9 Strength Theories of Lamina

Transcription

1 9 trength Theories of Lamina 9-

2 TRENGTH O ORTHOTROPIC LAMINA or isotropic materials the simplest method to predict failure is to compare the applied stresses to the strengths or some other allowable stresses. In this case there is no principal material direction so the material strengths are the same in all directions. or isotropic metals failure usually occurs by yielding and can be simply predicted by the maimum shear stress theory. or a plate in plane stress conditions the principal stresses I and II can be found half their difference can be compared to the yield strength of the metal. The criterion for failure is I II τ yield or I II yield which is represented graphically in ig. 9-. τ y II I igure 9-. ield criteria for biaial stress A better criterion for most ductile metals is the distortion energy theory that states failure occurs when + I II I II yield or orthotropic composite lamina these methods are not sufficient because the failure mechanisms and strength properties change with direction of loading. ailure usually does not occur by yielding but rather by fracture of one of the constituents or the fibermatri interface. Unlike isotropic materials, and ais of maimum stress does not 9-

3 necessarily coincide with direction of maimum strain. As a consequence the highest stress on body may not be the highest critical stress in the structure. The notation often used to describe strength properties of orthotropic lamina is shown in ig.9-. The ultimate tensile strength in the fiber direction is denoted by X, the ultimate longitudinal ' X X' iber Direction igure 9-. trength notation for orthotropic lamina compressive strength by X, the transverse tensile strength by, the transverse compressive strength by, and the shear strength by. In orthotropic composites the tensile and compressive strengths in the principal material directions usually do not have the same values. This is because the mechanism of failure can change from fiber fracture when in tension to fiber microbuckling and interfacial splitting when in compression. However, positive and negative shear strength in the principal material direction is the same. This can be seen in ig 9-3, where the fibers and matri eperience the same load Positive hear Negative hear igure 9-3. Comparison of positive and negative shear for stress in the principal material directions. 9-3

4 orientation regardless of the sign of the shear stress. The shear strength for off-ais shear, say at 45º to the principal material direction, can be different. This case is illustrated in ig In this case positive shear places the fibers in Positive hear Negative hear igure 9-4. Comparison of off-ais positive and negative shear for stress. compression and the matri and fiber-matri interface in tension. Negative shear places the fibers in tension and the matri and fiber-matri interface in compression. It is clear that the composite failure mechanisms will be different for the two directions of shear. BIAXIAL TRENGTH THEORIE tiffness and elastic properties in any direction can be obtained by applying tensor transformations. Tensor transformations of strengths are more difficult since the strength tensor is of higher order than stiffness tensor. There are also numerous failure modes that apply depending on the direction of load relative to the principal material directions. or isotropic materials the failure criterion is usually simply defined as yielding or fracture. or Orthotropic material there are considerably more possibilities. Maimum stress theory When stresses resolved on the principal material directions eceed a prescribed value, usually the lamina tensile strength in that direction, then failure has occurred. These prescribed values are N,, XN and X determined from uniaial tensile and compression tests and or N determined from in-plane shear tests (remember = N). ailure will not occur as long as X < < X < < < τ < (9.) 9-4

5 ailure will occur if any or all of the above are violated and there are no interaction between failure modes. If a stress is applied in an arbitrary direction,, say at θ º, from the fiber direction, as illustrated in ig. 9-5 then those stresses can be transformed to the principal direction to determine if the failure criteria of Eqn. (9.) has been eceeded. q Load Direction igure 9-5 Composite lamina loaded in arbitrary direction. Performing the transformation produces = cos θ = θ sin τ = cosθsinθ (9.) Eqn. (9.) then becomes X X < < cos θ cos θ < < sin θ sin θ < < cosθsinθ cosθsinθ (9.3) Each of these separate criteria can be plotted as a function of fiber orientation θ as is shown in ig. 9-. Any stress above the curves eceeds the safe stress for that particular orientation. rom this plot it can be seen that five separate criteria must be calculated and eamined. At small off-ais angles of the longitudinal strength increase with, hence, the composite is predicted to be stronger in off-ais loading. This prediction is contrary to actual behavior in which the off-ais tensile strength is never greater than on-ais strength. In this plot the individual curves meet at cusps suggesting local maima at a transition between modes. In reality no such local maima eists. Eamination of eperimental data also indicates that this method provides only a rough approimation to the failure stress. 9-5

6 X cos q X' cos q s cos q sin q ' sin q sin q igure 9- Maimum stress theory of lamina failure plotted for off-ais loading Maimum strain theory Material or their constituents undergo fracture when a critical separation is produced as a result of stress, therefore a more realistic criterion for failure is strain. This criterion can be written as q X X < ε < E E < ε < E E < γ < G G (9.4) Using Hooke s Law to epress strain in the principal material directions in terms stress in the principal material directions 9-

7 ε ε γ ν = E E ν = E E τ = G (9.5) Transforming to arbitrary off-ais directions Eqn.(9.5) becomes ε = θ ν θ ( cos sin ) E ε = θ ν θ ( sin cos ) E γ = ( cosθsinθ) G (9.) ubstituting back into Eqn.(9.4) X X < < cos θ ν sin θ cos θ ν sin θ < < sin θ ν cos θ sin θ ν cos θ < < cosθsinθ cosθsinθ (9.7) These values are very similar to those from the maimum stress theory. TRENGTH THEORIE WITH TRE INTERACTION In the maimum stress and maimum strain theories the stresses are assumed to act independently. To account for stress interactions various strength tensor theories were developed. The strength tensor in the most general form can be as the polynomial (in contracted notation) where i, j, k =, and α = β =. α ( i i ) ( ij i j) β γ + + ( ijk i j k) + = (9.0) or the orthotropic case γ = 0 and the following coefficients are also zero: 4, 5,, 4, 5,, 4, 5,, 34, 35, 3, 44, 45, 4 9-7

8 These assumptions and simplifications reduce the general polynomial to + = (9.) i i ij i j with the following non-zero coefficients:,,,,, Tsai-Hill failure criterion Using the Von Misses,(distortion energy) criterion Eqn. (9.) reduces to = (9.) ij i j which is the basis for the Tsai-Hill criterion. Epanding this tensor gives = (9.3) The constants,,, and can be related to the strength properties of the lamina in the principal material directions by considering uniaial loading to failure for all three stresses. irst consider the case, = X, = 0 and = 0, then Eqn (9.3) reduces to X = (9.4) Net consider the case =, = 0 and = 0, then Eqn. (9.3) becomes = (9.5) The last case to consider is =, and = = 0, then Eqn. (9.3) becomes = (9.) To find apply a balanced biaial stress, and equal to the longitudinal tensile strength, X and apply no shear stress = 0 then Eqn. (9.3) becomes X = (9.7) The strength coefficients can now be obtained from Eqns, (9.4),(9.5), (9.) and (9.7) 9-8

9 = X = = = X (9.8) rom Eqn. (93) the Tsai-Hill criterion can now be written + + = (9.9) X X Using Eqn. (9.) the Tsai-Hill criterion predicts the failure stress, for off-ais loading = 4 4 cos θ sin θ + + cos θsin θ X X (9.0) ig. 9-7 is a plot of Eqn. (9.0) for composite with a tensile, compressive and shear strengths of 80 ksi, 5 ksi and 9 ksi respectively in the principal material directions tress, s (psi) afe tress ailed igure 9-7 Predicted failure stress, iber Orientation, q using Tsai-Hill criterion 9-9

10 The failure stress in terms of y and τ y can be similarly determined. This plots agrees quite reasonably with eperimental data. The failure stress curve is smooth and continuous and always decreases with increasing degree of off-ais angle. This criterion accounts for the interaction between stress and reduces to the correct form for isotropic material. This criterion, however, does not address compressive failure. A modification to the Tsai-Hill criterion, now referred to as the Tsai-Wu criterion was developed to correct this limitation. Tsai-Wu failure criterion The Tsai-Wu criterion uses the tensor form of Eqn. (9.) epanded to = (9.) As was done to derive the constants for the Tsai-Hill criterion we apply a stress only in the longitudinal fiber direction,, that is equal to the longitudinal tensile strength X and the apply no transverse,, and shear,, stresses then Eqn (9.) reduces to X + = (9.) X We can also set the applied longitudinal stress,, to the compressive longitudinal strength, X (the value of X is positive but has a negative sign assigned to it) and the transverse, and shear stress to zero and get X + = (9.3) X Using Eqns. (9.) and (.3) the coefficients and can be found = X X (9.4a) = XX (9.4b) Likewise the following coefficients can be found (9.4c) = (9.4d) = (9.4e) 9-0

11 Applying the measured biaial tensile strength of the lamina, B to and only the coefficient becomes = B B B + + X X XX (9.4f) If biaial strength data is not available can be found empirically as = * where Using * is between -0.5 and 0. * = -0.5 the usual form of the Tsai-Wu criterion is (9.5) X X + + XX + XX + = AILURE ENVELOPE TRE PACE Plotting the biaial stresses that result in failure produced the so-called failure envelope. or isotropic brittle materials the failure is independent of the direction of loading and the failure envelope can be defined for the principal stresses, I and II and will occur if either stress eceeds the fracture strength, UT of the material. The maimum stress theory would satisfactorily failure of such a material. The failure envelope for this case is shown in ig tress combinations that fall outside the envelope constitute failure. ΙΙ s UT s UT s UT Ι s UT igure 9-8 ailure envelope for isotropic brittle material 9-

12 or isotropic ductile materials, such as most metals, failure is defined when yielding occurs,. In that case either the maimum shear stress theory (Tresca) or the yield distortion energy theory (Von Mises-Hencky) is satisfactory. ig. 9-9 shows the failure envelope for materials that are satisfactorily represented by the Tresca or Von Mises- Hencky criteria. ΙΙ Von Mises-Hencky Tresca Ι igure 9-9 ailure envelopes for isotropic ductile materials The failure envelope for orthotropic material obeying the maimum stress theory, Eqn. (9.) is shown in ig In this case failure is defined in terms of the stresses in the principal material directions. Unlike isotopic materials the failure stresses are different in the different principal material directions and in the tensile and compressive directions due to different failure mechanisms applying. The safe stresses fall within the envelope defined by the four equations = X = X = = (9.) The failure envelope for the maimum strain theory, Eqn. (9.4), shown in ig. 9-, is composed of the four equations = X +ν = X +ν = +ν = +ν (9.7) The maimum strain theory recognizes that strain controls fracture and takes into account the combined stresses in the principal material directions. 9-

13 X' X ' igure 9-0. ailure envelope for orthotopic material by the maimum stress theory X' X ' igure 9-. ailure envelope for orthotopic material by the maimum strain theory The maimum stress failure envelope is shown as the dashed line in this figure for comparison. The maimum strain failure envelope intersects the maimum stress failure envelope at the strength values where one of the stresses in the principal material directions is zero. or the Tsai-Hill or Tsai-Wu criteria the failure envelope is an ellipse. or the Tsai-Wu criterion Eqn. (9.) and be rearranged in quadratic form in terms of as the dependent variable = 0 (9.) ( ) ( ) 9-3

14 Plotting vs. of Eqn.(9.) produces the failure ellipse in stress space, shown in ig 9-. As with the other failure envelopes, points that lie inside the ellipse constitute a safe stress range and points that lie outside constitute failure. igure 9. ailure ellipse for Tsai-Wu criterion in stress space AILURE ELLIPE IN TRAIN PACE or highly anisotropic composites the ellipse is eaggerated in the major ais direction (i.e. cigar shaped) and then plotting the ellipse in strain space is preferred. Hooke's law is used to the failure criterion in strain space = Q ε (9.7) i ij j ubstituting Eqn. (9.7) into the stress criterion Eqn. (9.) gives QQ εε + Q ε = (9.8) ij ik jl k l i ij j using the following notation: QQ ij ik jl = Hkl and Q i ij = Hi Eqn. (9.8) can be written as In epanded form Eqn. (9.9) is H εε + Hε = (9.9) ij i j i i H ε + H ε + H ε + H εε + H ε + H ε = (9.30) where 9-4

15 H = Q + QQ + Q H = Q + Q Q + Q ( ) H = QQ + QQ + Q + Q Q H = Q H = Q + Q H = Q + Q olving Eqn. (9.30) for ε produces the quadratic equation ( ) ( ) ε H + ε H + ε H + ε H + ε H + ε H = 0 (9.3) Plotting ε vs. ε of Eqn (9.3) produces the failure ellipse in strain space of ig.9-3. Points that lie inside the ellipse constitutes a safe strain range and points outside constitutes failure. The aspect ratio of the strain space failure ellipse is smaller than the aspect ratio for the stress space ellipse. It may be noticed that the shear stress, or shear strain, ε is a constant in these equations. The maimum size of the ellipse occurs for = 0 or ε = 0. When = or ε = the failure strain in shear the ellipse shrinks to a point. ε ε igure 9. ailure ellipse for Tsai-Wu criterion in stress space 9-5

16 UING THE AILURE CRITERIA A A DEIGN TOOL X The ratios,,, τ y y are referred to a "strength ratios", R, Ry, R respectively. They can be regarded as factors of safety. X, and are fied for a given composite therefore will determine the safety factor. The strength tensor in terms of the strength ratio, R, is written as or on strain space as ( ) R ( ) R + = (9.3) i i ij i j ( Hε ) R ( H εε ) R + = (9.33) i i ij i j These can be solved as a quadratics roots R or R + and R or R - from the solutions + and a a b b 4 ac b b 4 ac The root R or R - from the conjugate solution predicts the factors of safety if all of the stresses are reversed in sign. Other strength theories can be found in the Addendum to this chapter. 9-

17 ADDENDUM TO CHAPTER 9 In addition to Tsai-Hill and Tsai-Wu there are numerous other stress interaction strength criteria. These are listed in Table 9A-. Table 9A- tress interaction strength criteria Theory Tensor Ashkenazi ij i j = 4 X U X Cowin ii + ij i j = X X XX ischer ij i j = K X XX Hoffman ii + ij i j = X X XX XX Malmeister ii + ij i j = XX X X X ( X ) + X X XX XX XX Marin ii + ij i j = X X XX X Tsai-Hill ij i j = X X Tsai-Wu ii + ij i j = * X X XX 9-7