非线性解析概要-Abaqus 6.9 Nonlinearity (1).docx

NonlinearityThis chapter discusses nonlinear structural analysis in Abaqus. The differencesbetween linear and nonlinear analyses are summarized below.Linear analysisAll the analyses discussed so far have been linear: there is a linear relationshipbetween the applied loads and the response of the system. For example, if a linearspring extends statically by 1 m under a load of 10 N, it will extend by 2 m when a loadof 20 N is applied. This means that in a linear Abaqus/Standard analysis the flexibilityof the structure need only be calculated once (by assembling the stiffness matrix andinverting it). The linear response of the structure to other load cases can be found bymultiplying the new vector of loads by the inverted stiffness matrix. Furthermore, thestructure's response to various load cases can be scaled by constants and/orsuperimposed on one another to determine its response to a completely new loadcase, provided that the new load case is the sum (or multiple) of previous ones. Thisprinciple of superposition of load cases assumes that the same boundary conditionsare used for all the load cases.Abaqus/Standard uses the principle of superposition of load cases in linear dynamicssimulations, which are discussed in Chapter 7, “Linear Dynamics.”Nonlinear analysisA nonlinear structural problem is one in which the structure's stiffness changes as itdeforms. All physical structures are nonlinear. Linear analysis is a convenientapproximation that is often adequate for design purposes. It is obviously inadequatefor many structural simulations including manufacturing processes, such as forging orstamping; crash analyses; and analyses of rubber components, such as tires or enginemounts. A simple example is a spring with a nonlinear stiffening response (see Figure8-1).Figure 8-1 Linear and nonlinear spring characteristics.ForceForceDisplacementLinear spring.DisplacementNonlinear spring.Stiffness is not constantStillness is constantSince the stiffness is now dependent on the displacement the initial flexibility can nolonger be multiplied by the applied load to calculate the spring's displacement for anyload. In a nonlinear implicit analysis the stiffness matrix of the structure has to beassembled and inverted many times during the course of the analysis, making it muchmore expensive to solve than a linear implicit analysis. In an explicit analysis theincreased cost of a nonlinear analysis is due to reductions in the stable timeincrement The stable time increment is discussed further in Chapter 9NonlineaeExplicit Dynamics.”Since the response of a nonlinear system is not a linear function of the magnitude ofthe applied load, it is not possible to create solutions for different load cases bysuperposition. Each load case must be defined and solved as a separate analysis. 8.1 Sources of nonlinearityThere are three sources of nonlinearity in structural mechanics simulations: Material nonlinearity. Boundary nonlinearity. Geometric nonlinearity.8.1.1 Material nonlinearityThis type of nonlinearity is probably the one that you are most familiar with and iscovered in more depth in Chapter 10, “Materials." Most metals have a fairly linearstress/strain relationship at low strain values; but at higher strains the material yields,at which point the response becomes nonlinear and irreversible (see Figure 8-2),StressFigure 8-2 Stress-strain curve for an elastic-plastic material under uniaxial tension.hMiial yield sneaRubber materials can be approximated by a nonlinear, reversible (elastic) response(see Figure 8-3).Figure 8-3 Stress-strain curve for a rubber-type material.StressStrainMaterial nonlinearity may be related to factors other than strain.Strain-rate-dependent material data and material failure are both forms of materialnonlinearity. Material properties can also be a function of temperature and otherpredefined fields.- 8.1.2 Boundary nonlinearityBoundary nonlinearity occurs if the boundary conditions change during the analysis.Consider the cantilever beam, shown in Figure 8-4, that deflects under an applied loaduntil it hits a "stop.”Figure 8-4 Cantilever beam hitting a stop.The vertical deflection of the tip is linearly related to the load (if the deflection issmall) until it contacts the stop. There is then a sudden change in the boundarycondition at the tip of the beam, preventing any further vertical deflection, and so theresponse of the beam is no longer linear. Boundary nonlinearities are extremelydiscontinuous: when contact occurs during a simulation, there is a large andinstantaneous change in the response of the structure.Another example of boundary nonlinearity is blowing a sheet of material into a mold.The sheet expands relatively easily under the applied pressure until it begins tocontact the mold. From then on the pressure must be