of radii for the loading noses and recommended Ee=E and v:=Ve=v,equation (10.2)reduces to equation (10.1). Poisson's ratio from compressive test. ulus,G,can be expressed as be found from the isotropic relationship 2(1+v) specimen dimensions standard D6272-02 describes a four-point bending test.Allowable ranges This test method involves three-point bending (fig.10.8),and a separate of plastics may be determined by the ASTM D790-03 test method [16] The flexural yield strength,flexural strength,and modulus of elasticity tensile compressive values of E and v.It is easily shown that when E 0.】)出0。)was1 of of of was found to give much better agreement with directly measured values This equation,which involves both tensile and compressive properties from compressive test,V:the Poisson's ratio from tensile test,and,ve the where E is the Young's modulus from tensile test,E the Young's modulus in tension and compression,Novak and Bert showed that the shear mod- using the Hooke's law for an isotropic material with different properties ented at 45 to the x,y axes.By equating these strain energy terms and the corresponding biaxial tensile and compressive principal stresses ori- pure shear along the x,y axes is equal to the strain energy associated with to a rotation of coordinates,the strain energy for an isotropic material in was based on the premise that since the elastic strain energy is invariant ences between tensile and compressive values of E and v.Their approach accurate calculation of G could be obtained by taking into account differ- twist versus torque for solid rod torsion tests.It was found that a more Directly measured values of G were determined from a plot of angle of pressive tests differ substantially from directly measured values of G. values of G found from applying equation(10.1)to either tensile or com- However,Novak and Bert [15]have reported that for some epoxies the definitions of those parameters.If desired,the shear modulus,G,can also E,and the Poisson's ratio,v,can then be determined from the standard measure the longitudinal and transverse strains.The Young's modulus, material,biaxial strain gages can be attached to the specimen so as to In either the tensile test or the compressive test of the neat resin matrix Principles of Composite Material Mechanics 102 4o1 it can be calcnlated from 10.2.3 FIGURE 10.8 to determine the matrix volume fraction,If the void fraction is desired, the matrix density pm can be determined,then equation(3.6)can be used weight and volume measurements on a separate neat resin matrix sample, and the fiber volume fraction can be calculated from v=Vi/Ve.From then measuring the weight,Wi,and volume,Vi of fibers remaining after resin removal,the fiber weight fraction can be calculated from wr=W/W, weight,W and volume,Vo of a composite sample before resin removal, burn-off in a furnace,in cases where it is safe to assume that the fibers are unaffected by the resin removal process.For example,by measuring the either chemical digestion(with acids or other chemicals)or ignition and involves removal of the matrix resin from the composite sample using matrix composites,ASTM standard D3171-99 [17]covers two basic approaches to the measurement of constituent volume fractions.Method I and for quality control during manufacturing of composites.For polymer Knowledge of the volume fractions of fiber and matrix materials(and also void fractions,if possible)is essential for use in micromechanical analysis Constituent Volume Fraction Measurement sion,impact,creep,and fatigue response are also given in ref.[1]. mechanical properties of other constituents such as sandwich core mate- rials and other constituent properties such as coefficient of thermal expan- are provided in tables in D790-03 [161.Test methods for measurement of Three-point bending specimen for flexural properties of neat resin or composite.(Prom ASTM Standard D790-03.Copyright ASTM International.Reprinted with permission.) Support span Mechanical Testing of Composites and Their Constituents