Predicting the rate-dependent loading paths to fracture in advanced high strength steels using an extended mechanical threshold model

Accepted Manuscript

Predicting the Rate-dependent Loading Paths to Fracture in Advanced High Strength Steels Using an Extended Mechanical Threshold Model Matthieu Dunand , Dirk Mohr PII: DOI: Reference:

S0734-743X(16)30941-1 10.1016/j.ijimpeng.2017.02.020 IE 2857

To appear in:

International Journal of Impact Engineering

Received date: Revised date: Accepted date:

18 November 2016 1 February 2017 25 February 2017

Please cite this article as: Matthieu Dunand , Dirk Mohr , Predicting the Rate-dependent Loading Paths to Fracture in Advanced High Strength Steels Using an Extended Mechanical Threshold Model, International Journal of Impact Engineering (2017), doi: 10.1016/j.ijimpeng.2017.02.020

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

AC

CE

PT

ED

M

AN US

CR IP T

Highlights  Formulated a new rate- and temperature dependent plasticity model  Performed low, intermediate and high strain rate experiments at different stress states  Employed a hybrid experimental-numerical method to determine the loading paths to fracture  Observed a significant increase in ductility in the high strain rate experiments

1

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Predicting the Rate-dependent Loading Paths to Fracture in Advanced High Strength Steels Using an Extended Mechanical Threshold Model

1

CR IP T

Matthieu Dunand1 and Dirk Mohr2 Department of Mechanical Engineering,

Massachusetts Institute of Technology, Cambridge MA, USA 2

Department of Mechanical and Process Engineering,

AN US

Swiss Federal Institute of Technology (ETH), Zürich, Switzerland

Abstract

Tensile experiments are carried out on a TRIP780 steel sheet at low ( ̇ intermediate ( ̇

) and high strain rates ( ̇

),

). The experimental program includes

M

notched as well as uniaxial tension specimens. Local displacements and surface strain fields are determined using Digital Image Correlation. Constitutive equations derived from

ED

Mechanical Threshold Stress (MTS) theory are proposed to describe rate-dependent behavior as well as the plastic anisotropy of the sheet material. Detailed finite element simulations of

PT

all experiments reveal that the model accurately predicts experimental results, including force displacement curves and local surface strain evolution. In particular, the behavior observed at

CE

large strains, beyond the onset of necking, is well predicted. Stress and strain histories where fracture initiates are extracted from the simulations to characterize the dependence of the material ductility on both loading velocity and stress state. The results clearly show that the

AC

fracture strains in high strain rate Hopkinson bar experiments are significantly higher than in static experiments. The same good agreement between the predicted and experimentallymeasured force-displacement curves is also obtained when using a Johnson-Cook plasticity model. However, the direct comparison of the strain and strain rate evolution inside the necks revealed that the results from hybrid experimental-numerical analysis are highly sensitive to the choice of the plasticity model formulation. Keywords: High strain rate, mechanical threshold stress, AHSS, viscoplasticity, ductile fracture

2

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

1. Introduction The plastic deformation behavior of Advanced High Strength Steels (AHSS) under uniaxial tension has been extensively studied during the last decade, indicating that this class of material experiences positive strain rate sensitivity (e.g. Khan et al., 2012). Experiments in the intermediate range of strain rates (1-10s-1) are typically carried out in servo-hydraulic testing machines (e.g. Huh et al., 2008) while high strain rate experiments (100-1000s-1) are

CR IP T

usually performed in Split Hokinson bar systems (e.g. Clausen et al., 2004; Van Slycken et al., 2007, Forni et al., 2016, Gruben et al., 2016) or direct impact setups (e.g. He et al., 2012). The effect of strain rate on the fracture of AHSS has received little attention in the open literature. Most experimental investigations are limited to characterizing uniaxial tension parameters such as the ultimate elongation (e.g. Olivier et al., 2007; Huh et al, 2008), thereby

AN US

putting aside the effect of stress state on ductile fracture. Results presented do not permit to draw a clear trend in the dependence of ultimate elongation to strain rate for AHSS steels. Kim et al. (2013) measured increasing ultimate elongations for increasing strain rates (ranging from 0.1s-1 to 200s-1) in case of a DP780 and a TRIP780 steel. Similarly, an elongation twice higher at 1000s-1 than at 0.001s-1 is found for a TRIP steel by Verleysen et al. (2011). On the

M

other hand, Wei et al. (2007) report decreasing ultimate elongations with increasing strain rates in case of two different TRIP-aided steels, while Curtze et al. (2009) find almost the

ED

same elongation at 10-3s-1 and 750s-1 for both DP600 and TRIP700 steels. It is emphasized that the ultimate elongation in a uniaxial tension experiment is a structural characteristic and

PT

not an intrinsic material property. It depends strongly on the specimen geometry (Verleysen et al., 2008; Sun et al., 2012) and is thus a questionable indicator of material ductility. A more

CE

reliable evaluation of the fracture strain in uniaxial tension experiments may be obtained by measuring the reduction of the cross-section area on fractured specimens. Based on this reduction-of-area method, Kim et al. (2013) report a decreasing fracture strain for increasing

AC

strain rates in case of a TRIP780 steel, and a non-monotonic dependence of ductility to strain rate for DP780 steel. In most fracture experiments on sheet materials, the localization of plastic deformation

through necking cannot be avoided and represents a major difficulty in the accurate characterization of the material state at fracture. After necking, the stress and strain fields within the specimen gage section become non-uniform and of three-dimensional nature (stresses in the thickness direction develop). Consequently, the stress history prior to fracture can no longer be estimated from the force history measurements using simple analytical formulas: stress and strain histories need to be determined in a hybrid experimental-numerical 3

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

approach (e.g. Roth and Mohr, 2014). In other words, a detailed finite element analysis of the experiment is required to identify the stress and strain fields up to the onset of fracture. The plasticity model used in the numerical simulation is one of the key elements contributing to the quality of the hybrid experimental-numerical results. A wide variety of constitutive equations describing the rate- and temperature-dependent plastic behavior of metals have been proposed. The goal of constitutive equations is to relate

CR IP T

the flow stress to strain, strain rate and temperature (and possibly other internal variables) in order to describe the basic mechanisms of strain hardening, strain rate sensitivity and thermal softening. Rate-dependent plasticity models can be categorized into two main branches. On the one hand phenomenological models are based on the separation of the effects of strain, strain rate and temperature (e.g. Johnson and Cook, 1983; Cowper and Symonds, 1952).

AN US

Strain hardening, strain rate sensitivity and thermal softening mechanisms are described by means of basic functions, depending respectively on strain, strain rate and temperature only, which may be combined in additive or multiplicative manners to build an ad-hoc model. A partial review of phenomenological basic functions can be found in Sung et al. (2010). Integrated phenomenological models in which variables are not separated have also been

M

proposed. They allow for the description of more complex mechanisms such as temperaturedependent strain hardening (e.g. Sung et al., 2010) and rate-dependent strain hardening (e.g.

ED

Khan et al., 2004). On the other hand, attempts have been made to incorporate a description of the governing mechanisms of plasticity into physics-based constitutive models (e.g. Zerilli and Armstrong, 1987; Follansbee and Kocks, 1988; Voyiadjis and Abed, 2005; Rusinek et al.,

PT

2007). Physics-based models make use of the theory of thermally-activated dislocation motion to relate flow stress, strain rate and temperature (Conrad, 1964; Kocks et al., 1975).

CE

Additional internal variables can also be introduced in the constitutive equations to account for history effects (Bodner and Partom, 1975; Durrenberger et al., 2008). Comparisons

AC

between phenomenological and physics-based models against experimental data tend to show that the latter offer better predictive capabilities when a wide range of strain rates and/or temperatures is considered (Liang and Khan, 1999; Abed and Makarem, 2012; Kajberg and Sundin, 2013). When dealing with sheet materials, plasticity model calibration is usually done based on uniaxial tension experiments in the range of uniform elongation, as experimental measurements give direct access to the flow stress, strain and strain rate. The material behavior at large strains (beyond uniform elongation) is then assumed by extrapolating the strain hardening at low strains with analytical functions (e.g. Hollomon, 1945; Swift, 1952; 4

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Voce, 1948). This extrapolation may lead to inaccurate predictions of the material behavior at large strains. Sung et al. (2010) calibrated six different hardening functions for a DP780 based on the uniaxial data in the range of uniform elongation. Even though all models fit the data correctly, significant differences are reported at large strains: most extrapolated models did not represent accurately the material behavior beyond uniform elongation. In this work, tensile experiments are carried out at low, intermediate and high strain rates

CR IP T

to characterize the rate-dependent behavior of a TRIP780 steel. In addition to uniaxial tension specimens, the experimental program includes tensile specimens with circular notches. Experiments at high strain rates are performed in a Split Hopkinson Pressure Bar system. Constitutive equations adapted from the Mechanical Threshold Stress (MTS) model (Follansbee and Kocks, 1988) are proposed to model the dependence of the material behavior

AN US

on strain rate, temperature as well as plastic anisotropy. After calibration using an inverse method, the predictive capabilities of the model are assessed over multiple strain rates and stress states through detailed finite element simulations of all the experiments. Good agreement is found between numerical and experimental results. A hybrid experimentalnumerical approach is taken to characterize the dependence of ductility to strain rate and

M

stress state. The results obtained with the MTS model are compared with earlier simulations of the same experiments that made use of an extended Johnson-Cook plasticity model (Roth

ED

and Mohr, 2014).

CE

PT

2. Experimental program

2.1. Material

AC

All experiments are performed on specimens extracted from 1.4mm thick TRIP780 steel sheets. This TRIP-assisted steel features a complex multiphase microstructure composed of ferrite, bainite and martensite. In addition, it contains about 6% of retained austenite which may undergo phase transformation upon mechanical loading. The chemical composition of the present TRIP780 material as measured by energy dispersive X-ray analysis is given in Table 1. The TRIP material exhibits in-plane anisotropy of the plastic flow. The Lankford ratios (r-values) are r0=0.67, r45=0.96 and r90=0.85 for tension along the rolling direction, the 45° direction and the transverse direction, respectively. However plastic yielding seems planar 5

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

isotropic as similar stress-strain curves are measured for tension along the rolling, 45° and transverse directions.

2.2. Tensile experiments at low and intermediate strain rate Tensile experiments are performed on straight specimens (Fig. 1a) and on flat notched specimens (Fig. 1c and 1d). The specimen loading axis is always oriented along the rolling

CR IP T

direction. All specimens are extracted from the sheet material using waterjet cutting. Notched specimens are 20mm wide and feature a b=10mm wide notched gage section. Specimens with two different notched radii are prepared: R=20mm (NT20) and R=6.65mm (NT6). Experiments are carried out on a servo-hydraulic testing machine under displacement control, specimens being held by friction in custom-made high pressure clamps. The force applied to

AN US

the specimen is measured by a conventional load cell, while displacements and strains in the gage section are measured by Digital Image Correlation (DIC), as detailed in Section 2.4. Two ranges of strain rates are investigated. For low strain rates, a constant crosshead velocity of 0.84mm/min is imposed to the straight specimens, while a velocity of 0.5mm/min is chosen for notched specimens. To reach intermediate strain rates, velocities of 22mm/s and 8.3mm/s

M

are selected for the straight and notched specimens, respectively. All experiments are carried out at room temperature. Note that the static load cell used is not correctly dimensioned for

ED

intermediate strain rate experiments, leading to noticeable oscillations in the force

PT

measurements (about 1.2% of the measured signal in amplitude).

2.3. Tensile experiments at high strain rate

CE

A modified Split Hopkinson Pressure Bar system (SHPB) is used to perform high strain rate tensile experiments, in which a tensile specimen is positioned between slender steel bars,

AC

named input and output bars, respectively. A compressive stress pulse is generated by impacting the input bar with a slender steel striker launched by a gas gun. In all experiments presented here, the stress pulse has a total duration of 550µs and a rise time of about 30µs, its magnitude being controlled by the striker velocity. A load-inversion device (Dunand et al., 2013) is used to transform the compressive loading pulse into tensile loading of the specimen gage section. During deformation of the specimen, the load is transmitted through the loadinversion device as a compressive stress wave into the output bars. In this setup, two output bars are used to minimize the presence of bending waves. The specimen is held by friction in the load-inversion device, the gripping pressure being applied by eight M5 screws. The force applied to the specimen gage section is derived from elastic waves propagating in the output 6

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

bars using standard formulas (Kolsky, 1949), while displacements and strains in the specimen gage section are measured by means of high speed imaging and digital image correlation. High strain rate experiments are performed on uniaxial tension specimens, as well as notched tensile specimens with notch radii of R=20mm and R=6.65mm. Note that the geometry of the uniaxial specimen, sketched in Fig. 1b, is different from the one used at low and intermediate strain rates (Fig. 1a). The gage section is shorter (l=15mm instead of 20mm)

CR IP T

and narrower (w=5mm instead of 10mm). The gage section geometry of the notched tensile specimens is the same in low and high strain rate experiments. A series of experiments are performed on the three geometries at room temperature with striker velocities ranging from 4m/s to 14m/s.

AN US

2.4. Local displacement measurements

In all tensile experiments, pictures of the specimen surface are continuously recorded to determine the surface displacement through digital image correlation. For that purpose a thin layer of white matt paint is applied to the specimen surface along with a black speckle pattern of a speckle size of about 70μm.

M

In low strain rate experiments, an AVT Pike F505B with 90mm macro lenses takes about 300 pictures (resolution of 2048x1024 pixels) before the onset of fracture. The camera is

ED

placed at 1.5m from the specimen surface, resulting in a square pixel edge length of 23μm. In intermediate and high strain rate experiments, a high speed video system (Phantom v7.3 with

PT

90mm macro lenses) is used. For intermediate strain rate experiments, it operates at a frequency of 1kHz with an exposure time of 50μs. A resolution of 800x600 pixels is chosen.

CE

The camera is positioned at a distance of about 400mm from the specimen surface (square pixel edge length of 55μm). In high strain rate experiments the acquisition frequency is set to

AC

100kHz with an exposure time of 2μs, and a spatial resolution of 560x32 pixels. The camera is positioned at a distance of about 550mm from the specimen surface (square pixel edge length of 70μm). The virtual extensometer function of the digital image correlation software VIC2D (Correlated Solutions, SC) is used to determine the relative displacement of two reference points located on the axis of symmetry of the specimen gage sections. For this, a quadratic transformation of a 23x23 pixel neighborhood of each point is assumed. Spline-cubic interpolation of the grey values is used to achieve sub-pixel accuracy. The position of the DIC reference points on the specimen surface is shown in Fig. 1 with solid red dots. For notched 7

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

tensile specimens, an initial spacing between the reference points of

is used to

compute the relative displacement imposed to the specimen, while surface strains at the center of the gage sections are calculated with an initial spacing of a value of

(resp.

. For uniaxial tension,

) is chosen at low and intermediate strain rates

(resp. high strain rate). Based on the relative displacement

of the two reference points, the

.

(1)

AN US

3. Experimental results

CR IP T

engineering axial surface strain is computed as

At least two repeats were performed of each type of experiment. The experimental results were highly repeatable as far as the force-displacement response is concerned. The forcedisplacement curves of parallel experiments lie perfectly on top of each other (except for the wave related minor oscillations in the dynamic experiments) which is attributed to the high

M

industrial quality of the TRIP780 material. In the notched experiments, the fracture displacement variations were less than  1.5% which is in line with earlier static fracture

ED

experiments on the same material (Dunand and Mohr, 2010). A higher fracture displacement variation of about  5% had been observed in the UT experiments. In the sequel, we show the

PT

results from a representative experiment of each type.

CE

3.1. Uniaxial tension

Figure 2 depicts the engineering stress-strain curves measured under uniaxial tension in the three ranges of strain rate. The instant of fracture is indicated with a solid dot. For low and

AC

intermediate strain rates (in black and red in Fig. 2, respectively) the onset of fracture corresponds to a sudden drop of the specimen load carrying capacity. In the high strain rate experiment, however, the force level decreases smoothly to zero. The instant of the onset of fracture is therefore identified from the DIC pictures as the instant at which a first crack appears on the specimen surface. In all three experiments, a maximum of force corresponding to the onset of localization of the plastic flow (i.e. necking) is observed. Note that the necking strain decreases as the strain rate increases. Necking occurs at an engineering strain of about 0.2 at low strain rate (black curve in Fig. 2) while it is only 0.14 at high strain rates (blue curve in Fig. 2). For each experiment, a pre-necking strain rate is defined as the time-average 8

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

of the axial plastic strain rate (measured by DIC) between initial yielding and the onset of necking. A pre-necking strain rate of ̇

is measured for the experiment at low

strain rate (black curve in Fig. 2), ̇ Fig. 2) and ̇

for intermediate strain rate (red curve in

for high strain rate (blue curve in Fig. 2). Note that after the onset of

necking, stress and strain fields are no longer uniform in the specimen gage section. Therefore engineering stress and strains should be interpreted as normalized forces and displacements,

CR IP T

which are structure-level measurements, rather that local stress and strain measures. The plastic behavior of the sheet material exhibits a weak dependency on the strain rate. Both the initial yield stress and the strain hardening are rate-sensitive. The material has the same initial yield stress but a higher hardening modulus at intermediate strain rates than at low strain rates. In addition, it has a higher initial yield stress but approximately the same

AN US

initial hardening modulus at high strain rates than at low strain rates. Before necking, the stress level is about 100MPa higher in the high strain rate experiment than at low strain rate. Note that a comparison of the post-necking part of the curves is not meaningful as different specimen geometries have been used.

M

3.2. Notched tension

Experimental results for the tensile specimen with 20mm notch radii and 6.65mm notch

ED

radii are presented in Figs. 3 and 4, respectively. Force-displacement curves for the three strain rates are depicted in Figs. 3a and 4a, while the evolution of the engineering axial

PT

surface strain at the center of the gage section is plotted in Figs. 3b and 4b. The onset of fracture is indicated by a solid dot on all curves. As for uniaxial tension experiments, it

CE

corresponds to a sharp drop of force in low and intermediate strain rate experiments (black and red curves in Figs. 3a and 4a, respectively), and to the appearance of the first surface

AC

crack in the high strain rate experiments (blue curves in Figs. 3a and 4a). Note that surface strain measurements are not available for the high strain rate experiment. Very high strains reached at the center of the gage section combined with high rates of deformation lead to cracking and unsticking of the painted pattern used for correlation. For the 20mm notched specimen, the relative velocities measured between the two DIC reference points (red points in Fig. 1c) are 0.5mm/min for the low strain rate experiment, 8mm/s for the intermediate strain rate experiment and 7.0m/s for the high strain rate experiment, respectively. For the 6.65mm notched specimen, the relative velocities are 0.5mm/min for the low strain rate experiment, 8mm/s for the intermediate strain rate 9

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

experiment and 4.5m/s for the high strain rate experiment, respectively. Note that the reference point locations are the same in all experiments. Regardless of the geometry or strain rate, all force-displacement curves feature a maximum of force which corresponds approximately to the onset through-thickness necking of the sheet (Dunand & Mohr, 2010). Because of the notched geometry, necking always occurs at the center of the gage section. After the onset of necking, the specimen experiences

CR IP T

a loss of load carrying capacity until fracture occurs. With the 20mm notched specimen (Fig. 3a), the loss of load carrying capacity is about 13% for all strain rates. In case of the 6.65mm geometry (Fig. 4a), however, the force drops before fracture increases with strain rate, from 6% at low strain rate to 15% at high strain rate. Note that the onset of necking has a nonmonotonic dependence to strain rate. For both geometries, necking occurs at a lower

AN US

displacement at intermediate strain rate than at low strain rate, and at a higher displacement at high strain rate than at intermediate strain rate.

The evolutions of axial surface strain, depicted in Figs. 3b and 4b for the 20mm and 6.65mm geometries, show a pronounced increase of axial surface strain rate, which corresponds to localization of the plastic flow. Note that the slope of the curves plotted in

M

Figs. 3b and 4b does not correspond, but is proportional to engineering strain rate as low and intermediate strain rate experiments are run at almost constant imposed velocity. The change

ED

of slope occurring earlier in the intermediate experiments (in red in Figs. 3b and 4b) confirms that necking happens at a lower displacement at intermediate strain rate than at low strain rate.

PT

A noticeable difference between the two geometries concerns the surface strain reached at the onset of fracture at the center of the gage section. With a notch radius of 20mm (Fig. 3b), the

CE

surface axial strain at fracture is lower by 7% at intermediate strain rate ( low strain rate (

)

) with a 6.65mm notch radius (Fig. 4b).

AC

than at low strain rate (

), whereas it is higher by 13% at intermediate strain rate (

) than at

As far as the failure mode is concerned, it is noted that all fractured specimens –

irrespective of the stress state and strain rate – exhibit a slanted fracture surface (Fig. 5a). The fracture surfaces (Fig. 5b) present the characteristic dimple patterns of ductile failure due to void growth and coalescence. However, it is important to note that the void volume fraction is low within the microstructure below the fracture surfaces (Fig. 5c). The observed features are consistent with a shear localization mechanism where accelerated void growth (and nucleation) takes place within a narrow shear band (of a thickness of a few grains only) before a crack is formed through void coalescence. 10

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

4. Extended Mechanical Threshold Stress (MTS) model

4.1. Constitutive equations A phenomenological approach is taken to model the material behavior. We have the Cauchy stress tensor

and the absolute temperature , in conjunction with a set of internal , the equivalent plastic strain rate ̇ ̅ and a hardening

variables: the plastic strain tensor

CR IP T

variable representing the material resistance to plastic deformation. Incremental plasticity based on an additive decomposition of the logarithmic strain tensor is employed

AN US

The constitutive equation for stress reads

(2)

(3)

is the fourth-order isotropic elasticity tensor, characterized by the Young’s modulus

where

M

and the Poisson ratio . The effect of thermal expansion is neglected in this work. The magnitude of the Cauchy stress is characterized by the equivalent stress

ED

̅

( )

(4)

̇ ̇

CE

PT

while the flow rule describing the evolution of the plastic strain tensor is defined as

where ̇ is a plastic multiplier, and

(5)

( ) represents the plastic potential. Note that when

AC

, Eq. 5 corresponds to an associated flow rule, while a non-associated flow is modeled

otherwise.

The equivalent plastic strain rate ̇ ̅ is defined as the work-conjugate to the equivalent

stress ̅ ̇̅ ̇

11

(6)

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

where

is the rate of plastic work. Upon evaluation of this relationship for the plastic flow

defined in Eq. 5, we obtain the following relationship between the equivalent plastic strain rate and the plastic multiplier ̇

(7) ̇.

The choice of an associated flow rule yields ̇ ̅

CR IP T

̇̅

The rate and temperature sensitivity of the material behavior is described through the functional relationship among the strain rate ̇ ̅ , the equivalent stress ̅, the deformation resistance , and the temperature , ̇̅

)

AN US

(̅

(8)

Finally, the material hardening during straining is described by the evolution equation (̅ ̇

)

(9)

Since this work focuses on monotonic loadings only, kinematic hardening is not considered.

M

In high strain rate deformation processes, the material is submitted to a temperature increase due to quasi-adiabatic heating. For the adiabatic case, the temperature rises according

ED

to

where

PT

̇

is the mass density,

CE

heat. Typically a value of

̅ ̇̅

(10)

the specific heat and

the ratio of plastic work converted to

is used (Rusinek and Klepaczko, 2009).

AC

4.2. Model specialization The constitutive model defined in Section 4.1 is fully characterized by the four functions

( ),

( ),

(̅

) and

(̅

). Specific parametric forms permitting to model

accurately the plastic behavior of the TRIP material are given thereafter.

4.2.1 Reduction from 3D to 1D: functions ( ) and

( )

For the present sheet material, nearly the same stress-strain curve is measured in uniaxial tension for different specimen orientations even though the Lankford coefficients are direction 12

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

dependent. As detailed in Mohr et al. (2010), we make use of a quadratic yield function, along with a non-associated flow rule based on a quadratic anisotropic flow potential. √(

)

(11)

√(

)

(12)

F and G are symmetric positive-semidefinite matrices, with is a hydrostatic stress or zero.

and

out-of-plane directions;

-

if and only if

(13)

represent the true normal stress in the rolling, transverse and

AN US

,

and

denotes the Cauchy stress vector in material coordinates,

,

The components

CR IP T

and

denotes the corresponding in-plane shear stress, while

and

represent the corresponding out-of-plane shear stresses. Note that in the specific case of quadratic stress and flow potentials defined by Eqs. 11 and 12, Stoughton (2002) demonstrated that the non-associated flow rule is consistent with laws of thermodynamics. As

,

and

.

ED

and

M

shown in Mohr et al. (2010), the matrices F and G have only three independent coefficients:

4.2.2 Relationship between the equivalent stress, strain rate and temperature

PT

A constitutive relation derived from the Mechanical Threshold Stress (MTS) theory (Follansbee and Kocks, 1988) is chosen to describe the relationship among the equivalent ̇̅

on the applied stress ̅, and temperature

. This physics-inspired

CE

plastic strain rate

plasticity model considers dislocation motion as the only source of plastic deformation and

AC

describes it as a thermally-activable process. Only the main governing equations are given thereafter. They describe a reduced version of the constitutive equations for a single active slip system of the polycrystal plasticity model of Balasubramanian and Anand (2002). The deformation resistance to dislocation motion

is introduced as temperature

independent internal variable. It is conceptually decomposed into two parts, (14) where

represents the part of resistance due to athermal obstacles (typically long range

obstacles such as grain boundaries or large precipitates) and 13

represents the part of the

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

resistance due to thermally activable obstacles which can be overcome by thermal fluctuations (typically short range obstacles such as forest dislocations). The free energy

that needs to be supplied by thermal fluctuations to overcome the

obstacles to dislocation motion is a function of stress. Denoting the stress in excess of the stress required to overcome athermal obstacles as {

̅

̅ ̅

(15)

,

Kocks et al. (1975) postulated the following parametric form, ( ̅)

̅

(

) ]

(16)

AN US

[

CR IP T

̅

to describe the relationship between the free energy required and the applied stress. denotes the (highest) activation free energy required to overcome short-range obstacles without the aid of stress ( ̅

). Knowing the required free energy, the relationship among

the strain rate, the applied stress and temperature can be established using Boltzmann

(̅

)

{

̇

{

where ̇ is a material constant and

̅

}

ED

̇̅

M

statistics and Orowan’s law (see Balasubramanian and Anand, 2002),

(17)

̅

PT

denotes Boltzmann’s constant.

CE

4.3. Strain hardening

Strain hardening of the material can be described either by an increase of

AC

and Anand, 1998), of

(e.g. Follansbee and Kocks, 1988) or both (e.g. Balasubramanian

and Anand, 2002) during plastic loading. Here we assume that both same rate: the ratio

(e.g. Kothari

⁄

and

increase at the

is kept constant throughout loading. Both resistances are

thus directly related to the total deformation resistance , and

Strain hardening is then defined for

(18) ( ̅ ) using a rate- and temperature-independent

saturation law, 14

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

( ) {

( ̅

with the deformation resistance

(19)

)

, its saturation value

, the modulus

and the hardening

exponent . In the present model, Eq. (19) corresponds to the hardening response at zero

CR IP T

temperature.

5. Finite element analysis

5.1. Finite element models

Simulations of all experiments are carried out using the commercial finite element

AN US

analysis package Abaqus/explicit. The above constitutive model is implemented as a user material through the VUMAT interface. The stresses and strains are computed in a corotational coordinate frame to account for finite rotations (Abaqus, 2012). The strain tensor obtained through co-rotational integration of strain increments is similar (and equal in most , with

denoting the right Cauchy-Green

M

cases) to the logarithmic strain tensor,

stretch tensor for a stationary coordinate frame. The corresponding stress tensor corresponds

ED

to a rotational pull back of the Cauchy stress tensor. Specimens are meshed with reduced integration solid elements with a temperature degree of freedom (type C3D8RT of the Abaqus library). Exploiting the symmetry of the notched

PT

specimen geometries, material properties and loading conditions, only one eighth of each notched specimen is modeled: the mesh represents the upper right quarter of the tensile

CE

specimens, with half the thickness. The distance between the center of the gage section and the upper mesh boundary corresponds exactly to half the length of the virtual extensometer

AC

that was used to measure displacements during the experiments. Note that the plastic localization observed experimentally also respects the specimen symmetries. Reduced meshes can therefore be used to model the post-necking behavior of notched tensile specimens. In uniaxial tension experiments, however, a slanted band of plastic localization may emerge that does not respect the initial symmetries. The meshes used for uniaxial tension therefore model the full specimen geometries, including the specimen shoulders. In all tensile experiments carried out, the plastic flow localizes before the onset of fracture, leading to steep gradients of stresses and strains along the in-plane and thickness directions. To capture accurately such behavior, a mesh featuring eight elements through the 15

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

half-thickness (0.7mm) and a comparable in-plane element dimension of 0.1mm at the center of the gage section is used. Simulations of each loading condition are performed. For that purpose half of the experimental velocity measured by DIC is uniformly imposed to the upper boundary of the mesh. A zero normal displacement condition is imposed to the nodes on the symmetry planes. Simulations are run slightly beyond the displacement at which fracture occurs in the experiment. To speed up calculations, uniform mass scaling is used in both the low and intermediate strain rate simulations. The material density is artificially adjusted to

CR IP T

reach the fracture displacement within 300,000 time increments. High strain rate experiments are simulated with the real material density (7.8g/cm3), resulting in an initial stable time increment of 1.1x10-8s and about 50,000 increments.

To account for both temperature increase due to plastic work and heat diffusion due to

AN US

temperature gradients, a coupled temperature-displacement analysis is carried out to simulate the intermediate and high strain rate experiments. In other words, both the equilibrium and heat transfer equations are solved simultaneously. Since most plastic deformation and thus most heat generation are localized at the center of the gage section, no heat flow is modeled on the upper boundary of the mesh. In addition, zero heat flow is assumed on the symmetry

M

planes. Furthermore, no heat transfer is considered at the specimen surface. A material specific heat of 490 J/K/kg and a conductivity of 45 J/s/m/K are chosen. All low strain rate

ED

experiments are modeled as isothermal.

PT

5.2. Plasticity model parameter identification The extended Mechanical Threshold Stress (MTS) model features 17 independent

CE

material parameters:

Two elastic constants: Young’s modulus



Three coefficients defining the equivalent stress function



Three coefficients defining the plastic flow potential



Five parameters *

AC



and Poisson’s ration ;

̇ + defining the rate and temperature sensitivity

function 

Four parameters *

+ defining the strain hardening function

Except for the elasticity constants, all parameters are formally summarized by the parameter vector

and identified through inverse identification. It is emphasized that the model features

only five parameters controlling the rate and temperature dependency. The frequently used 16

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

empirical Johnson-Cook model features for instance the same number of parameters among which at least three must be identified through mechanical experiments. The equivalent stress function

and plastic flow potential

are calibrated based on uniaxial tension experiments

carried out at low strain rate. Corresponding matrices F and G are calibrated following the procedure described in Mohr et al. (2010). Since the spatial resolution of high speed imaging system is not sufficient to measure the Lankford coefficients in high strain rate tension experiments, it is assumed that the anisotropy of the plastic flow observed at low strain rate is is therefore assumed rate-independent.

CR IP T

independent from the strain rate. The flow potential

An optimization-based inverse method is carried out to calibrate the coefficients of the strain hardening and rate sensitivity functions. The objective of the optimization process is to find the set of parameters that leads to the best prediction of the force-displacements curves of

AN US

the notched tensile specimen with a 6.65mm radius measured in low, intermediate and high strain rate experiments. For a given set of parameters , numerical simulations of the 6.65mm notched tensile specimen are carried out for the three different loading speeds. Then the three predicted force-displacement curves are compared to experimental measurements up to the

∑{

∫

( ),

where

for the set of parameters

( )

ED

( )

to evaluate the cost function

M

fracture displacement

is the number of calibration tests and

( )-

}

(20)

is a weighting factor used to focus the

PT

fitting process to the part of the force-displacement curves located between maximum force and the onset of facture. Before the force maximum, we use

, while

is used

CE

beyond this point. A downhill simplex minimization algorithm (Nelder and Mead, 1965) is used to find the set of parameters

that minimizes

, as it requires the evaluation of the

AC

error function only, but not its derivatives ( ).

(21)

A total of 640 iterations were required to obtain the coefficients given in Table 2. The corresponding force-displacement curves are shown in the left column of Fig. 5. Numerical predictions (solid black curves) are in very close agreement with the experimental results (black dots) for all three strain rates, including the results beyond the force maximum. The set of parameters given in Table 2 corresponds to a local minimum of the error function

. This set might not be the unique set of parameters minimizing 17

, and it is not

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

necessarily the best possible set (absolute minimum of activation energy of

). It is worth noting that the obtained

is coherent with the data provided by Frost and

Ashby (1982) for lattice-controlled dislocation glide in

-iron (ferrite), while it is

significantly lower than the theoretical value for obstacle-controlled glide. The fact that the identified activation energy is of the same order of magnitude as the theoretical estimates is reassuring. For a more quantitative comparison, we would need to embed our constitutive equations into a crystal plasticity framework (and apply these at the slip system level).

CR IP T

Experiments at very low and elevated homologous temperatures might also be needed to increase the reliability of the estimated split between athermal and thermally-activated resistance to slip.

AN US

5.3. Range of validity

It is useful to express the equivalent stress as a function of the equivalent plastic strain, the strain rate and temperature. Solving Eq. (17) for the equivalent stress yields ̅

( ̇̅

( ̅ )

( ̅ )

(22)

ED

M

with the function

)

[

(

̇ ̇̅

(23)

)] }

PT

{

controlling the temperature and rate-dependency. The stress-strain curve for ( ̅ ) (lower limit) and

CE

plastic loading is thus always sandwiched between the curves ( ̅ )

( ̅ ) (upper limit). This is illustrated by Fig. 6 which shows the stress-strain

AC

curves for isothermal loading at room temperature for different lowest strain rates (Fig. 6a), and for a constant strain rate of

for different temperatures (Fig. 6b).

is a monotonically

decreasing function in temperature and a monotonically increasing function in strain rate. The equivalence between temperature changes and strain rate changes is controlled through the material model parameter ̇ only. For instance, at a temperature of , increasing the temperature to the strain rate to about

and a strain rate of

has the same effect on the stress as decreasing

(for the present TRIP780 steel).

The model’s validity is limited to temperatures and strain rates that satisfy the condition ̇

.̅̇ /

. The maximum admissible strain rate is ̇ . This corresponds to the case 18

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

where the stress is sufficiently high (

) to overcome both short range and long

range obstacles without the help of thermal fluctuations. The strain rate at which plastic deformation takes place because thermal fluctuations only is temperature dependent, i.e. ̇̅ ̇

hence ̇

. .

/. For the present material model parameters, we have at room temperature, with the limit ̇ ̅

/

for

.

CR IP T

5.4. Plasticity model validation

and

Figures 5, 7 and 8 compare the results from numerical simulations (solid lines) of all tensile experiments with experimental measurements (solid dots). The instant of the onset of fracture is depicted by a vertical dashed line. Recall that only the force-displacement curves of

AN US

the 6.65mm notched tensile experiments (Fig. 5) had been used for calibrating the proposed plasticity model. All other results (Figs. 7 and 8) may be seen as plasticity model validations. Uniaxial engineering stress strain curves are shown in Fig. 7. The numerical engineering strain is computed according to Eq. (1) from the displacements of nodes positioned at the same location as DIC reference points, while the numerical engineering stress is obtained

M

from the force applied at the specimen boundaries normalized by the gage section initial cross section area. Experimental results at both intermediate (red curves in Fig. 7) and high strain

ED

rates (blue curves in Fig. 7) are predicted accurately with numerical results (solid lines) and experimental measurements (solid dots) lying almost on top of each other. Note that the

PT

reduction of the specimen’s load carrying capacity preceding fracture is accurately predicted in both cases. For low strain rates (black curves in Fig. 7), the stress-strain curve is well

CE

predicted in the range of uniform elongation (before the engineering stress reaches its maximum). However, the force level is underestimated in the post- necking range, resulting in

AC

7% difference between the simulation and experiment at the onset of fracture. In the left column of Fig. 8, we show the simulated force-displacement curves (solid

black lines) of the 20mm notch geometry for low (Fig. 8a), intermediate (Fig. 8b) and high (Fig. 8c) strain rates. The agreement with experimental results is good all the way to the onset of fracture. Irrespective of the loading speed, the relative difference between experimental and predicted forces does not exceed 3%. The comparison of the evolution of the surface axial strain at the center of the gage section of notched specimens with respect to the displacement (depicted in blue in Figs. 5 and 8) also shows a good agreement. Irrespective of the notch radius and the strain rate, the simulations are able to describe the characteristic increases in strain rates that have been observed in the experiments. Relative differences between 19

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

simulation and DIC strains in case of the 6.65mm notch geometry are about 5% at low strain rate (Fig. 5a) and 6% at intermediate strain rate (Fig. 5b) at the onset of fracture. For the 20mm notch geometry (Fig. 8), the computed strain matches almost exactly the DIC strain, with less than 1% relative error for both strain rates.

CR IP T

6. Loading paths to fracture

6.1. Evolution of the stress state

From each finite element simulation, the loading histories are extracted from the integration point where the equivalent plastic strain is the highest at the onset of fracture. The

AN US

onset of fracture is considered to be imminent when the imposed displacement corresponds to the experimentally-measured displacement at which a first crack appears at the specimen surface.

Figures 9 to 11 summarize the loading histories to fracture in all experiments. The evolutions of strain rate and stress state are shown in Figs. 9 and 10, respectively. Figure 11

M

depicts the evolution of the equivalent plastic strain versus stress state for all geometries and

PT

ED

loading conditions. Here, the stress state is characterized in terms of the tress triaxiality, (24)

̅

and the normalized third invariant of the stress deviator (Lode parameter)

CE

AC

where

(

Note that

)

(25)

̅

⁄ denotes the mean stress and ̅

lies in the range

.

the von Mises equivalent tensile stress.

corresponds to generalized shear (pure shear

with hydrostatic pressure/tension superposed), while

correspond to axisymmetric

stress states. In the specific case of a plane stress condition (represented by a dashed line in Fig. 10), uniaxial tension is characterized by (TPS) tension by

and

and

and transverse plane strain

.

Localized necking has a strong effect on the evolution of the local stress and strain fields. Figure 10 depicts the stress state trajectory in the (

) space for the three low strain rate

experiments. Solid dots in Figs. 10 and 11 mark the onset of fracture. Even though 20

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

experiments are performed on thin sheet specimens, the stress states at the onset of fracture (solid dots in Fig. 10) differ significantly from plane stress (represented by a dashed line in Fig. 10). After the onset of necking, the triaxiality increases (Fig. 11a) and the third invariant (Fig. 11b) decreases continuously. In the specific case of uniaxial tension at high strain rate, the triaxiality increases from down to

to

while the third invariant decreases from

. Overall, the stress state seems to move towards generalized shear in

all experiments (

in Fig. 10). This trend is attributed to the plane strain condition

CR IP T

imposed by the material surrounding the neck region. As the necks become deeper, the width of the plastically deforming neck region decreases which increases the severity of the imposed plane strain constrain onto the material at the center of the neck.

With regards to the stress state, it may be concluded that for a given range of strain rate,

AN US

the fracture strain decreases monotonically when moving from uniaxial tension to TPS tension. For example, at low strain rate loading (black curves in Fig. 11), the fracture strain is 60% higher in the 20mm notch specimen ( specimen (

) and 118% higher in the uniaxial

) than in the 6.65mm notch specimen (

). This observation is

fully consistent with the state-of-the-art of fracture initiation models (e.g. Mohr and Marcadet,

M

2015). The uncertainty in the reported fracture strains for notched specimens is expect to be less than  6% (see analysis by Dunand and Mohr (2010) for low strain rate loading). Given

ED

the axial engineering strain variations of  5% , the estimated repeatability-related uncertainty

PT

in the local fracture strains reported for uniaxial tension is about  7% .

6.2. Effect of loading velocity on strain to fracture

CE

Note that for a given geometry, the stress state evolution is more or less independent from the strain rate (Fig. 11). Differences in the fracture strain for a given specimen geometry are

AC

therefore attributed to the effect of the strain rate and temperature. The separation of the latter two is not possible based on the current experiments where nearly adiabatic conditions prevailed at intermediate and high strain rates. We therefore conclude only on the effect of the loading velocity which accounts for both the strain rate and temperature. Despite applying the loading at approximately constant boundary velocities, the local strain rate increases dramatically after the onset of necking: 

In the 20mm notch specimens, the equivalent plastic strain rate is 20 times larger at the onset of fracture than before the onset of necking. In the slow experiment, the strain rate increases from 5x10-4s-1 to 10-2s-1 (black line in Fig. 9b). The 21

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

increase ranges from 0.53s-1 to 10.5s-1 in the intermediate strain rate experiment (in red in Fig. 9b) and from 500s-1 to 10.2x103s-1 in the high strain rate experiment (in blue in Fig. 9b). Similar increases of strain rates are observed in the 6.65 notch experiment (Fig. 9c). 

In uniaxial tension experiments (Fig. 9a), the increase of strain rate due to necking is even more pronounced. In the low strain rate experiments (in black in Fig. 9a), the equivalent plastic strain rate is 40 times higher at the onset of fracture

CR IP T

( ̅ =3.2x10-2s-1) than prior to necking ( ̅ =8x10-4s-1).

The dependence of the fracture strain on the loading velocity appears to be stress state dependent:

In case of the 6.65mm notch specimen, it increases monotonically with the loading velocity, from

to

in an intermediate strain rate experiment, and to

AN US



in a high strain rate experiment; 

In the case of the 20mm notched tension specimens, however, the dependence is nonmonotonic. The fracture strain in the isothermal low strain rate experiment (

)

M

is slightly higher than that in the adiabatic intermediate strain rate experiment (



).

ED

), while it is highest in the adiabatic high strain rate experiment (

The non-monotonic effect of the loading speed is even more pronounced with the uniaxial tension specimen, in which the fracture strain in the low strain rate

experiment (

), while it is the highest in the high strain rate experiment

).

CE

(

) is 21% higher than that in the intermediate strain rate

PT

experiment (

AC

6.3. Comparison with Johnson-Cook results The notched experiments have also been modeled using a Johnson-Cook (J-C) plasticity

model with combined Swift-Voce hardening (Roth and Mohr, 2014). The force-displacement curves and the local strain evolution obtained with the J-C model are shown in the right columns of Figs. 5 and 8. The comparison with the left column plots (MTS model) demonstrates that the J-C model predictions are as good as the simulation results obtained with the extended MTS model. In other words, the experimental measurements at hand do not allow for discriminating between these two model approaches. Either modeling approach

22

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

appears to be acceptable when judged based on the present data. In other words, the potential advantages of the physically-sound basis of the MTS model cannot be demonstrated here. Both models make use of the same definition of the work-conjugate equivalent plastic strain definition which allows for a direct comparison of the obtained loading paths to fracture. In Figure 12, we superposed the loading paths to fracture for the J-C model (extracted from Roth and Mohr, 2014) with the MTS model results. For the NT6 specimens,

CR IP T

we observe that 1. the strain rate predicted by the MTS model is always lower than that predicted by the JC model. This suggest a lower degree of localization inside the neck in the case of the MTS model. Note that the surface strain history for an extensometer length of 1mm is identical in both simulations. In other words, the differences

AN US

prevail at a length scale that is smaller than the sheet thickness.

2. The strains to fracture predicted by the MTS model are also lower. For example, in the slow loading cases, the maximum strain obtained with the MTS model is 0.45, while a strain of 0.54 is reached in the JC simulation.

M

Similar differences are observed for the NT20 simulations, even though the differences in the fracture strains are less pronounced in the slow and fast loading cases. This is attributed to the

ED

fact that through-thickness necking is less pronounced in NT20 specimens as compared to NT6 specimens. The difference for the intermediate loading speed is similar to that observed in the NT6 specimens: the fracture strain predicted by the MTS model is only 0.67, while a

PT

strain of 0.84 is predicted by the JC model.

CE

The comparison of the MTS and JC results elucidates the high uncertainty in fracture strains that are identified through the present hybrid experimental-numerical approach. The comparison of computed and measured surface strain fields (e.g. Bertin et al., 2016) is

AC

expected to increase the reliability of the inverse plasticity model parameter identification. Additional accuracy is expected to be obtained from surface temperature field measurements, provided that experimental uncertainty related to deformation-induced emissivity changes are minimized. Without the availability of more reliable local loading path estimates, we can only conclude from this study that the ductility in Hopkinson bar experiments (high strain rates) is significantly higher than that observed in static experiments (low strain rates). Future research will need to separate the potentially competing effects of temperature and strain rate, as well as the possible influence of the stress state on the strain rate dependence, as suggested by the MTS results. 23

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

7. Conclusions An extended Mechanical Threshold Stress (MTS) model is formulated and used to

CR IP T

describe the temperature- and strain rate dependent response of advanced high strength steels. Low, intermediate and high strain rate experiments are performed on dogbone and notched specimens extracted from 1.4mm thick TRIP780 steel sheets. All material model parameters describing the MTS law are identified based on the experiments for a single notched specimen geometry (NT6). A positive, but rather weak, strain rate sensitivity of the plastic hardening

AN US

response is observed. Subsequently, the model is successfully validated for all other experiments based on a comparison of the force-displacement curves and a local 1mm long optical extensometer at the specimen center. It is worth noting that the material behavior before and after the onset of localized necking, is well captured.

A hybrid experimental-numerical approach is followed to evaluate the material loading

M

histories up to the onset of ductile fracture in all experiments. The results reveal that the equivalent plastic strain at the onset of fracture is significantly higher at high strain rates than

ED

at low strain rates. The loading paths to fracture obtained with the extended MTS model are also compared with the loading paths obtained with an extended Johnson-Cook (JC) model

PT

with combined Swift-Voce hardening (Roth and Mohr, 2014). Despite substantial differences in the model formulations, both approaches provide predictions that agree well with all

CE

experimental measurements. The extracted loading paths to fracture suggest the same trend when comparing the low and high velocity experiments, i.e. an increase in ductility as the loading velocity increases. However, the predicted strain rates (at the most highly strained

AC

element) are always higher when using the J-C model instead of the MTS approach. As a result, the estimated fracture strains are also higher when using the J-C model, to an extend that the ductility decrease at intermediate loading velocities observed with the MTS model, is no longer apparent in the results obtained with the J-C model. The identified differences in the MTS and J-C loading path predictions demonstrate that unique inverse model identification procedures must be developed first before drawing final conclusions with regards to the effect of the strain rate on the stress-state dependent ductility of advanced high strength steels.

24

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Acknowledgement The partial financial support of Industry Fracture Consortium is gratefully acknowledged. Professors Tomasz Wierzbicki (MIT), Lallit Anand (MIT) and Gérard Gary (Ecole Polytechnique) are thanked for valuable discussions. Mr. Philippe Chevalier (Ecole

References Abaqus (2011). "Reference manual v6.11". Abaqus Inc.

CR IP T

Polytechnique) is thanked for is assistance in carrying out the experimental work.

Abed, F. and F. Makarem (2012). "Comparisons of Constitutive Models for Steel Over a Wide Range of Temperatures and Strain Rates". Journal of Engineering Materials and

AN US

Technology-Transactions of the Asme 134(2).

Balasubramanian, S. and L. Anand (2002). "Elasto-viscoplastic constitutive equations for polycrystalline fcc materials at low homologous temperatures". Journal of the Mechanics and Physics of Solids 50(1): p. 101-126.

M

Bertin, M, Hild, F, Roux S, Mathieu F, Leclerc, H, Aimedieu, P (2016). Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment,

ED

Journal of strain analysis for engineering design 51 (2), 118-131. Bodner, S.R. and Y. Partom (1975). "Constitutive equations for elastic-viscoplastic strain-

PT

hardening materials". Journal of Applied Mechanics 42(2): p. 385-389.

CE

Conrad, H. (1964). "Thermally activated deformation of metals". J. Metals 16: p. 582. Cowper, G.R. and P.S. Symonds (1952). "Strain hardening and strain rate effects in the impact loading of cantilever beams", Brown University, Div. of Appl. Mech., Report

AC

no. 28.

Curtze, S., V.T. Kuokkala, M. Hokka, and P. Peura (2009). "Deformation behavior of TRIP and DP steels in tension at different temperatures over a wide range of strain rates". Materials Science and Engineering: A 507(1–2): p. 124-131. Dunand, M. and D. Mohr (2010). "Hybrid experimental-numerical analysis of basic ductile fracture experiments for sheet metals". International Journal of Solids and Structures 47(9): p. 1130-1143.

25

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Dunand, M., Gary, G. and D. Mohr (2013). "Load-Inversion Device for the High Strain Rate Tensile Testing of Sheet Materials with Hopkinson Pressure Bars". Experimental Mechanics 53. Durrenberger, L., A. Molinari, and A. Rusinek (2008). "Internal variable modeling of the high strain-rate behavior of metals with applications to multiphase steels". Materials Science and Engineering a-Structural Materials Properties Microstructure and

CR IP T

Processing 478(1-2): p. 297-304. Follansbee, P.S. and U.F. Kocks (1988). "A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable". Acta Metallurgica 36(1): p. 81-93.

Forni, D., Chiaia, B. & Cadoni, E. (2016). High strain rate response of S355 at high

AN US

temperatures. Materials & Design, 94, 467–478.

Gruben, G. et al. (2016). Low-velocity impact on high-strength steel sheets: An experimental and numerical study. International Journal of Impact Engineering, 88, pp.153–171. He, Z., Y. He, Y. Ling, Q. Wu, Y. Gao, and L. Li (2012). "Effect of strain rate on deformation

M

behavior of TRIP steels". Journal of Materials Processing Technology 212(10): p. 2141-2147.

ED

Hollomon, J.H. (1945). "Tensile deformation". Transaction of AIME 162: p. 268-290. Huh, H., S.B. Kim, J.H. Song, and J.H. Lim (2008). "Dynamic tensile characteristics of TRIP-

PT

type and DP-type steel sheets for an auto-body". International Journal of Mechanical Sciences 50(5): p. 918-931.

CE

Huh, J., H. Huh, and C.S. Lee "Effect of strain rate on plastic anisotropy of advanced high strength steel sheets".

International Journal of Plasticity. In press. DOI

AC

10.1016/j.ijplas.2012.11.012

Johnson, G.R. and W.H. Cook (1983). "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures". In: Proceedings of the 7th International Symposium on Ballistics, p. 541-547. Kajberg, J. and K.G. Sundin (2013). "Material characterisation using high-temperature Split Hopkinson pressure bar". Journal of Materials Processing Technology 213(4): p. 522531.

26

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Khan, A.S., M. Baig, S.-H. Choi, H.-S. Yang, and X. Sun (2012). "Quasi-static and dynamic responses of advanced high strength steels: Experiments and modeling". International Journal of Plasticity 30–31(0): p. 1-17. Khan, A.S., Y. Sung Suh, and R. Kazmi (2004). "Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys". International Journal of Plasticity 20(12): p. 2233-2248.

CR IP T

Kim, J.-H., D. Kim, H.N. Han, F. Barlat, and M.-G. Lee (2013). "Strain rate dependent tensile behavior of advanced high strength steels: Experiment and constitutive modeling". Materials Science and Engineering: A 559: p. 222-231.

Kocks, U.F., A.S. Argon, and M.F. Ashby (1975). "Thermodynamics and kinetics of slip", in Progress in Material Science, Pergamon: New York. p. 68-170.

AN US

Kolsky, H. (1949). "An Investigation of the Mechanical Properties of Materials at very High Rates of Loading". Proceedings of the Physical Society. Section B 62(11): p. 676. Kothari, M. and L. Anand (1998). "Elasto-viscoplastic constitutive equations for polycrystalline metals: Application to tantalum". Journal of the Mechanics and

M

Physics of Solids 46(1): p. 51-67.

Liang, R. and A.S. Khan (1999). "A critical review of experimental results and constitutive

ED

models for BCC and FCC metals over a wide range of strain rates and temperatures". International Journal of Plasticity 15(9): p. 963-980.

PT

Mohr, D., M. Dunand, and K.H. Kim (2010). "Evaluation of associated and non-associated quadratic plasticity models for advanced high strength steel sheets under multi-axial

CE

loading". International Journal of Plasticity 26(7): p. 939-956. Nelder, A. and R. Mead (1965). "A Simplex Method for Function Minimization". Computer

AC

Journal 7: p. 308-313.

Oliver, S., T.B. Jones, and G. Fourlaris (2007). "Dual phase versus TRIP strip steels: comparison of dynamic properties for automotive crash performance". Materials Science and Technology 23(4): p. 423-431. Roth, C.C. & Mohr, D. (2014). Effect of strain rate on ductile fracture initiation in advanced high strength steel sheets: Experiments and modeling. International Journal of Plasticity, 56, pp.19–44.

27

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Rusinek, A. and J.R. Klepaczko (2009). "Experiments on heat generated during plastic deformation and stored energy for TRIP steels". Materials & Design 30(1): p. 35-48. Rusinek, A., R. Zaera, and J.R. Klepaczko (2007). "Constitutive relations in 3-D for a wide range of strain rates and temperatures - Application to mild steels". International Journal of Solids and Structures 44(17): p. 5611-5634. Stoughton, T.B. (2002). "A non-associated flow rule for sheet metal forming". International

CR IP T

Journal of Plasticity 18(5-6): p. 687-714. Sun, X., A. Soulami, K.S. Choi, O. Guzman, and W. Chen (2012). "Effects of sample geometry and loading rate on tensile ductility of TRIP800 steel". Materials Science and Engineering: A 541(0): p. 1-7.

Sung, J.H., J.H. Kim, and R.H. Wagoner (2010). "A plastic constitutive equation

26(12): p. 1746-1771.

AN US

incorporating strain, strain-rate, and temperature". International Journal of Plasticity

Swift, H.W. (1952). "Plastic instability under plane stress". Journal of the Mechanics and Physics of Solids 1(1): p. 1-18.

M

Van Slycken, J., P. Verleysen, J. Degrieck, J. Bouquerel, and B.C. De Cooman (2007). "Dynamic response of aluminium containing TRIP steel and its constituent phases".

ED

Materials Science and Engineering: A 460-461: p. 516-524. Verleysen, P., J. Degrieck, T. Verstraete, and J. Van Slycken (2008). "Influence of Specimen

PT

Geometry on Split Hopkinson Tensile Bar Tests on Sheet Materials". Experimental Mechanics 48(5): p. 587-598.

CE

Verleysen, P., J. Peirs, J. Van Slycken, K. Faes, and L. Duchene (2011). "Effect of strain rate on the forming behaviour of sheet metals". Journal of Materials Processing

AC

Technology 211(8): p. 1457-1464.

Voce, E. (1948). "The relationship between stress and strain for homogeneous deformation". Journal of the Institute of Metals 74: p. 537-562.

Voyiadjis, G.Z. and F.H. Abed (2005). "Microstructural based models for bcc and fcc metals with temperature and strain rate dependency". Mechanics of Materials 37(2–3): p. 355-378.

28

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Wei, X., R. Fu, and L. Li (2007). "Tensile deformation behavior of cold-rolled TRIP-aided steels over large range of strain rates". Materials Science and Engineering: A 465(1-

AC

CE

PT

ED

M

AN US

CR IP T

2): p. 260-266.

29

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

Tables

Table 1. Chemical composition of the TRIP780 material in wt-% Al

Mn

Si

Mo

1.70

0.47

2.50

0.59

0.08

CR IP T

C

Table 2. Material parameters

[MPa]

[-]

185,000

0.3

Strain hardening parameters

AN US

Elastic constants [MPa] 503

[MPa]

1,383

[MPa]

41,505

[-] 2.513

Strain rate sensitivity parameters ̇ [s-1]

[-]

8.794E-20

1.911

1.861

ED

67,457

[-]

M

[J]

[-] 0.1949

Equivalent stress and flow potential parameters [-]

[-]

[-]

[-]

1.076

2.925

-0.462

-0.538

-0.538

[-]

[-]

[-]

[-]

[-]

[-]

0.873

1.071

-0.401

-0.599

-0.472

1

[-]

3.13

AC

1

[-]

CE

1

[-]

PT

[-]

30

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

AN US

CR IP T

Figures

(b)

AC

CE

PT

ED

M

(a)

(c)

(d)

Figure 1. Uniaxial tension specimens for (a) low and intermediate strain rate and (b) high strain rate experiments, tensile specimens with a notched radius of (c) 6.65mm and (d) 20mm. Red dots depict the location of the reference points used to measure displacements.

31

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

Dunand and Mohr (December 2016)

Figure 2. Experimental engineering stress-strain curves under uniaxial tension. Results for low strain rates are shown in black, intermediate strain rates in red and high strain rates in blue. Solid dots correspond to the onset of fracture.

32

ACCEPTED MANUSCRIPT

(b)

AC

CE

PT

ED

M

AN US

(a)

CR IP T

Dunand and Mohr (December 2016)

Figure 3. Experimental results of notched tensile specimens with a 20mm radius. (a) Force displacement curves and (b) Evolution of engineering axial surface strain at the center of the gage section. Results for low strain rates are shown in black, intermediate strain rates in red and high strain rates in blue. Solid dots correspond to the onset of fracture.

33

ACCEPTED MANUSCRIPT

(b)

AC

CE

PT

ED

M

AN US

(a)

CR IP T

Dunand and Mohr (December 2016)

Figure 4. Experimental results of notched tensile specimens with a 6.65mm radius. (a) Force displacement curves and (b) Evolution of engineering axial surface strain at the center of the gage section. Results for low strain rates are shown in black, intermediate strain rates in red and high strain rates in blue. Solid dots correspond to the onset of fracture.

34

ACCEPTED MANUSCRIPT

500mm

5mm

(b)

AC

CE

PT

ED

M

AN US

(a)

CR IP T

Dunand and Mohr (December 2016)

10mm

(c) Figure 5. Representative shear failure mode observed for a NT6 specimen: (a) side view of specimen after tensile loading along the horizontal direction, (b) SEM image of the fracture surface, (c) microstructure of the highly deformed microstructure below the fracture surface. 35

ACCEPTED MANUSCRIPT

CR IP T

Dunand and Mohr (December 2016)

(d)

(e)

AC

CE

PT

(b)

ED

M

AN US

(a)

(c) (f) Figure 6. Experimental (points) and simulation (solid curves) results for the notched tensile specimen with a radius, at low (a, d), intermediate (b, e) and high strain rates (c, e). Force displacement curves are in black and central logarithmic axial strain versus displacement curves in blue. The dashed lines depict the instant of fracture. The results shown in the left column are obtained with the extended MTS model (present work), while the figures shown in the right column are obtained with an extended Johnson-Cook model (figures extracted from Roth and Mohr (2014)). 36

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

AN US

CR IP T

̇

AC

CE

PT

ED

M

(a)

̇

Figure 7. Illustration of MTS model response for TRIP780 steel: (a) rate-dependent stressstrain response under isothermal conditions (room temperature); (b) temperature dependent stress-strain response at a constant strain rate of 1/s. 37

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

Dunand and Mohr (December 2016)

Figure 8. Experimental (points) and simulation (solid curves) results for uniaxial tension at low (black curve), intermediate (red curve) and high strain rate (blue curve). The dashed lines depict the instant of fracture.

38

ACCEPTED MANUSCRIPT

CR IP T

Dunand and Mohr (December 2016)

(d)

(e)

AC

CE

PT

(b)

ED

M

AN US

(a)

(c) (f) Figure 9. Experimental (points) and simulation (solid curves) results for the notched tensile specimen with a radius, at low (a, d), intermediate (b, e) and high strain rates (c, f). Force displacement curves are in black and central logarithmic axial strain versus displacement curves in blue. The dashed lines depict the instant of fracture. The results shown in the left column are obtained with the extended MTS model (present work), while the figures shown in the right column are obtained with an extended Johnson-Cook model (figures extracted from Roth and Mohr (2014)). 39

ACCEPTED MANUSCRIPT

CR IP T

Dunand and Mohr (December 2016)

AC

CE

PT

(b)

ED

M

AN US

(a)

(c) Figure 10. Numerical prediction of the evolution of equivalent plastic strain rate versus equivalent plastic strain at the location where fracture is assumed to initiate in (a) uniaxial tension, (b) 20mm notch tension and (c) 6.65mm notched tension experiments. Solid dots indicate the onset of fracture. 40

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

Dunand and Mohr (December 2016)

Figure 11. Loading path in the (

) plane for low strain rate experiments. Solid dots indicate

the instant of fracture. The black dashed line shows the relation between stress triaxiality and third stress invariant in case of plane stress condition.

41

ACCEPTED MANUSCRIPT

AN US

CR IP T

Dunand and Mohr (December 2016)

AC

CE

PT

ED

M

(a)

(b)

Figure 12. Loading paths to fracture at low (black lines), intermediate (red lines) and high strain rate (blue lines). (a) Evolution of equivalent plastic strain versus stress triaxiality and (b) evolution of equivalent plastic strain versus normalized third stress invariant in all experiments. Solid points depict the onset of fracture.

42

ACCEPTED MANUSCRIPT

Dunand and Mohr (December 2016)

NT6-error JC MTS JC

JC

AN US

MTS

CR IP T

MTS

(a)

JC

ED

M

MTS

MTS JC

PT

MTS

(b)

AC

CE

JC

Figure 13. Direct comparison of the loading paths to fracture for notched tension as determined with the Mechanical Threshold Stress (MTS) model and the extended JohnsonCook (JC) model.

43