0% 0% +台 X;0:C 100% 0% 0% 100% 45 台 0% 0% 0% 34% +×+ 66% 0% Figure 5.14 Laminate with Balanced Fabrics;Representation 2 Ny(=oy×h) Ty(=xw×h 61 Nx(=ox×h) [h a)stresses b)stress resultants Figure 5.15 Stresses and Stress Resultants of the laminate in its plane.This is called membrane loading.The mechanical loadings can take the form of stresses(o.T in Figure 5.15a)or stress resultants (NN.T in Figure 5.15b).The stress resultants are the products of the stresses with the thickness b of the laminate. Generally,three criteria should be considered by the designer for the ply configuration: 1.Support the loading without deterioration of the laminate 2.Limit the deformation of the loaded piece 3.Minimize the weight of the material used These criteria do not always work together.For example,searching for minimum thickness might not be compatible with high rigidity.Searching for high rigidity The laminate can also work in bending.This is studied in Chapters 12 and 17. 2003 by CRC Press LLC
of the laminate in its plane. This is called membrane loading. 6 The mechanical loadings can take the form of stresses (sx, sy, txy in Figure 5.15a) or stress resultants (Nx, Ny, Txy in Figure 5.15b). The stress resultants are the products of the stresses with the thickness h of the laminate. Generally, three criteria should be considered by the designer for the ply configuration: 1. Support the loading without deterioration of the laminate 2. Limit the deformation of the loaded piece 3. Minimize the weight of the material used These criteria do not always work together. For example, searching for minimum thickness might not be compatible with high rigidity. Searching for high rigidity Figure 5.14 Laminate with Balanced Fabrics; Representation 2 Figure 5.15 Stresses and Stress Resultants 6 The laminate can also work in bending. This is studied in Chapters 12 and 17. TX846_Frame_C05 Page 79 Monday, November 18, 2002 12:09 PM © 2003 by CRC Press LLC
,90° 0° plyn°1 10 plies;e=1.3 mm 212mm 16 plies;e=2.08 mm 2 [0±45/0/90]s 10 4(40%) 16-6(37%)→0/生45/0/90/t45/01s 2 Figure 5.16 Example of Representation might not be compatible with minimum weight.One will see in Section 5.4 guidelines for proportions values that allows a laminate with minimum laminate thickness to support specified mechanical loading without damage,. Once a laminate is defined (number of layers and orientations),one must respect the following conditions (without forgetting the technological minimum indicated at the end of the previous paragraph)as much as possible: ■9o°plies placed on the surface,then45°and-45°plies,when the pre- dominant stress resultant is oriented along the 0direction No more than 4 consecutive plies along the same direction 5.2.4.1 Example of Representation The plies are progressively terminated to obtain a gradual change in thickness (maximum 2 plies for each 6 mm interval).The symbols for the composition of the laminate are shown on plan view (see Figure 5.16). 5.2.4.2 The Case of Sandwich Structure The description of the sandwich material is done as in Figure 5.17. 5.3 FAILURE OF LAMINATES 5.3.1 Damages Figure 5.18 shows schematically different types of failure leading to damage of a laminate. The main modes of damage,when the loads exceed the critical limits,are illus- trated in Figure 5.19. 2003 by CRC Press LLC
might not be compatible with minimum weight. One will see in Section 5.4 guidelines for proportions values that allows a laminate with minimum laminate thickness to support specified mechanical loading without damage,. Once a laminate is defined (number of layers and orientations), one must respect the following conditions (without forgetting the technological minimum indicated at the end of the previous paragraph) as much as possible: 90∞ plies placed on the surface, then 45∞ and -45∞ plies, when the predominant stress resultant is oriented along the 0∞ direction No more than 4 consecutive plies along the same direction 5.2.4.1 Example of Representation The plies are progressively terminated to obtain a gradual change in thickness (maximum 2 plies for each 6 mm interval). The symbols for the composition of the laminate are shown on plan view (see Figure 5.16). 5.2.4.2 The Case of Sandwich Structure The description of the sandwich material is done as in Figure 5.17. 5.3 FAILURE OF LAMINATES 5.3.1 Damages Figure 5.18 shows schematically different types of failure leading to damage of a laminate. The main modes of damage, when the loads exceed the critical limits, are illustrated in Figure 5.19. Figure 5.16 Example of Representation TX846_Frame_C05 Page 80 Monday, November 18, 2002 12:09 PM © 2003 by CRC Press LLC
orientation of honey- material comb before expansion surface treatment (pnmary) material adhesive layer material Figure 5.17 Description of a Sandwich Material fiber rupture matrix rupture delamination Figure 5.18 Different Modes of Failure ●tension ●compression 小从名 ●shear;delamination Figure 5.19 Modes of Damage One cannot be satisfied with the classical maximum stress criterion Figure 5.20 shows a unidirectional laminate loaded successively in two different manners.In the two cases,the maximum normal stress has the same value denoted 2003 by CRC Press LLC
One cannot be satisfied with the classical maximum stress criterion Figure 5.20 shows a unidirectional laminate loaded successively in two different manners. In the two cases, the maximum normal stress has the same value denoted Figure 5.17 Description of a Sandwich Material Figure 5.18 Different Modes of Failure Figure 5.19 Modes of Damage TX846_Frame_C05 Page 81 Monday, November 18, 2002 12:09 PM © 2003 by CRC Press LLC
4Ob a） Figure 5.20 Stresses and Fiber Orientation as o.In the loading case (a),the unidirectional specimen will rupture when >Orupture along t This is the maximum stress criterion. In the loading case (b),the maximum normal stress occurs in a direction that is different from that of the fibers (one can obtain this by tracing the Mohr's circle as discussed previously).We have seen (Section 3.3.2)that the rupture resistance decreases.It is weaker than the situation of case (a).The unidirectional laminate therefore ruptures when Grupture along This phenomenon is more evident if the unidirectional laminate is loaded in a direction transverse to the fibers t.In this case,the laminate rupture resistance is that of the matrix,which is much less than that of the fibers. Taking into consideration the evolution of the rupture resistance with the loading direction,one can not use a simple maximum stress criterion as for the classical metallic materials. 5.3.2 Most Frequently Used Criterion:Hill-Tsai Failure Criterion? One can apply this criterion successively to each ply of the laminate,that is for each one of the orientations0°，90°，±45°that have been considered.As has been discussed in Chapter 3,the axes of a unidirectional ply are denoted as e for the direction along the fibers,and t for the transverse direction.The stresses are denoted as o in the fiber direction,o,in the direction transverse to the fibers,and T for the shear stress (see figure below). One denotes the Hill-Tsai number(see Figure 5.21)the number a such that If a<1:no ply rupture occurs. If a>1:rupture occurs in the ply considered.Generally,this deterioration is due to the rupture of the resin.The mechanical properties of a broken ply become almost negligible,except for those along the fiber direction (modulus of elasticity and rupture resistance) For more detailed study of this criterion,see Chapter 14. 2003 by CRC Press LLC
as s. In the loading case (a), the unidirectional specimen will rupture when s > srupture along This is the maximum stress criterion. In the loading case (b), the maximum normal stress occurs in a direction that is different from that of the fibers (one can obtain this by tracing the Mohr’s circle as discussed previously). We have seen (Section 3.3.2) that the rupture resistance decreases. It is weaker than the situation of case (a). The unidirectional laminate therefore ruptures when s < srupture along This phenomenon is more evident if the unidirectional laminate is loaded in a direction transverse to the fibers t. In this case, the laminate rupture resistance is that of the matrix, which is much less than that of the fibers. Taking into consideration the evolution of the rupture resistance with the loading direction, one can not use a simple maximum stress criterion as for the classical metallic materials. 5.3.2 Most Frequently Used Criterion: Hill–Tsai Failure Criterion 7 One can apply this criterion successively to each ply of the laminate, that is for each one of the orientations 0∞, 90∞, ±45∞that have been considered. As has been discussed in Chapter 3, the axes of a unidirectional ply are denoted as for the direction along the fibers, and t for the transverse direction. The stresses are denoted as s in the fiber direction, st in the direction transverse to the fibers, and tt for the shear stress (see figure below). One denotes the Hill–Tsai number (see Figure 5.21) the number a such that If a < 1: no ply rupture occurs. If a ≥ 1: rupture occurs in the ply considered. Generally, this deterioration is due to the rupture of the resin. The mechanical properties of a broken ply become almost negligible, except for those along the fiber direction (modulus of elasticity and rupture resistance) Figure 5.20 Stresses and Fiber Orientation 7 For more detailed study of this criterion, see Chapter 14. TX846_Frame_C05 Page 82 Monday, November 18, 2002 12:09 PM © 2003 by CRC Press LLC
Ot G: 2 2 GEO: ta 02= tet rupture rupture rupture rupture Figure 5.21 Hill-Tsai Number 5.3.2.1 Notes Attention:The rupture resistance oupure does not have the same value in tension and in compression (see,for example,Section 3.3.3).It is therefore useful to place in the denominators of the previous Hill-Tsai expression the rupture resistance values corresponding to the mode of loading (tension or compression)that appear in the numerator. Using this criterion,when one detects the rupture of one of the plies (more precisely the rupture of the plies along one of the four orientations),this does not necessarily lead to the rupture of the whole laminate.In most cases,the degraded laminate continues to resist the applied stress resultants. In increasing these stress resultants,one can detect which orientation can produce new rupture.This may-or may not-lead to complete rupture of the laminate.If complete rupture does not occur,one can still increase the admissible stress resultants."In this way one can use a multiplication factor on the initial critical loading to indicate the ratio between the first ply rupture and the ultimate rupture. As a consequence of the previous remark it appears possible to work with a laminate that is partially degraded.It is up to the designer to consider the finality of the application,to decide whether the partially degraded laminate can be used. One can make a parallel-in a gross way-with the situation of classical metallic alloys as represented in Figure 5.22. 5.3.2.2 How to Determine o u,To in Each Ply Consider for example the laminate shown in Figure 5.23,consisting of identical plies.The following characteristics are known: See Exercise 18.2.7. 2003 by CRC Press LLC
5.3.2.1 Notes Attention: The rupture resistance srupture does not have the same value in tension and in compression (see, for example, Section 3.3.3). It is therefore useful to place in the denominators of the previous Hill–Tsai expression the rupture resistance values corresponding to the mode of loading (tension or compression) that appear in the numerator. Using this criterion, when one detects the rupture of one of the plies (more precisely the rupture of the plies along one of the four orientations), this does not necessarily lead to the rupture of the whole laminate. In most cases, the degraded laminate continues to resist the applied stress resultants. In increasing these stress resultants, one can detect which orientation can produce new rupture. This may—or may not—lead to complete rupture of the laminate. If complete rupture does not occur, one can still increase the admissible stress resultants.8 In this way one can use a multiplication factor on the initial critical loading to indicate the ratio between the first ply rupture and the ultimate rupture. As a consequence of the previous remark it appears possible to work with a laminate that is partially degraded. It is up to the designer to consider the finality of the application, to decide whether the partially degraded laminate can be used. One can make a parallel–in a gross way–with the situation of classical metallic alloys as represented in Figure 5.22. 5.3.2.2 How to Determine s, st, tt in Each Ply Consider for example the laminate shown in Figure 5.23, consisting of identical plies. The following characteristics are known: Figure 5.21 Hill–Tsai Number 8 See Exercise 18.2.7. TX846_Frame_C05 Page 83 Monday, November 18, 2002 12:09 PM © 2003 by CRC Press LLC