# How Does Metal Bulk Work?

### Introduction

Most of the numerical simulations of the bulk metal forming processes nowadays are carried out using finite element method. It has been extensively applied to the metal forming industry to enable a more scientific approach to forming process development and optimization. In this section, a particular interest was paid on the critical challenges in finite element simulation with the aim to help the handbook users to identify the key issues in this area. Such issues include the material friction behaviors, mechanical properties, microstructure evolution, and fracture prediction.

### Friction in the Finite Element Simulation

In metal forming process, workpiece deformation is brought about during contact between a tool and a workpiece. This inevitably results in friction if there is any tangential force at the contacting surfaces. Numerous analytical models have been proposed to describe the frictional behaviors between the workpiece and die; below are the some of the most popular ones used in the finite element simulation software. Their areas of application are also presented.

#### Coulomb Friction Model

Coulomb friction model is used when contact occurs between two elastically deforming objects (could include an elastic-plastic object, if it is deforming elastically), or an elastic object and a rigid object. Generally, it is used to model sheet metal forming processes. The frictional force in the Coulomb law model is defined by

$$ f=\mu p $$

(21)

where *f* is the frictional stress, *p* is the interface pressure between two bodies, and *μ* is the friction coefficient. However, the use of coulomb friction model gives occasion of an overestimation of the friction stresses at the tool-workpiece interface, as the normal pressure is often considerably greater than the yield stress of the material. Consequently, the friction stress becomes greater than the yield stress of the material in pure shear.

#### Shear Friction Model

The constant shear friction is used mostly for bulk metal forming simulations. The frictional force in the constant shear model is defined by

$$ f= mk $$

(22)

where *f* is the frictional stress, *k* is the shear yield stress (of the workpiece), and *m* is the friction factor. This states that the friction is a function of the yield stress of the deforming body. A general benchmark of friction coefficients for different bulk metal forming processes are listed in Table 10. It is important to note that the lubricant used on the tooling may play a large role in the value of friction stress. The friction will in turn affect the metal flow at contact surfaces.

**Table 10 Typical friction coefficient values using constant shear friction model**

Full size table

#### Hybrid Friction Model

A hybrid friction model (combination of the coulomb and the constant shear models) is often used when rolling or elastoplastic deformation (for springback) is considered. The general function is as Eq. 23.

$$ \left\{\begin{array}{c}\hfill \tau =\mu p\ \left(\mu p< mk\right)\hfill \\ {}\hfill \tau = mk\ \left(\mu p\ge mk\right)\hfill \end{array}\right. $$

(23)

This model describes that at low normal pressure, the friction is proportional to the normal pressure whereas at high normal pressure, the friction is proportional to the workpiece shear stress. The frictional behavior is illustrated in Fig. 24.

**Fig. 24**

Illustration of hybrid coulomb and constant shear friction model

Full size image

#### General Friction Model (by Wanheim and Bay)

The general friction model proposed by Bay and Wanheim (1976) is a very popular friction model which also takes into consideration the different friction behaviors under low/high normal pressure.

$$ \tau = f\alpha k $$

(24)

where *τ* and *k* represent the friction stress and shear yield stress of the material, respectively. *f* is the friction factor and *α* is the real contact area ratio. This equation can also be treated as a combination of coulomb and constant shear models as at high normal pressure, both *f* and *α* stay constant, which implies a constant *m* value:

$$ \frac{\tau }{k}= f\alpha =m $$

(25)

At low normal pressure, the real contact area ratio α exhibits a linear relationship with normal pressure *p*. With constant *f* and *k*, *τ* has a linear relationship with *p*, which turns into a typical coulomb friction model. An illustration of the general friction model showing such a combined nature can be found in Fig. 25.

**Fig. 25**

Illustration of the general friction model function

Full size image

#### Advanced Friction Models

There are numerous advanced friction models to accurately capture the interaction between the workpiece and die under varying processing conditions. These models take into consideration the influence of the time, interface pressure, interface temperature, and surface stretch of the deforming workpiece or even a combination of these. Normally, the empirical models for the pressure, strain rate, and sliding-velocity-dependent friction coefficients can be obtained, as listed below.

*Pressure*-*dependent friction coefficient*:

$$ m={m}_0\left(1-{e}^{-\alpha p}\right)\ \mathrm{or}/\mathrm{and}\ \mu ={\mu}_0\left(1-{e}^{-\alpha p}\right) $$

(26)

where *p* is the normal pressure and α is a constant with typical values ranging from 0.012 to 0.06.

*Strain*-*rate*-*dependent friction coefficient*:

$$ m={m}_0\left(1+\alpha \dot{\overline{\varepsilon}}\right)\ \mathrm{or}/\mathrm{and}\ \mu ={\mu}_0\left(1+\alpha \dot{\overline{\varepsilon}}\right) $$

(27)

where \( \dot{\overline{\varepsilon}} \) is the effective strain rate and *α* is a constant with typical values ranging from 0.0012 to 0.0045.

*Sliding*-*velocity*-*dependent friction coefficient*:

$$ m={m}_0{\left|{\upsilon}_s\right|}^{\alpha }\ \mathrm{or}/\mathrm{and}\ \mu ={\mu}_0{\left|{\upsilon}_s\right|}^{\alpha}\sigma = E\varepsilon $$

(28)

where *υ*
*s*
is the sliding velocity and α is a constant with typical values ranging from 0.0016 to 0.014.

These models provide additional options for users to accurately capture the friction behaviors during bulk metal forming process if such a process exhibits a strong sensitivity to friction. However, most conventional processes are not extremely sensitive to friction, and the typical values listed above may be adequate for initial process design and load prediction.

### Material Mechanical Properties Representation

The material mechanical properties here are mainly the material’s flow stress during the forming processes. It is a fundamental parameter to determine toque and power of metal forming equipment and is defined as a stress that results in the material flow in a one-dimensional stress state. Initially we focus on the models that describe the material flow stress curves. Subsequently, multiple-dimensional stresses are introduced, with their anisotropy (orientation-dependence) considered in terms of yield function.

#### Elastic Region

A classic ductile metal stress-strain curve resembles that in Fig. 26, in which the two distinct material behaviors, elastic and plastic regions, are shown. For elastic region (till yield stress), Hooke’s law applies to relate the stress σ with strain ε,

**Fig. 26**

Stress–strain curve of a typical ductile metal

Full size image

$$ \sigma = E\varepsilon $$

(29)

where *E* is the modulus of elasticity, or Young’s modulus. As it is a linear relationship, the modulus can be determined from the slope of the engineering stress–strain curve in the elastic region.

#### Power Law

In the plastic region, work hardening is observed for plastic materials. Work hardening is the strengthening of a metal by plastic deformation. This strengthening occurs because of dislocation movements and dislocation pile-up within the crystal structure of the material. Perhaps the most common mathematical description of such work hardening phenomenon is the power law, which is an empirical stress–strain relationship obtained by fitting an exponential curve to the experimental data points of the flow stress curve.

$$ \sigma =K{\varepsilon}^n+{\sigma}_0 $$

(30)

where *K* is the strength coefficient and *n* is the strain-hardening exponent coefficient, while *σ*
0 can be seen as the initial yield value. Figure 27 provides an example of the power law fitting of Inconel 718 alloy flow stress data.

**Fig. 27**

Flow stress of INCONEL 718 (annealed)

Full size image

#### Linear Hardening

Another popular (but to a lesser extent accurate) work hardening rule is the linear hardening law. It assumes that the flow stress is proportional to the strain in the plastic region with hardening coefficient of *H*, which in principle is smaller than the value of the material Young’s modulus. Such rough approximation is especially helpful when limited material data are available (i.e., only yield stress, UTS, and elongation are provided). The linear hardening rule is presented as

$$ \sigma ={\sigma}_0+ H\varepsilon $$

(31)

where both *σ*
0 and *H* are dependent on temperature and the dominating atom content (in steel, the dominating atom content is carbon percentage). Figure 28 gives an illustration of the linear hardening flow stress curve.

**Fig. 28**

Illustration of the linear hardening law

Full size image

In this law, if the linear hardening coefficient *H* is set to 0, the material behavior becomes elastic-perfect plastic, which is a special case of the linear hardening law.

#### Johnson-Cook Flow Stress Model

JC flow stress model is perhaps the most famous empirical flow stress model that takes strain rate and temperature effects into consideration. Such rate-dependent inelastic behavior of solids is called viscoplasticity in continuum mechanics theory. The general form of a JC flow stress law looks like

$$ \sigma =\left(A+B{\varepsilon}^n\right)\;\left(1+C \ln \left({\dot{\varepsilon}}^{\ast}\right)\right)\;\left(1-{\left({T}^{*}\right)}^m\right) $$

(32)

where *ε* is the equivalent plastic strain, \( {\dot{\varepsilon}}^{*} \) is the normalized plastic strain rate, *T** is the normalized equivalent temperature, and *A*, *B*, *C*, *n*, and *m* are material constants.

The normalized strain rate and temperature in the equation above are defined as

$$ {\dot{\varepsilon}}^{*}=\frac{{\dot{\varepsilon}}^{*}}{{\dot{\varepsilon}}_0^{*}}\ \mathrm{and}\ {\mathrm{T}}^{*}=\frac{\left(\mathrm{T}-{T}_0\right)}{\left({T}_{\mathrm{m}}-{T}_0\right)} $$

(33)

where \( {\dot{\varepsilon}}_0^{*} \) is the effective plastic strain rate of the quasi-static reference test used to determine the yield and hardening parameters *A*, *B*, and *n. T*
0 is a reference temperature, and *T*
m is a reference melting temperature. For conditions where *T** < 0, t is assumed that *m* = 1.

It is also worth noting that in JC flow stress rule, if the influences of the strain rate (second term in the equation) and temperature (third term in the equation) are ignored, the equation becomes a classic power law mentioned before.

#### Microstructure-Based Flow Stress Model

This flow rule or the so-called Taylor equation (Taylor 1934) is relatively new to the industry. However, it is very popular among the academia in solid mechanics research field, which provides a good link between the microscopic and macroscopic levels by explaining the strain-hardening phenomenon using the dislocation pile-up theory. A general form of the equation is

$$ \sigma ={\sigma}_0+\alpha Gb\sqrt{\rho } $$

(34)

where α is a dimensionless coefficient, *G* is the shear modulus, *b* is the Burgers vector, and *ρ* is the dislocation density. The strain hardening occurs when the dislocations are piling-up during plastic deformation, by which their density increases and results in increment of material flow stress. The relationship between plastic deformation and dislocation density needs to be provided alongside this Equation.

#### Poisson’s Ratio

The equations above are all for one-dimensional stress state. If multiaxial stress state is considered, the material may behave differently from that of the one-dimensional. In the elastic region, for instance, the material may show different Young’s modulus if uniaxially loaded in different directions. This is so-called anisotropy in elasticity. Most of the metals exhibit such anisotropy in elastic region but not in a large amount. More often, when a piece of metal is tensile loaded in one dimension, the other two dimensions will shrink accordingly to accommodate such shape change. Such phenomenon is called Poisson’s effect, with ratio of transverse to axial strain named as Poisson’s ratio. Most materials have Poisson’s ratio values ranging between 0.0 and 0.5. Majority of steels and rigid polymers when used within their elastic limits (before yield) exhibit Poisson ratio values of about 0.3, increasing to 0.5 for plastic deformation. Rubber has a Poisson ratio of nearly 0.5.

#### Yield Function: Von Mises

Yield function is a function describing the material yield point when triaxial stress is presented. For uniaxial loading, the yield point can be easily identified in the stress–strain curve as the transition point between linear elastic and plastic regions. When triaxial stress state is considered, if the material is isotropic, the yield condition should be based on the von Mises yield stress value *σy*. When the von Mises stresses or equivalent tensile stress *σv* > *σy*, then the material plastically deforms. If *σv* < = *σy*, then the material only deforms elastically.

$$ {\sigma}_{\mathrm{v}}=\overline{\sigma}=\frac{1}{2}\left({\left({\sigma}_{\mathrm{x}\mathrm{x}}-{\sigma}_{\mathrm{y}\mathrm{y}}\right)}^2+{\left({\sigma}_{\mathrm{y}\mathrm{y}}-{\sigma}_{\mathrm{z}\mathrm{z}}\right)}^2+{\left({\sigma}_{\mathrm{x}\mathrm{x}}-{\sigma}_{\mathrm{z}\mathrm{z}}\right)}^2+6{\left({\sigma}_{\mathrm{x}\mathrm{y}}+{\sigma}_{\mathrm{y}\mathrm{z}}+{\sigma}_{\mathrm{z}\mathrm{x}}\right)}^2\right)=\sqrt{\frac{1}{2}\left({\left({\sigma}_{\mathrm{x}}-{\sigma}_{\mathrm{y}}\right)}^2+{\left({\sigma}_{\mathrm{y}}-{\sigma}_{\mathrm{z}}\right)}^2+{\left({\sigma}_{\mathrm{x}}-{\sigma}_{\mathrm{z}}\right)}^2\right)} $$

(35)

where *σ*
1
*σ*
2
*σ*
3 are the principle stresses (normal stresses in the directions without any shear stresses) and

$$ \left\{\begin{array}{c}\hfill {\sigma}_{\mathrm{v}}\le {\sigma}_{\mathrm{y}},\mathrm{elastic}\ \mathrm{deformation}\hfill \\ {}\hfill {\sigma}_{\mathrm{v}}>{\sigma}_{\mathrm{y}},\mathrm{plastic}\ \mathrm{deformation}\hfill \end{array}\right. $$

It is worth noting that in this case, if uniaxial stress is considered, *σ*
1 ≠ 0, *σ*
2 = *σ*
3 = 0, therefore the von Mises criterion simply reduces to *σ*
1 = *σ*
*y*
, which is the yield point of uniaxial loading.

#### Yield Function: Hill’s

The quadratic Hill’s yield criterion has the form

$$ F{\left({\sigma}_{\mathrm{xx}}-{\sigma}_{\mathrm{yy}}\right)}^2+G{\left({\sigma}_{\mathrm{yy}}-{\sigma}_{\mathrm{zz}}\right)}^2+H{\left({\sigma}_{\mathrm{xx}}-{\sigma}_{\mathrm{zz}}\right)}^2+2L{\sigma}_{\mathrm{yz}}^2+2M{\sigma}_{\mathrm{zx}}^2+2N{\sigma}_{\mathrm{xy}}^2=1 $$

(36)

Here F, G, H, L, M, and N are constants that are needed to be determined experimentally. The quadratic Hill yield criterion depends only on the deviatoric stresses and is volumetric stress independent. It predicts the same yield stress in tension and compression. It is especially useful when the material has strong texture, for instance, for rolled plates or hot-extruded billets. In materials science, texture is the distribution of crystallographic orientations of a polycrystalline sample. A workpiece in which these orientations are fully random is said to have no texture. If the crystallographic orientations are not random, but have some preferred orientation, then the sample has a weak, moderate, or strong texture depending on the number of crystallites sharing the sample orientation. The material properties show a strong anisotropy because of the texture, and the phenomenon of anisotropic yield can be accounted for using Hill’s yield criterion. The disadvantage associated with the criterion is that there are quite a number of coefficients (six in the quadratic form) to be determined (hence multiple tests are required) before it can be applied to finite element simulation .

#### Lankford Coefficient

Earlier we briefly discussed the anisotropy in elasticity and anisotropy in yield. The anisotropy in plasticity is in no way simpler than that of elasticity or yield. A commonly used plastic anisotropy indicator is the Lankford coefficient (also called Lankford value or R-value). This scalar quantity is used extensively as an indicator of the formability of recrystallized low-carbon steel sheets. Its definition follows:

If *x* and *y* are the coordinate directions in the plane of rolling
and *z* is the thickness direction, then the Lankford coefficient (R-value) is given by

$$ R=\frac{\varepsilon_{xy}^p}{\varepsilon_z^p} $$

(37)

where *ε*
*p*
*xy*
is the plastic strain in-plane and *ε*
*p*
*z*
is the plastic strain through the thickness. In practice, the *R*-value is usually measured at 20 % elongation in a tensile test.

For sheet metals, the *R*-values are usually determined for three different directions of loading in-plane (0°, 45°, and 90° to the rolling
direction) and the normal R-value is taken to be the average

$$ R=\frac{1}{4}\left({R}_0+2{R}_{45}+{R}_{90}\right) $$

(38)

The planar anisotropy coefficient or planar R-value is a measure of the variation of *R* with angle from the rolling
direction. This quantity is defined as

$$ {R}_p=\frac{1}{2}\left({R}_0-2{R}_{45}+{R}_{90}\right) $$

(39)

It has been widely recognized that anisotropy is closely linked with the material microstructure. The evolution of material microstructure during bulk forming process may greatly influence the end product’s structural integrity. Hence, understanding the microstructural behavior, before and during the forming process, is one of the main focuses.

### Simulation of Microstructure Evolution During Bulk Metal Forming Processes

The finite element simulation of microstructure evolution is a very hot topic in research communities. Divided opinions exist in many areas, even on the definitions of some fundamental mechanisms. Therefore, in this section, only the most popular definitions and generalized equation forms are provided.

To start, an introduction on the metal microstructure has to be provided. Metal alloys are unusually polycrystalline solids, which consist of many crystallites that are small, often microscopic crystals that are held together through highly defective boundaries. Metallurgists often refer to these crystallites as grains (grain size ∼30 μm). Figure 29 provides an example of the grain structure in the steel.

**Fig. 29**

Optical micrograph of AA6061 aluminum alloy showing polycrystalline structure

Full size image

They are normally of different orientations and separated by grain boundaries. Grain boundaries are interfaces where crystals of different orientations meet. Grains in the metal change shapes and orientations during forming process, which in turn influence the material mechanical properties. Modeling
such changes may not be easy as there are thousands of grains in the workpiece and capture of individual behavior becomes computationally impossible. Therefore, such changes are always modeled using statistical methods. To represent the grain structure before deformation, methods like *cellular automata* (Wolfram 1983) and *voronoi tessellation* (Voronoi 1908) are commonly employed. A value of average grain size is of course another option (although very rough).

For the grain structure evolution during deformation, numerous phenomenological models have been developed in this area, and controversies exist on the definitions of various recrystallization mechanisms. However, the computational algorithms behind them are similar: in each time step, local temperature, strain, strain rate, and evolution history, the mechanism of evolution is determined, and then the corresponding grain variables are computed and updated. In the condition that all the phenomena can be divided into the following three microstructural evolution groups, then in each group the corresponding mathematical function can be used to describe such evolution.

*Dynamic recrystallization* (*DRX*) occurs during deformation and when the strain exceeds the critical strain. The driving force is dislocations annihilation.

*Static recrystallization* occurs after deformation and when the strain is less than the critical strain. The driving force for static recrystallization is dislocations annihilation. The recrystallization begins in a nuclei-free environment.

*Grain growth* occurs before recrystallization begins or after recrystallization is completed. The driving force is the reduction of grain boundary energy.

#### Dynamic Recrystallization

The dynamic recrystallization is a function of strain, strain rate, temperature, and initial grain size, which change in time. It is very difficult to model dynamic recrystallization concurrently during forming as this has the possibility of creating numerical instability. Instead, the dynamic recrystallization is computed in the group immediately after the deformation stops. The average temperature and the strain rate of the deformation period are used as inputs of the Equations.

##### Activation Criteria

The onset of DRX usually occurs at a critical stain *ε*
*c*

$$ {\varepsilon}_{\mathrm{c}}={\mathrm{a}}_2{\varepsilon}_{\mathrm{p}} $$

(40)

where *ε*
*p*
denotes the stain corresponding to the flow stress maximum:

$$ {\varepsilon}_p={a}_1{d}_0^{n_1}{\dot{\varepsilon}}^{m_1}{e}^{\left({Q}_1/ RT\right)}+{c}_1 $$

(41)

in which *d*
0 is the initial grain size, *R* is the gas constant, *T* is the temperature in Kelvin, and *Q* is activation energy.

##### Kinetics

The Avrami equation (Avrami 1939) is used to describe the relation between the dynamically recrystallized fraction *X* and the effective strain.

$$ {X}_{drex}=1- \exp \left[-{\beta}_d{\left(\frac{\varepsilon -{\mathrm{a}}_{10}{\varepsilon}_{\mathrm{p}}}{\varepsilon_{0.5}}\right)}^{h_d}\right] $$

(42)

where ε0.5 denotes the strain for 50 % recrystallization:

$$ {\varepsilon}_{0.5}={a}_5{d}_0^{n_5}{\dot{\varepsilon}}^{m_5}{e}^{\left({Q}_5/ RT\right)}+{c}_5 $$

(43)

##### Grain Size

The recrystallized grain size is expressed as a function of initial grain size, strain, strain rate, and temperature

$$ {d}_{rex}={a}_8{d}_0^{h_8}{\varepsilon}^{n_8}{\dot{\varepsilon}}^{m_8}{e}^{\left({Q}_8/ RT\right)}+{c}_8 $$

(44)

$$ \left(\mathrm{if}\ {d}_{rex}\ge {d}_0\ \mathrm{then}\ {d}_{rex}=0\right) $$

#### Static Recrystallization

When deformation stops, the strain rate and critical strain are used to determine whether static recrystallization should be activated. The static recrystallization is terminated when this element starts to deform again.

##### Activation Criteria

When strain rate is less than \( {\dot{\varepsilon}}_{sr} \), static recrystallization occurs after deformation.

$$ {\dot{\varepsilon}}_{sr}= Aexp\left({b}_1-{b}_2{d}_0-{Q}_2/T\right) $$

(45)

##### Kinetics

The model for recrystallization kinetics is based on the modified Avrami equation.

$$ {X}_{srex}=1- \exp \left[-{\beta}_s{\left(\frac{t}{{\mathrm{t}}_{0.5}}\right)}^{h_s}\right] $$

(46)

where *t*
0.5 is an empirical time constant for 50 % recrystallization:

$$ {t}_{0.5}={a}_3{d}^{h_3}{\varepsilon}^{n_3}{\dot{\varepsilon}}^{m_3}{e}^{\left({Q}_3/ RT\right)} $$

(47)

##### Grain Size

The recrystallized grain size is expressed as a function of initial grain size, strain, strain rate, and temperature

$$ {d}_{rex}={a}_6{d_0}^{h_6}{\varepsilon}^{n_6}{\dot{\varepsilon}}^{m_6}{e}^{\left({Q}_6/ RT\right)}+{c}_6 $$

(48)

$$ \left(\mathrm{if}\ {d}_{rex}\ge {d}_0\ \mathrm{then}\ {d}_{rex}=0\right) $$

#### Grain Growth

Grain growth takes place before recrystallization starts or after recrystallization finishes.

The kinetics is described by equation

$$ {d}_r={\left[{d}_{rex}^m+{a}_9 texp\left(\frac{Q_9}{ RT}\right)\right]}^{1/m} $$

(49)

##### Retained Strain

When recrystallization of a certain type is incomplete, the retained strain available for following another type of recrystallization can be described by a uniform softening method:

$$ {\varepsilon}_i=\left(1-\lambda {X}_{rex}\right){\varepsilon}_{i-1} $$

(50)

##### Temperature Limit

The temperature limit is the lower boundary of all grain evolution mechanisms. Below this temperature, no grain evolution occurs.

##### Average Grain Size

The mixture law is employed to calculate the recrystallized grain size for incomplete recrystallization:

$$ d={X}_{rex}{d}_{rex}+\left(1-{X}_{rex}\right){d}_0 $$

(51)

Based on the abovementioned equations, the evolution of the microstructure during bulk forming process can be estimated.

### Fracture Prediction in Bulk Metal Forming

Perhaps one of the most important questions that mechanical engineers would like to ask is when the materials will fracture/damage during the forming process. The answer to this question depends on the geometry of the workpiece, the boundary condition, and the material properties. As the first two factors have already been taken into consideration during simulation using finite element method, the focus here will be placed on the materials. There are many numerical models available which intend to provide the material damage criteria under different loading conditions. They may consider damage as a progressive process with initiation and evolution at different stages of loading. Indeed, in most of the metals, the ductile damage dominates, which is a process due to nucleation, growth, and coalescence of voids in ductile metals. However, in our case, the damage is treated as an instantaneous event with a single-value indicator to determine that particular damage for ease of application. Below listed are some of the most commonly used ones in the bulk forming areas:

#### Maximum Principle Stress/Ultimate Tensile Strength

Perhaps the easiest criterion that one can immediately think of is the comparison between the current stress (*σ*) state and the maximal principle stress or UTS (*σ*
*UTS*
). The critical value is given by the ratio between them as

$$ \alpha =\frac{\sigma }{\sigma_1}\le {\alpha}_c\ \mathrm{or}\ \alpha =\frac{\sigma }{\sigma_{UTS}}\le {\alpha}_c $$

(52)

#### Cockcroft and Latham

This is the most commonly used fracture criterion with bulk deformation (Cockcrof and Latham 1968), which states

$$ {\displaystyle {\int}_0^{\varepsilon_f}{\sigma}^{*}d\overline{\varepsilon}\le C} $$

(53)

where *σ** is the maximum tensile stress in the work piece, *ε*
*f*
is the strain at fracture, and *C* is the C&L constant. This method has been used successfully to predict fracture in edge cracking in rolling
and free-surface cracking in upset forging
under conditions of cold working.

#### Rice and Tracy

This model is defined as a function of mean stress and effective stress. α is the model coefficient.

$$ {\displaystyle {\int}_0^{\varepsilon_f}{e}^{\frac{\alpha {\sigma}_m}{\overline{\sigma}}}d\overline{\varepsilon}\le C} $$

(54)

#### Brozzo

Brozzo model is defined as a function of principal stress and mean stress:

$$ {\displaystyle {\int}_0^{\varepsilon_f}\begin{array}{l}\frac{2{\sigma}^{*}}{3\left({\sigma}^{*}-{\sigma}_m\right)}d\overline{\varepsilon}\le C\\ {}\end{array}} $$

(55)

The disadvantage associated with above criteria is that all these methods predict the damage based on a certain critical value, which can only be determined experimentally. Moreover, the damage point in the experiment is not easy to identify as it is a progressive process. To make the situation even worse, the critical value varies from material to material, and sometimes different setups may also contribute to such deviation. Therefore, these fracture/damage values can only be used as a rough guideline for the process design.

In this section, we had a brief overview of the challenges in utilization and application of FE simulation to design and optimize the forming processes. Issues such as the material friction behaviors, mechanical properties, microstructural evolution, and fracture prediction were covered. Surely the challenges in the simulation are far greater than those presented here. However, this provides a flavor to the readers of this handbook on the complexity the bulk metal forming processes have in terms of modeling and simulation .

The key difference between bulk deformation and sheet metal forming is that **in bulk deformation, the work parts have a low area to volume ratio whereas, in sheet metal forming, the area to volume ratio is high.**

Deformation processes are important in transforming one shape of a solid material into another shape. Usually, the initial shape is a simple one. We can deform it using tools to obtain the desired shape. Moreover, this process is important to increase the tolerance of a solid material.

### CONTENTS

1. Overview and Key Difference

2. What is Bulk Deformation

3. What is Sheet Metal Forming

4. Side by Side Comparison – Bulk Deformation vs Sheet Metal Forming in Tabular Form

5. Summary

## What is Bulk Deformation?

Bulk deformation is the metal forming operation where a significant change in shape occurs via plastic deformation in metallic parts. Usually, the initial shapes of the material can be cylindrical bars, billets, rectangular billets, slabs, etc. In this process, we plastically deform these structures in cold or warm/hot conditions to get the desired shape. The bulk deformation process is important in manufacturing complicated shapes having good mechanical properties. Furthermore, in this process, we can observe a considerable increase in the surface-to-volume ratio. We can list the characteristic of bulk deformation as follows:

• The plastic deformation of the workpiece is large, making a considerable change in shape and cross-section

• Generally, the permanent elastic deformation is larger than the elastic deformation of the workpiece.

The operational steps of bulk deformation are as follows:

1. Forging – squeezing and shaping the initial structure between two dies

2. Rolling – squeezing a slab r plate-like initial structure between two rotating rolls to reduce the height

3. Extrusion – squeezing the initial shape through a shaped die so that the shape of the workpiece changes into the shape of the die

4. Wire and bar drawing – changing the shape of wire-like and bar-like structures

## What is Sheet Metal Forming?

Sheet metal forming is a metal forming operation in which the geometry of a piece of sheet undergoes modification upon the addition of a force. Here, no removal of the material is done. Furthermore, the applied force has to be larger than the yield strength of the metal. It causes the metal to undergo plastic deformation. Using this method, we can either bend or stretch a metal sheet into the desired shape.

Moreover, this process includes plastically deforming a sheet metal into a complex 3D configuration. Generally, this method does not make any significant change in thickness and surface characteristics of the sheet. The characteristics of this method are as follows:

• Workpiece – a sheet or a part fabricated from a sheet

• Changes the shape but not the cross-section

• Sometimes, the permanent plastic deformation and elastic deformation are comparable. Thus, elastic recovery is significant

## What is the Difference Between Bulk Deformation and Sheet Metal Forming?

Bulk deformation is the metal forming operation where a significant change in shape occurs via plastic deformation in metallic parts, while sheet metal forming is a metal forming operation in which the geometry of a piece of sheet undergoes modification upon the addition of a force. The key difference between bulk deformation and sheet metal forming is that in bulk deformation, work parts have a low area to volume ratio whereas, in sheet metal forming, the area to volume ratio is high.

Moreover, the initial shape of the workpiece can be billet, rod, slab, etc. in bulk deforming process while, in sheet metal forming process, the initial shape is a sheet.

The below infographic provides more description of the difference between bulk deformation and sheet metal forming.

## Summary – Bulk Deformation vs Sheet Metal Forming

Bulk deformation and sheet metal forming are important deformation process for metal workpieces. The key difference between bulk deformation and sheet metal forming is that in bulk deformation, the work parts have a low area to volume ratio whereas, in sheet metal forming, the area to volume ratio is high.

##### Reference:

1. “Read ‘Unit Manufacturing Processes: Issues and Opportunities in Research’ at NAP.edu.” National Academies Press: OpenBook, Available here.

2. “Mechanical Engineering – Design For Manufacturing.” NPTEL, IIT Bombay, Available here.

##### Image Courtesy:

1. “Huge drop hammers work day and night forming sheet metal parts for United Nations bombers and fighters at the North American Aviation plant, Inglewood, Calif” By Alfred T. Palmer – from the United States Library of Congress’s Prints and Photographs division (Public Domain) via Commons Wikimedia