**1.3. Classification of sandwich sheets**

86 Metal Forming – Process, Tools, Design

cracking. [6]

shear fixed)

**1.2. Failure-modes of sandwich sheets** 

noise-absorbent behavior are highly useful in the engine bay. [5]

So the thin outer sheets are chosen of 1.4301, the supporting layer of DC06 and the vibration damping layer of a viscoelastic adhesive. At first, the metal layers of both sides are cladded 1. In step no. 2 they are bonded with a very thin adhesive layer (see chapter 5.4). To achieve advantages concerning the forming process (see chapter 4.1), a metal interlayer is included.

Sheets with vibration damping qualities can be made of steel sheets enclosing a viscoelastic plasticcorelayer (see Figure 17). [2] The vibration energy of the oscillating coversheet is converted into heat.[3] [4] For automotive lightweight constructions, sandwich sheets with a

Parts of sandwich sheets with viscoelastic layers excel with a higher security against

Due to the relocatability of the outer and the inner sheet, leakage of e.g. an oil pan in contact with the subsoil or in case of a crash is unlikely for this sandwich in contrast to a comparable single layer sheet. The cover-sheets remain undamaged, but the bond can fail during forming.

Under load, during and after the forming process special effects and failures occur. [7]

**Figure 2.** Failure modes of during forming of sandwich sheets (viscoelastic, shear transmitting and

Mainly displacement and delamination of the cover sheets occur as failure modes (see Figure 2). Wrinkling at the inside surface is detected especially by vibration-damping composite sheets with long non-deformed legs and small thickness of the inner layer. Plastics and adhesives creep under load. [8] Because of the residual stresses in the cover sheets or due to temperature influences, the composite delaminates often after hours or days, respectively.[9] To decay noises effectively, the viscoelastic interlayer has to be as thin as possible (chapter 5). In contrast to this, lightweight sandwich sheets achieve great The joining process of several layers has a great influence of the formability and the damping behavior of the sandwich. Layers can be connected viscoelasticly by using an adhesive film. The cover sheets are allowed to slide on each other. For that displacement only a negligible shear force is necessary. This viscoelastic bond is weak. Shear transmitting bond lines like 1 or 2 component-adhesives are ductile. Parameters of the roll bonding process and surface treatments are shown in [10]. Layers which are connected shear fixed (cladded) can´t slide on each other. [15] In a forming process the sheet behaves like a single sheet, but after forming, astonishing effects due to different Young's moduli and hardness occur.

**Figure 3.** Classification of joining with respect to transmitting shear force

**Figure 4.** Tensile shear test of sandwich sheets; Strip production line similar to [3]

$$k\_{f\\_lin} = R\_{p0,2} + \varphi m \qquad \qquad m = \frac{(1+\varepsilon\_{gl})R\_m - R\_{p0,2}}{\varepsilon\_{gl} - \frac{R\_{p0,2}}{E}} \tag{1}$$

$$k\_{f\_\*\mathcal{SM}ft} = b(c + \ln(1 + \epsilon))^d \qquad \qquad n = \varphi\_v \approx \ln(1 + A\_g) \tag{2}$$

$$c = \sqrt[n]{\frac{R\_{p\alpha z}}{R\_m}} \quad d = \text{ ( $c + n$ )} \qquad b = R\_m \* \frac{e^n}{d^d} \tag{3}$$



#### **2.2. Tensile shear test**

During the forming process the cover layers of the sandwich slide on each other. By shifting the cover plates the adhesive layer is sheared. Tensile shear tests provide the stressdisplacement behavior of the adhesive for a constant shear rate at room temperature. They can be transferred directly to the forming process [6]. Different adhesives applied in liquid state were investigated. In addition, sandwich sheets of Bondal® [5] CB (Car Body) from ThyssenKrupp are considered. As seen in Figure 6, the specimens were prepared according to DIN 53 281 [19] respectively [20] but also [21]. The thicknesses of the cover sheets �� = 0,�5 ��� and of the adhesive layer ���� = 0,05 ��� are given by the composite material. In Figure 6 three tensile shear tests of sandwich sheets with different testing velocities are shown. The shear stress is calculated from the initial overlapping length *Lü* and the gauged force *F*.

Forming of Sandwich Sheets Considering Changing Damping Properties 91

the force, as well as by the initial overlapping length. There are different modes of delamination. G. Alfano and M. A. Crisfield e.g. established an interaction model "mixedmode" which summarizes many fracture criteria proposed in literature. [22] To describe the elasticity of the adhesive, the traction/relative displacement law is used. The damage law with nominal stress of the normal mode and the both other directions is chosen. Numerical results of the uniaxial tensile shear tests fit to the experiments. Special attention for forming sandwich sheets is laid on the failure of the adhesive bond. As seen in Figure 7 the elastic and damage behavior is calculated exactly, but the maximal shear force and displacement

To predict the suitability of vibration-damping sandwich sheets for forming, material properties and interactions are calibrated with the simulation of the die-bending tests. V-die bending with different temperatures are considered in [23] and of velocity in [24]. Therefore

Optimized adhesive parameters for numerical calculation are applied on the shear test (see Figure 7). With bilinear, quadrilateral elements the material is performed. Reduced integration plane stress space with hourglass control (CPS4R) is used. Hard pressureoverclosure behavior is approximated with the penalty method. The penetration distance is proportional to the contact force. Contact is implemented frictionless to assume that surfaces

In Figure 8 the v-die bending process of a symmetrical sandwich with a length of 2L is considered. The stress, strain and displacement of both cover sheets out of DC06 with a thickness of s = 0,7 mm are determinant for two zones, the bending area and the adjacent

specimens with a special grid on the surface of edge were prepared.

in contact slide freely and isotropic with a friction-coefficient of ��� = 0,1.

depart up to 9 %.

**Figure 7.** Numerical calculation of tensile shear tests

**3. Forming tests: Die-bending** 

**Figure 6.** Tensile shear test with Bondal CB (cover sheets of DC06), specimen geometry

The adhesive layer fails by reaching the maximum force. For composites with a shear transmitting adhesive a shear stroke of Δl��� ≈ 0,22 mm was determined at a maximum shear stress of � ≈ 5,5 � ��� (see Figure 6). These values were computed by arithmetic mean. With higher velocity, the shear stress increase, but the tolerable displacement drops. The specific stiffness of the adhesive ( � ��� <sup>=</sup> ���� �� ), a measurement to describe the increase of the shear stress, rises.

Delamination and displacement of laminated metal sheet can be analyzed by using finite elements. In Figure 7 the experimental and numerical tensile shear tests of the adhesive film are shown. In literature, several investigations determined the stress distribution depending on the overlapping lengths [8]. As the experimental tensile shear test, the force F and displacement �� are gauged with numerical calculations. The shear stress is determined by the force, as well as by the initial overlapping length. There are different modes of delamination. G. Alfano and M. A. Crisfield e.g. established an interaction model "mixedmode" which summarizes many fracture criteria proposed in literature. [22] To describe the elasticity of the adhesive, the traction/relative displacement law is used. The damage law with nominal stress of the normal mode and the both other directions is chosen. Numerical results of the uniaxial tensile shear tests fit to the experiments. Special attention for forming sandwich sheets is laid on the failure of the adhesive bond. As seen in Figure 7 the elastic and damage behavior is calculated exactly, but the maximal shear force and displacement depart up to 9 %.

**Figure 7.** Numerical calculation of tensile shear tests

#### **3. Forming tests: Die-bending**

90 Metal Forming – Process, Tools, Design

During the forming process the cover layers of the sandwich slide on each other. By shifting the cover plates the adhesive layer is sheared. Tensile shear tests provide the stressdisplacement behavior of the adhesive for a constant shear rate at room temperature. They can be transferred directly to the forming process [6]. Different adhesives applied in liquid state were investigated. In addition, sandwich sheets of Bondal® [5] CB (Car Body) from ThyssenKrupp are considered. As seen in Figure 6, the specimens were prepared according to DIN 53 281 [19] respectively [20] but also [21]. The thicknesses of the cover sheets �� = 0,�5 ��� and of the adhesive layer ���� = 0,05 ��� are given by the composite material. In Figure 6 three tensile shear tests of sandwich sheets with different testing velocities are shown. The shear stress is calculated from the initial overlapping length *Lü* and

**Figure 6.** Tensile shear test with Bondal CB (cover sheets of DC06), specimen geometry

The adhesive layer fails by reaching the maximum force. For composites with a shear transmitting adhesive a shear stroke of Δl��� ≈ 0,22 mm was determined at a maximum

mean. With higher velocity, the shear stress increase, but the tolerable displacement drops.

Delamination and displacement of laminated metal sheet can be analyzed by using finite elements. In Figure 7 the experimental and numerical tensile shear tests of the adhesive film are shown. In literature, several investigations determined the stress distribution depending on the overlapping lengths [8]. As the experimental tensile shear test, the force F and displacement �� are gauged with numerical calculations. The shear stress is determined by

��� <sup>=</sup> ����

��� (see Figure 6). These values were computed by arithmetic

�� ), a measurement to describe the increase of

**2.2. Tensile shear test** 

the gauged force *F*.

shear stress of � ≈ 5,5 �

the shear stress, rises.

The specific stiffness of the adhesive ( �

To predict the suitability of vibration-damping sandwich sheets for forming, material properties and interactions are calibrated with the simulation of the die-bending tests. V-die bending with different temperatures are considered in [23] and of velocity in [24]. Therefore specimens with a special grid on the surface of edge were prepared.

Optimized adhesive parameters for numerical calculation are applied on the shear test (see Figure 7). With bilinear, quadrilateral elements the material is performed. Reduced integration plane stress space with hourglass control (CPS4R) is used. Hard pressureoverclosure behavior is approximated with the penalty method. The penetration distance is proportional to the contact force. Contact is implemented frictionless to assume that surfaces in contact slide freely and isotropic with a friction-coefficient of ��� = 0,1.

In Figure 8 the v-die bending process of a symmetrical sandwich with a length of 2L is considered. The stress, strain and displacement of both cover sheets out of DC06 with a thickness of s = 0,7 mm are determinant for two zones, the bending area and the adjacent area. By the punch radius r and the bending angle α, the bending area is defined. In contrary to homogeneous sheets, the adjacent area of the bending area shows a significant stress distribution. This leads to a plasticization of the primarily adjacent area. The residual stress also leads to delamination even after weeks or even under influence of temperature.

**Figure 8.** Pilot project: Experiment v-die bending, v-die bending with cover sheet material DC06

The adhesive layer transmits shear-force of the bending area into the adjacent area. For distances t = [1, 2, 3…] mm from the bending axis the displacement is measured in the simulation and experimental tests. Accurate preparing of the specimens for die bending has a great influence on applicability of the results. After polishing the surfaces of the edges, a micro grid is scribed with a depth of tdia = 3 µm. At the inflection line of the chamfer with the width bdia = 5 µm the light reflects and a sharp line can be seen. For scribing, a diamond with a point angle of �� = 130° is used. Figure 9 shows calculated and experimental displacements of cover sheets after bending. With the influence of manufacturing, the enlarged displacements of experimental tests are explained. The simulation-model is verified.

Forming of Sandwich Sheets Considering Changing Damping Properties 93

Normally, only the edges of a formed part can be seen. Thus, the displacement of the sheets in the bending area is not detectable. At first the displacement of the edges depends on the side length L = [40, 35, 30, and 25] mm, see Figure 10. As expected, increasing side length

**Figure 10.** Numerical results of die-bending; displacement Δl over bending radius α for variation of side length L (a), elongation of the external fiber of the both layers with bending axis at distance

Also the strain of the upper and under fiber of the under layer in bending direction is shown. With larger side lengths the strain increases and a special peek of strain occurs in the bending center. The neutral axis of the lower cover-layer moved depending on the side length in direction to the upper fiber. In the upper layer, the neutral axis moved to the midst of the sandwich sheet, too. Further strain distributions of other fibers, the influence of friction and several displacements dependent on the distance from the bending axis in the bending and adjacent area are shown in [25]. Also the influence of adhesive in contrast to tow single sheets, which slide frictionless on each other, can be seen. So a cutback of the edge displacement from Δl = 0,37 mm to 0,09 mm (L = 40 mm) is achieved with a viscoelastic adhesive. From this it follows that the shear transmitting interlayer doesn't fail at the edges. But at the beginning of the adjacent area, the displacement crosses Δl = 0,15 mm by already a

Even sandwich sheets with large side lengths of for example L = 40 mm can gain an inner failures during forming. Simulations with adjusted layer-thickness are shown in chapter 6.

Correct use of simulation tools demands experiences in element types, material laws, contact properties, solver and integration step characteristics to evaluate numerical results.

To reduce the calculation time a plastomechanical preliminary design is established (see Figure 11). Several numerical calculations describe the bending behavior of three-layer sandwich sheets with viscoelastic interlayers. Numerical investigations are made by M.

Moreover the calculation requires high calculation time and performance. [26]

causes decreasing displacement.

t = 40 mm (b)

bending angle α = 30°.

**4. Plastomechanical preliminary design** 

**Figure 9.** Pilot project v-die bending: calculated and experimental displacement of cover sheets, material DC06 (b)

Normally, only the edges of a formed part can be seen. Thus, the displacement of the sheets in the bending area is not detectable. At first the displacement of the edges depends on the side length L = [40, 35, 30, and 25] mm, see Figure 10. As expected, increasing side length causes decreasing displacement.

**Figure 10.** Numerical results of die-bending; displacement Δl over bending radius α for variation of side length L (a), elongation of the external fiber of the both layers with bending axis at distance t = 40 mm (b)

Also the strain of the upper and under fiber of the under layer in bending direction is shown. With larger side lengths the strain increases and a special peek of strain occurs in the bending center. The neutral axis of the lower cover-layer moved depending on the side length in direction to the upper fiber. In the upper layer, the neutral axis moved to the midst of the sandwich sheet, too. Further strain distributions of other fibers, the influence of friction and several displacements dependent on the distance from the bending axis in the bending and adjacent area are shown in [25]. Also the influence of adhesive in contrast to tow single sheets, which slide frictionless on each other, can be seen. So a cutback of the edge displacement from Δl = 0,37 mm to 0,09 mm (L = 40 mm) is achieved with a viscoelastic adhesive. From this it follows that the shear transmitting interlayer doesn't fail at the edges. But at the beginning of the adjacent area, the displacement crosses Δl = 0,15 mm by already a bending angle α = 30°.

Even sandwich sheets with large side lengths of for example L = 40 mm can gain an inner failures during forming. Simulations with adjusted layer-thickness are shown in chapter 6.

#### **4. Plastomechanical preliminary design**

92 Metal Forming – Process, Tools, Design

material DC06 (b)

area. By the punch radius r and the bending angle α, the bending area is defined. In contrary to homogeneous sheets, the adjacent area of the bending area shows a significant stress distribution. This leads to a plasticization of the primarily adjacent area. The residual stress

also leads to delamination even after weeks or even under influence of temperature.

**Figure 8.** Pilot project: Experiment v-die bending, v-die bending with cover sheet material DC06

**Figure 9.** Pilot project v-die bending: calculated and experimental displacement of cover sheets,

experimental tests are explained. The simulation-model is verified.

The adhesive layer transmits shear-force of the bending area into the adjacent area. For distances t = [1, 2, 3…] mm from the bending axis the displacement is measured in the simulation and experimental tests. Accurate preparing of the specimens for die bending has a great influence on applicability of the results. After polishing the surfaces of the edges, a micro grid is scribed with a depth of tdia = 3 µm. At the inflection line of the chamfer with the width bdia = 5 µm the light reflects and a sharp line can be seen. For scribing, a diamond with a point angle of �� = 130° is used. Figure 9 shows calculated and experimental displacements of cover sheets after bending. With the influence of manufacturing, the enlarged displacements of

> Correct use of simulation tools demands experiences in element types, material laws, contact properties, solver and integration step characteristics to evaluate numerical results. Moreover the calculation requires high calculation time and performance. [26]

> To reduce the calculation time a plastomechanical preliminary design is established (see Figure 11). Several numerical calculations describe the bending behavior of three-layer sandwich sheets with viscoelastic interlayers. Numerical investigations are made by M.

$$
\Delta l = \alpha \ast 0.5 \ast (\mathbf{s}\_a + \mathbf{s}\_l) \tag{10}
$$

$$d\alpha\_r = \int\_{\mathbf{x}=0}^{\mathbf{x}=r\alpha} \frac{2 \ast \sigma\_{ad}}{E \ast \mathbf{s}} d\mathbf{x} = \frac{2 \ast \sigma\_{ad}}{E \ast \mathbf{s}} R\_n \alpha \tag{11}$$

$$\sigma\_{\mathbf{a}\_\*el} = \frac{6R\_n}{s^2} \left( \frac{2}{3} \left( \frac{R\_{p0,2}}{E} \right)^3 \left( E + \frac{m}{2} \right) - \left( \frac{R\_{p0,2}}{E} \right)^2 R\_{p0,2} \right) + \frac{s}{2r}m + 1,\\ 5(R\_{p0,2} - \frac{R\_{p\mathbf{a},2}}{E}m) \tag{12}$$

$$
\alpha\_{sb} \approx 3 \frac{\kappa\_{p\_{0,2}}}{E} \* \frac{\kappa\_n}{s} \* \alpha \tag{13}
$$

$$a\_d = a\_{sb\_\ast a} - a\_{sb\_l} \tag{14}$$

$$\alpha\_{d\_\text{\\_mat}} \approx 3a \frac{R\_{p\_{\text{0},2}}}{E} \left( \frac{s\_l R\_{bend} + s\_l s\_l + s\_{ad} s\_l - s\_a R\_{bend}}{s\_l s\_a} \right) \tag{15}$$

$$\alpha\_{d\_{\text{-}}th} \approx \frac{3a}{s} \Big( \frac{R\_{\text{po},a}}{E\_a} \ast (R\_{\text{bend}} + \mathbf{s}\_{ad} + \mathbf{1}, \mathbf{5} \ast \mathbf{s}) - \frac{R\_{\text{po},\mathbf{z}\_l}}{E\_l} (R\_{\text{bend}} + \mathbf{0}, \mathbf{5} \ast \mathbf{s}) \Big) \tag{16}$$

$$
\alpha\_{d\_{\text{-}}mat\_{\text{-}}th} \approx 3a \frac{R\_{\text{po},2}}{E} \left(\frac{\text{s}^2 + s\_{ads}\text{s}}{\text{s}^2}\right) \tag{17}
$$


• The delamination can be prevented by pre-bending with a smaller radius. With the expansion of the radius in the forming process, the different angle of the cover sheets is used.

Forming of Sandwich Sheets Considering Changing Damping Properties 97

2 ,

= + ⋅ + ⋅ +⋅ (20)

<sup>2</sup> <sup>2</sup> <sup>2</sup> 0.2,

*p i i i n innen i n i*

3

, 12 *i*

Substituting the bending moment under load *MB* with the elastic bending moment *MB-el*, the

2 2 , , , ,

*M EI k*

*B el i n i i zi in*

, , <sup>2</sup> , , 1 3

<sup>=</sup> ⋅ ⋅ +⋅

, , , , ,, ,,

*bl i n bl i n Rin min n d r*

κακ

*<sup>b</sup> <sup>M</sup> E y dy E I k r r*

*z i*

( )

,

*i n*

, , , , <sup>1</sup> <sup>2</sup>

*Rin s Rin k*

( )

−

*i n*

1 2

*<sup>s</sup> <sup>k</sup>*

*i*

*Rin*

κ

0

1

*min <sup>n</sup> r*

α

⋅

1 3 *bl*

− +

*i*

4 3 2

*p i i i*

*s s R*

2

(21)

*k*

, , 1 3 *i n*

*min*

*r* + ⋅

*b s <sup>I</sup>* <sup>⋅</sup> <sup>=</sup> (22)

( ) ,

= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ +⋅ (23)

( ) , ,

*Bin*

*i zi in*

( ) , ,

( )

*B BkBk Bk <sup>m</sup> <sup>B</sup>*

01 2 3

+ ⋅+ ⋅ + ⋅ = − <sup>−</sup> + ⋅

( )

2

1

<sup>1</sup> 1 3

 α

= ⋅ =− ⋅ ⋅ (25)

2 3

*k E*

(24)

(26)

Integration of the bending moment leads to dependency of the k-factor:

0.2,

*i pi*

1

1

= −⋅ ⋅

*<sup>A</sup> <sup>E</sup>*

2,

*i*

With the moment of inertia Iz, i of a rectangular cross-section:

radius of curvature *К* can be determined for the unloaded case.

−

So, the remaining radius *αbl* is calculated:

α

α

( )

*M AA A k A k mI*

1

*<sup>m</sup> AR b <sup>E</sup>*

= ⋅− ⋅

1

*A r <sup>E</sup>*

*Bin i in in in i in i Zi*

, , 0, , 1, , , 2, , ,

0, , , ,

*i n innen i n i*

*i*

*i*

<sup>2</sup> <sup>2</sup> 0.2,

*i*

*i*

43 2

*p i i i*

*s s R*

2 0.2, 1, , , ,

= −⋅ ⋅ +

*<sup>R</sup> <sup>s</sup> <sup>A</sup> r s <sup>E</sup>*

3 2

=− ⋅ ⋅ + ⋅

#### **4.2. Sandwich sheets with shear transmitting interlayer**

As described by the tensile shear tests in chapter 2.2, the metal layers can slide on each other under shear force. The adhesive layer has been considered as viscoelastic. Now, sandwich sheets with shear transmitting interlayers are considered. The neutral axis of the cover lower layer moved depended on the side length (chapter 3) and the shear stress � of the adhesive. In the upper layer, the neutral axis moved to the midst of the sandwich sheet, too.

So the shear force of the adhesive, which superposes the bending moment M� with the force Fü, constitutes the shifting of the axis. The shifting of the neutral axis is expressed by the kfactor (Figure 13). The compressive strain and elongation of the cover sheets changes. Part (b) of Figure 13 shows the outer cover sheet of a symmetrical shear-fixed sandwich. Strain modes of a sandwich layer, which transmits no Fü = 0, k� = k� = 0 (c), tension (a, b) or even compression forces (d, e), are shown.

**Figure 13.** Principle of k-factor for each layer (a-e), geometric of a sandwich with different layerthicknesses (f) and a layer under a bending moment (g)

The strain distribution of sandwiches with i metal layers, which may have different thicknesses si, can be described by the k-factor. For each metal layer with the thickness si, the neutral axis Rn i is computed dependent on the thickness of the sandwich s:

$$r\_{m,i,n} = r\_{inner,i,n} + \left(1 + k\_{i,n}\right) \cdot \frac{s\_i}{2} \tag{18}$$

The bending radius r����� refers to the inside of the sandwich. Considering this, the true bending moment M�,�,� is:

$$M\_{B,i,n} = \underbrace{\int\_0^{\frac{s\_i}{2}} \sigma\_{plast,i,n} \cdot b \cdot y \cdot dy}\_{\text{tension}} + \underbrace{\int\_0^0 \sigma\_{elast-plast,i,n} \cdot b \cdot y \cdot dy}\_{\text{compression}} + \underbrace{\int\_0^{\frac{s\_i}{2}} \sigma\_{elast-plast,i,n} \cdot b \cdot y \cdot dy}\_{\text{compression}} \tag{19}$$

Integration of the bending moment leads to dependency of the k-factor:

96 Metal Forming – Process, Tools, Design

compression forces (d, e), are shown.

thicknesses (f) and a layer under a bending moment (g)

bending moment M�,�,� is:

( )

*M b* σ

*<sup>i</sup> i n*

*<sup>s</sup> <sup>k</sup>*

⋅ −

,

*tension*

used.

• The delamination can be prevented by pre-bending with a smaller radius. With the expansion of the radius in the forming process, the different angle of the cover sheets is

As described by the tensile shear tests in chapter 2.2, the metal layers can slide on each other under shear force. The adhesive layer has been considered as viscoelastic. Now, sandwich sheets with shear transmitting interlayers are considered. The neutral axis of the cover lower layer moved depended on the side length (chapter 3) and the shear stress � of the adhesive. In the upper layer, the neutral axis moved to the midst of the sandwich sheet, too.

So the shear force of the adhesive, which superposes the bending moment M� with the force Fü, constitutes the shifting of the axis. The shifting of the neutral axis is expressed by the kfactor (Figure 13). The compressive strain and elongation of the cover sheets changes. Part (b) of Figure 13 shows the outer cover sheet of a symmetrical shear-fixed sandwich. Strain modes of a sandwich layer, which transmits no Fü = 0, k� = k� = 0 (c), tension (a, b) or even

**Figure 13.** Principle of k-factor for each layer (a-e), geometric of a sandwich with different layer-

neutral axis Rn i is computed dependent on the thickness of the sandwich s:

<sup>1</sup> <sup>2</sup> <sup>0</sup>

<sup>0</sup> <sup>1</sup> <sup>2</sup>

The strain distribution of sandwiches with i metal layers, which may have different thicknesses si, can be described by the k-factor. For each metal layer with the thickness si, the

,, ,, , ( ) 1

The bending radius r����� refers to the inside of the sandwich. Considering this, the true

= ⋅ <sup>⋅</sup> <sup>⋅</sup> <sup>+</sup> <sup>⋅</sup> <sup>⋅</sup> <sup>⋅</sup>

*m i n inner i n i n*

, , , , , ,

*Bin plast i n elast plast i n*

2 *i*

( )

*<sup>i</sup> i n*

*<sup>s</sup> <sup>k</sup>*

−⋅+

,

 σ*y dy b y dy*

*compression*

−

(19)

*<sup>s</sup> rr k* = ++ ⋅ (18)

**4.2. Sandwich sheets with shear transmitting interlayer** 

$$M\_{B,i,n} = A\_i \left( A\_{0,i,n} + A\_{1,i,n} \cdot k\_{i,n} + A\_{2,i} \cdot k\_{i,n}^2 \right) + m\_i \cdot I\_{Z,i} \frac{1 + \mathfrak{Z} \cdot k\_{i,n}^2}{r\_{m,i,n}} \tag{20}$$

$$\begin{aligned} A\_i &= R\_{p0,2,i} \cdot \left( 1 - \frac{m\_i}{E\_i} \right) \cdot b \\ A\_{0,i,n} &= \frac{s\_i^2}{4} - \frac{1}{3} \cdot \left( \frac{R\_{p0,2,i}}{E\_i} \right)^2 \cdot \left( r\_{imnn,i,n} + \frac{s\_i}{2} \right)^2 \\ A\_{1,i,n} &= -\frac{1}{3} \cdot \left( \frac{R\_{p0,2,i}}{E\_i} \right)^2 \cdot \left( r\_{imnn,i,n} + \frac{s\_i}{2} \right) \cdot s\_i \\ A\_{2,i} &= \frac{s\_i^2}{4} - \frac{1}{3} \cdot \left( \frac{R\_{p0,2,i}}{E\_i} \cdot \frac{s\_i}{2} \right)^2 \end{aligned} \tag{21}$$

With the moment of inertia Iz, i of a rectangular cross-section:

$$I\_{z,i} = \frac{b \cdot s\_i^3}{12} \tag{22}$$

Substituting the bending moment under load *MB* with the elastic bending moment *MB-el*, the radius of curvature *К* can be determined for the unloaded case.

$$\mathcal{M}\_{B-el,i,n} = \frac{b}{r\_{\mathcal{R},i,n}} \cdot E\_i \cdot \int\_{-\frac{S\_i}{2}(1+k\_{i,n})}^{\frac{S\_i}{2}(1-k\_{i,n})} y^2 \cdot dy = \frac{1}{r\_{\mathcal{R},i,n}} \cdot E\_i \cdot I\_{z,i} \cdot \left(1 + 3 \cdot k\_{i,n}^2\right) \tag{23}$$

$$\kappa\_{\mathcal{R},i,n} = \frac{M\_{\mathcal{B},i,n}}{E\_i \cdot I\_{z,i} \cdot \left(1 + 3 \cdot k\_{i,n}^2\right)}\tag{24}$$

So, the remaining radius *αbl* is calculated:

$$\alpha\_{bl,i,n} = \int\_0^{r\_{m,i,n} \cdot \alpha\_n} \kappa\_{bl,i,n} \cdot d\alpha = \left(1 - \kappa\_{R,i,n} \cdot r\_{m,i,n}\right) \cdot \alpha\_n \tag{25}$$

$$\alpha\_{bl} = 1 - B \frac{\left(B\_0 + B\_1 \cdot k + B\_2 \cdot k^2 + B\_3 \cdot k^3\right)}{\left(1 + 3 \cdot k^2\right)} - \frac{m}{E} \tag{26}$$

$$\begin{aligned} \mathbf{B} &= \frac{R\_{p0,2}}{E \cdot I\_Z} \cdot \left(1 - \frac{m}{E}\right) \cdot b \\ \mathbf{B}\_0 &= \frac{s^2}{4} \cdot \left(r\_{inner} + \frac{s}{2}\right) - \frac{1}{3} \cdot \frac{R\_{p0,2}^2}{E^2} \cdot \left(r\_{inner} + \frac{s}{2}\right)^3 \\ \mathbf{B}\_1 &= \frac{s^3}{8} - \frac{1}{2} \cdot \frac{R\_{p0,2}^2}{E^2} \cdot \left(r\_{inner} + \frac{s}{2}\right)^2 \cdot s \\ \mathbf{B}\_2 &= \left(1 - \frac{1}{3} \cdot \frac{R\_{p0,2}^2}{E^2}\right) \cdot \frac{s^2}{4} \cdot \left(r\_{inner} + \frac{s}{2}\right) \\ \mathbf{B}\_3 &= \left(1 - \frac{1}{3} \cdot \frac{R\_{p0,2}^2}{E^2}\right) \cdot \frac{s^3}{8} \end{aligned} \tag{27}$$

Forming of Sandwich Sheets Considering Changing Damping Properties 99

**[-]** 

30% plasticizer 0,8 20 50 100

�� = � � ����(�) (32)

nitrile rubber 0,8 330 20 1000

**E-Modulo [N/mm2]** 

**Tamb [°C]**  **f [1/s]** 

��(���) (30)

��� (31)

friction of the solid. Compared to synthetic materials, steel converts a lower amount of vibration energy per oscillation into heat. The decay will take more time compared to a sandwich sheet. Using the poisson's ratio ν, the shear modulus is determined according to:

� = �

����� = ��

Complex shear modulus �<sup>∗</sup> = �� +����� with its amount |�∗| = �(��)� + (���)� is highly dependent on temperature and amplitude. Often in a range of about 60 °C a constant

**δ Material** ��� �

**Steel** 0,0006 – 0,0001 Polyvinyl chloride 1,8 30 92 20 **Aluminum** 0,0001 – 0,001 Polystyrene 2,0 300 140 2000 **Gray iron** 0,01 – 0,02 Polyisobutylene 2,0 6 20 3000

**Bitumen** 0,2 – 0,4 hard rubber 1,0 200 60 40

**Table 2.** Loss factor tan δ for different materials under ambient temperature of Tamb = 20 °C [34] and

Figure 15 shows the relation of the loss factor tan δ and the shear modulus, which leads to

The loss factor is the ratio of storage modulus ��� and loss modulus ��� [3]:

value is assumed.

**Figure 14.** Offset between stress and strain over time

**Damping mat** 0,2 - 1 Polyvinyl chloride with

Thus, the storage modulus �� can be detremined as:

**Material loss factor tan** 

Material properties according to [33]

following equation:

**Film of** 

$$
\Delta l\_{j,n} = \Delta l\_{j,n-1} + \left(\alpha\_{bl,i,n} - \alpha\_{bl,i-1,n}\right) \cdot r\_{m,n} - \left(1 + k\_i\right) \cdot \frac{s\_i}{2} \cdot \alpha\_{bl,i,n} - \left(1 - k\_{i-1}\right) \cdot \frac{s\_{i-1}}{2} \cdot \alpha\_{bl,i-1,n} \tag{28}
$$

$$
\Delta \alpha\_{j,n} = \Delta \alpha\_{j,n-1} + \left(\kappa\_{R,i-1,n} - \kappa\_{R,i,n}\right) \cdot r\_{m,n} \cdot \alpha\_n \tag{29}
$$

In a multilayer sandwich sheet, the displacement Δl and the angle of delamination Δα between two layers can be calculated as shown in equation (28), (29). The determination of the k-factor and the following proposal list to avoid or minimize the failure modes at the edges are according to [25] :


### **5. Acoustical calculations**

#### **5.1. The loss factor, a measurement for damping behavior of a material**

The transmission of structure-borne noise depends highly on the behavior of a material. Through the temporal offset of the shear stress and strain, the vibration energy is converted into heat energy. [32], [33], [34]

A gauge of the internal damping of a material, the absorption capacity of vibrations, is the loss factor tan δ. With increasing loss factor the material behavior approaches a Newtonian fluid with viscosity. Even if the metal sheet is not damped by a polymeric coating or interlayer, the vibration of the sheet decays after a certain time. This effect is due to internal friction of the solid. Compared to synthetic materials, steel converts a lower amount of vibration energy per oscillation into heat. The decay will take more time compared to a sandwich sheet. Using the poisson's ratio ν, the shear modulus is determined according to:

$$G = \frac{E}{2 \cdot (1 + \theta)}\tag{30}$$

The loss factor is the ratio of storage modulus ��� and loss modulus ��� [3]:

$$\tan \delta = \frac{\sigma'}{\sigma''} \tag{31}$$

Complex shear modulus �<sup>∗</sup> = �� +����� with its amount |�∗| = �(��)� + (���)� is highly dependent on temperature and amplitude. Often in a range of about 60 °C a constant value is assumed.

**Figure 14.** Offset between stress and strain over time

98 Metal Forming – Process, Tools, Design

0.2

(27)

α

*p*

*E*

*innen*

( ) ( ) ( ) <sup>1</sup> , , 1 , , , 1, , , , <sup>1</sup> , 1, 1 1

<sup>−</sup> Δ =Δ + − ⋅ − + ⋅ ⋅ − − ⋅ ⋅ − − − − (28)

*rk k*

 κ α

*<sup>j</sup>*, , 1 , 1, , , , *n jn Ri n Rin mn n* ( ) *r* Δ =Δ + − ⋅ ⋅ − − (29)

*j n j n bl i n bl i n m n i bl i n i bl i n*

In a multilayer sandwich sheet, the displacement Δl and the angle of delamination Δα between two layers can be calculated as shown in equation (28), (29). The determination of the k-factor and the following proposal list to avoid or minimize the failure modes at the

2 2 *i i*

> α

<sup>2</sup> <sup>3</sup> <sup>2</sup>

 

1 4 23 2

*<sup>R</sup> ss s B r <sup>r</sup>*

= ⋅ + −⋅ ⋅ +

*innen*

*innen innen*

0.2

*<sup>R</sup> s s B rs E*

*p*

8 2 2

= −⋅ ⋅ + ⋅

<sup>2</sup> <sup>2</sup> <sup>3</sup>

<sup>2</sup> <sup>2</sup>

34 2

<sup>2</sup> <sup>3</sup>

0 2

1

 

*<sup>R</sup> <sup>m</sup> B b EI E*

= ⋅− ⋅ <sup>⋅</sup>

0.2

*p Z*

1 2

1

2 2

<sup>1</sup> <sup>1</sup>

3 2

 α

αα

α

edges are according to [25] :

• Enlarge the number of metal layers • Reduce the thicknesses of the outer layers

• Choose a small hardening coefficient

**5. Acoustical calculations** 

into heat energy. [32], [33], [34]

<sup>1</sup> <sup>1</sup>

*<sup>R</sup> <sup>s</sup> <sup>B</sup>*

= −⋅ ⋅

• Keep the required overlapping length, respectively the side length • Choose an adhesive depending on the shear force which is calculated

**5.1. The loss factor, a measurement for damping behavior of a material** 

The transmission of structure-borne noise depends highly on the behavior of a material. Through the temporal offset of the shear stress and strain, the vibration energy is converted

A gauge of the internal damping of a material, the absorption capacity of vibrations, is the loss factor tan δ. With increasing loss factor the material behavior approaches a Newtonian fluid with viscosity. Even if the metal sheet is not damped by a polymeric coating or interlayer, the vibration of the sheet decays after a certain time. This effect is due to internal

• Choose a material for outer layers with low yield strength

 

0.2

*<sup>R</sup> s s B r E*

= −⋅ ⋅ ⋅ +

*p*

0.2

*p*

*E*

3 8

*s s l l*

 κ


**Table 2.** Loss factor tan δ for different materials under ambient temperature of Tamb = 20 °C [34] and Material properties according to [33]

Figure 15 shows the relation of the loss factor tan δ and the shear modulus, which leads to following equation:

$$G' = G \cdot \cos(\delta) \tag{32}$$

Thus, the storage modulus �� can be detremined as:

$$G\_2' = G \cdot \cos(\arctan(\tan \delta\_2))\tag{33}$$

$$f\_{(t)} = M\ddot{\mathbf{y}} + \mathbf{C}\dot{\mathbf{y}} + K\mathbf{y} \tag{34}$$


$$0 = \ddot{\mathbf{y}} + \frac{c}{m}\dot{\mathbf{y}} + \frac{k}{m}\mathbf{y} \tag{35}$$

$$
\omega\_0^2 = \frac{k}{m} \tag{36}
$$

$$D\omega\_0 = \frac{c}{m} \tag{37}$$

$$0 = \ddot{\mathbf{y}} + D\omega\_0 \dot{\mathbf{y}} + \omega\_0^2 \mathbf{y} \tag{38}$$

$$\mathcal{Y}\_{\{\mathbf{t}\}} = \circlearrowleft\_{0} \cdot e^{-D\omega\_{0} \cdot \mathbf{t}} \cdot \sin(\omega\_{d} \cdot \mathbf{t} + \varphi\_{0}) \tag{39}$$

$$\delta = \tan \delta\_{\text{Sandwich}} \cdot \pi \cdot f \tag{40}$$

$$Q = \frac{1}{\tan \delta\_{\text{Sandwch}}} \tag{41}$$

$$D' = \frac{\tan \delta\_{\text{Sandtwtic}}}{2} \tag{42}$$

$$
\omega\_d = \sqrt{\omega\_0^2 - \delta^2} \tag{43}
$$

$$D = \frac{\delta}{\omega\_0} \tag{44}$$

$$M = B\frac{\partial \Phi}{\partial \chi} = \Sigma\_1^3 M\_{ll} + \Sigma\_1^3 F\_l H\_{lo} \tag{45}$$

$$
tan\delta\_{\text{Sandwch}} = \tan\delta\_2 \cdot \frac{h \cdot g}{[[1 + (1 + \iota \tan\delta\_2) \cdot g]^2 + g \cdot h \cdot [1 + g \cdot (1 + \tan\delta\_2^2)]]} \tag{46}
$$

$$\frac{1}{a^2} = \frac{{B\_1}^{'} + {B\_3}^{'}}{a^2} \cdot \left(\frac{1}{E\_1 \cdot d\_1} + \frac{1}{E\_3 \cdot d\_3}\right) \tag{47}$$

$$g = \frac{G\_2'}{d\_2 \cdot k^2} \cdot \left(\frac{1}{E\_1 \cdot d\_1} + \frac{1}{E\_3 \cdot d\_3}\right) \tag{48}$$

$$a \approx d\_2 + \frac{(d\_1 + d\_3)}{2} \tag{49}$$

$$k = \left(\frac{\omega^2 \cdot m'}{B}\right)^{\frac{1}{4}}\tag{50}$$

$$
\mathbf{m}' = \rho \cdot \mathbf{S} \tag{51}
$$

$$S = lg(d\_1 + d\_2) \tag{52}$$

$$B' = (B'\_{\,1} + B'\_{\,3}) \cdot \left(1 + \frac{g \cdot h}{1 + g \cdot (1 + l \cdot \tan \delta\_2)}\right) \tag{53}$$

$$B'\_{\,\,l} = \frac{1}{\iota z} \cdot E\_{\,l} \cdot d\_{\,l}^3 \tag{54}$$

between adhesive thickness, surface roughness Rmax and bond strength is shown according to [8]. It is recommended that the adhesive thickness is equal to the surface roughness Rmax. A smaller thickness avoids a complete coating. To achieve improvements in damping behavior the surface roughness and the thickness of the adhesive layer has to be reduced.

Forming of Sandwich Sheets Considering Changing Damping Properties 105

layer thicknesses (Figure 20) are calculated. The same material as in chapter 3 and 6.1 is

For this great side length, the displacement after forming is very small and no failure mode occurs for all five constellations. The symmetrical sandwich (ݏଵ ൌ ݏଶ ൌ Ͳǡ݉݉) sheet no. 1 shows less spring back than all of the other constellations with 4 metal layers. The highest value for spring back shows no. 4 with increasing layer thickness. The remaining stress which can cause failure (chapter 1.2) decreases by using unsymmetrical thicknesses (no. 2, 3 and 4). As proposed in chapter 4.2, the number of layers influences the strain distribution. Minimal elongation shows specimen no. 3 and compression no. 5. Because of the minor stiffness no. 5 tends to buckling. This is an initial point for inner failures. The different

The normal plastic strain component in bending direction depends on the thickness of the metal layers. The thinner the inner cover sheet is, the minor is the plastic strain and the displacement of the edges (see no. 2-4). But a thin inner layer tends to buckling. To get the lowest normal stress in bending direction, the thickness should be increased as seen in configuration no. 4. Also the spring-back of the undamaged sandwich depends on the layer-

An example for an application of a commercial three-layer sandwich sheet in the automotive industry is shown in [25]. For this profile, formed by rolling the failures displacement,

At the Chair of Forming Technologies at the University of Siegen vibrations damping

During forming, failure modes like delamination, displacement and buckling of the cover sheets occur. The mechanical properties of the metal layers are determined by uniaxial

delamination and buckling could be predicted and verified with experimental tests.

composite sheets were investigated regarding their forming limits.

used. For this calculation, the adhesive thickness is neglected.

**Figure 20.** Numerical results of die-bending for five multilayer sandwich sheet

spring back of each layer can be absorbed by the adhesive.

configuration.

**7. Conclusions and forecast** 
