What is Nonlinear Finite Element Analysis in Solid Mechanics?

Reality is Nonlinear

Structural analysis is nonlinear in nature because the real-world is nonlinear. However, sometimes the structural engineers can do good approximations by approaching the problem with a linear analysis and obtaining good results. But also, it may yield to wrong results if significant nonlinear behaviors occur in the structure.

In the beginning, in the 1970s, most FEA software did not have nonlinear capabilities. This feature was implemented around 1977, when database technology was introduced. To obtain a solution, a nonlinear analysis requires an iterative and incremental process. However, the first implementations of nonlinear capabilities did not use automatic methods, and the user’s intervention was required at every iteration.

Nowadays, the very advanced capabilities of nonlinear solvers are based on algorithms that use automated iteration methods, with convergence criteria. To understand and use finite element nonlinear solutions, the FEA analyst must understand the deep interaction and mutual enrichment that exist between the physical aspects of a problem and its mathematical formulation.

The Three Types of Nonlinearity

In structural analysis, a nonlinear effect can occur because of three types of nonlinearity:

Geometric nonlinearity: if a continuous body undergoes large deformations, the strain-displacement relations become nonlinear. Moreover, under large deformations, the stiffness of the system will change with deformation, making the problem nonlinear.
Material nonlinearity: if a material does not follow Hooke’s law, nonlinear material models must be used.
Boundary nonlinearity: the most frequent boundary nonlinearities are encountered in contact problems.

Nonlinear Geometric

Under large deformation, the deformed structure has a different geometry, implying a changing stiffness. The stiffness matrix [K] is a function of displacements {u}.

{P} = [K(u)]{u}

Nonlinear Material

The stiffness response depends on deformation. The stiffness matrix [K] is a function of displacements {u}.

{P} = [K(u)]{u}

Boundary Nonlinearity

Boundary conditions changing with deformation: the size c of the contact surface and the contact load Rc depend on deformation and load. The stiffness matrix [K] and loading {P} are functions of displacements {u}.

{P(u)} = [K(u)]{u}

What is a Nonlinear System?

Mathematically speaking, a nonlinear system is one whose behavior is not equal to the sum of its parts. Therefore, the behavior of a nonlinear system does not satisfy the principle of superposition. In a linear analysis, it is assumed that the response of the structure (deformation, internal loads, or stresses, etc.) is linearly proportional to the applied loads. However, in real life, this response may not be linearly proportional to the applied load and then the structure must be analyzed using nonlinear assumptions. In linear static analysis, the stiffness [K] of the analyzed structure is assumed to be constant.

In the real world, it is very likely that a structure will behave in a nonlinear manner, for geometrical, material, or boundary reasons. Indeed, the stiffness of the structure is based on its geometry and material properties. In a linear analysis, these parameters are assumed to be unchanged while the loading is applied. In a nonlinear analysis, these changes are taken into account, and the stiffness matrix is updated using the deformed structure’s configuration, after each incremental load application.

Characteristics of a Nonlinear System

LOAD-DISPLACEMENT RELATION

The stiffness of the analyzed structure is not constant and varies with the applied loads. The displacements are large (translations and rotations) and are not related to the original stiffness of the structure.

STRESS-STRAIN RELATION

Stresses and strains are not related to a linear function.

SCALABILITY

The results of a nonlinear analysis cannot be scaled.

SUPERPOSITION

The principle of superposition cannot be applied. If a load P1 produces a displacement d1 and a load P2 a displacement d2, then the load P1 + P2 will not cause a displacement d1 + d2.

INITIAL STATE OF STRESS

The initial state of stress (residual stresses, temperature, pre-stressing) may be extremely important in the overall response.

LOAD HISTORY

The structure’s response is related to the load history: it is influenced by the loading sequence. When several subcases are applied in sequence in the structure, the end of a subcase is the initial condition for the next subcase.

REVERSIBILITY

The deformation of the structure is not fully reversible once the applied loads are removed.

SOLUTION SETTINGS

The external loads are applied in small increments, and iterations are performed to ensure that equilibrium is satisfied at each load increment. Solution monitoring by the user is required to ensure convergence. The computing time is usually large. While linear problems always have a unique solution, a nonlinear problem might not. In fact, the iterative and incremental processes used to solve nonlinear problems may not converge and may even produce an incorrect solution at convergence.

Do You Really Need to Conduct a Nonlinear Analysis?

A nonlinear analysis requires more resources in terms of disk space and computing time, so it is important to ensure that it is really necessary.

If you are planning to run a nonlinear analysis, you should answer a few simple questions to decide whether you really need to, or whether a linear analysis will suffice. If you answer "yes" to one of the following questions, you should go with a nonlinear analysis:

Does the structure deform significantly?

Does the structure exhibits stress stiffening (tension-bending coupling in a membrane under pressure for example)?

Do the stresses exceed the proportional limit?

Does the stiffness of the structure change when a loading is applied?

Do you expect to capture contact conditions (engaged or disengaged) between some components of you model?

What Do you Need to Learn to Perform Nonlinear Analysis?

For the FEA learning process of nonlinear analysis, it is important to learn in detail the three types of nonlinearity: geometric, material and contact. It is strongly recommended to learn them separately.

It is also important to learn how the FE solvers compute nonlinear problems. The FE analyst must understand the various methods used by the solvers in order to select the proper method which better suit to her/his problem.

"Chapter 21 – Nonlinear Static Analysis" & "Chapter 23 – Nonlinear Buckling Analysis" from “Practical Finite Element Analysis for Mechanical Engineers – First Edition” cover all the aspects of nonlinear analysis for solving solid mechanics and structural problems.

These chapters cover in detail the methods used to solve the three common nonlinearities mentioned above. The methods, guidelines and recommendations which permit to build reliable nonlinear analysis are presented and practical examples illustrate each type of nonlinearity.

See below, the detailed table of content for these two chapters.

Table of Content of Chapters About Nonlinear Analysis

Select a chapter below

Nonlinear Static Analysis
Nonlinear Buckling Analysis

Chapter 21 - NONLINEAR STATIC ANALYSIS

21.1 OVERVIEW
21.2 WHAT IS A NONLINEAR SYSTEM?
21.3 CHARACTERISTICS OF A NONLINEAR ANALYSIS
21.3.1 LOAD-DISPLACEMENT RELATION
21.3.2 STRESS-STRAIN RELATION
21.3.3 SCALABILITY
21.3.4 SUPERPOSITION
21.3.5 INITIAL STATE OF STRESS
21.3.6 LOAD HISTORY
21.3.7 REVERSIBILITY
21.3.8 SOLUTION SETTINGS
21.4 GEOMETRIC NONLINEARITY
21.4.1 SOURCES OF GEOMETRICAL NONLINEARITY
21.4.2 HOW DOES NONLINEAR GEOMETRY WORK?
21.4.3 DO YOU REALLY NEED A NONLINEAR GEOMETRIC ANALYSIS?
21.4.4 THE FOLLOWER LOAD CONCEPT
21.4.5 SMALL OR LARGE STRAIN?
21.4.6 EXAMPLE OF GEOMETRIC NONLINEARITY
21.5 MATERIAL NONLINEARITY
21.5.1 YIELD CRITERIA
21.5.2 HARDENING RULES
21.5.3 MATERIAL MODELS
21.5.4 ENGINEERING STRESS-STRAIN OR TRUE STRESS-STRAIN?
21.5.5 HOW DOES NONLINEAR MATERIAL WORK?
21.5.6 DO YOU REALLY NEED A NONLINEAR MATERIAL ANALYSIS?
21.6 BOUNDARY NONLINEARITY
21.6.1 LOAD VARIATION
21.6.2 CONSTRAINT VARIATION
21.6.3 CONTACTS
21.7 CHOOSING THE RIGHT ELEMENTS FOR A NONLINEAR ANALYSIS
21.8 HOW DO FEA SOFTWARE COMPUTE NONLINEAR PROBLEMS?
21.8.1 CHARACTERIZATION AND FORMULATION OF A NONLINEAR PROBLEM
21.8.2 NEWTON-RAPHSON METHOD
21.8.3 MODIFIED NEWTON-RAPHSON METHOD
21.8.4 NEWTON-RAPHSON METHOD EXAMPLES
21.8.5 COMPUTATIONAL METHODS IN NONLINEAR ANALYSIS
21.8.6 EQUILIBRIUM PATH AND CRITICAL POINTS
21.8.7 ADAPTIVE SOLUTION STRATEGIES
21.8.8 STIFFNESS MATRIX UPDATE STRATEGIES
21.8.9 CHOOSING THE INCREMENTAL LOAD STEP
21.8.10 ARC-LENGTH METHODS
21.8.11 LINE SEARCH PROCEDURES
21.8.12 CONVERGENCE CRITERIA
21.8.13 HOW TO DEAL WITH CONVERGENCE ISSUES
21.8.14 SUMMARY OF ITERATIVE SOLUTION SCHEMES
21.8.15 HOW TO SELECT THE RIGHT ITERATIVE SOLUTION SCHEME
21.8.16 SUMMARY OF THE NONLINEAR SOLUTION STRATEGY
21.9 GENERAL RECOMMENDATIONS FOR NONLINEAR ANALYSIS
21.9.1 UNDERSTAND THE NONLINEAR FEATURES
21.9.2 UNDERSTAND YOUR PROBLEM AND STRUCTURAL BEHAVIOR
21.9.3 UNDERSTAND THE DIFFERENCE BETWEEN A LINEAR SUBCASE AND A NONLINEAR SUBCASE
21.9.4 SIMPLIFY YOUR MODEL
21.9.5 USE AN ADEQUATE MESH AND ELEMENT TYPES
21.9.6 APPLY LOADING GRADUALLY
21.9.7 READ THE OUTPUT
21.9.8 NUMBER OF INCREMENTS
21.9.9 CONVERGENCE PROBLEMS
21.9.10 KEEP AN EYE ON YOUR MATERIAL DEFINITION
21.10 COMMON MISTAKES IN NONLINEAR ANALYSIS
21.11 EXAMPLES OF NONLINEAR STATIC ANALYSIS
21.11.1 GEOMETRIC NONLINEARITY AND HISTORY PATH
21.11.2 CUMULATIVE EFFECT OF A NONLINEAR ANALYSIS
21.11.3 INFLUENCE OF THE INCREMENTAL LOAD STEP ON RESULTS
21.11.4 MATERIAL NONLINEARITY: ELASTOPLASTIC PLATE
21.11.5 HIGHLY NONLINEAR PROBLEM

Chapter 23 - NONLINEAR BUCKLING ANALYSIS

23.1 OVERVIEW
23.2 WHY PERFORM A NONLINEAR BUCKLING ANALYSIS?
23.3 THE STABILITY PATH AND THE CONVERGED SOLUTION
23.4 NONLINEAR BUCKLING PROCEDURE
23.5 POST-BUCKLING
23.6 ESSENTIAL STEPS IN NONLINEAR BUCKLING ANALYSIS
23.7 EXAMPLES OF NONLINEAR BUCKLING ANALYSIS
23.7.1 NONLINEAR BUCKLING OF A CURVED PANEL
23.7.2 SNAP-THROUGH: NEWTON-RAPHSON VS ARC-LENGTH