 21

Applying Loads An important aspect of structural analysis is the formulation of the forces applied to the modeled structure. The freedom is available to use custom expressions, predefined or user-defined coordinate systems, and even variables from other modeling interfaces. Loads can be applied in the structural mechanics interfaces on the body, face, edge, or point levels. Add The Solid Mechanics User Interface ( ) to the Model Builder, then right-click the node to select Body Load, Face Load, Edge Load, or Point Load from the context menu. This guide includes a detailed description of the functionality for each physics interface. In this section: Units, Orientation, and Visualization Load Cases Singular Loads Moments in the Solid Mechanics User Interface Pressure Acceleration Loads Temperature Loads Thermal Expansion Total Loads Units, Orientation, and Visualization USING UNITS Enter loads in any unit, independently of the base SI unit system in the model, because COMSOL Multiphysics automatically converts any unit to the base SI unit system. To use the feature for automatic unit conversion, enter the unit in square brackets, for example, 100[lbf/in^2]. PREDEFINED AND CUSTOM COORDINATE SYSTEMS In this module, different predefined coordinate systems are available when loads are specified. There is always the global coordinate system. Depending on the dimensionality of the part being worked with, there can also be predefined coordinate systems such as and the local tangent and normal coordinate system for boundaries. 22 CHAPTER 2: STRUCTURAL MECHANICS MODELING

Custom coordinate systems are also available and are useful, for example, to specify a load in any direction without splitting it into components. Right-click the Definitions node ( ) in the Model Tree, to select a Coordinate System from the context menu. Some coordinate systems can have solution dependent axis directions. If you use such a system for defining a load, the directions of the load follow the moving coordinate axis directions if the Include geometric nonlinearity check box is selected under the Study settings section of the current study step. In the COMSOL Multiphysics Reference Manual: Using Units Coordinate Systems Load Cases For a Stationary study, you can define load cases and constraint cases. A load or constraint can be assigned to a load or constraint case, and then used conditionally. Using Load Cases in the COMSOL Multiphysics Reference Manual For an example about how to set up expressions for controlling position and distribution of loads using load cases, see Pratt Truss Bridge: Model Library path Structural_Mechanics_Module/Civil_Engineering/ pratt_truss_bridge. Singular Loads In reality, loads always act on a finite area. However, in a model a load is sometimes defined on a point or an edge, which leads to a singularity. The reason for this is that points and lines have no area, so the stress becomes infinite. Because of the stress singularity, there are high stress values in the area surrounding the applied load. The size of this area and the magnitude of the stresses depend on both the mesh and the APPLYING LOADS 23