When subjected to blast loading, metal plates will most likely experience permanent deformations. It is therefore important to understand the material behaviour during plastic deformation. It is also necessary to be familiar with the interaction of the plastic deformation process and how the material is formed. In this section, forming limits and forming limit diagrams are therefore introduced. The following theory and equations are described by Hosford et. al [49].
During manufacturing, metal plates may be subjected to sheet-forming operations.
The deformation is characterized by biaxial stretching and failure normally occurs when a sharp localized neck develops on the surface. It is important not to confuse localized necking with diffuse necking, which occurs when the load reaches its maximum value.
θ
2
′1
2
b
l
t
Figure 3.11:Localized neck and its coordinate axes.
Figure 3.11 depicts a thin sheet specimen subjected to uniaxial tension in the 1-direction.
It is from now on presumed that the material obeys the power law hardening rule, σeq=K pn. The strain at the onset of diffuse necking is expressed asε∗1 =n, where the star (*) indicates the critical strain at instability. After being exposed to a
consider-expression for the critical localized necking in uniaxial tension is derived and denoted ε∗1=2n. This demonstrates a factor two increase of the critical strain for localised neck-ing compared with diffuse neckneck-ing.
Due to the narrowness of the neck, the strain parallel to the neckdε20 must be zero. An expression fordε20is given as,
dε20=dε1cos2θ+dε2si n2θ=0 (3.15)
where dε1and dε2is the incremental major and minor principal strains in the plane of the sheet.
By assuming a constant strain ratioβ=εε21 during stretching, Equation (3.15) becomes,
ε1cos2θ+βε1sin2θ=0 (3.16) or
tanθ= 1
p−β (3.17)
Equation (3.17) indicates thatθonly has a real value ifβis less than zero. This means that localized necking can not occur if the strain rate is constant or positive. Assuming a constant strain ratio is equivalent to assuming a constant stress ratioα=σσ21. When the stress becomes more biaxial,αincreases. By obtaining an expression ofβbased on α,β=(2α−1)(2−α), it is seen thatβbecomes less negative asαincreases. This implies an increase ofθas the stress becomes more biaxial.
The strain ratio influences the critical strain of necking. By applying the consistency criterion for volume, an expression relatingβ,dε1,dσ1andσ1is derived,
dσ1
σ1 =(1+β)dε1 (3.18)
By applying the power law, the condition for localized necking is obtained,
ε∗1= n
1+β (3.19)
Whenβ=12, the critical strain becomes equal to 2n. For plane strain, the critical strain decreases ton.
Swift [49] showed that diffuse necking can occur when,
ε∗1= 2n(1+β+β2)
(β+1)(2β2−β+2) (3.20)
If the loading is applied with constantαand consequently constantβ, localized neck-ing can not occur and stretchneck-ing continues until the sheet fractures. However, sheet materials are never completely homogeneous andαandβdo change during stretching.
Due to small inhomogeneities in geometry and material properties, local changes in the strain path occur and this leads to localized necking for positive values ofε2and hence forβ>0. This is the fundamental principle of the Marciniak-Kucsynski analysis[49].
t
bb
a a 1
2 t
a(a)A sheet with a narrow groove.
0.0 0.1 0.2 0.3 0.4 0.5 0.1
0.2 0.3 0.4 0.5
0.0
Minor strain ε
2Ma jor strain ε
1(b)Strain path change within the groove.
Figure 3.12a illustrates a sheet with a narrow groove perpendicular to the major princi-pal stress direction used for the Marciniak-Kucsynski analysis. The stress ratioαaand strain ratioβain regionaare assumed constant, whileαbandβbin regionbvary dur-ing plastic deformation. The strain in regionbparallel to the groove is restricted by the uniform strain rate in regiona, such thatε2b=ε2a.
When applying equilibrium requirements across the groove, the principal tractions in-side and outin-side the groove should be equal,t1a=t1b. This is also expressed as,
σ1aha=σ1bhb (3.21)
The localization problem is solved by an incremental-iterative solution technique that requires the equilibrium across the groove, the constitutive relations and the compati-bility equations to be satisfied throughout the deformation process. When deformation evolves,∆ε1abecomes smaller and smaller compared with∆ε1b. This corresponds to a∆ε2b=∆ε2a=β∆ε1athat decreases compared with∆ε1band the flow in the groove proceeds towards plane strain. Localized necking occurs when the strain ratio reaches a critical value.
β=∆ε3b
∆ε3a =βcr (3.22)
This critical value corresponds to excessive thinning inside the groove.
The Marciniak and Kuczynski procedure can be used to calculate the shape of the right side of the forming limit diagram(FLD). Figure 3.12b depicts the strain paths in region aandband the forming limit curve. The dashed and thin lines represent the strain paths in regionaandb, respectively. The forming limit curve (FLC), which is displayed as the thick line, connects with the end points of the strain paths in regiona. Initially, the strain paths in regionaandbare similar. As the deformation process evolves, the two strain paths start to diverge and the strain path in regionbtends towards plane strain with increasing strain rate. The forming limit is reached when the strain ratioβ is equal to the critical value. It is important to know that the forming limit is based on the critical strains in regiona.
3.4.1 Forming Limit Diagrams (FLD)
The major and minor principal strains (ε1,ε2) can be experimentally determined for different loadings along various paths. By plotting the strains in a diagram, a form-ing limit diagram (FLD) is constructed. This diagram can give indications of potential problems in the sheet forming and thereby prevent production failures. Figure 3.13 il-lustrates the difference between the experimental forming limit curve (FLC) and the theoretical curves for diffuse and localized necking.
By plotting local strains and comparing them with the FLD, potential trouble spots can be determined. If the measured strains are close to the FLC, there is a probability of weakness in the material. If the strains are located underneath the experimental FLC, localized necking is not a problem. A placement over the FLC indicates that the sheet may be exposed to local necking or fracture. By identifying the nature of the problem, it is possible to alter the sheet forming process such that production failures are reduced [49][50][51]. By creating a forming limit diagram for the experimental data obtained in this thesis, it is possible to investigate if the FLD provides reliable results.
−20 −10 10 20 30 Minor engineering strainε2
Majorengineeringstrainε1
20 40 60
Diffuce necking (theory)
Diffuce necking (theory) Local necking (theory)
Experimental forming limit
Figure 3.13:Experimental FLD with theoretical curves [49].